# Sharpe ratio

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Here, based on the assumption that the monthly returns are serially uncorrelated (Sharpe, 1994), the annual standard deviation √ of returns is 12 times the monthly standard√deviation. Hence, overall, the annualized Sharpe ratio would be 12 times the monthly Sharpe ratio. In general, if you calculate your average and standard deviation of returns based on a certain trading period T, whether T is a month, a day, or an hour, and you want to annualize these quantities, you have to first find out how many such trading periods there are in a year (call it NT ). Then Annualized Sharpe Ratio = NT × Sharpe Ratio Based on T P1: JYS c03 JWBK321-Chan September 24, 2008 13:52 Printer: Yet to come Backtesting 45 For example, if your strategy holds positions only during the NYSE market hours (9:30–16:00 ET), and the average hourly returns is R, and the standard deviation of the hourly returns is s, then the annu√ alized Sharpe ratio is 1638 × R/s. This is because NT = (252 trading days) × (6.5 trading hours per trading day) = 1,638.

If you are trading just gold futures, then the market index should be gold spot price, rather than a stock index. The Sharpe ratio is actually a special case of the information ratio, suitable when we have a dollar-neutral strategy, so that the benchmark to use is always the risk-free rate. In practice, most traders use the Sharpe ratio even when they are trading a directional (long or short only) strategy, simply because it facilitates comparison across different strategies. Everyone agrees on what the riskfree rate is, but each trader can use a different market index to come up with their own favorite information ratio, rendering comparison difficult. (Actually, there are some subtleties in calculating the Sharpe ratio related to whether and how to subtract the risk-free rate, how to annualize your Sharpe ratio for ease of comparison, and so on. I will cover these subtleties in the next chapter, which will also contain an example on how to compute the Sharpe ratio for a dollar-neutral and a long-only strategy.)

I will cover these subtleties in the next chapter, which will also contain an example on how to compute the Sharpe ratio for a dollar-neutral and a long-only strategy.) If the Sharpe ratio is such a nice performance measure across different strategies, you may wonder why it is not quoted more often instead of returns. In fact, when a colleague and I went to SAC Capital Advisors (assets under management: \$14 billion) to pitch a strategy, their then head of risk management said to us: “Well, a high Sharpe ratio is certainly nice, but if you can get a higher return instead, we can all go buy bigger houses with our bonuses!” This reasoning is quite wrong: A higher Sharpe ratio will actually allow you to make more profits in the end, since it allows you to trade at a higher leverage. It is the leveraged return that matters in the end, not the nominal return of a trading strategy.

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Market Sense and Nonsense by Jack D. Schwager

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The potential for a mismatch between Sharpe ratio rankings and investor preferences has led to the creation of other return/risk measures that seek to address the flaws of the Sharpe ratio. Before we review some of these alternative measures, we first consider the question: What are the implications of a negative Sharpe ratio? Although it is commonplace to see negative Sharpe ratios reported for managers whose returns are less than the risk-free return, negative Sharpe ratios are absolutely meaningless. When the Sharpe ratio is positive, greater volatility (as measured by the standard deviation), a negative characteristic, will reduce the Sharpe ratio, as it logically should. When the Sharpe ratio is negative, however, greater volatility will actually increase its value—that is, the division of a negative return by a larger number will make it less negative. Comparisons involving negative Sharpe ratios can lead to absurd results. An example is provided in Table 8.3.

(In his article, Ziemba uses zero as the benchmark value.) Unlike the Sortino ratio, the SDR Sharpe ratio (with the benchmark set to the average) can be directly compared with the Sharpe ratio.8 The SDR Sharpe ratio (with any of the standard choices for a benchmark value) is preferable to the Sharpe ratio because it accounts for the very significant difference between the risk implications of downside deviations versus upside deviations as viewed from the perspective of the investor. The SDR Sharpe ratio is also preferable to the Sortino ratio because it is an almost identical calculation,9 but with the important advantage of being directly comparable with the widely used Sharpe ratio. Also, by comparing a manager’s SDR Sharpe ratio versus the Sharpe ratio, an investor can get a sense of whether the manager’s returns are positively or negatively skewed.

This ratio provides the same fix as the Sortino ratio, and it has the advantage of an additional adjustment that allows for direct comparisons of its values with Sharpe ratio values. Similar to the Sortino ratio, the SDR Sharpe ratio also uses the compounded return instead of the arithmetic average return. Since the SDR Sharpe ratio will provide nearly identical rankings as the Sortino ratio and has the advantage of allowing for comparisons with the Sharpe ratio for the same manager, it seems the better choice for any investor. Using both ratios would be redundant. Gain-to-pain ratio (GPR). Similar to the Sortino and SDR Sharpe ratios, the GPR penalizes a manager only for losses (zero percent is also a common choice for minimum acceptable return or benchmark in the Sortino and SDR Sharpe ratios). The GPR weights losses proportionate to their magnitude, while the Sortino and SDR Sharpe ratios magnify the weight of larger losses.

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Portfolio Design: A Modern Approach to Asset Allocation by R. Marston

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7 To assess the return on stocks versus the return on bonds, the first step is to adjust each return for risk. The Sharpe ratio adjusts each return by first subtracting the risk-free rate, rF , then dividing by the standard deviation. If rj is the return on asset j and σ j is its standard deviation, then the Sharpe ratio for asset j is [rj − rF ]/sj To calculate the Sharpe ratio, we use the arithmetic returns and standard deviations in Table 2.1. It’s possible to define a Sharpe ratio using geometric returns, but only if the standard deviation is defined over a similar long horizon.8 Using the returns in Table 1, the Sharpe ratios are defined by Stocks : [0.113 − 0.047]/0.146 = 0.45 Bonds : [0.063 − 0.047]/0.095 = 0.17 The Sharpe ratio for stocks is more than twice the size of the ratio for bonds. So, after adjusting for the higher risk of stocks, the (excess) return on stocks is much larger than bonds.

The concept that may be new to many readers is alpha∗ . This is a measure of the excess return on the portfolio itself. Suppose that an investment advisor has measured the return on a portfolio in terms of its Sharpe ratio and wants to compare it with some benchmark. (In a later chapter, the performance of the Yale portfolio is compared with the benchmark of university portfolios as a whole). The advisor could simply compare the Sharpe ratios of the portfolio and benchmark. Sharpe ratios rather than alphas are appropriate because it is the total return and total risk that is being assessed. Alpha∗ provides a way of comparing the Sharpe ratios by measuring the excess return earned by the portfolio relative to its benchmark. To measure alpha∗ , the risk level of the benchmark has to be reduced to that of the portfolio. This measure is explained further in the appendix.12 With these tools, we can begin studying the assets that will be included in the portfolio.

The hedge for the world index lost 1.3 percent per year on average over this period. Table 7.6 also reports the standard deviations and Sharpe ratios of the hedged investments. With standard deviations so low, the foreign bond P1: a/b c07 P2: c/d QC: e/f JWBT412-Marston T1: g December 8, 2010 17:47 140 Printer: Courier Westford PORTFOLIO DESIGN returns generally have Sharpe ratios higher than that of the un-hedged foreign bond series. In some cases, these Sharpe ratios are also higher than those of the U.S. Treasury bond indexes. Consider the world index hedged for exchange risk. By hedging, the return falls from 8.1 percent to 6.8 percent, but the standard deviation falls from 9.6 percent to 3.2 percent. The Sharpe ratio rises sharply to 0.86, far above that of the U.S. bond indexes. Hedging the currency risk does raise the correlation between the foreign bond and U.S. bonds and stocks.

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Understanding Asset Allocation: An Intuitive Approach to Maximizing Your Portfolio by Victor A. Canto

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Ideally, one would then search for those asset classes that would add alpha (that is, excess returns to a portfolio) without increasing beta (that is, the risk of the portfolio in relation to the benchmark). Now, let’s apply the Sharpe ratio. Once more, the Sharpe ratio divides a portfolio’s excess returns (returns less risk less Treasury bill returns) by its volatility. In effect, the Sharpe ratio treats each asset class as a separate portfolio, focusing on the standard deviations that measure total risk. If a portfolio in question represents an individual’s entire investment, then volatility matters and the Sharpe ratio is a fitting comparison tool. As such, the Sharpe ratio provides an appropriate way to compare and evaluate the size, style, and location choices within our SAA portfolio. For Table 6.5, I applied the Sharpe ratio simply by calculating the ratio of the risk-adjusted portfolio returns to their standard deviation, the idea being that the portfolio with the highest return-tostandard-deviation ratio offers the highest reward-to-risk ratio.

Value of \$1 Return Top 2,919.50 30.5% Second 365.57 21.7% Third 92.48 16.3% Fourth/Median 31.89 12.2% Fifth 11.44 8.4% Sixth 1.75 1.9% Seventh 0.24 –4.7% Strategic Asset Allocation Based On… Period Sharpe Ratio \$72.59 15.35% Yearly Sharpe Ratio \$34.39 12.5% Market Weights \$28.76 11.8% Source: MSCI, Research Insight, and Ibbotson Associates 112 UNDERSTANDING ASSET ALLOCATION Table 6.5 Risk measurements: 1975–2004. CAPM Beta Jensen’s Alpha T-Statistics Sharpe Ratio Small Cap 1 5.64% 2.07 0.65 Large Cap 1 0.00% Growth 1.06 –1.46% 1.87 0.43 Value 0.93 0.81% 1.67 0.60 International 0.62 –1.27% 0.04 0.29 0.53 Strategic Asset Allocation Based On… Period Sharpe Ratio 0.66 0.03% 24.09 0.72 Yearly Sharpe Ratio 0.52 0.02% 24.58 0.63 Market Weights 0.54 0.1% 27 0.57 Comparing the Historical- and Market-Based Allocations As I pointed out in Chapter 1, “In Search of the Upside,” financial economics developments over the past three decades provide us with the necessary tools to develop risk-adjusted returns in a rigorous and systematic way.

This splits the index into two mutually exclusive groups designed to track two of the predominant investment styles in the U.S. equity market.”4 These styles are value and growth, a distinction William Sharpe found valid.5 The Sharpe ratio, named after William Sharpe (the 1990 Nobel Prize in Economics winner), divides a portfolio’s excess return (return less riskless T-bill return) by its volatility. In effect, the Sharpe ratio treats each asset class as a separate portfolio, focusing on the standard deviations that measure total risk. If the portfolio in question represents the entire investment of an individual, volatility matters—and the Sharpe ratio is an appropriate comparison Chapter 2 The Case for Cyclical Asset Allocation 21 tool. As such, the Sharpe ratio provides an apt way to compare and evaluate the size, style, location, and balance of portfolios. The Optimal Value Stocks/Growth Stocks Mix The style portfolios I produced are reported in Table 2.3.

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High-Frequency Trading: A Practical Guide to Algorithmic Strategies and Trading Systems by Irene Aldridge

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If returns of the trading strategy can be assumed to be normal, Jobson and Korkie (1981) showed that the error in Sharpe ratio estimation is normally distributed with mean 0 and standard deviation s = [(1/T)(1 + 0.5SR2 )]1/2 For a 90 percent confidence level, the claimed Sharpe ratio should be at least 1.645 times greater than the standard deviation of the Sharpe ratio errors, s. As a result, the minimum number of evaluation periods used for Sharpe ratio verification is Tmin = (1.6452 /SR2 )(1 + 0.5SR2 ) The Sharpe ratio SR used in the calculation of Tmin , however, should correspond to the frequency of estimation periods. If the annual Sharpe ratio claimed for a trading strategy is 2, and it is computed based on the basis of monthly data, then the corresponding monthly Sharpe ratio SR is 2/(12)0.5 = 0.5774. On the other hand, if the claimed Sharpe ratio is computed based on daily data, then the corresponding Sharpe ratio SR 60 HIGH-FREQUENCY TRADING TABLE 5.2 Minimum Trading Strategy Performance Evaluation Times Required for Veriﬁcation of Reported Sharpe Ratios Claimed Annualized Sharpe Ratio No. of Months Required (Monthly Performance Data) No. of Months Required (Daily Performance Data) 0.5 1.0 1.5 2.0 2.5 3.0 4.0 130.95 33.75 15.75 9.45 6.53 4.95 3.38 129.65 32.45 14.45 8.15 5.23 3.65 2.07 is 2/(250)0.5 = 0.1054.

For any other portfolio, trading strategy, or individual security A, the higher the Sharpe ratio, the closer the security is to the efficient frontier. Sharpe himself came up with the metric when developing a portfolio optimization mechanism for a mutual fund for which he was consulting. Sharpe’s mandate was to develop a portfolio selection framework for the 55 Evaluating Performance of High-Frequency Strategies Return Sharpe ratio of the market portfolio A E [RM ] E [RA ] − RF E [RM ] − RF RF Sharpe ratio of instrument A σM σA Risk FIGURE 5.2 Sharpe ratio as a mean-variance slope. The market portfolio has the highest slope and, correspondingly, the highest Sharpe ratio. fund with the following constraint: no more than 5 percent of the fund’s portfolio could be allocated to a particular financial security. Sharpe then created the following portfolio solution: he first ranked the security universe on what now is known as Sharpe ratio, then picked the 20 securities with the best performance according to the Sharpe ratio measure, and invested 5 percent of the fund into each of the 20 securities.

Length of the Evaluation Period for High-Frequency Strategies Most portfolio managers face the following question in evaluating candidate trading strategies for inclusion in their portfolios: how long does one need to monitor a strategy in order to gain confidence that the strategy produces the Sharpe ratio advertised? Some portfolio managers have adopted an arbitrarily long evaluation period: six months to two years. Some investors require a track record of at least six years. Yet others are content with just one month of daily performance data. It turns out that, statistically, any of the previously mentioned time frames is correct if it is properly matched with the Sharpe ratio it is intended to verify. The higher the Sharpe ratio, the shorter the strategy evaluation period needed to ascertain the validity of the Sharpe ratio. If returns of the trading strategy can be assumed to be normal, Jobson and Korkie (1981) showed that the error in Sharpe ratio estimation is normally distributed with mean 0 and standard deviation s = [(1/T)(1 + 0.5SR2 )]1/2 For a 90 percent confidence level, the claimed Sharpe ratio should be at least 1.645 times greater than the standard deviation of the Sharpe ratio errors, s.

Commodity Trading Advisors: Risk, Performance Analysis, and Selection by Greg N. Gregoriou, Vassilios Karavas, François-Serge Lhabitant, Fabrice Douglas Rouah

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As a result of the recurring frequency of down markets since the collapse of Long-Term Capital Management (LTCM) in August 1998, VaR has played a paramount role as a risk management tool and is considered a mainstream technique to estimate a CTA’s exposure to market risk. 377 378 PROGRAM EVALUATION, SELECTION, AND RETURNS With the large acceptance of VaR and, specifically, the modified VaR as a relevant risk management tool, a more suitable portfolio performance measure for CTAs can be formulated in term of the modified Sharpe ratio.1 Using the traditional Sharpe ratio to rank CTAs will underestimate the tail risk and overestimate performance. Distributions that are highly skewed will experience greater-than-average risk underestimation. The greater the distribution is from normal, the greater is the risk underestimation. In this chapter we rank 30 CTAs according to the Sharpe ratio and modified Sharpe ratio. Our results indicate that the modified Sharpe ratio is more accurate when examining nonnormal returns. Nonnormality of returns is present in the majority of CTA subtype classifications. LITERATURE REVIEW Many CTAs produce statistical reports that include the traditional Sharpe ratio, which can be misleading because funds will look better in terms of risk-adjusted returns.

For two CTAs the same models offered a better representation for the period after the breakpoint (January 2000), while for the third CTA a different ARMA model appears to offer better results. CHAPTER 22 Risk-Adjusted Returns of CTAs: Using the Modified Sharpe Ratio Robert Christopherson and Greg N. Gregoriou any institutional investors use the traditional Sharpe ratio to examine the risk-adjusted performance of CTAs. However, this could pose problems due to the nonnormal returns of this alternative asset class. A modified VaR and modified Sharpe ratio solves the problem and can provide a superior tool for correctly measuring risk-adjusted performance. Here we rank 30 CTAs according to the Sharpe and modified Sharpe ratio and find that larger CTAs possess high modified Sharpe ratios. M INTRODUCTION The assessment of portfolio performance is fundamental for both investors and funds managers, as well as commodity trading advisors (CTAs).

We use this comparison to see if there exist any differences between groups in terms of the Sharpe and modified Sharpe ratio. We use the Extreme metrics software available on the www.alternativesoft.com web site to compute the results using a 99 percent VaR probability, and we assume that we are able to borrow at a risk-free rate of 0 percent. The difference between the traditional and modified Sharpe ratio is that, in the latter, the standard deviation is replaced by the modified VaR in the denominator. The traditional Sharpe ratio, generally defined as the excess return per unit of standard deviation, is represented by this equation: Sharpe Ratio = Rp − RF σ where RP = return of the portfolio RF = risk-free rate and s = standard deviation of the portfolio (22.1) 380 PROGRAM EVALUATION, SELECTION, AND RETURNS A modified Sharpe ratio can be defined in terms of modified VaR: Modified Sharpe Ratio = Rp − RF MVaR (22.2) The derivation of the formula for the modified VaR is beyond the scope of this chapter.

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Expected Returns: An Investor's Guide to Harvesting Market Rewards by Antti Ilmanen

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The coincidence of losses could reflect similar fundamental (tail) risk exposures but also liquidity-induced and positioning-induced correlation spikes (the latter were especially apparent in 1998, 2007, and 2008). • I show empirically that many strategies with the best Sharpe ratios since the 1980s are of the second type. Their covariance with MU is high: they tend to lose money in bad times. This feature can largely explain their high Sharpe ratios. It is harder to explain the performance of government bonds since the 1980s as an equilibrium outcome, given the combination of a high Sharpe ratio and a wonderful diversification/hedging role. As noted, the high in-sample Sharpe ratios of Treasuries over that time frame likely reflect windfall gains from unanticipated yield declines. Note that the Sharpe ratios of Treasuries over periods ending in the early 1980s were very poor, reflecting windfall losses from unanticipated yield increases—a mirror image of the situation from the early 1980s to the present

[3] If we make predictions of one-year returns, we may be able to make statistically significant inferences about excess returns using 20 years of historical data; whether such short-term forecasts are economically significant is a topic of debate. The Sharpe ratio is closely related to statistical significance. It is the ratio of mean excess return over its volatility, while statistical signifcance uses the same two variables plus sample size. (The standard error of a mean is σ/√N where σ is standard deviation and N is sample size. The denominator of a Sharpe ratio only includes σ.) Given a 20-year sample period, an asset’s annual excess return is statistically significantly above zero if the Sharpe ratio exceeds 0.44 (i.e., the critical value of a two-sided 5% significance level is 1.95 and 1.9/√20 ≈ 0.44). However, classical assumptions include an unbiased sample period, normally distributed asset returns, and constant expected returns—and the empirical validity of each of these is questionable.

[2] Long data histories are necessary to make statistically significant inferences of one-year returns, even assuming constant expected returns and no structural changes (see Note 3 in Chapter 2). Assessing the statistical significance of multi-year horizon returns would require much longer data histories. [3] Sharpe ratio and relative return investors: The information ratio is the benchmark-oriented relative return manager’s counterpart to an absolute return manager’s Sharpe ratio. Information ratio is the mean excess return of a portfolio vs. its benchmark, divided by the tracking error (i.e., volatility of that excess return). Sharpe ratio is sometimes computed as the ratio of total return and its volatility, which is plain wrong; the return on a money market asset should be subtracted from total returns. The ratio of total return and its volatility is sometimes called the information ratio for hedge funds, on the premise that their benchmark is zero (motivated by the absolute return mandate).

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Market Risk Analysis, Quantitative Methods in Finance by Carol Alexander

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If the autocorrelation in excess returns is positive then (I.6.63) is greater than the square root of h, so the denominator in the Sharpe ratio will increase and the Sharpe ratio will be reduced. Conversely, if the autocorrelation is negative the Sharpe ratio will increase. Example I.6.12: Adjusting a Sharpe ratio for autocorrelation Ex post estimates of the mean and standard deviation of the excess returns on a portfolio, based on a historical sample of daily data, are 0.05% and 0.75%, respectively. Estimate the Sharpe ratio under the assumption that the daily excess returns are i.i.d. and that there are 250 trading days per year. Now suppose that daily returns have an autocorrelation of 0.2. What is the adjusted Sharpe ratio that accounts for this autocorrelation? Solution We have the ordinary Sharpe ratio, 005% × 250 125% = = 10541 √ 1186% 075% × 250 But calculating the adjustment (I.6.63) to account for a positive √ autocorrelation of 0.2 in the spreadsheet gives an annualizing factor of 19.35 instead of 250 = 1581 for the standard deviation.

However, let us compare the Sharpe ratios of the two investments. We calculate the mean and standard deviation of the excess returns and divide the former by the latter. The results are shown in Table I.6.5. Hence, according to the Sharpe ratio, A should be preferred to B! Introduction to Portfolio Theory 259 Table I.6.5 Sharpe ratio and weak stochastic dominance Portfolio Expected excess return Standard deviation Sharpe ratio A B 8.0% 9.80% 0.8165 10.0% 13.56% 0.7372 The Sharpe ratio does not respect even weak stochastic dominance, and the example given above can be extended to other RAPMs derived in the CAPM framework. For this reason they are not good metrics to use as a basis for decisions about uncertain investments. I.6.5.3 Adjusting the Sharpe Ratio for Autocorrelation The risk and return in an RAPM are forecast ex ante using a model for the risk and the expected return.

This gives an autocorrelation adjusted Sharpe ratio, SR = ASR1 = 125% 005% × 250 = = 08612 075% × 1935 1451% Clearly the adjustment to a Sharpe ratio for autocorrelation can be very significant. Hedge funds, for instance, tend to smooth their reported returns and in so doing can induce a strong positive autocorrelation in them. Taking this positive autocorrelation into account will have the effect of reducing the Sharpe ratio that is estimated from reported returns. I.6.5.4 Adjusting the Sharpe Ratio for Higher Moments The use of the Sharpe ratio is limited to investments where returns are normally distributed and investors have a minimal type of risk aversion to variance alone, as if their utility function is exponential. Extensions of the Sharpe ratio have successfully widened its application to non-normally distributed returns but its extension to different types of utility function is more problematic.

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Efficiently Inefficient: How Smart Money Invests and Market Prices Are Determined by Lasse Heje Pedersen

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WARREN BUFFETT: THE ULTIMATE VALUE AND QUALITY INVESTOR Warren Buffett has become one of the world’s richest people based on his investment success over the past half century. How large a Sharpe ratio does it take to become the richest person in the world? Most investors guess that Warren Buffett must have realized a Sharpe ratio well north of 1 or even 2, perhaps based on Sharpe ratios promised by aggressive fund managers. The truth is that Buffett’s firm Berkshire Hathaway has delivered a Sharpe ratio of 0.76 from 1976 to 2011. While this is lower than some might have expected, it is nevertheless an extremely impressive number. Buffett’s Sharpe ratio is double that of the overall stock market over the same time period, which means that Buffett has delivered twice as much return per unit of risk. While some stocks or funds have clearly delivered higher Sharpe ratios over a shorter time period (which could be just luck), Buffett’s Sharpe ratio is the highest of any U.S. stock or any U.S. mutual fund that has been around for at least 30 years.2 How has Buffett done it?

We see that the strategies’ realized volatilities closely match the 10% ex ante target, varying from 9.5% to 11.9%. More importantly, all the time series momentum strategies have impressive Sharpe ratios, reflecting a high average excess return above the risk-free rate relative to the risk. Comparing the strategies across trend horizons, we see that the long-term (12-month) strategy has performed the best, the medium-term strategy has done second best, and the short-term strategy, which has the lowest Sharpe ratio of the three strategies, still has a high Sharpe ratio of 1.3. Comparing asset classes, commodities, fixed income, and currencies have performed a little better than equities. Figure 12.2. Performance of time series momentum by individual asset and trend horizon. This figure shows the Sharpe ratios of the time series momentum strategies for each commodity futures (dark grey), currency forward (light grey), equity futures (light blue), and fixed-income futures (dark blue).

The standard deviation σ can often be estimated with more precision: It is the square root of the variance σ2, which is estimated as the squared deviations around the arithmetic average, 2.4. TIME HORIZONS AND ANNUALIZING PERFORMANCE MEASURES Performance measures depend on the horizon over which they are measured. For instance, table 2.1 shows that a strategy that has an annual Sharpe ratio of 1 has very different Sharpe ratios if measured over other time horizons, such as a Sharpe ratio of 2 over a four-year period and a mere 0.06 over a trading day. Hence, when we talk about performance measures, we need to be clear about the horizon. Furthermore, when we compare the performance of two different strategies or hedge funds, we need to make sure that the performance measures are calculated over the same time horizon. It is therefore useful to have a standard measurement horizon and, to accomplish this, performance measures are often annualized.

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Red-Blooded Risk: The Secret History of Wall Street by Aaron Brown, Eric Kim

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It makes more sense to speak of individual bets with upsides and downsides measured in dollars, not dollars per unit of investment over time. A Sharpe ratio of 0.1 produces a 1 percent excess return; a Sharpe ratio of 0.2 gives about 4 percent; a Sharpe ratio of 0.5 gives about 25 percent. Since the high Sharpe ratio strategies have limited capacity but grow so quickly that initial investment is almost irrelevant, they are appropriate for people investing their own money. Sharpe ratios around 0.5 make good hedge fund strategies. The lower Sharpe ratios are useful for large institutions with cheap capital and high risk tolerance. Card counters gravitated to very high Sharpe ratio strategies. When I say these have limited capacity, I don’t mean there are always opportunities to invest a small amount in them.

It’s still important to distinguish the difference, however, because the effect that the new generation of quants had on events was quite different from the effect from the old generation. Sharpe Ratios and Wealth Returning to the old school, the two main branches found different kinds of market opportunities, distinguished by Sharpe ratio. We’re going to get a bit mathematical again, but you don’t need the numbers to follow the argument. The Sharpe ratio of a strategy is defined as the return of the strategy minus what you could make investing the same capital in risk-free instruments, divided by the standard deviation of the return. It is a measure of risk-adjusted return. A strategy with an annualized Sharpe ratio of 1 will make more than the risk-free rate about five years out of six. A strategy with an annualized Sharpe ratio of 2 will make more than the risk-free rate about 39 years out of 40. However, it’s hard to find Sharpe ratios near or above 1 in high-capacity, liquid strategies that are inexpensive to run.

However, it’s hard to find Sharpe ratios near or above 1 in high-capacity, liquid strategies that are inexpensive to run. You don’t need a Sharpe ratio near or at 1 to get rich. For a Kelly investor, the long-term growth of capital above the risk-free rate is approximately equal to the Sharpe ratio squared (it’s actually always higher than this, substantially so for high Sharpe ratios, but that doesn’t affect the points I want to make). A Sharpe ratio of 1 means growing at 100 percent per year—that is, doubling your capital. A Sharpe ratio of 2 means growing at 400 percent per year. (For the purists, the actual figures are 173 percent and 5,360 percent.) Clearly both of those strategies will have to hit some kind of short-term limit. In fact, Sharpe ratios above one are usually meaningless; it doesn’t make sense to speak about the long-term growth rate of something that can’t grow at that rate for long.

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The New Science of Asset Allocation: Risk Management in a Multi-Asset World by Thomas Schneeweis, Garry B. Crowder, Hossein Kazemi

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As discussed previously, the CAPM purports that the expected return of a security stems more from the covariance of the security with the market portfolio than from the stand-alone risk of the individual asset. The Sharpe Ratio has other well-known shortcomings, including: ■ ■ ■ In periods of historical negative returns, the strict Sharpe comparisons have little value. The Sharpe Ratio should be based on expected return and risk; however, in practice, actual performance over a particular period of time is often used. In periods of negative mean return, an asset may have a lower negative return as well as a lower standard deviation and yet report a lower Sharpe Ratio (e.g., more negative) than an alternative asset with a greater negative return and with a higher relative standard deviation. Gaming the Sharpe Ratio. A manager with a high Sharpe Ratio will get a close look from institutional investors even if the absolute returns are less than stellar.

The Sharpe Ratio is computed as: Si = (Ri − Rf ) σi where R̄i is the estimated mean rate of return of the asset, Rf is the risk-free rate of return, and σi is the estimated standard deviation. This measure can be taken to show return obtained per unit of risk. While the Sharpe Ratio does offer the ability to rank assets with different return and risk (measured as standard deviation), its use may be limited to comparing portfolios that may realistically be viewed as alternatives to one another. First, the Sharpe Ratio has little to say about the relative return to risk of individual securities. There is simply too much randomness in the price movement of individual securities to make the Sharpe Ratio of any real use at the individual asset level. Moreover, the Measuring Risk 27 Sharpe Ratio does not take into account that the individual assets may themselves be used to create a portfolio. As discussed previously, the CAPM purports that the expected return of a security stems more from the covariance of the security with the market portfolio than from the stand-alone risk of the individual asset.

While we attempt to summarize some of the issues in many of the traditional model approaches to asset allocation, the size of the asset allocation problem overwhelms any individual approach if for the simple reason that there are too many individuals, each with their own unique set of investment concerns. CLASSIC SHARPE RATIO For much of this and the previous chapter we have emphasized the wide range of risks involved in asset allocation and security return estimation; however, for many, when the choice is between two (or more) assets, one way of ranking investments (the Sharpe Ratio) is based on simplifying risk into a single parameter (e.g., standard deviation). This ratio essentially divides the return of the security (after first subtracting the risk-free rate of return) by the price risk (standard deviation of return) of the security. The higher the ratio, the more favorable the assumed risk-return characteristics of the investment. The Sharpe Ratio is computed as: Si = (Ri − Rf ) σi where R̄i is the estimated mean rate of return of the asset, Rf is the risk-free rate of return, and σi is the estimated standard deviation.

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Models. Behaving. Badly.: Why Confusing Illusion With Reality Can Lead to Disaster, on Wall Street and in Life by Emanuel Derman

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But you will earn a higher reward only on average, not every time. That’s why it’s risky. So how do you tell if you are a smart investor? According to the EMM the test is the value of your portfolio’s Sharpe ratio. If your portfolio contains securities whose realized Sharpe ratios turn out to be greater than the market’s average Sharpe ratio, you have been smart. Of course, only if the EMM weren’t quite right could you hope to find exceptional Sharpe ratios. The theory of efficient markets has become so much a part of accepted market lore that professional money managers measure and report their Sharpe ratios with the hope of demonstrating their talent. Judging Investments by Their Sharpe Ratios Suppose that over the past 10 years a security A has produced an average annual return μA = 15%, with high returns of 20% and low returns of 10%, so that the annual volatility σA is 5%, corresponding to return fluctuations of ±5 percentage points about the mean.

If there is only one kind of risk, you are better off exposing yourself to it through security A than through security B. In this example we have calculated historically realized Sharpe ratios, looking backward rather than forward in time. AN ASIDE: THE PLEASURE PREMIUM The Sharpe ratio X tells you how much reward you should expect from a given chunk of risk. Risk is a fundamental quality, something you can choose to expose yourself to or try to avoid, a primitive that cannot be reduced to something simpler. The demand for risk reflects psychology and the physiology beneath it. Humans will expect different degrees of return from a given amount of risk at different times. When they become more risk-averse and demand more return from accepting the same risk, stock prices fall and the Sharpe ratio rises; conversely, when they become willing to accept less return, the ratio will fall and prices will rise.

If you consider a market with two securities, Apple stock and Apple call options, then you have two ways to expose yourself to Apple risk: via the stock or via the option. The market will be in equilibrium when the price of the option and the price of the stock are such that each security’s Sharpe ratio—each security’s expected excess return per unit of its risk—is the same for both, so that investors will have no reason to prefer one security over the other as a route to taking on Apple risk. Until that is the case, investors will preferentially buy the more efficient security and sell the less efficient one until their prices adjust to demand and they eventually provide the same risk premium. That’s equilibrium. By equating the Sharpe ratio of the stock and the Sharpe ratio of the option, Black and Scholes were able to derive and, a few years later, solve an equation for the model value of the call option. Furthermore, because the option must at all times have the same risk premium as the stock, you can replace the option at any instant by an equivalent investment in stock.

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Quantitative Value: A Practitioner's Guide to Automating Intelligent Investment and Eliminating Behavioral Errors by Wesley R. Gray, Tobias E. Carlisle

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By investigating the price metrics along these lines, we seek to find any problems not uncovered by the raw or adjusted performance measures. William Sharpe created the Sharpe ratio in 1966, intending it to be used to measure the risk-adjusted performance of mutual funds.5 Sharpe was interested in the extent to which managers took on extra risk to generate additional return. He wanted to find some measure that would adjust the return for the risk taken to generate it. He created the Sharpe ratio, which does this by examining the historical relationship between excess return—the return in excess of the risk-free rate—and volatility, which stands in for risk. The higher the Sharpe ratio, the more return is generated for each additional unit of volatility, and the better the price metric. The Sortino ratio, like the Sharpe ratio, measures risk-adjusted return. The difference is that the Sortino ratio only measures downside volatility, while the Sharpe ratio measures both upside and downside volatility.

Table 7.3 shows that the enterprise multiples have the top risk-adjusted performance, whether we examine the results using the Sharpe ratio or the Sortino ratio. The enterprise multiple (EBIT variation) monthly Sharpe ratio of 0.58 is the highest, and its monthly Sortino ratio of 0.89 is also the highest. This means the enterprise multiple (EBIT variation) metric offers the best risk/reward ratio, whether we define risk as volatility (Sharpe ratio) or just downside volatility (Sortino ratio). The EBITDA variation also stands out with favorable Sharpe and Sortino ratios of 0.53 and 0.82, respectively. TABLE 7.3 Risk Measures for the Value Decile of All Price Ratios The BM ratio has the worst risk-adjusted performance, with a Sharpe ratio of 0.33 and a Sortino ratio of 0.50. The enterprise multiples also perform well relative to all other metrics in drawdown risk.

The Graham portfolio averaged 21 positions for the full period, but Figure 1.3 illustrates that the portfolio was frequently heavily concentrated in only very few stocks, and was fully invested in only one security in 2004. In practice, portfolio risk considerations would prevent us from investing “all in” on one stock. TABLE 1.2 Performance of Graham's Simple Quantitative Value Strategy (1976 to 2011) Graham S&P 500 TR CAGR 17.80% 11.05% Standard Deviation 23.92% 15.40% Downside Deviation 16.26% 11.15% Sharpe Ratio 0.59 0.42 Sortino Ratio (MAR = 5%) 0.88 0.60 Worst Drawdown −54.61% −50.21% Worst Month Return −28.84% −21.58% Best Month Return 40.79% 13.52% Profitable Months 59.95% 61.57% Rolling 5-Year Win — 90.35% Rolling 10-Year Win — 95.53% FIGURE 1.3 Graham Strategy Portfolio Holdings over Time (1976 to 2011) Table 1.2 sets out the performance statistics for Graham's simple quantitative strategy over the period from 1976 to 2011.

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Trend Following: How Great Traders Make Millions in Up or Down Markets by Michael W. Covel

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This whole episode generated great feedback on my blog at michaelcovel.com, with Bhardwaj joining in to make his defense: “If my affiliation is the only criticism that you have of the results, I am vindicated. So stop taking about who I work for and start justifying the industry wide Sharpe Ratio of 0.09 to your invstors [sic]. You have been stealing investor money for too long, 2-20 for trend following really?????” Performance data for trend following traders, month-by-month performance, is there for all to see (Appendix B and Iasg.com). If those numbers are considered “stealing” to Bhardwaj, I can’t convince him to see another light. After word Further, Bhardwaj thinks the Sharpe ratio is an appropriate measure of trend following traders. It is not (see Chapter 3, “Performance Data”). Trend following trader David Harding has written on the Sharpe ratio: “The Sharpe ratio appears at first blush to reward returns (good) and penalize risks (bad). Upon closer inspection, things are not so simple.

As a result, their performance records have sometimes been far less impressive than the old pro trend followers. 281 The Sharpe ratio appears at first blush to reward returns (good) and penalize risks (bad). Upon closer inspection, things are not so simple. The standard deviation takes into account the distance of each return from the mean, positive or negative. By this token, large positive returns increase the perception of risk as though they could as easily be negative, which for a dynamic investment strategy may not be the case. Large positive returns are penalized and therefore the removal of the highest returns from the distribution can increase the Sharpe ratio: a case of “reductio ad absurdum” for Sharpe ratio as a universal measure of quality. David Harding, Winton Capital, www.hedgefundsreview.com 282 Everyone wants to invest when you’re at new highs and making 50 percent a year.

Large positive returns are penalized, and thus the removal of the highest returns from the distribution can increase the Sharpe ratio: a case of reductio ad absurdum for Sharpe ratio as a universal measure of quality!” Other readers on my blog responded to Bhardwaj: “Of course Geetesh Bhardwaj’s affiliation is significant. Vanguard is famous for taking the position that actively managed funds are a waste of time. That is why the vast majority of their assets under management are in indexed funds. So is it surprising that their marketing department hired an economics major to write reports that show active management in a bad light? Don’t hold it against Geetesh. His previous job being a vice president of a disaster like AIG can’t look good on a resume. He’s probably lucky to be working at all.” Another reader responded: “The Sharpe ratio of CTAs [trend following traders] does not need to be ‘explained.’

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Beyond the Random Walk: A Guide to Stock Market Anomalies and Low Risk Investing by Vijay Singal

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However, it is important to take risk into account because industry-momentum-based trading strategies are clearly riskier than holding a 91 92 Beyond the Random Walk broader market portfolio. The Sharpe ratio is used to compare the risk-adjusted returns.3 Sharpe ratios are reported below based on annual returns for the best-case scenarios, an average return of 4 percent for short-term Treasury bills during 1997–2001, and standard deviations (reported in parentheses) in Table 5.4. 25-week estimation period and 5-week holding period 5-week estimation period and 5-week holding period 5-week estimation period and 1-week holding period S&P 500 holding return 0.91 0.76 1.05 0.49 Since higher Sharpe ratios indicate superior investment, the best results are for the one-week holding period, with a ratio of 1.05. The five-week holding period with a twenty-five-week estimation period is only slightly lower at 0.91.

If an optimal currency portfolio, consisting of the German mark, Japanese yen, Swiss franc, and the British pound with the U.S. dollar as the risk-free asset, is formed, then it has been found that the portfolio would have generated an average excess return of 2.79 percent per year over the period November 1989 through June 1999. The Sharpe ratio for this portfolio is 0.69 compared with a Sharpe ratio of 0.53 for the U.S. Treasury index, 0.49 for an unhedged global Treasury index, 0.80 for a hedged global Treasury index, and 0.95 for the S&P 500. The Sharpe ratio for the S&P 500 is unusually high because of the high returns earned by stocks during this period. Overall, the evidence suggests that holding currencies can be a superior form of investment than several other forms of investment, after accounting for risk, even during the 1990s. Strategies based on the forward rate bias are implemented and tested later with more current data. 265 266 Beyond the Random Walk Explanations The previous section established that the forward rate is biased in the sense that, on average, the actual future spot rate is not equal to the forward rate.

Beyond the Random Walk Table 11.6 Bias in Currency Forward Rates The annual return over the period January 2000 to June 2002 is 15.6 percent, slightly better than the 13.4 percent return in Table 11.5. The excess return is 11.5 percent with a standard deviation of 10.7 percent and a Sharpe ratio of 1.07. The excess return is impressive, as is the risk-adjusted return as measured by the Sharpe ratio. Overall, implementation of the trading strategy reveals annual excess returns of 9.3 percent and 11.5 percent. The strategy seems to be eminently successful with Sharpe ratios of 0.85 and 1.07. Qualifications The evidence presented in this chapter and the trading strategy recommendations based on the evidence rely on past data. Since future market conditions and market patterns may be completely different, and particularly as no valid and rational explanation for the forward rate bias is known, the forward rate bias may cease to exist without notice.

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Mathematics of the Financial Markets: Financial Instruments and Derivatives Modelling, Valuation and Risk Issues by Alain Ruttiens

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When it is the case, above performance measures cannot hold, in particular the Sharpe ratio and its variants. The Sortino Ratio The Sortino ratio is built in a similar way as the Sharpe ratio, except that, instead of dividing by the standard deviation of a series of past returns, the divisor is the downside semi-standard deviation σd, that is, the standard deviation of the past negative returns. This makes sense when the portfolio is involving asymmetric instruments like options (cf. Chapter 4, Sec-tion 4.3.7): Example. On the same data as for the previous Sharpe ratio example in Section 14.1.3, that is a fund invested in the S&P 500, data of year 2009, with a p.a. standard deviation of negative returns (downside semi-standard deviation) of 19.76%, the result is instead of 0.59 for the usual Sharpe ratio. The Omega Ratio5 The Omega Ω(L) is the probability weighted ratio of returns above some threshold L, to returns below this threshold.

For an example, with two securities, see Chapter 4, Section 4.3.3. 14.1.3 Risk versus return ratios, or performance measures Absolute Performance Measures The Sharpe Ratio Following the Portfolio Theory, which is elaborated on the basis of the return – risk paradigm, investors or traders/speculators are actually concerned by both returns and associated risks. With this respect, it makes sense to assess the attractiveness of a return by considering the risk associated with. The simplest way of doing this is by dividing the return by the risk. However, a risk-less investment, in a non-defaultable government bond, pays a risk-free rate for σ = 0. Hence, it makes sense to consider that in a risky investment (r, σ) it should be the excess return only, that is, r − rf, that pays for the supported risk. Hence the Sharpe ratio: Practically speaking, for a given period of past data leading to r and σ measures, the rf rate must be of a non-defaultable government bill or bond of maturity coinciding with the same period of time as used for r and σ.

Hence the Sharpe ratio: Practically speaking, for a given period of past data leading to r and σ measures, the rf rate must be of a non-defaultable government bill or bond of maturity coinciding with the same period of time as used for r and σ. The data for r and rf being usually expressed on a p.a. basis, σ must also be computed on a p.a. basis. Example. For a fund passively invested in the S&P 500 in 2009, the computed return and risk were 17.96% p.a. and 27.04% p.a. respectively (based on daily closing prices). The corresponding 12-month T-Bill was 2.004%. The Sharpe ratio is Note that in the fund industry, it is hard to achieve a Sharpe ratio above 1, which may be viewed as a reference level. The Treynor Ratio In the funds industry, the performance objective of a portfolio P is generally referring to a benchmark, typically an equity index. It makes thus sense to rate the portfolio return rP to the index return, through its βP (cf. Chapter 4, Section 4.3.4), hence the Treynor ratio: Jensen's Alpha With respect to the CAPM (cf.

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The Crisis of Crowding: Quant Copycats, Ugly Models, and the New Crash Normal by Ludwig B. Chincarini

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It was as if Meriwether and his 140 employees had a magic money tree. TABLE 2.1 LTCM Returns versus Standard Asset Classes The Sharpe ratio is an important, risk-adjusted tool for comparing the performances of different investments or portfolio managers. The higher the ratio, the better the portfolio manager. (See Box 2.1.) Table 2.1 shows that, before 1998, the LTCM fund had a Sharpe ratio five times that of the standard returns of U.S. Treasury bills and bonds. Box 2.1 The Sharpe Ratio The Sharpe ratio measures the return of a portfolio minus the risk-free rate divided by the portfolio’s standard deviation. It is a risk-adjusted return measure that assists in comparing different portfolios or investments, even in the presence of leverage. If portfolio A has a higher Sharpe ratio than portfolio B, then there is no amount of leverage that can make portfolio B as good as A.

In these cases, a better measure is the Sortino ratio, which is similar to the Sharpe ratio but divides excess return by the semistandard deviation rather than the standard deviation. Consider a golf analogy. Suppose an average amateur golfer plays with a top pro golfer such as Phil Mickelson. On every hole, Mickelson drives his ball further than the amateur does. Think of this as the portfolio’s net return. It’s important, but it’s not enough to judge who’s a better golfer. Distance matters, but so does accuracy. How often do drives land in the fairway? The Sharpe ratio is essentially drive distance divided by the variation of the drive distance from the center of the fairway. It lets us compare two golfers. It probably goes without saying that Phil Mickelson had a lower variation than the average amateur, and a much higher Sharpe ratio. He is clearly the better golfer.

In each fund’s first three years, LTCM had an average monthly return of 2.31% and a monthly volatility of 2.59%. JWMP had a 0.87% monthly return and a 1.28% monthly standard deviation. The new firm’s risk was about half that of LTCM, and its returns were between a third and a half of its predecessor’s. LTCM’s Sharpe ratio was 2.54 during its first three years, much higher than JWMP’s 1.63. JWMP existed from December 1999 to April 2009. It performed reasonably well during its first eight years, which led up to the 2008 financial crisis. Before 2008, the fund’s average annual return was 8.55%, with a standard deviation of 3.81%. The fund’s Sharpe ratio was 1.33. In comparison, the S&P 500 had a Sharpe ratio of 0.00 over the same period. The fund’s best monthly return was 3.64%; its worst monthly return was −2.99%. TABLE 14.1 The LTCM Spinoff and Copycat Returns Table 14.1 shows JWMP’s performance and that of other well-known relative-value hedge funds and major asset classes, such as the S&P 500.

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Python for Finance by Yuxing Yan

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.] >>> The output shows that the function value is 3, and it is achieved by assigning x as 0. [ 214 ] Chapter 8 Constructing an optimal portfolio In finance, we are dealing with the trade-off between risk and return. One of the widely used criteria is the Sharpe ratio, which is defined as follows: 6KDUSH ( 5 5I VS (27) The following program would maximize the Sharpe ratio by changing the weights of the stock in the portfolio. We have several steps in the program: the input area is very simple, just several tickers in addition to the beginning and ending dates. Then, we define four functions: converting daily returns into annual ones, estimate a portfolio variance, estimate the Sharpe ratio, and estimate the nth weight when n-1 weights are given: from matplotlib.finance import quotes_historical_yahoo import numpy as np import pandas as pd import scipy as sp from scipy.optimize import fmin # Step 1: input area ticker=('IBM','WMT','C') # tickers begdate=(1990,1,1) # beginning date enddate=(2012,12,31) # ending date rf=0.0003 # annual risk-free rate In the second part of the program, we define a few functions: download data from Yahoo!

Our major function would start from Step 3 as shown in the following code: # Step 3: generate a return matrix (annul return) n=len(ticker) # number of stocks x2=ret_annual(ticker[0],begdate,enddate) for i in range(1,n): x_=ret_annual(ticker[i],begdate,enddate) x2=pd.merge(x2,x_,left_index=True,right_index=True) # using scipy array format R = sp.array(x2) print('Efficient porfolio (mean-variance) :ticker used') print(ticker) [ 216 ] Chapter 8 print('Sharpe ratio for an equal-weighted portfolio') equal_w=sp.ones(n, dtype=float) * 1.0 /n print(equal_w) print(sharpe(R,equal_w)) # for n stocks, we could only choose n-1 weights w0= sp.ones(n-1, dtype=float) * 1.0 /n w1 = fmin(negative_sharpe_n_minus_1_stock,w0) final_w = sp.append(w1, 1 - sum(w1)) final_sharpe = sharpe(R,final_w) print ('Optimal weights are ') print (final_w) print ('final Sharpe ratio is ') print(final_sharpe) From the following output, we know that if we use a naïve equal-weighted strategy, the Sharpe ratio is 0.63. However, the Sharpe ratio for our optimal portfolio is 0.67: Constructing an efficient frontier with n stocks Constructing an efficient frontier is always one of the most difficult tasks for finance instructors since the task involves matrix manipulation and a constrained optimization procedure. One efficient frontier could vividly explain the Markowitz Portfolio theory.

Estimate the correlation coefficient between IBM, DELL, and W-Mart. 21. Why is it claimed that the sn.npv() function from SciPY() is really a Present Value (PV) function? 22. Design a true NPV function using all cash flows, including today's cash flow. 23. The Sharpe ratio is used to measure the trade-off between risk and return: Sharpe = R − Rf σ Here, R is the expected returns for an individual security, and R f is the expected risk-free rate. σ is the volatility, that is, standard deviation of the return on the underlying security. Estimate Sharpe ratios for IBM, DELL, Citi, and W-Mart by using their latest five-year monthly data. [ 122 ] Visual Finance via Matplotlib Graphs and other visual representations have become more important in explaining many complex financial concepts, trading strategies, and formulae.

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Frequently Asked Questions in Quantitative Finance by Paul Wilmott

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It is calculated as where µ is the return on the strategy over some specified period, r is the risk-free rate over that period and σ is the standard deviation of returns. The Sharpe ratio will be quoted in annualized terms. A high Sharpe ratio is intended to be a sign of a good strategy. If returns are normally distributed then the Sharpe ratio is related to the probability of making a return in excess of the risk-free rate. In the expected return versus risk diagram of Modern Portfolio Theory the Sharpe ratio is the slope of the line joining each investment to the risk-free investment. Choosing the portfolio that maximizes the Sharpe ratio will give you the Market Portfolio. We also know from the Central Limit Theorem that if you have many different investments all that matters is the mean and the standard deviation. So as long as the CLT is valid the Sharpe ratio makes sense. The Sharpe ratio has been criticized for attaching equal weight to upside ‘risk’ as downside risk since the standard deviation incorporates both in its calculation.

The most popular is the Sharpe ratio. Example One stock has an average growth of 10% per annum, another is 30% per annum. You’d rather invest in the second, right? What if I said that the first had a volatility of only 5%, whereas the second was 20%, does that make a difference? Long Answer Performance measures are used to determine how successful an investment strategy has been. When a hedge fund or trader is asked about past performance the first question is usually “What was your return?” Later maybe “What was your worst month?” These are both very simple measures of performance. The more sensible measures make allowance for the risk that has been taken, since a high return with low risk is much better than a high return with a lot of risk. Sharpe Ratio The Sharpe ratio is probably the most important non-trivial risk-adjusted performance measure.

The Sharpe ratio has been criticized for attaching equal weight to upside ‘risk’ as downside risk since the standard deviation incorporates both in its calculation. This may be important if returns are very skewed. Modigliani-Modigliani Measure The Modigliani-Modigliani or M2 measure is a simple linear transformation of the Sharpe ratio:M2 = r + ν × Sharpe where ν is the standard deviation of returns of the relevant benchmark. This is easily interpreted as the return you would expect from your portfolio if it were (de)leveraged to have the same volatility as the benchmark. Sortino Ratio The Sortino ratio is calculated in the same way as the Sharpe ratio except that it uses the square root of the semi-variance as the denominator measuring risk. The semi variance is measured in the same way as the variance except that all data points with positive return are replaced with zero, or with some target value.

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Inside the House of Money: Top Hedge Fund Traders on Profiting in a Global Market by Steven Drobny

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Big Bet Performance Analysis Hedge fund managers are often judged by their Sharpe ratios, which are calculated as the fund’s return minus the risk-free rate divided by the volatility of returns. The Sharpe ratio is also known as the reward-tovolatility ratio and provides a sense of the quality of the managers’ returns per unit of risk. It allows for some element of comparison across managers. Hedge funds often strive for a Sharpe ratio of at least 1.0 such that their 70% 60% Probability 50% 40% 30% 20% 10% 0% –50% –40% –30% –20% –10% 0% Return FIGURE A.1 Hypothetical Big Bet Portfolio 10% 20% 30% 40% 50% WHY GLOBAL MACRO IS THE WAY TO GO 345 performance is commensurate with the amount of risk assumed. Hedge funds that produce a Sharpe ratio well over 1.0 attract investor interest while those with a Sharpe ratio well below 1.0 do not.

In most scenarios, however, performance tends to cluster around the 10 percent return investors were told to expect. (See Figure A.2.) Multibet Performance Analysis Checking the performance of the multibet approach, we find the strategy had a volatility of about 20 percent, resulting in a Sharpe ratio of 0.25.While this figure is not particularly impressive, it represents a drastic improvement to the 0.1 Sharpe ratio produced by the single big bet approach. Both the big bet strategy and the multibet strategy produced average annual returns of 10 percent, but the multibet approach produced much less volatility and thus a higher Sharpe ratio. In sum, the smoother ride of the diversified portfolio might have compelled investors to stick around. HOW TO PRODUCE SUPERIOR RISK-ADJUSTED RETURNS By simply increasing the number of independent bets from one to five per year, the same better-than-average hedge fund manager increased the qual- WHY GLOBAL MACRO IS THE WAY TO GO 347 ity of returns considerably.

But that’s the beauty of 62 INSIDE THE HOUSE OF MONEY being able to take on leverage.When we allocate 20 percent of our risk to fixed income, it doesn’t mean we only put 20 percent of our assets into fixed income.There are all kinds of interesting things you can do in fixed income with leverage and still only utilize 20 percent of your capital. For example, you could put 40 percent of your capital into shorterduration bonds.When using leverage, you want the highest Sharpe ratio because you’re borrowing money against your investment, and the best Sharpe ratios are found in the two years and under the sector of fixed income. On an absolute return basis, two years and under bonds are not going to pay as much as a 10-year bond because the yields are usually lower. But the risk-to-return ratio is also very different.You could be five times levered in the two-year and get a higher payout with the same risk as a 10-year bond because of duration.

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Optimization Methods in Finance by Gerard Cornuejols, Reha Tutuncu

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This quantity is precisely the reward-to-variability ratio introduced by Sharpe to measure the performance of mutual funds [15]. Now more commonly known as the Sharpe measure, or Sharpe ratio, this quantity measures the expected return per unit of risk (standard deviation) for a zero-investment strategy. The portfolio that maximizes the Sharpe ratio is found by solving the following problem: maxx µT x−rf (xT Qx)1/2 Ax = b Cx ≥ d. (5.3) In this form, this problem is not easy to solve. Although it has a nice, polyhedral feasible region, its objective function is somewhat complicated, and worse, is possibly non-concave. Therefore, (5.3) is not a convex optimization problem. The standard strategy to find the portfolio maximizing the Sharpe ratio, often called the optimal risky portfolio, is the following: First, one traces out the efficient frontier on a two dimensional return vs. standard deviation graph.

We present these conditions next. xR is an optimal solution of problem (5.1) if and only if there exists λR ∈ <, γE ∈ <m , and γI ∈ <p satisfying the following conditions: QxR − λR µ − AT γE − C T γI = 0, µT xR ≥ R, AxR = b, CxR ≥ d, λR ≥ 0, λR (µT xR − R) = 0, γI ≥ 0, γIT (CxR − d) = 0. 5.2 (5.2) Maximizing the Sharpe Ratio Consider the setting in the previous subsection. Let us define the function σ(R) : [Rmin , Rmax ] → < as σ(R) := (xTR QxR )1/2 , where xR denotes the unique solution of problem (5.1). Since we assumed that Q is positive definite, it is easily shown that the function σ(R) is strictly convex in its domain. As mentioned before, the efficient frontier is the graph E = {(R, σ(R)) : R ∈ [Rmin , Rmax ]}. 5.2. MAXIMIZING THE SHARPE RATIO 61 We now consider a riskless asset whose expected return is rf ≥ 0. We will assume that rf < Rmin , which is natural since the portfolio xmin has a positive risk associated with it while the riskless asset does not.

Asset Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 3 3 4 4 5 6 7 8 9 11 11 . . . . . . . . 13 13 14 17 18 18 21 24 27 . . . . . 29 29 30 31 34 36 iv CONTENTS 3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Quadratic Programming: Theory and Algorithms 4.1 The Quadratic Programming Problem . . . . . . . 4.2 Optimality Conditions . . . . . . . . . . . . . . . . 4.3 Interior-Point Methods . . . . . . . . . . . . . . . . 4.4 The Central Path . . . . . . . . . . . . . . . . . . . 4.5 Interior-Point Methods . . . . . . . . . . . . . . . . 4.5.1 Path-Following Algorithms . . . . . . . . . . 4.5.2 Centered Newton directions . . . . . . . . . 4.5.3 Neighborhoods of the Central Path . . . . . 4.5.4 A Long-Step Path-Following Algorithm . . . 4.5.5 Starting from an Infeasible Point . . . . . . 4.6 QP software . . . . . . . . . . . . . . . . . . . . . . 4.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 5 QP 5.1 5.2 5.3 5.4 5.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Models and Tools in Finance Mean-Variance Optimization . . . . . . . . . . . . . . . . Maximizing the Sharpe Ratio . . . . . . . . . . . . . . . Returns-Based Style Analysis . . . . . . . . . . . . . . . Recovering Risk-Neural Probabilities from Options Prices Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Stochastic Programming Models 6.1 Introduction to Stochastic Programming . 6.2 Two Stage Problems with Recourse . . . . 6.3 Multi Stage Problems . . . . . . . . . . . . 6.4 Stochastic Programming Models and Tools 6.4.1 Asset/Liability Management . . . . 6.4.2 Corporate Debt Management . . . . . . . . . . . . . . . . . . . . . in Finance . . . . . . . . . . . . 7 Robust Optimization Models and Tools in Finance 7.1 Introduction to Robust Optimization . . . . . . . . . 7.2 Model Robustness . . . . . . . . . . . . . . . . . . . . 7.2.1 Robust Multi-Period Portfolio Selection . . . . 7.3 Solution Robustness . . . . . . . . . . . . . . . . . . 7.3.1 Robust Portfolio Selection . . . . . . . . . . . 7.3.2 Robust Asset Allocation: A Case Study . . . 7.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 . . . . . . . . . . . . 43 43 44 45 48 49 49 50 53 55 56 56 57 . . . . . 59 59 60 63 65 68 . . . . . . 71 71 72 74 76 76 78 . . . . . . . 83 83 83 84 88 88 90 92 CONTENTS 8 Conic Optimization 8.1 Conic Optimization Models and Tools in Finance 8.1.1 Minimum Risk Arbitrage . . . . . . . . . . 8.1.2 Approximating Covariance Matrices . . . . 8.2 Exercises . . . . . . . . . . . . . . . . . . . . . . .

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The Investopedia Guide to Wall Speak: The Terms You Need to Know to Talk Like Cramer, Think Like Soros, and Buy Like Buffett by Jack (edited By) Guinan

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In most cases, retained earnings are the larger of the two figures. Related Terms: • Capital Structure • Equity • Retained Earnings • Common Stock • Preferred Stock Sharpe Ratio What Does Sharpe Ratio Mean? A ratio developed by Nobel laureate William F. Sharpe that is used to measure risk-adjusted performance. rp − rf The Sharpe ratio is calculated by sub= tracting the risk-free rate, such as that σp of the 10-year U.S. Treasury bond, from Where : the rate of return of a portfolio and rp = Expected portfolio return then dividing the result by the stanrf = Rissk free rate dard deviation of the portfolio returns. σp = Portfolio standard deviation Investopedia explains Sharpe Ratio The Sharpe ratio indicates whether a portfolio’s returns are due to smart investment decisions or are a result of excess risk. This measurement is very useful because although one portfolio or fund can reap higher returns than its peers, it is a good investment only if those higher returns are not a result of taking on too much additional risk.

Investopedia explains Alpha (1) Alpha is one of five technical risk measures that are used in modern portfolio theory (MPT); the others are beta, standard deviation, R-squared, and the Sharpe ratio. These indicators help investors determine the risk-reward profile of a mutual fund. Simply stated, alpha often is considered to represent the value that a portfolio manager adds to or subtracts from a fund’s return. A positive alpha of 1.0 means the fund has outperformed its benchmark index by 1%. Conversely, a similar negative alpha would indicate an underperformance of 1%. (2) If a CAPM analysis estimates that a portfolio should earn 10% on the basis of the risk of that portfolio yet the portfolio actually earns 15%, the portfolio’s alpha would be 5%. The 5% is the excess return above the predicted CAPM return. Related Terms: • Beta • Capital Asset Pricing Model—CAPM • R-Squared • Sharpe Ratio • Standard Deviation American Depositary Receipt (ADR) What Does American Depositary Receipt (ADR) Mean?

This measurement is very useful because although one portfolio or fund can reap higher returns than its peers, it is a good investment only if those higher returns are not a result of taking on too much additional risk. The greater a portfolio’s Sharpe ratio is, the better its risk-adjusted performance has been. A variation of the Sharpe ratio is the Sortino ratio, which removes the effects of upward price movements on standard deviation to measure only return against downward price volatility. 270 The Investopedia Guide to Wall Speak Related Terms: • Portfolio • Risk-Free Rate of Return • Total Return • Risk • Standard Deviation Short (or Short Position) What Does Short (or Short Position) Mean? (1) The sale of a borrowed security, commodity, or currency with the expectation that the asset will fall in value. (2) In the context of options, it is the sale (also known as “writing”) of an options contract.

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Finance and the Good Society by Robert J. Shiller

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The manager can take home high management fees for all the years that the risks do not show up, and then walk away when the catastrophe finally comes. For years, nance students have been taught to use the Sharpe ratio to evaluate whether a portfolio manager is really beating the market. The Sharpe ratio, named after Stanford University nance professor William Sharpe, is the average excess return over the historical life of the manager’s portfolio above the return of the market of all possible investments as a whole divided by the standard deviation of the return over the historical life of the manager’s portfolio. A high Sharpe ratio is taken as a sign of a good investment manager. If the manager is outperforming the market consistently, then the numerator of the ratio should be large. But if the manager is taking signi cant risks to achieve a high return relative to the market, that will show up in the denominator as high variability in the manager’s portfolio return, and thus bring down the Sharpe ratio.

The private investment company Integral Investment Management, managed by former biologist Conrad Seghers, advertised, according to a Wall Street Journal story, an extremely high Sharpe ratio but disclosed that it was pursuing some unusual derivatives activities.17 According to Goetzmann and his co-authors, Integral was coming close to the optimal Sharpe ratio manipulation because of massive sales of out-of-the-money puts on U.S. equity indices and a short call position implicit in the hedge fund fees. The manipulation worked, and Integral managed to persuade the Art Institute of Chicago to invest \$43 million of its endowment in Integral and related funds. After the stock market collapse in 2001, and at least a \$20 million loss on its investment, the Art Institute sued Integral. Integral was caught on a number of securities law violations, but it was not penalized for its unusual Sharpe ratio strategy.18 This example illustrates why investing cannot be done by the numbers alone.

But if the manager is taking signi cant risks to achieve a high return relative to the market, that will show up in the denominator as high variability in the manager’s portfolio return, and thus bring down the Sharpe ratio. But the Sharpe ratio is not necessarily a reliable indicator of a manager’s performance, as the risks do not necessarily show up in a high standard deviation of returns for the portfolio over most of its life. If there is no news about the risks, then prices will not change, until the catastrophe comes. Consider for example the risk of investing in politically unstable economies. Those investments are inherently very risky. But suppose the country was Egypt under Hosni Mubarak. The end of his regime came with shocking suddenness in 2011. It can be traced to an outbreak of riots in neighboring Tunisia.

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The Power of Passive Investing: More Wealth With Less Work by Richard A. Ferri

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William Sharpe developed a similar formula for evaluating risk-adjusted returns of portfolios. Ironically, rather than using his own beta formula as the denominator in the equation, Sharpe used a portfolio’s standard deviation of return. Perhaps this was because Treynor beat him to the punch. Sharpe’s formula became known as the Sharpe Ratio. An interesting 1966 paper published by Sharpe in the Journal of Business evaluated the performance of 34 mutual funds over a period from 1954–1963 using the Sharpe ratio; the Treynor Ratio; and a third factor, fund expenses.12 Sharpe’s intent was to compare the three methods and perhaps determine which was better at determining skill among mutual fund managers. Sharpe found sufficient evidence that all three ratios had some predictability for selecting funds relative to each other, although no one method isolated funds that consistently outperformed the market as measured by the DJIA (Sharpe doesn’t disclose why he chose this limited market indicator when the more comprehensive S&P 500 existed).

Sharpe acknowledged that the DJIA had no transaction cost or administrative expenses; however, he also noted that the fund returns were calculated without deducting their sales commission, which for most was 8.5 percent. Here are the results: The market as measured by the DJIA was less than 11 active funds and better than the remaining 23 funds. Basically, there was one winning fund for every two losing funds, a win-loss ratio of 1 to 2. The Sharpe Ratio for the Dow was 0.67 while the average ratio for the 34 funds was only 0.63. This means the Dow had a better return per unit of risk than the average mutual fund in the study. The Treynor Ratio returned results similar to the Sharpe Ratio. Sharpe also tested fees as a predictor of return. He makes this important observation about fees near the conclusion of the paper: While it may be dangerous to generalize the results found during one ten-year period, it appears that the average mutual fund selects a portfolio at least as good as the Dow-Jones Industrials, but that the results actually obtained by the holder of mutual fund shares (after the costs associated with the operation of the fund have deducted) fall somewhat short of those from the Dow-Jones portfolio.

These funds can experience higher share-price volatility than diversified funds because sector funds are subject to issues specific to a given sector. Securities and Exchange Commission (SEC) The federal government agency that regulates mutual funds, registered investment advisors, the stock and bond markets, and broker-dealers. The SEC was established by the Securities Exchange Act of 1934. Sharpe ratio A measure of risk-adjusted return. To calculate a Sharpe ratio, an asset’s excess return (its return in excess of the return generated by risk-free assets such as Treasury bills) is divided by the asset’s standard deviation. It can be calculated compared to a benchmark or an index. short sale The sale of a security or option contract that is not owned by the seller, usually to take advantage of an expected drop in the price of the security or option.

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The Invisible Hands: Top Hedge Fund Traders on Bubbles, Crashes, and Real Money by Steven Drobny

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Basically, our risk process mimics our investment process in that we work down from macro factor diversification to micro factor risk, allowing us to integrate the equity and commodity strategies that comprise our overall portfolio. Sharpe Ratio The Sharpe, or reward-to-variability, ratio is a measure of the excess return (or risk premium) per unit of risk in an investment asset or a trading strategy. The Sharpe ratio is used to measure the return of an asset relative to the level of risk taken. When comparing two assets, an investor can compare the expected returns E[R] against the relative benchmarks with return Rf. The asset with the higher Sharpe ratio gives more return for the same risk. How do you deal with the closet dollar exposure? We hedge our dollar factor exposure actively. For example, we are currently running pretty heavy overall risk, but because our dollar factor started to go to 50 percent of the portfolio, we cut this risk materially by selling a basket of currencies versus the dollar.

I would need to hear the story behind the numbers: how money was made, how much risk was taken, etc. So I would like to see a combination of people whose thinking is logical, consistent, and sounds as if they would be alpha extracting over time (see box). Then I would want the track record to verify it. Alpha Versus Beta In modern portfolio theory (MPT), there are five basic statistical measurements: beta, alpha, standard deviation (volatility), R-squared (correlation), and the Sharpe ratio (return/risk). Beta measures both the correlation and volatility of a fund or security to a benchmark. For example, if a fund has a beta of 2.0 in relation to the S&P 500, the fund’s returns are on average double those of the S&P. If a fund has a beta of -0.5, the fund’s returns are on average half those of the S&P, and in the opposite direction. Alpha is a risk-adjusted measure of the excess return of a fund or manager relative to an applicable benchmark.

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Hedge Fund Market Wizards by Jack D. Schwager

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For the near nine-year period since its inception, Cornwall Capital has realized an average annual compounded net return of 40 percent (52 percent gross).2 The annualized standard deviation has been relatively high at 32 percent (37 percent gross). Cornwall’s Sharpe ratio of 1.12 (1.23 gross) represents very good performance based on this widely used return/risk measure, but greatly understates the true return/risk performance of the fund. Cornwall is the poster child for the inadequacy of the Sharpe ratio if applied to managers with non-normal return distributions. The crux of the problem is that the Sharpe ratio uses volatility as the proxy for risk. Because of the asymmetric design of its trades, Cornwall’s volatility consists mostly of upside volatility. In other words, Cornwall’s volatility is high because they have many instances of very large gains. However, I have yet to meet an investor who finds large gains to be a problem. The risk measure of the Sharpe ratio, the standard deviation, will penalize exceptionally large gains.

Since its inception in 2004, Benedict’s fund has realized an average annualized compounded net return of 11.5 percent (19.3 percent gross). If this return does not sound sufficiently impressive, keep in mind that it was achieved with an extremely low annualized volatility of 5.8 percent and, even more impressive, a maximum drawdown of less than 5 percent. Benedict’s return/risk numbers are exemplary. His Sharpe ratio is very high at 1.5. The Sharpe ratio, however, understates Benedict’s performance because this statistic does not distinguish between upside and downside volatility, and in Benedict’s case, most of the limited volatility is on the upside. Benedict’s Gain to Pain ratio is an extremely high 3.4. (See Appendix A for an explanation of the Gain to Pain ratio.) Benedict won’t allow family or friends to invest in his fund, as one of his friends since childhood told me.

And I found that advice markedly stupid because I knew I could find a way to try any number of combinations and not overfit the data. You get new out-of-sample data every day. If you are rigorous about acknowledging what that new data is telling you, you can really get somewhere. It may take a while. If you are trading the system, and it is not performing in line with expectations over some reasonable time frame, look for overfit and hindsight errors. If you are expecting a Sharpe ratio above 1, and you are getting a Sharpe ratio under 0.3, it means that you have made one or more important hindsight errors, or badly misjudged trading costs. I was using the data up to a year before the current date as the training data set, the final year data as the validation data set, and the ongoing realtime data as the test. Effectively, the track record became the test data set. I understand why you would look for a method that was not trend following, given your aversion to following the herd, but why would you inherently avoid a mean reversion approach?

Evidence-Based Technical Analysis: Applying the Scientific Method and Statistical Inference to Trading Signals by David Aronson

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This discussion here is conﬁned to the sampling distribution of the mean because the performance statistic used to evaluate TA rules in this book is a rule’s mean rate of return. However, there are many other performance measures that might be used: the Sharpe ratio,30 the proﬁt factor,31 the mean return divided by the Ulcer Index,32 and so forth. It should be pointed out that the sampling distributions of these alternative performance statistics would be different from the sampling distribution of the mean. It should also be pointed out that the methods used in this book to generate the sampling distribution of the mean may be of limited value in generating sampling distributions for performance statistics with elongated right tails. This can occur with performance statistics that involve ratios such as the Sharpe ratio, the mean-return-to-Ulcer-Index ratio, and the proﬁt factor. However, this problem can be mitigated by taking the log of the ratio to shorten the right tail.

There are data suggesting that trend following in stocks is indeed a less rewarding enterprise.103 Lars Kestner compared the performance of trend-following systems for a portfolio of futures and a portfolio of stocks. The futures considered were 29 commodities in 8 different sectors.104 The stocks were represented by 31 large-cap stocks in 9 different industry sectors,105 and 3 stock indices. Risk-adjusted performance (Sharpe ratio) was computed for 5 different trend-following systems,106 over the period January 1, 1990 through December 31, 2001. The Sharpe ratio averaged over the ﬁve trend-following systems in futures was 384 METHODOLOGICAL, PSYCHOLOGICAL, PHILOSOPHICAL, STATISTICAL FOUNDATIONS .604 versus .046 in stocks. These results support the notion that futures trend followers are earning a risk premium that is not available to trend followers in stocks. See Figure 7.15. Liquidity Premium and the Gains to Counter Trend Trading in Stocks The stock market offers risk-taking stock traders a different form of compensation.

The denominator is stated as an absolute value (no sign) so the value of the proﬁt factor is always positive. A value of 1 would indicate the rule is break even. A superior way to calculate the proﬁt factor to transform it to have a natural zero point is to take the log of the ratio. The Ulcer Index is an alternative and possibly superior measure of risk that considers the magnitude of equity retracements, which are not directly considered by the Sharpe ratio. The standard deviation, the risk measure employed by the Sharpe ratio, does not take into account the sequence of winning and losing periods. For a deﬁnition of the Ulcer Index, see P.G. Martin and B.B. McCann, The Investor’s Guide to Fidelity Funds (New York: John Wiley & Sons, 1989), 75–79. A similar concept, the return to retracement ratio, is described by J.D. Schwager, Schwager on Futures—Technical Analysis (New York: John Wiley & Sons, 1996).

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Nerds on Wall Street: Math, Machines and Wired Markets by David J. Leinweber

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Prognostications here include: an increase in the complexity of derivative and structured products driven by the demands of alpha-seeking strategies; some products’ requirement of willingness to commit capital in innovative ways; and increased trading interest in risk classes, over individual securities. *The Sharpe ratio is a measure of management skill that adjusts pure alpha (value added) by the variability of that value added. Details of the Sharpe ratio can be found at http:// en.wikipedia.org/wiki/Sharpe_ratio. Algorithm Wars 81 Both articles forecast an increasingly risk-centric view of trading. IBM opines, “As the industry matures, many traditional activities will come under increasing pressure and new value engines will emerge. Activities under pressure are unnecessary bundles and transaction businesses.Value engines will be risk assumption and risk mitigation.”

In Chapter 6, the last of this part, “Stupid Data Miner Tricks,” we see how with the right mix of hubris, stupidity, and CPU cycles, it is possible to do some real damage to your financial health. In investing, as in the bomb squad, knowing what not to do is extremely worthwhile. *The Sharpe ratio is a measure of management skill that adjusts pure alpha (value added) by the variability of that value added. The others (Jensen & Treynor) are refinements based on characteristics of the portfolio, such as beta. They are less commonly used. Details are here http://en.wikipedia. org/wiki/Sharpe_ratio. Chapter 4 Where Does Alpha Come From? Life Is Alpha. The Rest Is Details. —POPULAR T-SHIRT AT HEDGE FUND EVENTS T here was a time not too long ago when, if you posed the question “Where does alpha come from?” to a roomful of academic financial economists, most of them would complain: “It’s a trick question!

This is particularly true for equities. Financial models never capture every aspect of market participants’ motivations. Varied outcomes likely. Simple games like tic-tac-toe can be modeled exactly. One action always leads to another. This is clearly not the case in trading. Performance feedback and reinforcement. Performance measurement is natural for trading agents. For alpha-seeking algos, metrics like the Sharpe ratio* fit. Pure execution algos use implementation cost or VWAP shortfall. Layered behaviors. Agents should have default behaviors that complete their tasks and avoid errors. Basic behavior is at the lower layers, more sophisticated behavior above. Some of these agents will be programs, and some will be people. We can call these people “the employed traders of 2015.” Markets in 2015, Focus on Risk Two recent prognostications on markets in 2015 are remarkably similar.

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Derivatives Markets by David Goldenberg

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This leads to a ﬁnal way of understanding the generation of EMMs in a world of nonreplicability. 17.3 MPRS AND EMMS, ANOTHER VERSION OF FTAP 2 We start with the condition for no-arbitrage. It is shown in mathematical ﬁnance that there are equivalent ways to ensure the existence of an EMM for the discounted underlying price process (no-arbitrage). One important necessary and sufﬁcient condition is the existence of a ‘Market Price of Risk’ (MPR), which is also called the Sharpe Ratio. We mentioned this important concept brieﬂy as a relative risk measure in Chapter 15. Under the appropriate technical assumptions, EMMs are in one-to-one correspondence with Sharpe ratios. This is because the market price of risk for security i, (MPRi )=(i–r)/i . is the generator of EMMs. In terms of riskneutral probabilities and the actual probabilities, the mechanism of the transformation from pi to q*i is Girsanov’s Theorem, provided the assumptions underlying Girsanov actually hold.

We can further simplify the [Risk Premia Cancellation Condition] by substituting into it, CC = SS ⎛ ⎞ = ⎜ C ⎟ SS ⎝ S ⎠ 624 OPTIONS which says, after a little algebra, that, C C C = S S S or, C = S RC RS [Risk Premia Cancellation Condition, Returns] Note that, just as RH=H/H0, by exactly the same argument, RC=C/C0 and RS=S /S0. This equilibrium condition, [Risk Premia Cancellation Condition, Returns], between the risk premium on the option and the risk premium on the underlying stock says that, in order for the hedge portfolio to be riskless, the Sharpe ratios of the option and the stock must be equal. The Sharpe ratio is a standard portfolio risk measure deﬁned as risk-premium to standard deviation and was introduced in Chapter 15. We have also called it the Market Price of Risk (MPR). As we have just demonstrated, [Risk Premia Cancellation Condition, Returns] is a consequence of the fact that it is possible to replicate the call option in the (BOPM, N=1). If this condition is necessary and sufﬁcient for replication of any contingent claim (which it is by the revised version of FTAP2 in section 17.3), then this is just another way to say that the (BOPM, N=1) is complete.

Even from the point of view of a risk-averse investor, we know that if the stock is assumed to be risk neutral then the option must also be risk neutral, because different risks cannot co-exist in the riskless hedge portfolio (see Chapter 17). If they did, there would be no way to cancel them out, as they must in order to generate a riskless hedge. By risk is meant relative risk, which is also called the ‘Sharpe ratio’. Once again, what do we know? We know that the no-arbitrage, replicable option price, C0, is given under the risk-neutralized stock price measure (pr,1–pr). That is, E r (C1( )|C 0 ) = 1+ r ′ C0 EQUIVALENT MARTINGALE MEASURES 527 Evaluating E r(C1()|C0) we obtain, E r (C1( )|C 0 ) = p rCu + (1 − p r )C d Therefore, E r (C1( )|C 0 ) prCu + (1 − p r )C d = C0 C0 = 1+ r ′ or, since pr=p′, C0 = p′ 1 − p′ * Cu + * Cd 1+ r ′ 1+ r ′ We can also get the same result using the state price representation of C0 as, C 0 = P0 ( ADu ( )) * C u + P0 ( ADd ( )) * C d = pr 1 − pr C + * u * Cd 1+ r ′ 1+ r ′ This says that the current no-arbitrage option price is the weighted sum of its payoffs times the no-arbitrage prices of the primitive AD securities.

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The Bogleheads' Guide to Investing by Taylor Larimore, Michael Leboeuf, Mel Lindauer

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In "Rebalancing Diversified Portfolios of Various Risk Profiles," (Article 14 of the October 2001 issue of the journal of Financial Planning), author Cindy Sin-Yi Tsai, CFA, looked at a number of rebalancing methods: a Never rebalancing • Monthly rebalancing • Quarterly rebalancing • Rebalance if more than 5 percent from target at month's end Rebalance if more than 5 percent from target at quarter's end Often, investors who don't rebalance are simply letting their winners run in the belief that doing so would produce much higher returns. Contrary to what these investors might believe, the article reported that the increased returns were actually found to be small or even nonexistent when compared with the additional risk (as measured by the volatility) taken on by those investors who didn't rebalance. In addition, the study showed that portfolios that were never rebalanced had the lowest Sharpe ratios of all the rebalancing methods studied. Since the Sharpe ratio measures the additional return an investor receives for taking on more risk, this lower ratio indicates that investors who didn't rebalance were not being compensated for the additional risk they were taking. This study's results came to the same conclusion as did Jack Bogle when he previously reported on the results from his 25-year rebalancing study in his classic 1993 book Bogle on Mutual Funds.

Investing is about probabilities-and the probability is good that by using lowcost mutual funds, we will outperform the majority of other investors. The Financial Research Corporation conducts research for industry insiders. One of their most important studies was to determine which of eleven common predictors of future mutual fund performance really worked. These predictors were Morningstar ratings; past performance; expenses; turnover; manager tenure; net sales; asset size; alpha; beta; standard deviation (SD); and the Sharpe ratio. Their study's conclusion: The expense ratio is the only reliable predictor of future mutual fund performance. In another study, Standard and Poor's examined all diversified U.S. stock funds in nine different Morningstar style-box categories. The study, reported in the September 2003 issue of Kiplinger magazine, divided the funds in each Style Box into two groups: funds with above-average costs, and funds with below-average costs.

Often called your sleep factor. Rollover: A tax-free transfer of assets from one retirement plan to another. Roth IRA: A tax-favored retirement plan. Contributions are not deductible, but earnings are tax-free during accumulation and also when withdrawn. Sector/specialty fund: A mutual fund that invests in a narrow segment of the market, such as health, technology, utilities, or real estate. Sharpe ratio: A measure of risk-adjusted performance developed by Nobel Laureate William Sharpe. Spousal IRA: An IRA established for a nonworking spouse. Standard deviation: A statistical measure of volatility. Taxable account: An account in which the securities are subject to annual federal taxes. Tax-deferred accounts: An account in which federal income taxes are deferred until withdrawn. Total return: The most complete measure of a fund's gain or loss.

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No One Would Listen: A True Financial Thriller by Harry Markopolos

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Walter “Bud”Haslett, CFA; LLC; Suite 455; NJ 08065; Tel#: or [Bud’s firm runs \$ hundreds of millions in options related strategies and he knows all of the math.] c. Joanne Hill, Ph.D.; Vice-President and global head of equity derivatives research, Goldman Sachs (NY), ; One New York Plaza, New York, NY 10004; 24. Red Flag # 28: BM’s Sharpe Ratio of 2.55 (Attachment 1: Fairfield Sentry Ltd. Performance Data) is UNBELIEVABLY HIGH compared to the Sharpe Ratios experienced by the rest of the hedge fund industry. The SEC should obtain industry hedge fund rankings and see exactly how outstanding Fairfield Sentry Ltd.’s Sharpe Ratio is. Look at the hedge fund rankings for Fairfield Sentry Ltd. and see how their performance numbers compare to the rest of the industry. Then ask yourself how this is possible and why hasn’t the world come to acknowledge BM as the world’s best hedge fund manager?

Among all the funds on the database in that same period, the Madoff/ Fairfield Sentry fund would place at number 16 if ranked by its absolute cumulative returns. Among 423 funds reporting returns over the last five years, most with less money and shorter track records, Fairfield Sentry would be ranked at 240 on an absolute return basis and come in number 10 if measured by risk-adjusted return as defined by its Sharpe ratio. What is striking to most observers is not so much the annual returns—which, though considered somewhat high for the strategy, could be attributed to the firm’s market making and trade execution capabilities—but the ability to provide such smooth returns with so little volatility. The best known entity using a similar strategy, a publicly traded mutual fund dating from 1978 called Gateway, has experienced far greater volatility and lower returns during the same period.

The best known entity using a similar strategy, a publicly traded mutual fund dating from 1978 called Gateway, has experienced far greater volatility and lower returns during the same period. The capital overseen by Madoff through Fairfield Sentry has a cumulative compound net return of 397.5 percent. Compared with the 41 funds in the Zurich database that reported for the same historical period, from July 1989 to February 2001, it would rank as the best performing fund for the period on a risk-adjusted basis, with a Sharpe ratio of 3.4 and a standard deviation of 3.0 percent. (Ranked strictly by standard deviation, the Fairfield Sentry funds would come in at number three, behind two other market neutral funds.) Questions Abound Bernard Madoff, the principal and founder of the firm, who is widely known as Bernie, is quick to note that one reason so few might recognize Madoff Securities as a hedge fund manager is because the firm makes no claim to being one.

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New Market Wizards: Conversations With America's Top Traders by Jack D. Schwager

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Second, I felt that, over the long run, I could probably make more money on my own—although that certainly wasn’t true over the short run. How do you define success in trading? I sincerely believe that the person who has the best daily Sharpe ratio at the end of the year is the best trader. [The Sharpe ratio is a statistical performance measure that normalizes return by risk, with the variability of returns being used to measure risk. Thus, for example, assume Trader A and Trader B managed identical-sized funds and made all the same trades, but Trader A always entered orders for double the number of contracts as Trader B. In this case, Trader A would realize double the percentage return, but because risk would also double, the Sharpe ratio would be the same for both traders.* Normally, the Sharpe ratio is measured using monthly data. Thus, only equity variability that occurs on a month-to-month basis would be considered.

In the area of The Silence of the Turtles / 139 futures traders, one reference source I used was the quarterly summary provided by Managed Accounts Reports. This report summarizes the performance of a large number of commodity trading advisors (CTAs), providing a single synopsis sheet for each advisor. At the bottom of each sheet is a summary table with key statistics, such as average annual percentage return, largest drawdown, Sharpe ratio (a return/risk measure), percentage of winning months, and the probabilities of witnessing a 50 percent, 30 percent, and 20 percent loss with the given CTA. To be objective, I flipped through the pages, glancing only at the tables (not the names at the top of the sheets) and checking off the names of those advisors whose exceptional performance seemed to jump off the page. By the end of this process, I had checked off eighteen of the more than one hundred CTAs surveyed.

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Weapons of Math Destruction: How Big Data Increases Inequality and Threatens Democracy by Cathy O'Neil

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The culture of Wall Street is defined by its traders, and risk is something they actively seek to underestimate. This is a result of the way we define a trader’s prowess, namely by his “Sharpe ratio,” which is calculated as the profits he generates divided by the risks in his portfolio. This ratio is crucial to a trader’s career, his annual bonus, his very sense of being. If you disembody those traders and consider them as a set of algorithms, those algorithms are relentlessly focused on optimizing the Sharpe ratio. Ideally, it will climb, or at least never fall too low. So if one of the risk reports on credit default swaps bumped up the risk calculation on one of a trader’s key holdings, his Sharpe ratio would tumble. This could cost him hundreds of thousands of dollars when it came time to calculate his year-end bonus. I soon realized that I was in the rubber-stamp business.

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More Money Than God: Hedge Funds and the Making of a New Elite by Sebastian Mallaby

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Conceivably, the Tiger cubs might have achieved higher returns by taking extra risk, in which case there would be nothing to brag about. Thanks to the Nobel laureate William Sharpe, we have a way of testing whether this was so: If you divide the Tiger cubs’ returns by their volatility, you get a Sharpe ratio of 1.42—that is, a risk-adjusted return that is superior by far to any of the benchmarks. For instance, Hennessee’s general hedge-fund index had a Sharpe ratio of just 0.59. The comparison makes it difficult to resist the conclusion that the Tiger cubs learned something from Robertson. Let’s try to resist a little longer. There are ways for hedge funds to game the Sharpe ratio by behaving like undercapitalized insurance companies.2 For example, a fund can sell options that insure against extreme swings in the market. For months and maybe years, the insurer will collect a steady stream of premiums from these options, delivering consistent, market-beating returns; but one day the extreme market swing will occur, at which point the fund will go bankrupt.

For example, Mark Wehrly, Farallon’s general counsel, reports that Farallon borrows about \$25 for every \$100 in equity. Mark Wehrly, interview with the author, July 25, 2008. 20. Robert Howard and Andre F. Perold, “Farallon Capital Management: Risk Arbitrage” (Harvard Business School case study 9-299-020, November 17, 1999). According to this HBS study, the Sharpe ratios for two Farallon funds between 1990 and 1997 were 1.38 and 1. 75. The S&P 500 had a Sharpe ratio of 0.50. 21. Enrique Boilini, who led Farallon’s investment in Alpargatas, recalls that Gabic, a similar textile company, did not attract the interest of a foreign hedge fund, with the result that its factories were liquidated and all its workers lost their jobs. In turning Alpargatas around, Farallon worked with Texas Pacific Group, another U.S. investor.

Along with profits and transparency, the event-driven merchants promised consistency. They used very little leverage, which in the wake of Long-Term’s blowup was a selling point in itself; partly as a result, their returns were almost miraculously steady.19 Farallon’s consistency was legendary: Between 1990 and 1997, there was not a single month in which the fund lost money. As a result, Farallon’s Sharpe ratio, a measure of returns adjusted for risk, was roughly three times higher than that of the broad stock market, making it an overwhelmingly attractive place for endowments to park savings.20 Even during the height of the dot-com madness, Steyer sailed along serenely. He did not ride the bubble like Stan Druckenmiller. He did not get run over by it like Julian Robertson. Instead, he applied his methods to analyzing the epic takeover battles of the era, hedging out the market risk as he did so.

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Fooled by Randomness: The Hidden Role of Chance in Life and in the Markets by Nassim Nicholas Taleb

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Differences between returns: See Ambarish and Siegel (1996). The dull presenter was actually comparing “Sharpe ratios,” i.e., returns scaled by their standard deviations (both annualized), named after the financial economist William Sharpe, but the concept has been commonly used in statistics and called “coefficient of variation.” (Sharpe introduced the concept in the context of the normative theory of asset pricing to compute the expected portfolio returns given some risk profile, not as a statistical device.) Not counting the survivorship bias, over a given twelve-month period, assuming (very generously) the Gaussian distribution, the “Sharpe ratio” differences for two uncorrelated managers would exceed 1.8 with close to 50% probability. The speaker was discussing “Sharpe ratio” differences of around .15! Even assuming a five-year observation window, something very rare with hedge fund managers, things do not get much better.

Trend Commandments: Trading for Exceptional Returns by Michael W. Covel

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You will never have, nor will you ever, produce returns that exhibit a normal distribution. You will never produce the mythologically consistent returns that many believe to exist. When trend followers hit home runs from the likes of Barings Bank, Long-Term Capital Management, and the 2008 market crash, they are targeting unknowable extreme occurrences that happen to occur with a probability greater than expected. The Sharpe ratio is oversold. It can give a false sense of precision and lead people to make predictions unwisely.2 Those occurrences are fat tails—in statistician speak. Trend following’s nature of riding a trend to the end when it bends, and then cutting losses very fast, puts you in a position to benefit when the next unexpected flood rolls in. Trend following’s alpha comes from letting winners run on the right-hand side of a fat tail and cutting losses short on the left-hand side.

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Extreme Money: Masters of the Universe and the Cult of Risk by Satyajit Das

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Sharpe Practice Investors use Sharpe or information ratios to measure investment performance. Assume risk-free government securities yield 5 percent and hedge fund A has returns of 20 percent with a volatility of 10 percent while hedge fund B has returns of 15 percent and a volatility of 5 percent. The Sharpe ratios are respectively: Hedge Fund A = [20%—5%] / 10% = 1.5 Hedge Fund B = [15%—5%] / 5% = 2.0 Despite lower absolute performance, B provides investors with greater returns relative to risk. Sharpe ratios are ex post (based on actual risk) rather than ex ante (expected risk). Actual returns should be compared to expected risk at the time the position was taken. Insufficient attention is paid to the asymmetry of hedge fund returns, which do not follow the familiar bell-shaped normal distribution.

, 301 Hasset, Kevin, 97 Havel, Václav, 359 Hawala, 22, 24 Hawkin, Greg, 248 Hawking, Stephen, 126 Hayek, Frederick, 103 HE (home equity), 181 Heathrow Airport, 161 Hederman, Abbot Mark Patrick, 361 Hedge Fund Alley, 239 hedge funds, 73, 77, 80, 260 Alfred Winslow Jones, 240 Amaranth, 250, 252 Centaurus Energy, 319 clientele, 247-248, 250 compensation, 314 fees, 245 formula for, 239 Fortress, 318 George Soros, 240 Hedgestock, 252, 261-262 Hyman Minsky, 260-262 leverage, 254 markets, 241 Porsche, 257-260 returns, 243-244, 255-257 Sharpe ratios, 246-247 strategies, 241-243 structure of CDOs, 195 Hedgestock, 261-262 hedge funds, 252 hedging, 235 aspect of Black-Scholes model, 122 derivatives, 216-217 Heine, Heinrich, 38, 64 Heisenberg, Werner, 101 Heller, Walter, 129 Hellman, Lillian, 350 HELOC (home equity line of credit), 181 Hennessy, Peter, 278 Heritage Foundation, 350 Herodotus, 74 Hertz, 155 Hewlett-Packard (HP), 122 Heyman, William, 270 Hickman, W.

See also options profits, 121 risk, 124 intellectual property rights, securitization of, 168 interest rates cutting of, 340-341 lowering of, 348 International Accounting Standards Board, 289 International Grain Council, 334 International Institute of Finance (IIF), 289 International Monetary Fund (IMF), 96 international reply coupons (IRCs), 33 Internet bubble (1990s), 54 stocks, 58 InterNorth, 55 interviews on financial TV shows, 94 invention of money, 24-25 investment banks, 57, 309 leveraged buyouts (LBOs), 147-148 percentage of jobs in, 313 separation from commercial banks, 66 investments alternative, 252 exotic products, 73-74 hedge funds Amaranth, 250-252 clientele, 247-250 fees, 245 Hedgestock, 252 markets, 241 returns, 243-244 Sharpe ratios, 246-247 strategies, 241-243 incentives, 348 IO (interest only) bonds, 178 Ireland, 83, 344 Irish Times, The, 356 Iron Chef, The, 168 Irving, John, 29 Ising model, 204 It’s A Wonderful Life, 65, 180 Italy, derivatives, 215-216 ITT Corporation, 60 J Jackson, Marjorie, 156 Jackson, Michael, 21 Jackson, Tony, 363 James, Oliver, 274 Japan debt, 357 financialization, 38-39 housewife traders, 40-41 lost decades, 357 retirement, 49-50 six sigma, 60 Jefferson County, Alabama, 211-214 Jefferson, Thomas, 91 Jenkins, Simon, 302 Jenson, Michael, 120, 138-141 Jiabao, Wen, 86-87 Jian, Ma, 295 Jintao, Hu, 363 jobbers, 53 jobs certifications, 309-310 finance, 307-308 Jobs, Steve, 164 John F.

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Statistical Arbitrage: Algorithmic Trading Insights and Techniques by Andrew Pole

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Stop loss rules and bet sizing significantly impact outcome characteristics of a strategy, making more general absolute statements unwise. It is worth noting the following observation. Suppose we pick only those trades that are profitable round trip. Daily profit variation will still, typically, be substantial. Experiments with real data using a popcorn process model show that the proportion of winning days can be as low as 52 percent for a strategy with 75 percent winning bets and a Sharpe ratio over 2. Reversion defined as any movement from today’s price in the direction of the center of the price distribution includes overshoot cases. The scenario characterizes as reversionary movement a price greater than the median that moves to any lower price—including to any price lower than the median, the so-called overshoot. Similarly, Quantifying Reversion Opportunities 117 movement from a price below the median that moves to any higher price is reversionary. 7.2.1 Frequency of Reversionary Moves For any price greater than the median price, Pt = pt > m: Pr[Pt+1 < pt ] = FP (pt ) where FP (·) denotes the distribution function of the probability distribution from which prices are assumed to be generated.

See also Return decline catastrophe process, 194–198 catastrophe process forecasts, 198–200 catastrophe process theoretical interpretation, 205–209 Cuscore statistics and, 200–205, 211–221 risk management and, 209–211 trend change identification, 200–205 Revealed reversion, see Expected revealed reversion Reverse bets, 11 Reversion, law of, 67–89, 113–114, 139–140 first-order serial correlation and, 77–82 inhomogeneous variances and, 74–77 interstock volatility and, 67, 99–112, 164–165 looking several days ahead and, 87–89 nonconstant distributions and, 82–84 in nonstationary process, 136–137 serial correlation, 138–139 75 percent rule and, 68–74 in stationary random process, 114–136 temporal dynamics and, 91–98 U.S. bond futures and, 85–87 Reynders Gray, 26 Risk arbitrage, competition and, 160–161 Risk control, 26–32 event correlations, 31–32 forecast variance, 26–28 market exposure, 29–30 market impact, 30–31 229 Index Risks scenarios, 141–154 catastrophe process and, 209–211 correlation during loss episodes, 151–154 event risk, 142–145 new risk factors, 145–148 redemption tension, 148–150 Regulation Fair Disclosure (FD), 150–151 Royal Dutch Shell (RD)–British Petroleum (BP) spread, 46–47 S&P (Standard & Poor’s): S&P 500, 28 futures and exposure, 21 Sample distribution, 123 Santayana, George, 5n2 SARS (severe acute respiratory syndrome), 175 Securities and Exchange Commission (SEC), 3, 150–151 Seismology analogy, 200n1 September 11 terrorist attacks, 175 Sequentially structured variances, 136–137 Sequentially unstructured variances, 137 Serial correlation, 138–139 75 percent rule, 68–74, 117 first-order serial correlation and, 77–82 inhomogeneous variances and, 74–77, 136–137 looking several days ahead and, 87–89 nonconstant distributions and, 82–84 U.S. bond futures and, 85–87 Severe acute respiratory syndrome (SARS), 175 Shackleton, E. H., 113 Sharpe ratio, 116 Shaw (D.E.), 3, 189 Shell, see Royal Dutch Shell (RD)–British Petroleum (BP) spread Sinusoid, 19–20, 170 Spatial model analogy, 200n1 Specialists, 3, 156–157 Speer, Leeds & Kellog, 189 Spitzer, Elliot, 176, 180 Spread margins, 16–18. See also specific companies Standard & Poor’s (S&P): S&P 500, 28 futures and exposure, 21 Standard deviations, 16–18 Stationarity, 49, 84–85 Stationary random process, reversion in, 114–136 amount of reversion, 118–135 frequency of moves, 117 movements from other than median, 135–136 Statistical arbitrage, 1–7, 9–10 Stochastic resonance, 20, 50, 58–59, 169, 204 Stochastic volatility, 50–51 Stock split, 13n1 Stop loss, 39 Structural change, return decline and, 179–180 Structural models, 37–66 accuracy issues, 59–61 classical time series models, 47–52 doubling and, 81–83 exponentially weighted moving average, 40–47 factor model, 53–58, 63–66 stochastic resonance, 58–59 Stuart, Alan, 63 Student t distribution, 75, 124–126, 201 Sunamerica, Inc.

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Python for Data Analysis by Wes McKinney

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Finance: In [105]: import pandas.io.data as web In [106]: data = web.get_data_yahoo('SPY', '2006-01-01') In [107]: data Out[107]: <class 'pandas.core.frame.DataFrame'> DatetimeIndex: 1655 entries, 2006-01-03 00:00:00 to 2012-07-27 00:00:00 Data columns: Open 1655 non-null values High 1655 non-null values Low 1655 non-null values Close 1655 non-null values Volume 1655 non-null values Adj Close 1655 non-null values dtypes: float64(5), int64(1) Now, we’ll compute daily returns and a function for transforming the returns into a trend signal formed from a lagged moving sum: px = data['Adj Close'] returns = px.pct_change() def to_index(rets): index = (1 + rets).cumprod() first_loc = max(index.notnull().argmax() - 1, 0) index.values[first_loc] = 1 return index def trend_signal(rets, lookback, lag): signal = pd.rolling_sum(rets, lookback, min_periods=lookback - 5) return signal.shift(lag) Using this function, we can (naively) create and test a trading strategy that trades this momentum signal every Friday: In [109]: signal = trend_signal(returns, 100, 3) In [110]: trade_friday = signal.resample('W-FRI').resample('B', fill_method='ffill') In [111]: trade_rets = trade_friday.shift(1) * returns We can then convert the strategy returns to a return index and plot them (see Figure 11-1): In [112]: to_index(trade_rets).plot() Figure 11-1. SPY momentum strategy return index Suppose you wanted to decompose the strategy performance into more and less volatile periods of trading. Trailing one-year annualized standard deviation is a simple measure of volatility, and we can compute Sharpe ratios to assess the reward-to-risk ratio in various volatility regimes: vol = pd.rolling_std(returns, 250, min_periods=200) * np.sqrt(250) def sharpe(rets, ann=250): return rets.mean() / rets.std() * np.sqrt(ann) Now, dividing vol into quartiles with qcut and aggregating with sharpe we obtain: In [114]: trade_rets.groupby(pd.qcut(vol, 4)).agg(sharpe) Out[114]: [0.0955, 0.16] 0.490051 (0.16, 0.188] 0.482788 (0.188, 0.231] -0.731199 (0.231, 0.457] 0.570500 These results show that the strategy performed the best during the period when the volatility was the highest.

First, I’ll load historical prices for a portfolio of financial and technology stocks: names = ['AAPL', 'GOOG', 'MSFT', 'DELL', 'GS', 'MS', 'BAC', 'C'] def get_px(stock, start, end): return web.get_data_yahoo(stock, start, end)['Adj Close'] px = DataFrame({n: get_px(n, '1/1/2009', '6/1/2012') for n in names}) We can easily plot the cumulative returns of each stock (see Figure 11-2): In [117]: px = px.asfreq('B').fillna(method='pad') In [118]: rets = px.pct_change() In [119]: ((1 + rets).cumprod() - 1).plot() For the portfolio construction, we’ll compute momentum over a certain lookback, then rank in descending order and standardize: def calc_mom(price, lookback, lag): mom_ret = price.shift(lag).pct_change(lookback) ranks = mom_ret.rank(axis=1, ascending=False) demeaned = ranks - ranks.mean(axis=1) return demeaned / demeaned.std(axis=1) With this transform function in hand, we can set up a strategy backtesting function that computes a portfolio for a particular lookback and holding period (days between trading), returning the overall Sharpe ratio: compound = lambda x : (1 + x).prod() - 1 daily_sr = lambda x: x.mean() / x.std() def strat_sr(prices, lb, hold): # Compute portfolio weights freq = '%dB' % hold port = calc_mom(prices, lb, lag=1) daily_rets = prices.pct_change() # Compute portfolio returns port = port.shift(1).resample(freq, how='first') returns = daily_rets.resample(freq, how=compound) port_rets = (port * returns).sum(axis=1) return daily_sr(port_rets) * np.sqrt(252 / hold) Figure 11-2.

Cumulative returns for each of the stocks When called with the prices and a parameter combination, this function returns a scalar value: In [122]: strat_sr(px, 70, 30) Out[122]: 0.27421582756800583 From there, you can evaluate the strat_sr function over a grid of parameters, storing them as you go in a defaultdict and finally putting the results in a DataFrame: from collections import defaultdict lookbacks = range(20, 90, 5) holdings = range(20, 90, 5) dd = defaultdict(dict) for lb in lookbacks: for hold in holdings: dd[lb][hold] = strat_sr(px, lb, hold) ddf = DataFrame(dd) ddf.index.name = 'Holding Period' ddf.columns.name = 'Lookback Period' To visualize the results and get an idea of what’s going on, here is a function that uses matplotlib to produce a heatmap with some adornments: import matplotlib.pyplot as plt def heatmap(df, cmap=plt.cm.gray_r): fig = plt.figure() ax = fig.add_subplot(111) axim = ax.imshow(df.values, cmap=cmap, interpolation='nearest') ax.set_xlabel(df.columns.name) ax.set_xticks(np.arange(len(df.columns))) ax.set_xticklabels(list(df.columns)) ax.set_ylabel(df.index.name) ax.set_yticks(np.arange(len(df.index))) ax.set_yticklabels(list(df.index)) plt.colorbar(axim) Calling this function on the backtest results, we get Figure 11-3: In [125]: heatmap(ddf) Figure 11-3. Heatmap of momentum strategy Sharpe ratio (higher is better) over various lookbacks and holding periods Future Contract Rolling A future is an ubiquitous form of derivative contract; it is an agreement to take delivery of a certain asset (such as oil, gold, or shares of the FTSE 100 index) on a particular date. In practice, modeling and trading futures contracts on equities, currencies, commodities, bonds, and other asset classes is complicated by the time-limited nature of each contract.

pages: 505 words: 142,118

A Man for All Markets by Edward O. Thorp

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The annualized return of 7.77 percent and the annualized standard deviation of 15.07 percent for the S&P 500 during this period are somewhat below its long-term values. The unlevered annualized return for XYZ before fees, at 18.21 percent, is more than double that of the S&P; the riskiness, as measured by the standard deviation, is 6.68 percent. The ratio of (annualized) return to risk for XYZ at 2.73 is more than five times that of the S&P. Estimating 5 percent as the average three-month T-bill rate over the period, the corresponding Sharpe ratios are 0.18 for the S&P versus 1.98 for XYZ. The graph in Appendix E, XYZ Performance Comparison, displays two major “epochs.” The first, from August 12, 1992, to early October 1998, shows a steady increase. The second epoch, from then until September 13, 2002, has a higher rate of return, including a remarkable six-month spurt just after the collapse (after four years) of the large hedge fund called, ironically, Long-Term Capital Management.

Table 9: Consumer Price Index Year Index Year Index Year Index 1913 9.9 1934 13.4 1955 26.8 1914 10.0 1935 13.7 1956 27.2 1915 10.1 1936 13.9 1957 28.1 1916 10.9 1937 14.4 1958 28.9 1917 12.8 1938 14.1 1959 29.2 1918 15.0 1939 13.9 1960 29.6 1919 17.3 1940 14.0 1961 29.9 1920 20.0 1941 14.7 1962 30.3 1921 17.9 1942 16.3 1963 30.6 1922 16.8 1943 17.3 1964 31.0 1923 17.1 1944 17.6 1965 31.5 1924 17.1 1945 18.0 1966 32.5 1925 17.5 1946 19.5 1967 33.4 1926 17.7 1947 22.3 1968 34.8 1927 17.4 1948 24.0 1969 36.7 1928 17.2 1949 23.8 1970 38.8 1929 17.2 1950 24.1 1971 40.5 1930 16.7 1951 26.0 1972 41.8 1931 15.2 1952 26.6 1973 44.4 1932 13.6 1953 26.8 1974 49.3 1933 12.9 1954 26.9 1975 53.8 1976 56.9 1989 124.0 2002 179.9 1977 60.6 1990 130.7 2003 184.0 1978 65.2 1991 136.2 2004 188.9 1979 72.6 1992 140.3 2005 195.3 1980 82.4 1993 144.5 2006 201.6 1981 90.9 1994 148.2 2007 207.3 1982 96.5 1995 152.4 2008 215.3 1983 99.6 1996 156.9 2009 214.5 1984 103.9 1997 160.5 2010 218.1 1985 107.6 1998 163.0 2011 224.9 1986 109.6 1999 166.6 2012 229.6 1987 113.6 2000 172.2 2013 233.0 1988 118.3 2001 177.1 US Department of Labor Bureau of Labor Statistics Washington, DC 20212 Consumer Price Index All Urban Consumers—(CPI-U) US City Average All Items 1982–84=100 * * * * For an insightful discussion of why the inflation index from the 1970s may be much too low as a result of a series of government revisions in the method of calculation, and the consequences to investors and consumers, see “Fooling with Inflation” by Bill Gross (June 2008) at www.pimco.com. For updated Consumer Price Index numbers and for month-by-month values, go to www.bls.gov/cpi or do the usual Google search. Appendix B * * * HISTORICAL RETURNS Table 10: Historical Returns on Asset Classes, 1926–2013 Series Compound Annual Return* Average Annual Return** Standard Deviation Real (after inflation) Compound Annual Return* Sharpe Ratio† Large Company Stocks 10.1% 12.1% 20.2% 6.9% 0.43 Small Company Stocks 12.3% 16.9% 32.3% 9.1% 0.41 Long-Term Corporate Bonds 6.0% 6.3% 8.4% 2.9% 0.33 Long-Term Government Bonds 5.5% 5.9% 9.8% 2.4% 0.24 Intermediate-Term Government Bonds 5.3% 5.4% 5.7% 2.3% 0.33 US Treasury Bills 3.5% 3.5% 3.1% 0.5% ——— Inflation 3.0% 3.0% 4.1% ——— ——— * Geometric Mean ** Arithmetic Mean † Arithmetic From: Ibbotson, Stocks, Bonds, Bills and Inflation, Yearbook, Morningstar, 2014.

of the impact Market impact refers to the fact that “market orders” to buy are, on average, filled at or above the last previous price and “market orders” to sell tend to be filled at or below the last previous price. gained 9 percent The accounting period with an odd length of five months arose here for PNP because the fiscal year end for PNP changed in 1987 from October 31 to December 31. statistics confirmed Common metrics include the Sharpe ratio, the Sortino ratio, the distribution of drawdowns, and the MAR ratio (annual return divided by maximum drawdown). See, for instance, the three-part series by William Ziemba in Wilmott magazine: “The Great Investors,” March, May and July 2006. or losing quarters For comparison, the S&P 500 was down in eleven of the thirty-two full quarters and small company stocks lost in thirteen. CHAPTER 16 associates, and clients Den of Thieves by James B.

All About Asset Allocation, Second Edition by Richard Ferri

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These funds can experience higher share-price volatility than diversified funds because sector funds are subject to issues specific to a given sector. Securities and Exchange Commission (SEC) The federal government agency that regulates mutual funds, registered investment advisors, the stock and bond markets, and broker-dealers. The SEC was established by the Securities Exchange Act of 1934. Sharpe Ratio A measure of risk-adjusted return. To calculate a Sharpe ratio, an asset’s excess returns (its return in excess of the return generated by risk-free Glossary 329 assets such as Treasury bills) is divided by the asset’s standard deviation. It can be calculated compared to a benchmark or an index. Short Sale The sale of a security or option contract that is not owned by the seller, usually to take advantage of an expected drop in the price of the security or option.

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The Little Book of Hedge Funds by Anthony Scaramucci

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A quick note—do not put too much emphasis on statistical techniques. As history has proven, statistical and quantitative techniques have done far more harm than good to both capital markets and hedge fund investors (think LTCM). It’s hard to convince yourself that levering any investment strategy 100:1 is safe unless you are both egregiously arrogant and have developed such sophisticated models that nothing can go wrong. After all, what good is a Sharpe Ratio of 4 for three years when you lose 100 percent of your money in the fourth year? That said, the application of data analysis can be helpful when applied by thoughtful, humble minds. The operational due diligence process is focused on making sure the manager does not or cannot do anything completely stupid on the business side to blow up his business. Remember, most hedge fund blowups have occurred due to operational issues not bad investment bets (think Madoff or Beacon Hill).

pages: 206 words: 70,924

The Rise of the Quants: Marschak, Sharpe, Black, Scholes and Merton by Colin Read

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Sharpe also began to study pensions in the post-CAPM portion of his career. In his research, he continued to look into ways in which theoretical concepts can be reduced to methodologies that can be applied by practitioners. For instance, he produced a discrete-time binomial option pricing procedure that offered a readily applicable procedure for BlackScholes securities pricing, which will be covered in the next part of this book. He also developed the Sharpe ratio, a measure of the risk of a mutual or index fund versus its reward. Sharpe continued to work to make financial concepts more democratic and more accessible. He helped develop Financial Engines, an Internetbased application to deliver investment advice online. 78 The Rise of the Quants Ever concerned about the practitioner’s side of finance, Sharpe began to consult with investment houses, first Merrill Lynch and then Wells Fargo.

A Primer for the Mathematics of Financial Engineering by Dan Stefanica

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For notation purposes, assume that asset 4 is the asset with uncorrelated return, i.e., Pi,4 = 0, for i = 1 : 3. Let Wi be the weight of asset i in the BFinding efficient portfolios is one of the fundamental problems answered by the modern portfolio theory of Markowitz and Sharpe; see Markowitz [17] and Sharpe [25] for seminal ~apers. Of all the efficient portfolio~, the portfolio with the higheflt Sharpe ratio E~l~?, l.e., the expected return above the rIsk free rate r f normalized by the standard deviation 0-( R) of the return, is called the market portfolio (or the tangency portfolio) and plays an important role in the Capital Asset Pricing Model (CAPM). 262 CHAPTER 8. LAGRANGE MULTIPLIERS. NEWTON'S METHOD. portfolio, for i = 1 : 4. From (8.49) and (8.50), it follows that E[R] var(R) = WlJLl + W2JL2 + W3JL3 + W4JL4; The gradient W1 + W2 + W3 + W4 W1JL1 + W2JL2 + W3JL3 + W4JL4 = 1; = JLp.

Monte Carlo Simulation and Finance by Don L. McLeish

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If the market portfolio m has standard deviation σm and mean ηm , then the line L is described by the relation η=r+ ηm − r σ. σm For any investment with mean return η and standard deviation of return σ to be competitive, it must lie on this eﬃcient frontier, i.e. it must satisfy the relation η − r = β(ηm − r), where β = σ or equivalently σm (2.19) η−r (ηm − r) . = σ σm This is the most important result in the capital asset pricing model. The excess return of a stock η − r divided by its standard deviation σ is supposed constant, and is called the Sharpe ratio or the market price of risk. The constant β called the beta of the stock or portfolio and represents the change in the expected portfolio return for each unit change in the market. It is also the ratio of the standard deviations of return of the stock and the market. Values of β > 1 indicate a stock that is more variable than the market and tends to have higher positive and negative returns, whereas values of β < 1 are investments that are more conservative and less volatile than the market as a whole.

pages: 313 words: 101,403

My Life as a Quant: Reflections on Physics and Finance by Emanuel Derman

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We lived in a campus dorm and luxuriated in our freedom from corporate life, running on the MIT track in the late afternoons and eating in Cambridge in the evenings. Myers's course focused on the Capital Asset Pricing Model, and I was captivated by the apparent similarity between financial theory and thermodynamics. I saw a perhaps-too-facile correspondence between heat and money, temperature and risk, and entropy and the Sharpe ratio, but have never since figured out how to exploit it. The course was brief and intense and required more work than we put into it. One of the lecturers was Terry Marsh, now a Professor at Berkeley and a founding partner of the financial software firm Quantal. At that time he was just beginning to make his reputation, and I was always happy to run into him years later at professional finance meetings or when I gave a seminar at the Haas business school at Berkeley.

pages: 318 words: 87,570

Broken Markets: How High Frequency Trading and Predatory Practices on Wall Street Are Destroying Investor Confidence and Your Portfolio by Sal Arnuk, Joseph Saluzzi

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This typically involves: • Large technological expenditures in hardware, software and data • Latency sensitivity (order generation and execution taking place in sub-second speeds) • High quantities of orders, each small in size • Short holding periods, measured in seconds versus hours, days, or longer • Starts and ends each day with virtually no net positions • Little human intervention Will Psomadelis, Head of Trading at Schroeder Investment Management, in a paper titled “High Frequency Trading - Credible Research Tells the Real Story,” found that HFT “returns are abnormally high, with Sharpe ratios often in the order of nine or double digits. Well-known names in the HFT space include GETCO, Infinium, and Optiver.”1 Although initial perceptions focused on how HFT has shrunken spreads and generated liquidity that investors could embrace, in recent years those perceptions have turned quite negative. HFT’s reputation has gotten so poor that lobbyists hired by HFT firms have tried to change the name or discourage its use in the media.

pages: 338 words: 106,936

The Physics of Wall Street: A Brief History of Predicting the Unpredictable by James Owen Weatherall

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Ed Thorp also played a significant role in the early development of the idea; for more on his contribution, see Thorp (2004). “. . . a variety of computer programs known as genetic algorithms”: For more on genetic algorithms, see, for instance, Mitchell (1998). For Packard’s early contributions, see Packard (1988, 1990). “. . . over the firm’s first fifteen years . . .”: More specifically, this person told me that the company had a Sharpe ratio of 3. 7. Tyranny of the Dragon King “Didier Sornette looked at the data again”: The opening story, which plays out throughout the chapter, is a dramatization, but the basic details are correct. In late summer 1997, Sornette observed a pattern in U.S. financial data that he had previously argued could be used to predict financial crashes; he contacted his colleagues Olivier Ledoit and Anders Johansen and proceeded as described here.

Investment: A History by Norton Reamer, Jesse Downing

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Furthermore, the best performers do change through time, since the market does not reward the same strategy all the time. That said, to provide some empirical insight on the matter of the more recent performance of these strategies, the aggregate risk and return ﬁgures from 1994 to 2011, assembled by KPMG and the Centre for Hedge Fund Research (table 8.1), show that relative value and event-driven funds have been the strongest performers on a risk-adjusted basis (as measured by their Sharpe ratios).25 By contrast, short bias funds have tended to have the least attractive risk-reward characteristics, returning just over 1 percent per year but table 8.1 Statistics for Hedge Fund Strategies equity emerging event hedge m a rkets driven cta a nd relative m a rket m acro va lue neutr a l short bi as Annualized Mean 10.58 9.60 10.32 8.39 8.23 5.73 1.04 Annualized Std 9.49 14.25 6.97 6.69 4.35 3.30 18.96 Annualized Sharpe 0.74 0.42 0.97 0.72 1.06 0.65 –0.13 Source: Rober t Mirsky, Anthony Cowell, and Andrew Baker, “The Value of the Hedge Fund Industr y to Investors, Markets, and the Broader Economy,” KPMG and the Centre for Hedge Fund Research, Imperial College, London, last modified April 2012, http://www.kpmg.com/KY/en /Documents/the-value-of-the-hedge-fund-industr y-par t-1.pdf, 11.

How I Became a Quant: Insights From 25 of Wall Street's Elite by Richard R. Lindsey, Barry Schachter

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Peter Bossaerts at Caltech tried to validate the Capital Asset Pricing Model (CAPM) using MBA students at Stanford and Yale as traders. The environment met all the CAPM assumptions: All participants had same time horizon and the same knowledge of the securities. The market contained a riskless security, the security return distributions were normal and all the available assets were tradable and held by the investors, and so on. Bossaerts expected the students to trade their portfolios to attain the highest Sharpe ratio, which he had set to be that of the capitalization weighted market. He assumed the students would drive their portfolios to the zero line, as shown in Figure 4.1. However, the results were all over the place when they traded using continuous market mechanisms like those used by the NYSE, Nasdaq, and London’s SETS (Stock Exchange Electronic Transfer Service). Indeed, the students, who were well versed in finance, left potentially profitable arbitrage opportunities on the table because, when trading one security for another, they could not be certain of filling the second part of the trade at an advantageous price.

pages: 461 words: 128,421

The Myth of the Rational Market: A History of Risk, Reward, and Delusion on Wall Street by Justin Fox

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“I found that his spectacular level of performance was due to his spectacular level of market risk,” Treynor recalled decades later. He presented his conclusions to a room full of Yale trustees and alumni up from Wall Street. “I looked around that room and all I could see after my pitch was angry faces,” he said. “Yale discarded my recommendation completely.” Years later, the “Treynor ratio” became a much-used measure of investment manager performance. Even better known is the “Sharpe ratio,” a similar gauge—originally termed “reward-to-volatility”—that fellow CAPM pioneer Bill Sharpe introduced in a paper in 1966. Most famous of all is probably “alpha,” devised by Michael Jensen for his 1968 Chicago Ph.D. dissertation on mutual fund performance. Alpha is a portfolio’s performance minus the performance of a hypothetical benchmark portfolio of equivalent risk. This metric sounds complicated, but it delivers a wonderfully simple result.