# zero-coupon bond

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The Concepts and Practice of Mathematical Finance by Mark S. Joshi

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In effect, a pound a year from now is therefore worth less than a pound today. The interest paid on a riskless loan expresses this. We can quantify precisely how much less by using risk-free bonds. A zero-coupon bond with principal £1 maturing in a year is precisely the same as receiving the sum of £1 in a year. We can therefore change the timing of a cashflow through the use of zero-coupon bonds. (A cashflow is a flow of money that occurs at some time.) If we are to receive a definite cashflow of LX at time T, then that is the same as being given X zero-coupon bonds today, and we can convert it into a cashflow today by simply selling X zero-coupon bonds of maturity T. The two cashflows at time 7' will then cancel each other. If the market value of a T-maturity bond is P(T), then £X at time 7' is equivalent to £XP (T ) today.

This means that we have precisely hedged the forward contract at zero cost, so the contract must be worth zero or there would be an arbitrage. If we have a forward contract struck at K', we can decompose it as a forward contract struck at K, with K as above, and the right to receive £(K - K') at time T. The right to receive £(K - K') is the same as holding K - K' zero-coupon bonds. Note that if K < K', we are really borrowing K' - K zero-coupon bonds. The forward contract struck at K has zero value so the value of the contract must be the value of the zero-coupon bonds, that is e' T (K - K') = (eGr-d)T So - K)(2.3) and we are done. The second part of the theorem motivates a definition. The forward price of a stock for a contract at time T is e(' -d)T So. 2.7 Mathematically defining arbitrage 27 2.7 Mathematically defining arbitrage We have seen that arbitrage can price various simple contracts precisely in a way that allows for no doubt in the price, and the price is independent of our views on how asset prices will evolve.

If we call a marketmaker and ask to buy or sell, he will always quote a pair of prices straddling the theoretical price; thus there is always a spread around the theoretical curve. 13.4.2 Gilts The lowest yielding instruments are, of course, the riskless ones - for example, UK government bonds which are generally known as gilts. The UK government does not generally issue zero-coupon bonds so all we can observe in the market is the 13.4 Curves and more curves 315 price of coupon-bearing bonds. However, a coupon-bearing bond is decomposable into a sum of zero-coupon bonds. This is clear if we remember that the bond is really just a sequence of cashflows. The cashflows are the coupon at each couponpayment date and the repayment of the principal at maturity. Any cashflow is just a zero-coupon bond with expiry equal to the timing of the flow and notional equal to the size of the cashflow. This means that we can attempt to fit a theoretical discount curve for zerocoupon bonds to the observed prices of UK gilts.

Mathematics for Finance: An Introduction to Financial Engineering by Marek Capinski, Tomasz Zastawniak

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There are many kinds of bonds like treasury bills and notes, treasury, mortgage and debenture bonds, commercial papers, and others with various particular arrangements concerning the issuing institution, duration, number of payments, embedded rights and guarantees. 2.2.1 Zero-Coupon Bonds The simplest case of a bond is a zero-coupon bond , which involves just a single payment. The issuing institution (for example, a government, a bank or a company) promises to exchange the bond for a certain amount of money F , called the face value, on a given day T , called the maturity date. Typically, the life span of a zero-coupon bond is up to one year, the face value being some round ﬁgure, for example 100. In eﬀect, the person or institution who buys the bond is lending money to the bond writer. Given the interest rate, the present value of such a bond can easily be computed.

Of course, if the interest rates are independent of maturity, then this formula is the same as (10.1). 230 Mathematics for Finance Remark 10.1 To determine the initial term structure we need the prices of zero-coupon bonds. However, for longer maturities (typically over one year) only coupon bonds may be traded, making it necessary to decompose coupon bonds into zero-coupon bonds with various maturities. This can be done by applying formula (10.3) repeatedly to ﬁnd the yield with the longest maturity, given the bond price and all the yields with shorter maturities. This procedure was recognised by the U.S. Treasury, who in 1985 introduced a programme called STRIPS (Separate Trading of Registered Interest and Principal Securities), allowing an investor to keep the required cash payments (for certain bonds) by selling the rest (the ‘stripped’ bond) back to the Treasury. Example 10.9 Suppose that a one-year zero-coupon bond with face value \$100 is trading at \$91.80 and a two-year bond with \$10 annual coupons and face value \$100 is trading at \$103.95.

It is convenient to think of this account as a tradable asset, which is indeed the case, since the bonds themselves are tradable. A long position in the money market involves buying the asset, that is, investing money. A short position amounts to borrowing money. First, consider an investment in a zero-coupon bond closed prior to maturity. An initial amount A(0) invested in the money market makes it possible to purchase A(0)/B(0, T ) bonds. The value of each bond will fetch B(t, T ) = e−(T −t)r = ert e−rT = ert B(0, T ) 44 Mathematics for Finance at time t. As a result, the investment will reach A(t) = A(0) B(t, T ) = A(0)ert B(0, T ) at time t ≤ T . Exercise 2.35 Find the return on a 75-day investment in zero-coupon bonds if B(0, 1) = 0.89. Exercise 2.36 The return on a bond over six months is 7%. Find the implied continuous compounding rate. Exercise 2.37 After how many days will a bond purchased for B(0, 1) = 0.92 produce a 5% return?

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Derivatives Markets by David Goldenberg

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This is what you would receive if you invested in a zero-coupon bond with face value Ft,T payable at time T. So instead of borrowing the value of the underlying commodity, we invest (lend) in such a bond, B. A Long Spot Position–A Long Forward Position=Investing in a Zero Coupon Bond with face value equal to the forward price. 82 FORWARD CONTRACTS AND FUTURES CONTRACTS Rearranging B., this says that, C. A Long Spot Position=A Long Forward Position+Investing in a Zero Coupon Bond with face value equal to the forward price. Rearranging this, we get the same result as in A because—(investing in a zerocoupon bond with face value equal to the forward price) is the same as borrowing by issuing that zero-coupon bond. That is, D. A Long Forward Position=A Long Spot Position–Investing in a Zero-Coupon Bond with face value equal to the forward price, or E.

The answer simply requires us to apply the discount factor which we know is e–r . The current price of the bond is therefore, B(t,T)=e–r*\$1 =\$e–r. We can use this result to price any zero-coupon bond. Suppose that its face value (value at maturity) is \$F, it has years to maturity and the annualized, continuously compounded rate is r % per year. Then the current price must be e–r*F. n CONCEPT CHECK 4 Given the information above, suppose that F=\$1,000, =2 months, and r=2%. a. What is the current price of the corresponding zero-coupon bond? Here is another interesting example. Suppose that the zero-coupon bond has face value equal to the current forward price of an underlying commodity, Ft,T . That is, this zero-coupon bond will have as its payoff the forward price Ft,T . An economic situation that corresponds to this payoff is that of the short in a forward contract.

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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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The price relationship is rather straightforward for a zero-coupon bond of price B0-cpn. As an example, a 5-year zero-coupon bond @ 5% is estimated from Eq. 1.7, that is in discrete compounding: in continuous compounding: B0-cpn = 100 / (1 + 0.05)5 = 78.35 B0-cpn = 100 * e-0.05*5 = 77.88, supposing the rate is 5% in both cases. These relationships indicate that investing in the bond at its present value brings \$100 to the investor at maturity, the return of such an investment corresponding to the interest rate of the zero-coupon bond. For a classic bullet coupon bond, we can extrapolate the above result by considering that a coupon bond on n installments may be viewed as a sum of a series of n zero-coupon bonds, that is, for a bond involving n semi- or annual coupons: one zero-coupon bond for each of coupon payments, until the n−1th installment: their maturities correspond to those of the interest payments; each single repayment is equal to the coupon; one zero-coupon bond for the last (nth) installment, at maturity, corresponding to the payment of the last coupon plus the reimbursement of the principal.

In practice, however, the use of convexity can be more problematic than the use of duration in the case of lack of market liquidity, affecting the market bond price. Here are some properties of convexity: As yields decrease, both duration and convexity increase, and conversely. Among bonds with equal duration: the higher the coupon, the higher the convexity; the zero-coupon bond has the smallest convexity. This can easily be checked by building (B, y) curves for a zero-coupon bond and for various coupon bonds of same duration: we see that the flattest curve is the one of the zero coupon. Among bonds with same maturity, the zero-coupon bond has not only the greatest duration but also the greatest convexity. Beyond its role of improving the sensitivity calculation from only the use of duration, the convexity may also play some role in selecting bonds for a portfolio. Suppose that a portfolio manager needs to buy a bond with a given duration and has a choice between two bonds, A of lower convexity and B with higher convexity.

Let us now shift both cash flows PV and FV by + a time T. Their durations are now valuing T and t+T respectively. Buying a forward or future contract of maturity T on a zero-coupon bond maturing at t after T can be viewed as the combination of one short cash flow PV, corresponding to the payment of the contract at its maturity T, plus one long cash flow FV, at time t later – see Figure 3.8. Figure 3.8 A single cash flow valuing FV after time t Hence, the duration Dfwd is the sum of both durations of PV (as a negative cash flow) and FV: The extension to a coupon bond is straightforward, since a coupon bond can be split into a series of zero-coupon bonds. The duration Dopt of bond options (cf. Chapter 11, Section 11.2) will understandably involve the duration DB of its underlying bond, the delta Δ of the option (i.e., the quantity of underlying used to hedge the option position, cf.

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The Oil Factor: Protect Yourself-and Profit-from the Coming Energy Crisis by Stephen Leeb, Donna Leeb

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In addition, the dollar income that you are getting from your bonds becomes worth more for the same reasons that cash gains in value. Zero coupon bonds have even more potential. These are bonds that don’t pay coupons at regular intervals during their life span. Rather, you buy them at a discount to par and they are guaranteed to mature at par. For instance, you could buy a zero coupon bond that is guaranteed to pay you \$10,000 in fifteen years. Your purchase price, say, is \$2,000. During inflationary times, this isn’t so attractive, because in fifteen years \$10,000 might be worth next to nothing, maybe even less than the \$2,000 you put up initially. You gain little in real terms or even lose. But during deflation, they are suddenly a great deal because that guaranteed \$10,000 at the end of the rainbow keeps gaining in value. Typically during times of deflation, the gains from zero coupon bonds are 50 percent higher than the gains from regular coupon-paying bonds.

Stick to government bonds and ultra-high-quality corporate bonds. For regular bonds, our first choice is the Fidelity Investment Grade Bond Fund (1-800-544-8888), a well-managed fund that invests in high-grade bonds. As for zero coupon bonds, we’d recommend the American Century Zero-Coupon Bond Funds (1-800-345-2021). Key Points: • In the coming years of economic and market volatility, deflationary scares will be the counterpoint to inflationary pressures. Investors need to hold some deflation insurance at all times. When our oil indicator flashes a negative signal, emphasize deflation plays more heavily. • Deflation plays include T-bills, regular bonds, and zero coupon bonds. Zeros will appreciate most sharply during deflationary interludes but during inflationary stretches will offer nothing; T-bills and coupon-paying bonds provide steady income.

Cash, Bonds, and Zeros Historically the only investments that perform well during the kind of economy-ravaging deflation that would occur this time around are fixed income instruments such as cash and bonds—and in particular zero coupon bonds. The most analogous period is 1929-32, and as figure 17a, “Bonds in the Depression,” shows, fixed income investments were the only shelter. More recently, deflationary fears arose when oil prices surged and acted as the catalyst that punctured the tech bubble. The sharp fall in the market threatened to cause an economic meltdown. And from mid-1999 through early 2003, bonds rose 40 percent, while zero coupon bonds scored 100 percent gains. Let’s look in more detail at various deflation hedges. The first is cash, by which we mean money put into very short-term money market accounts, and preferably those guaranteed by the government or that invest only in government securities.

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Financial Modelling in Python by Shayne Fletcher, Christopher Gardner

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The second from last line of the implementation is the code representation of the above formula. class fixed leg payoff: def call (self, t, controller): event = controller.get event() flow = event.flow() id = event.reset id() obs = flow.observables()[id] model = controller.get model() env = controller.get environment() fixed rate = obs.coupon rate() requestor = model.requestor() state = model.state().fill(t, requestor, env) cpn = fixed rate*flow.notional()*flow.year fraction()\ *controller.pay df(t, state) return cpn Note that we delegate to the controller for the actual calculation of the zero coupon bond. The implementation of the pay df method on the controller class is given below: def pay df(self, t, state): if t < 0: historical df = self. model.state().create variable() historical df = self. historical df return historical df else: flow = self. event.flow() fill = self. model.fill() requestor = self. model.requestor() T = self. env.relative date(flow.pay date())/365.0 return fill.numeraire rebased bond(t, T, flow.pay currency()\ , self. env, requestor, state) endif In a pattern that should be familiar, the fil component of the model is called upon to perform the calculation of the numeraire-rebased zero coupon bond. It should also be noted that the implementation returns a value for discount factors in the past; the value being determined by the historical df argument passed in at construction time of the controller.

Indeed Boost.Python offers many more features to help the C++ programmer to seamlessly expose C++ classes to Python and embed Python into C++. 218 Financial Modelling in Python Note that, as expected, the stochastic discount factor is a Q-martingale, in fact it is an exponential martingale, whereas the zero coupon bond price is not a Q-martingale because, as can be seen below, its SDE has a non-zero drift. d P(t, T ) = P(t, T ) (r (t)dt + (φ(t) − φ(T )) C(t)dW (t)) . (C.11) For non path-dependent pricing problems it is normally convenient to work in the so-called forward QT -measure. In this measure the numeraire at time t is simply P(t, T) and Girsanov’s theorem implies that W̄ (t), as define below, is a QT -Brownian motion d W̄ (t) = dW (t) + (φ(T ) − φ(t)) C(t)dt. Substitution of equation (C.12) into equation (C.10) yields t P(t, T ) P(0, T ) C(s)d W̄ (s) = exp − φ(T ) − φ(T ) P(t, T ) P(0, T ) 0 2 t 1 − φ(T ) − φ(T ) C(s)2 ds , ∀t ≤ T ≤ T. 2 0 (C.12) (C.13) In other words, the numeraire-rebased zero coupon bond in the forward QT -measure is a QT martingale.

The requestor component encapsulates the need for a model pricer to gain access to both primary and secondary information: in essence a model pricer makes ‘requests’ of the model for this information. In the case of the Hull–White model there are only a few pieces of information required: a discount factor, a localvolatility and a term volatility. In the language of Appendix C, the t 2 term volatility is simply 0 C (s)ds and the local volatility is φ(t) − φ(T ). Note that, taken together with the relevant discount factors, any zero coupon bond can be written in terms of the local volatility and the term volatility. What we actually store in the environment for the term volatility is the following t 2 0 C (s)ds . t 0 exp(2λs)ds (8.1) The reason for this is that the above variable is more natural to use when calibrating the model to market prices. The requestor for the Hull–White model can be found in the ppf.model.hull white.requestor module as detailed below: class requestor: def discount factor(self, t, ccy, env): key = "zc.disc."

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The Essays of Warren Buffett: Lessons for Corporate America by Warren E. Buffett, Lawrence A. Cunningham

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Neither our bonds nor those of certain other companies that issued similar bonds last year (notably Loews and Motorola) resemble the great bulk of zero-coupon bonds that have been issued in recent years. Of these, Charlie and I have been, and will continue to be, outspoken critics. As I will later explain, such bonds have often been used in the most deceptive of ways and with deadly consequences to investors. But before we tackle that subject, let's travel back to Eden, to a time when the apple had not yet been bitten. If you're my age you bought your first zero-coupon bonds during World War II, by purchasing the famous Series E U.S. Savings Bond, the most widely-sold bond issue in history. (After the war, these bonds were held by one out of two U.S. households.) Nobody, of course, called the Series E a zero-coupon bond, a term in fact that I doubt had been invented. But that's precisely what the Series E was.

Further illuminating the folly of junk bonds is an essay in this collection by Charlie Munger that discusses Michael Milken's approach to finance. Wall Street tends to embrace ideas based on revenue-generating power, rather than on financial sense, a tendency that often perverts good ideas to bad ones. In a history of zero-coupon bonds, for example, Buffett shows that they can enable a purchaser to lock in a compound rate of return equal to a coupon rate that a normal bond paying periodic interest would not provide. Using zero-coupons thus for a time enabled a borrower to borrow more without need of additional free cash flow to pay the interest expense. Problems arose, however, when zero-coupon bonds started to be issued by weaker and weaker credits whose free cash flow could not sustain increasing debt obligations. Buffett laments, "as happens in Wall Street all too often, what the wise do in the beginning, fools do in the end."

We suggest this cause: many of the foolish buyers, and their advisers, were trained by finance professors who pushed beloved models (efficient market theory and modern portfolio theory) way too far, while they ignored other models that would have warned of danger. This is a common type of "expert" error .... H. Zero-Coupon Bonds25 Berkshire issued \$902.6 million principal amount of ZeroCoupon Convertible Subordinated Debentures, which are now listed on the New York Stock Exchange. Salomon Brothers handled the underwriting in superb fashion, providing us helpful advice and a flawless execution. Most bonds, of course, require regular payments of interest, usually semi-annually. A zero-coupon bond, conversely, requires no current interest payments; instead, the investor receives his yield by purchasing the security at a significant discount from maturity value. The effective interest rate is determined by the original issue price, the maturity value, and the amount of time between issuance and maturity.

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Frequently Asked Questions in Quantitative Finance by Paul Wilmott

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If you knew what this function was you would be able to value fixed-coupon bonds of all maturities by using the discount factor to present value a payment at time T to today, t. Unfortunately you are not told what this r function is. Instead you only know, by looking at market prices of various fixed-income instruments, some constraints on this r function. As a simple example, suppose you know that a zero-coupon bond, principal \$100, maturing in one year, is worth \$95 today. This tells us that Suppose a similar two-year zero-coupon bond is worth \$92, then we also know that This is hardly enough information to calculate the entire r(t) function, but it is similar to what we have to deal with in practice. In reality, we have many bonds of different maturity, some without any coupons but most with, and also very liquid swaps of various maturities. Each such instrument is a constraint on the r(t) function.

The risk-neutral forward curve evolves according to dF (t; T) = m(t, T) dt + ν(t, T) dX. Zero-coupon bonds then have value given by the principal at maturity is here scaled to \$1. A hedging argument shows that the drift of the risk-neutral process for F cannot be specified independently of its volatility and so This is equivalent to saying that the bonds, which are traded, grow at the risk-free spot rate on average. A multi-factor version of this results in the following risk-neutral process for the forward rate curve In this the dXi are uncorrelated with each other. Brace, Gatarek and Musiela The Brace, Gatarek & Musiela (BGM) model is a discrete version of HJM where only traded bonds are modelled rather than the unrealistic entire continuous yield curve. If Zi(t) = Z (t; Ti) is the value of a zero-coupon bond, maturing at Ti, at time t, then the forward rate applicable between Ti and Ti+1 is given by where τ = Ti+1 − Ti.

The assumption that there are no arbitrage opportunities in the market is fundamental to classical finance theory. This idea is popularly known as ‘there’s no such thing as a free lunch.’ Example An at-the-money European call option with a strike of \$100 and an expiration of six months is worth \$8. A European put with the same strike and expiration is worth \$6. There are no dividends on the stock and a six-month zero-coupon bond with a principal of \$100 is worth \$97. Buy the call and a bond, sell the put and the stock, which will bring in \$ − 8 − 97 + 6 + 100 = \$1. At expiration this portfolio will be worthless regardless of the final price of the stock. You will make a profit of \$1 with no risk. This is arbitrage. It is an example of the violation of put-call parity. Long Answer The principle of no arbitrage is one of the foundations of classical finance theory.

A Primer for the Mathematics of Financial Engineering by Dan Stefanica

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The convexity C of a bond with price B and yield y is 1 82 B C=B8 y 2· 2.8. NUMERICAL IMPLEMENTATION OF BOND MATHEMATICS 73 From (2.62) and (2.64), we conclude that y = r(O, T). In other words, the yield of a zero coupon bond is the same as the zero rate corresponding to the maturity of the bond. This explains why the zero rate curve r(O, t) is also called the yield curve. As expected, the duration of a zero coupon bond is equal to the maturity of the bond. From (2.58) and (2.64), we obtain that D = - ~ 8B = _ _ 1_ (-T Fe- yT ) = T Fe- yT B' 8y . The convexity of a zero coupon bond can be computed from (2.60) and (2.64): 1 1 82B C = - = -(T2 Fe- yT ) B 8y2 Fe- yT (2.60) = T2. Using (2.56), it is easy to see that C _ - ",n t2 -yt· L..,1i=1 i Ci t e B . (2.61) 2.8 The following approximation of the percentage change in the price of the bond for a given a change in the yield of the bond is more accurate than (2.59) and will be proved in section 5.6 using Taylor expansions: flB 13 ~ - Dfly + 1 2,C(fly)2.

(2.61) 2.8 The following approximation of the percentage change in the price of the bond for a given a change in the yield of the bond is more accurate than (2.59) and will be proved in section 5.6 using Taylor expansions: flB 13 ~ - Dfly + 1 2,C(fly)2. Numerical implementation of bond mathematics When specifying a bond, the maturity T of the bond, as well as the cash flows Ci and the cash flows dates ti, i = 1 : n, are given. The price of the bond can be obtained from formula (2.53), i.e., B 2.7.1 = Zero Coupon Bonds i=l A zero coupon bond is a bond that pays back the face value of the bond at maturity and has no other payments, i.e., has coupon rate equal to 0. If F is the face value of a zero coupon bond with maturity T, the bond pricing formula (2.53) becomes B = F e -r(O,T)T, (2.62) ° where B is the price of the bond at time and r (0, T) is the zero rate corresponding to time T. If the instantaneous interest rate curve r(t) is given, the bond pricing formula (2.54) becomes (2.63) provided that the zero rate curve r(O, t) is known for any t > 0, or at least for the cash flow times, i.e., for t = ti, i = 1 : n.

Bonds. 2.1 Double integrals. . . . . . . . . . 2.2 Improper integrals . . . . . . . . . . . . . . 2.3 Differentiating improper integrals . . . . . . 2.4 Midpoint, Trapezoidal, and Simpson's rules. 2.5 Convergence of Numerical Integration Methods 2.5.1 Implementation of numerical integration methods 2.5.2 A concrete example. . 2.6 Interest Rate Curves . . . . . 2.6.1 Constant interest rates 2.6.2 Forward Rates. . . . . 2.6.3 Discretely compounded interest 2.7 Bonds. Yield, Duration, Convexity . . 2.7.1 Zero Coupon Bonds. . . . . . . 2.8 Numerical implementation of bond mathematics 2.9 References 2.10 Exercises . 3 Probability concepts. Black-Scholes formula. Greeks and Hedging. 3.1 Discrete probability concepts. . . . . . . . . 3.2 Continuous probability concepts. . . . . . . 3.2.1 Variance, covariance, and correlation 3.3 The standard normal variable 3.4 Normal random variables . . . 3.5 The Black-Scholes formula. . 3.6 The Greeks of European options. 3.6.1 Explaining the magic of Greeks computations 3.6.2 Implied volatility . . . . . . . . . . . . 3.7 The concept of hedging. ~- and r-hedging . 3.8 Implementation of the Black-Scholes formula. 3.9 References 3.10 Exercises. . . . . . . . . . . . . . . . . . . . 4 45 45 48 51 52 56 58 62 64 66 66 67 69 72 73 77 78 81 81 83 85 89 91 94 97 99 103 105 108 110 111 Lognormal variables.

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Asset and Risk Management: Risk Oriented Finance by Louis Esch, Robert Kieffer, Thierry Lopez

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As for the second factor, it can be assumed that for equal levels of maturity, the rate is the same for all securities in accordance with the law of supply and demand. In reality, the coupon policies of the various issuers introduce additional differences; in the following paragraphs, therefore, we will only be dealing with zero-coupon bonds whose rate now depends only on their maturities. This simpliﬁcation is justiﬁed by the fact that a classic bond is a simple ‘superimposition’ of zero-coupon securities, which will be valuated by discounting of the various ﬁnancial ﬂows (coupons and repayment) at the corresponding rate.14 We are only dealing with deterministic structures for interest rates; random cases are dealt with in Section 4.5. If we describe P (s) as the issue price of a zero-coupon bond with maturity s and R(s) as the rate observed on the market at moment 0 for this type of security, called the spot rate, these two values are clearly linked by the relation P (s) = (1 + R(s))−s .

There may be premiums (positive or negative) on issue and/or on repayment. The bonds described above are those that we will be studying in this chapter; they are known as ﬁxed-rate bonds. There are many variations on this simple bond model. It is therefore possible for no coupons to be paid during the bond’s life span, the return thus being only the difference between the issue price and the redemption value. This is referred to as a zero-coupon bond .1 This kind of security is equivalent to a ﬁxed-rate investment. There are also bonds more complex than those described above, for example:2 • Variable rate bonds, for which the value of each coupon is determined periodically according to a parameter such as an index. 1 A debenture may therefore, in a sense, be considered to constitute a superimposition of zero-coupon debentures. Read for example Colmant B., Delfosse V. and Esch L., Obligations, Les notions ﬁnancières essentielles, Larcier, 2002.

For a ﬁxed period of time (such as one year), it is possible to use a rate of return equivalent to the return on one equity: Pt + Ct − Pt−1 Pt−1 This concept is, however, very little used in practice. 4.1.2.1 Actuarial rate on issue The actuarial rate on issue, or more simply the actuarial rate (r) of a bond is the rate for which there is equality between the discounted value of the coupons and the repayment value on one hand and the issue price on the other hand: P = T Ct (1 + r)−t + R(1 + r)−T t=1 Example Consider for example a bond with a period of six years and nominal value 100, issued at 98 and repaid at 105 (issue and reimbursement premiums 2 and 5 respectively) and a nominal rate of 10 %. The equation that deﬁnes its actuarial rate is therefore: 98 = 10 10 10 10 10 10 + 105 + + + + + 1+r (1 + r)2 (1 + r)3 (1 + r)4 (1 + r)5 (1 + r)6 This equation (sixth degree for unknown r) can be resolved numerically and gives r = 0.111044, that is, r = approximately 11.1 %. The actuarial rate for a zero-coupon bond is of course the rate for a risk-free investment, and is deﬁned by P = R(1 + r)−T Bonds 117 The rate for a bond issued and reimbursable at par (P = N V = R), with coupons that are equal (Ct = C for all t) is equal to the nominal rate: r = rn . In fact, for this particular type of bond, we have: P = T C(1 + r)−t + P (1 + r)−T t=1 =C (1 + r)−1 − (1 + r)−T −1 + P (1 + r)−T 1 − (1 + r)−1 =C 1 − (1 + r)−T + P (1 + r)−T r From this, it can be deduced that r = C/P = rn . 4.1.2.2 Actuarial return rate at given moment The actuarial rate as deﬁned above is calculated when the bond is issued, and is sometimes referred to as the ex ante rate.

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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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However, because calculating a bond’s YTM is complex and involves trial and error, it usually is done with a programmable business calculator. Related Terms: • Bond • Interest Rate • Yield • Coupon • Par Value Z zero-couPon bond What Does Zero-Coupon Bond Mean? A debt security that does not pay interest (a coupon) but is traded at a deep discount and paid in full at face value upon maturity; also called an accrual bond. Investopedia explains Zero-Coupon Bond Some zero-coupon bonds are issued as such, whereas others are bonds that have been stripped of their coupons by a financial institution and then repackaged as zero-coupon bonds. Because they offer the entire payment at maturity, zero-coupon bonds tend to fluctuate in price more than coupon bonds do. Related Terms: • Bond • Discount Rate • Maturity • Coupon • Face Value 327 This page intentionally left blank Index Note: page numbers in bold indicate definition 10-K/10-Q report, 1 401(k) plan, 1-2 403(b) plan, 2 ABS.

The effective annual rate of return after considering the effect of compounding interest; APY assumes that funds will remain in the investment vehicle for a full 365 days and = (1 + periodic rate)# Periods - 1 is calculated as follows: Investopedia explains Annual Percentage Yield (APY) APY is similar to the annual percentage rate insofar as it standardizes varying interest rate agreements into an annualized percentage number. For example, suppose you are considering whether to invest in a one-year zero-coupon bond that pays 6% at maturity or a high-yield money market account that pays 0.5% per month with monthly compounding. At first glance, the yields appear identical— 12 months multiplied by 0.5% equals 6%—but when the effects of compounding are included, it can be seen that the second investment actually yields more: 6.17% (1.005^(12 – 1) = 0.0617). Related Terms: • Certificate of Deposit—CD • Compound Annual Growth Rate—CAGR • Compounding • Money Market Account • Yield Annuity What Does Annuity Mean?

This also is referred to as the coupon rate or coupon percent rate. Investopedia explains Coupon For example, a \$1,000 bond with a coupon of 7% will pay \$70 a year. It is called a coupon because some bonds literally have coupons attached to them. Holders receive interest by stripping off the coupons and redeeming them. This is less common today as more records are kept electronically. Related Terms: • Bond • Premium • Zero-Coupon Bond • Interest Rate • Yield Covariance What Does Covariance Mean? A measure of the degree by which the returns on two risky assets move in tandem. A positive covariance means that asset returns move together; a negative covariance means the returns move inversely. One method of calculating covariance is by looking at return surprises (deviations from expected return) in each scenario. Another method is to multiply the correlation between the two variables by the standard deviation of each variable. 56 The Investopedia Guide to Wall Speak Investopedia explains Covariance Financial assets that have a high covariance with each other will not provide very much diversification.

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Traders, Guns & Money: Knowns and Unknowns in the Dazzling World of Derivatives by Satyajit Das

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The Japanese investors were keen on the high rates; specifically, they wanted to buy zero coupon bonds. Now in a normal bond you get regular interest payments and in a zero coupon bond, you get all your interest at the end. For example, let’s DAS_C08.QXP 8/7/06 222 4:49 PM Page 222 Tr a d e r s , G u n s & M o n e y say you own a \$100 bond that pays 10% for ten years. Normally you would get \$10 interest each year and get your \$100 back after ten years. In a zero coupon bond you don’t get any interest but at the end of ten years you get back \$259. The \$259 is the \$100 you invested plus \$159 of interest, which is \$10 for each of the ten years and the interest on the interest. The two bonds are exactly the same in terms of the return you get, but there are interesting differences. Zero coupon bonds are very sensitive to changes in interest rates, which the Japanese investors liked.

There was also a tax advantage: if you got normal interest rates then you paid tax on them but with a zero coupon bond, you paid \$100 today for a payment of \$259 in ten years. Was the \$159 income or something else? It all depended where you were. In Japan, at the time, the \$159 was treated as a capital gain and wasn’t taxed. This made it even more attractive for the investors – tax The MoF became free income. Understandably, Japanese investors concerned about this were keen on US\$ zero coupon bonds and blatant form of tax bought a lot of them. The MoF became avoidance. The solution concerned about this blatant form of tax was very Japanese. avoidance. The solution was very Japanese: the MoF let it be known that they preferred that investors limit their purchases of dollar zero coupon bonds. The investors complied. It was the way Japan Inc. worked.

This was shares without tears, investment without fear. The deal was an ingenious collage. The investor was buying a zero coupon bond, a bond that paid no interest. The investor bought it at a discount to its face value. For example, the investor would pay \$74 to buy a DAS_C04.QXP 8/7/06 258 4:51 PM Page 258 Tr a d e r s , G u n s & M o n e y bond worth \$100. Over five years, this was the same as getting 6.00% pa. The \$26 discount was the interest. In the capital guaranteed note, the investor paid \$100 anyway. The \$26 interest was used to buy a call option on the stocks, which provided the upside for the investor should the stock market rally. If it fell then the zero coupon bond matured and guaranteed the return of principal to the investor. The structure worked well. Billions were sold. The dealers gouged the investors on the interest rate on the zero and the option but it still worked.

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

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So which yield are we plotting at the 10-year point in figure 14.1? To clarify this question, fixed-income traders often look at the term structure of zero-coupon bond yields, i.e., the yield on a bond where C = 0 so that its entire value comes from the face value, which is paid at a single point in time. Traders observe zero-coupon bonds both by looking at the prices of such traded bonds and by inferring the zero-coupon bond yields from the prices of coupon bonds. Indeed, a coupon bond can be viewed as a portfolio of zero-coupon bonds—one for each coupon payment and one for payment of the face value. Hence, coupon bond values can be derived from zero-coupon bond yields, and vice versa. Figure 14.1. The yield curve, also called the term structure of interest rates. Bond Returns and Duration Having understood bond prices and bond yields, we just need to understand bond returns—i.e., how much money one can make in percentage from holding a bond.

Hence, the price sensitivity to yield changes is negative, and its absolute value is called the duration, D: By the magic of fixed-income mathematics, the duration can be shown (by differentiating equation 14.1) to be equal to the weighted-average time to maturity of all the remaining cash flows (coupons and face value) where each weight wti is the fraction of the bond’s present value being paid at that time Equation 14.4 explains the term “duration”: Dt is a weighted average of the times ti – t to the remaining cash flows. For instance, the duration of a 5-year zero-coupon bond is naturally equal to its time to maturity, 5. The magic is that Dt is also given by equation 14.3, that is, it also tells us how sensitive a bond price is to changes in its yield. Hence, equations 14.3 and 14.4 together tell us that the prices of longer term bonds are more yield sensitive than those of shorter term bonds. With this definition of duration, we can compute the price change ΔP that occurs with a sudden change in yield, ΔYTMt: Here, the last equality introduces the “modified duration,” .

If the YTM changes, then this yield change leads to an additional effect given via the modified duration (computed next time period at the current yield): If the yield rises as in figure 14.2, then the bond return will be reduced during this period, as seen in equation 14.7. If this happens, however, then the expected return going forward will be higher, as the bond now earns a higher yield. Indeed, if a zero-coupon bond is held to maturity, its return will still average its original YTM. Yield and Return of a Leveraged Bond Traders are often interested in their excess return over the risk-free rate and, correspondingly, a bond’s yield above the short rate. Indeed, bonds are often leveraged, that is, bought with borrowed money (where the bond is used as collateral) and the bond’s excess return is effectively the return of such a leveraged position.

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Money Mavericks: Confessions of a Hedge Fund Manager by Lars Kroijer

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Because Bure had so many bad associations, it was also likely that brand-name Swedish investment companies would shy away from having Bure appear in their holdings. The rights issue was structured as follows: for every share you held in the company you would be given one new share, two warrants with a five-year exercise period and a strike price of 0.75 SEK, and a zero-coupon bond with a par value of 2.50 SEK that would mature five years after issue (zero-coupon bonds don’t pay interest but trade at a discount to the eventual payment that reflects the time to maturity and credit risk). Jesus, it was confusing. After the recent mess, who would understand that, much less want to invest in it? After spending a couple of weeks analysing Bure, we came to the view that the company was trading at about a 45 per cent discount to the net asset value.

Although high, this discount level is not unusual for a European holding company, particularly when the values in the non-quoted businesses are not transparent and there is turmoil in the organisation. Two things about Bure intrigued us. The new and convoluted capital structure left shareholders confused and we felt there was a good chance that one of these three instruments (share, warrant or zero-coupon bond) would be severely mispriced. Also, we felt that the new management was open and willing to acknowledge that nothing was sacred in trying to turn the business round. I asked if this included the possibility that the organisation might be worth more if it ceased to exist and they replied, ‘Theoretically, yes’. Then again, without a dominant shareholder to support management, a shareholder-friendly attitude was clearly the order of the day.

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A Random Walk Down Wall Street: The Time-Tested Strategy for Successful Investing by Burton G. Malkiel

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Of course, the actual long-run rate of inflation may be considerably greater than 2 percent. But the 4 percent real return they promise gives a reasonably generous margin of safety. In my view, there are four kinds of bond purchases that you may want especially to consider: (1) zero-coupon bonds (which allow you to lock in high yields for a predetermined length of time); (2) no-load bond mutual funds (which permit you to buy shares in bond portfolios); (3) tax-exempt bonds and bond funds (for those who are fortunate enough to be in high tax brackets); and (4) U.S. Treasury inflation-protection securities (TIPS). Zero-Coupon Bonds Can Generate Large Future Returns Suppose you were told you could invest \$10,000 now and be guaranteed by the government that you would get back double that amount in about fifteen years. The ability to do so is possible through the use of zero-coupon securities.

A FITNESS MANUAL FOR RANDOM WALKERS Exercise 1: Gather the Necessary Supplies Exercise 2: Don’t Be Caught Empty-Handed: Cover Yourself with Cash Reserves and Insurance Cash Reserves Insurance Deferred Variable Annuities Exercise 3: Be Competitive—Let the Yield on Your Cash Reserve Keep Pace with Inflation Money-Market Mutual Funds (Money Funds) Bank Certificates of Deposit (CDs) Internet Banks Treasury Bills Tax-Exempt Money-Market Funds Exercise 4: Learn How to Dodge the Tax Collector Individual Retirement Accounts Roth IRAs Pension Plans Saving for College: As Easy as 529 Exercise 5: Make Sure the Shoe Fits: Understand Your Investment Objectives Exercise 6: Begin Your Walk at Your Own Home—Renting Leads to Flabby Investment Muscles Exercise 7: Investigate a Promenade through Bond Country Zero-Coupon Bonds Can Generate Large Future Returns No-Load Bond Funds Are Appropriate Vehicles for Individual Investors Tax-Exempt Bonds Are Useful for High-Bracket Investors Hot TIPS: Inflation-Indexed Bonds Should You Be a Bond-Market Junkie? Exercise 8: Tiptoe through the Fields of Gold, Collectibles, and Other Investments Exercise 9: Remember That Commission Costs Are Not Random; Some Are Lower than Others Exercise 10: Avoid Sinkholes and Stumbling Blocks: Diversify Your Investment Steps A Final Checkup 13.

Financial innovation over the same period has been equally rapid. In 1973, when the first edition of this book appeared, we did not have money-market funds, NOW accounts, ATMs, index mutual funds, ETFs, tax-exempt funds, emerging-market funds, target-date funds, floating-rate notes, volatility derivatives, inflation protection securities, equity REITs, asset-backed securities, Roth IRAs, 529 college savings plans, zero-coupon bonds, financial and commodity futures and options, and new trading techniques such as “portfolio insurance” and “flash trading,” to mention just a few of the changes that have occurred in the financial environment. Much of the new material in this book has been included to explain these financial innovations and to show how you as a consumer can benefit from them. This tenth edition also provides a clear and easily accessible description of the academic advances in investment theory and practice.

Mathematical Finance: Core Theory, Problems and Statistical Algorithms by Nikolai Dokuchaev

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The new market can be considered as a multistock market model with N stocks (N−1 options plus the original stock). Is this market arbitrage-free? (Hint: consider first N=2 and Ti≥T.) 5.13 Bond markets Bonds are being sold an initial time for a certain price, and the owners are entitled to obtain certain amounts of cash (higher than this initial price) in fixed time (we restrict our consideration to zero-coupon bonds only). Therefore, the owner can have fixed income. Typically, there are many different bonds on the market with different times of maturity, and they are actively traded, so the analysis of bonds is very important for applications. For the bond-and-stock market models introduced above, we refer to bonds as a riskfree investment similar to a cash account. For instance, it is typical for the Black-Scholes market model where the bank interest rate is supposed to be constant.

To ensure that the process θ(t) is finite and the model is arbitrage-free, some special conditions on a must be imposed such that equation (5.25) is solvable with respect to θ. To satisfy these restrictions, the bond market model deals with ã being linear functions of σ. In addition, we have feature (ii): the process (ã, σ) must be chosen to ensure that the price process is bounded (for instance, a.s. if Si(t) is the price for a zero-coupon bond with the payoff 1 at terminal (maturing) time T). Consider the case when the bank interest rate r(t) is non-random and known. Let P(t) be the price of a bond with payoff 1 at terminal time T (said to be the maturity time). Clearly, the only price of the bond that does not allow arbitrage for seller and for buyer is In this case, investment in the bond gives the same profit as investment in the cash account.

The choice of this measure may be affected by risk and risk premium associated with particular bonds. (For instance, some bonds are considered more risky than others; to ensure liquidity, they are offered for some lower price, so the possible reward for an investor may be higher.) Models for bond prices are widely studied in the literature (see the review in Lambertone and Lapeyre, 1996). An example: a model of the bond market Let us describe a possible model of a market with N zero-coupon bonds with bond prices Pk(t), where © 2007 Nikolai Dokuchaev and where is a given set of maturing times, Continuous Time Market Models 105 We consider the case where there is a driving n-dimensional Wiener process w(t). Let be a filtration generated by this Wiener process. We assume that the process r(t) is (To cover some special models, we do not assume that r(t)≥0.) In addition, adapted to we assume that we are given an and bounded process q(t) that takes values in Rn.

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

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In fact, it is impossible to model one bond without modeling all of them. A five-year bond and a three-year bond have other commonalities, too.You can think of a five-year Treasury bond that pays interest every six months as a collection of ten zero-coupon bonds with maturities spread six months apart over the next five years. Similarly, a three-year Treasury bond is a collection of six zero-coupon bonds respectively maturing every successive six months over the next three years. Decomposed in this way, the bonds' ingredients are shared: Both contain the first six zero-coupon bonds. Therefore, in modeling the three-year bond, you are also implicitly modeling parts of the five-year bond. In essence, Ravi's model allowed impermissible violations of the law of one price that lies beneath all rational financial modeling.

Stocks are relatively simple; they guarantee no future dividend payments and have no natural termination date, so their future prices are unconstrained. Treasury bonds are much more intricate: Because they promise to repay their principal when they mature, their price on that date is constrained to be par. Furthermore, since all Treasury bonds can be decomposed into a sum of more primitive zero-coupon bonds of varying maturities, they are all interrelated. My new boss Ravi had heuristically modified the Black-Scholes stock option model to make it work, at least approximately, for short-dated Treasury bond options. He had written a computer program to implement it, and the bond options desk now priced and hedged their options by means of it. As they got more experienced at using it, Peter Freund's desk discovered that Ravi's model was fine for short-terns options but questionable for longer-term ones; it suffered from a variety of theoretical inconsistencies stemming from its inadequate modeling of the longterm behavior of bond prices.

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How the City Really Works: The Definitive Guide to Money and Investing in London's Square Mile by Alexander Davidson

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Globally, net issues of international debt security markets reached US \$2,733 billion in 2006, up from US \$1,850 billion in 2005, with the UK contributing US \$414 billion, up from US \$362 billion, marking a decline in UK market share from 20 to 15 per cent, according to the May 2007 IFSL report. ____________________________________________ CREDIT PRODUCTS 93  The United States gained in market share from 11 to 18 per cent over the same period, leapfrogging the UK as its net issues rose from US \$205 billion to US \$505 billion. Spain, France, Italy and others lost market share, while the Netherlands and Ireland gained. Zero-coupon bonds Zero-coupon bonds do not pay interest in their life. Investors buy them at a deep discount from par value, which they receive in full when the bond reaches maturity. The bonds give investors predictability, but the price swings easily with interest rate changes and the market is fairly illiquid. If investors should sell before maturity, they may not make a proﬁt. Gains are subject to capital gains tax.

The investor would then be left with money to reinvest in a world where interest rates are low. To compensate for this reinvestment risk, the callable bond will often pay a high coupon. 12 Credit products Introduction In this chapter, we will cover credit products, as distinct from the interest rate products covered in Chapter 11. We will focus on corporate bonds, international debt securities, junk bonds, asset-backed securities, zero-coupon bonds and equity convertibles. We will consider credit derivatives. Overview Credit products are integral to ﬁnancial markets and help to fuel merger and acquisition activity, which, as we saw in Chapter 7, can keep equity market activity high. A predator will often ﬁnance a company takeover partly out of cheap debt, which has helped to keep credit markets buoyant. Credit products may be seen as parts of a larger whole.

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Predator's Ball by Connie Bruck

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Hot on the heels of Murdoch-Metromedia came Storer Communications, at \$1.93 billion further testament to this kind of investment faith. Like Metromedia, Storer did not have enough money to meet its fixed charges out of cash flow, and like Metromedia it contained a healthy quantity of zero-coupon bonds (one third of the total) in its mix of securities. The zero-coupon bond was vital to these deals. Sold at a discount from its face value, it requires no interest payments (hence, “zero-coupon”) until maturity, when the annual accrued interest and the principal are paid out. Drexel had pioneered the heavy use of zero-coupon bonds in junk deals (especially in the communications industry) where the company in the foreseeable future could not make its interest payments. The day of reckoning for many of these deals, with securities issued in 1985 and 1986, would be years away.

But Drexel was already working on one of Sharon Steel’s famous 3(a)9 swaps (unregistered exchange offers), in an effort to avert Chapter 11. Moreover, since 1982 the firm had floated nearly a half-billion dollars for various companies of Posner’s, in three private placements (of securities that are not publicly registered and therefore can be issued more quickly and require less disclosure) and two public deals. One of them, issued for DWG back in 1982—\$50 million of zero-coupon bonds (which are sold at a discount and pay no interest until the annual accreted interest is paid at maturity)—had a short maturity, due in 1986. Among the heaviest buyers of Posner’s paper, furthermore, were members of Milken’s select coterie, those he most protected. In one \$25 million issue for a Posner company in 1982, for example, according to a November 1984 article in Forbes magazine, Fred Carr (First Executive) and Carl Lindner (American Financial) bought the entire issue.

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Investing Demystified: How to Invest Without Speculation and Sleepless Nights by Lars Kroijer

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Also be careful in thinking that adding these kinds of bonds provide you with additional safety; they are typically a poor diversifier of risk as they tie back to the same creditworthiness as the domestic government bonds. Matching time horizon In the discussion above, short-term bonds are the minimal risk asset. This is because longer-term bonds have greater interest risk (the fluctuation in the value of the bond as a result of fluctuations in the interest rate). Consider the example of a one-month zero-coupon bond and a 10-year zero-coupon bond that trade at 100 (zero-coupon bonds don’t pay interest, only the principal back at maturity). Now suppose annual interest rates go from zero to 1% suddenly. What happens to the value of the bonds? The one-month bond declines a little in value to reflect an interest rate of 1%, while the 10-year bond declines to a value of around 90.5 to reflect the higher interest rate. Clearly something that can go from 100 to 90.5 fairly quickly (rate changes are rarely that dramatic) is riskier, even if your chance of eventually being paid in full has not changed.

The Intelligent Asset Allocator: How to Build Your Portfolio to Maximize Returns and Minimize Risk by William J. Bernstein

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Implementing Your Asset Allocation Strategy 165 Retirement—The Biggest Risk of All This book is focused primarily on the investment process, particularly the establishment and maintenance of efficient allocations. Asset allocation in retirement is no different, except that you will primarily be using your withdrawals to control your allocations, as opposed to deposits and rebalancing. However, there is a risk peculiar to retirement called “duration risk.” In order to explore this, let’s start with the simplest and least risky of all investments, a one-year Treasury bill. A bill is in reality a zero-coupon bond, bought at a discount. For example, a 5% bill will sell at auction for \$0.9524 and be redeemed at par (\$1). If a few seconds after it is issued yields suddenly rise to 10%, the bill falls in price to \$0.9091, with an immediate loss of 4.55% in value. But if our investor holds the bill to maturity, he or she will receive the full 5% return, the same as if there had been no yield rise and price fall.

However, a bond is a very different beast than a T-bill: It throws off coupons that can be reinvested at the higher yield. Because of this, the recovery from disaster takes considerably less than 30 years. In fact, it only takes our hapless bondholder 10.96 years to break even. This 10.96-year period is known in financial circles as the duration of the security, and for a coupon-bearing bond it is always less than the maturity, sometimes considerably so. (For a zero-coupon bond, maturity and duration are the same.) There are lots of other definitions of duration, some dizzyingly complex, but “point of indifference” is the simplest and most intuitive. (The other useful definition is the ratio of price-to-yield 166 The Intelligent Asset Allocator change. That is, our 30-year bond will decrease 10.96% in price with each 1% increase in yield.) Duration is also an excellent measure of the risk of an investment.

Value stock: A security that sells at a discount to its intrinsic value. Value stocks are often identified by low price-book and priceearnings ratios. Variance: A measure of the scatter of numbers around their average value; the square root of the variance is the standard deviation (SD). Like SD, the variance of a security’s or portfolio’s returns is a proxy for its risk, or volatility. Yield: The percentage of a security’s value paid as dividends. Zero-coupon bond: A bond in which no periodic coupon is paid; principal and reinvested interest are paid in toto at maturity. Bibliography Preface Brinson, Gary P., Hood, L. Randolph, and Beebower, Gilbert L., “Determinants of Portfolio Performance.” Financial Analysts Journal, July/August 1986. Brinson, Gary P., Singer, Brian D., and Beebower, Gilbert L., “Determinants of Portfolio Performance II: An Update.”

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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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Calculate the discount rate for a k-year bond on day t, where the maturity is in between the maturity of two other bonds with maturity t1 and t2, using the following formula: (D.2) With the new discount factor, we can immediately compute the price of the zero-coupon bond on the next business day. For example, suppose we wish to compute the return of the 10-year zero-coupon bond from day t to day t+1.1 The price of bond with maturity m on day t would be given by .2 The same bond's price one day later would be given by , where is calculated on day t+1 according to the interpolation above. The return of the bond for that constant maturity series is given by: (D.3) Consider an example of this methodology. Suppose that on April 5, 1989, and April 6, 1989, the 10-year and 9-year zero-coupon bond yields were 9.34, 9.372 and 9.374, 9.407, respectively. Table D.1 uses the methodology described above to compute the price of the 10-year on April 5, 1989, the price of the 10-year minus 1-day on April 6, 1989, and the daily return of the 10-year from April 5 to April 6.

In fact, the spread of these quantitative techniques may have gradually diminished attractive opportunities in bond arbitrage. Box 2.2 Salomon Arb Group Interview Question Question: Your portfolio group strongly believes that the yield curve is going to flatten very soon. It could be that short-term rates will rise or long-term rates will fall or some combination of the two. Suppose also that you have three instruments available: a 30-year zero-coupon bond, a 1-year Treasury bill, and a cash account. Suppose the modified duration of the 30-year is 28 and the modified duration of the 1-year is 1. What strategy should you pursue to benefit from your beliefs? Suggested Solution: The investor would ideally like to have no interest-rate exposure, but take a view on the flattening yield curve. Thus, one would like to hedge parallel yield curve shifts, but take advantage of the nonparallel moves.

As of November 2007 Lehman had \$691 billion in assets. Of these, \$301 billion were in collateralized lending agreements (e.g., repos and such); the firm also had \$258 billion in collateralized financing. Lehman was a net lender of cash. Some traders believe Lehman may have taken initial margin from its prime broker business and used that cash in reverse repos. 5. In a 30-year liability with a given interest rate, the liability is like a zero-coupon bond with a duration of 30. , where y is the bond yield and P is the bond price. 6. Duration is a bond portfolio management concept that expresses how much a portfolio’s value will move for a given change in interest rates. If interest rates go down, a position with long duration makes money. If interest rates go up, the position loses money. 7. When a government issues a bond, it is essentially a fixed-rate payer and has a short or negative duration.

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J.K. Lasser's Your Income Tax by J K Lasser Institute

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Current reporting also applies to persons who separate or strip interest coupons from a bond and then retain the stripped bond or stripped coupon; the accrual rule applies to the retained obligation. - - - - - - - - - - Caution Reporting Zero Coupon Bond Discount Zero coupon bond discount is reported annually as interest over the life of the bond, even though interest is not received. This tax cost tends to make zero coupon bonds unattractive to investors, unless the bonds can be bought for IRA and other retirement plans that defer tax on income until distributions are made. Zero coupon bonds also may be a means of financing a child’s education. A parent buys the bond for the child. The child must report the income annually, and if the income is not subject to the parent’s marginal tax bracket under the “kiddie tax” (Chapter 24), the income subject to tax may be minimal. The value of zero coupon bonds fluctuates sharply with interest rate changes. This fact should be considered before investing in long-term zero coupon bonds.

This fact should be considered before investing in long-term zero coupon bonds. If you sell zero coupon bonds before the maturity term at a time when interest rates rise, you may lose part of your investment. - - - - - - - - - - For short-term nongovernmental obligations, OID is generally taken into account instead of acquisition discount, but an election may be made to report the accrued acquisition discount. See IRS Publication 550 for details. Basis in the obligation is increased by the amount of acquisition discount (or OID for nongovernmental obligations) that is currently reported as income. Interest deduction limitation for cash-basis investors. A cash-basis investor who borrows funds to buy a short-term discount obligation may not fully deduct interest on the loan unless an election is made to report the accrued acquisition discount as income.

OID arises when a bond is issued for a price less than its face or principal amount. OID is the difference between the principal amount (redemption price at maturity) and the issue price. For publicly offered obligations, the issue price is the initial offering price to the public at which a substantial amount of such obligations were sold. All obligations that pay no interest before maturity, such as zero coupon bonds, are considered to be issued at a discount. For example, a bond with a face amount of \$1,000 is issued at an offering price of \$900. The \$100 difference is OID. Generally, part of the OID must be reported as interest income each year you hold the bond, whether or not you receive any payment from the bond issuer. This is also true for certificates of deposit (CDs), time deposits, and similar savings arrangements with a term of more than one year, provided payment of interest is deferred until maturity.

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Fortune's Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street by William Poundstone

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Every six months, when an interest payment was due, the holder would detach a coupon and redeem it for the interest payment. After all the coupons were detached and the bond reached its maturity date, the bond certificate itself would be submitted for return of the principal. Regan’s idea was to buy new treasury bonds, immediately detach the coupons, and sell the pieces of paper separately. People or companies that expected to need a lump sum down the road could buy a “stripped,” zero-coupon bond maturing at the time they needed the money. It would be cheaper than a whole bond because they wouldn’t be paying for income they didn’t need in the meantime. Other people might want the current income but not care about the future lump-sum payment. They would buy the coupons. An even bigger selling point of Regan’s idea was a loophole in the tax law. Most of the pieces of paper from a dismembered bond would sell for a small fraction of their face value.

Most of the pieces of paper from a dismembered bond would sell for a small fraction of their face value. This was as it should be. A zero-coupon \$10,000 bond that matures in thirty years is not worth anywhere near \$10,000 now. Since there are no interest payments, the buyer can profit only by capital gains. That is possible only if the buyer pays much less than \$10,000 for the bond now. Fair enough. Buy a \$10,000 bond, strip off the coupons, and resell the zero-coupon bond for, say, \$1,000. This, it was theorized, ought to give you the right to claim a \$9,000 capital loss on your current year’s taxes. At any rate, nothing in the tax code said how taxpayers were supposed to figure the cost basis of the various parts of the bond. The law said nothing because no one in Congress had thought of stripping treasury bonds at the time the laws were written. Regan took the idea to Michael Milken.

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Wall Street: How It Works And for Whom by Doug Henwood

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The more reality INSTRUMENTS financial innovations adjustable rate convertible notes • adjustable rate preferred stock • adjustable/variable rate mortgages • All-Saver certificates • Annericus trust • annuity notes • auction rate capital notes • auction rate notes/debentures • auction rate preferred stock • bull and bear CDs • capped floating rate notes • collateralized connmercial paper • collateralized nnortgage obligations/real estate mortgage investment conduits • collateralized preferred stock • commercial real estate-backed bonds • commodity-linked bonds • convertible adjustable preferred stock • convertible exchangeable preferred stock • convertible mortgages/reduction option loans • convertible reset debentures • currency swaps • deep discount/zero coupon bonds • deferred interest debentures • direct public sale of securities • dividend reinvestment plan • dollar BILS • dual currency bonds • employee stock ownership plan (ESOP) • Eurocurrency bonds • Euronotes/Euro-commercial paper • exchangeable auction rate preferred stock • exchangeable remarketed preferred stock • exchangeable variable rate notes • exchange-traded options • extendible notes • financial futures • floating rate/adjustable rate notes • floating rate extendible notes • floating rate, rating sensitive notes • floating rate tax-exempt notes • foreign-currency-denominated bonds • foreign currency futures and options • forward rate agreements • gold loans • high-yield (junk) bonds • increasing rate notes • indexed currency option notes/ principal exchange linked securities • indexed floating rate preferred stock • indexed sinking fund debentures • interest rate caps/collars/floors • interest rate futures • interest rate reset notes • interest rate swaps • letter of credit/surety bond support • mandatory convertible/equity contract notes • master limited partnership • medium-term notes • money market notes • mortgage-backed bonds • mortgage pass-through securities • negotiable CDs • noncallable long-term bonds • options on futures contracts • paired common stock • participating bonds • pay-in-kind debentures • perpetual bonds • poison put bonds • puttable/adjustable tender bonds • puttable common stock • puttable convertible bonds • puttable-extendible notes • real estate-backed bonds • real yield securities • receivable-backed securities • remarketed preferred stock • remarketed reset notes • serial zero-coupon bonds • shelf registration process • single-point adjustable rate stock • Standard & Poor's indexed notes • state rate auction preferred stock • step-up put bonds • stock index futures and options • stripped mortgage-backed securities • stripped municipal securities • stripped U.S.

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A Demon of Our Own Design: Markets, Hedge Funds, and the Perils of Financial Innovation by Richard Bookstaber

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Three years out of school he had been terminated from trading positions at both Morgan Stanley and First Boston. Then in July 1991 he started work as a trader on Kidder, Peabody’s STRIPS (separate trading 39 ccc_demon_033-050_ch03.qxd 7/13/07 2:42 PM Page 40 A DEMON OF OUR OWN DESIGN of registered interest and principal securities) desk. The STRIPS desk takes Treasury bonds and strips apart their coupons to sell as individual “strips” or zero coupon bonds, and also works in the reverse, pulling together zero coupon bonds from various sources to rebuild or reconstitute Treasuries. Jett lost money in his first month trading at Kidder, was close to flat the following months, and received a negative performance review for the year. He could see the writing on the wall for a third failure in his trading career. So he resourcefully developed a trading strategy to improve his performance.

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Capital Ideas: The Improbable Origins of Modern Wall Street by Peter L. Bernstein

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There are markets for options (puts and calls) and markets for futures, and markets for options on futures. There is program trading, index arbitrage, and risk arbitrage. There are managers who provide portfolio insurance and managers who offer something called tactical asset allocation. There are butterfly swaps and synthetic equity. Corporations finance themselves with convertible bonds, zero-coupon bonds, bonds that pay interest by promising to pay more interest later on, and bonds that give their owners the unconditional right to receive their money back before the bonds come due. The world’s total capital market of stocks, bonds, and cash had ballooned from only \$2 trillion in 1969 to more than \$22 trillion by the end of 1990; the market for stocks alone had soared from \$300 billion to \$55 trillion.

See also Wells Fargo Bank Barr Rosenberg Associates (BARRA) Battle for Investment Survival, The (Loeb) “Behavior of Stock Prices, The” (Fama) Bell Journal Bell Laboratories Beta: see Risk, systematic “Beta Revolution: Learning to Live with Risk” Black Monday (October, 1987, crash) Black/Scholes formula Block trading Boeing Bond(s) convertible discount rates and government high-grade interest rates international junk liquidity maturity risk treasury: see Bond(s), government zero-coupon Bond market Boston Company Brokerage commissions. See also Transaction costs Brownian motion “Brownian Motion in the Stock Market” (Osborne) Butterfly swaps Buy and hold strategy California Public Employees Retirement System Calls: see Options Capital cost of optimal structure of preserving strategy “Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk” (Sharpe) Capital Asset Pricing Model (CAPM) non-stock applicability risk/return ratio in time analysis and Capital gains tax Capital Guardian Capital markets theory competition and corporate investment and debt/equity ratios and research CAPM: see Capital Asset Pricing Model CDs CEIR Center for Research in Security Prices (CRSP).

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Last Man Standing: The Ascent of Jamie Dimon and JPMorgan Chase by Duff McDonald

Jay Light noticed the same thing that Mike Ingrisani had at Browning—Dimon had a powerful independent streak, and often a different grasp of what a manager’s priorities should be in case studies. He bore down on fundamental issues such as expense strategy and risk management. One day, in a class discussion of various fixed income investments, Light challenged Dimon on the concept of investing in a long-term zero coupon bond that nevertheless had a 15 percent yield-to-maturity. (In other words, although the bond offered no annual interest payments, it was selling at a price that would offer a 15 percent annualized return at maturity.) Dimon launched a bomb into the middle of class: “If you don’t see the merits of investing in a 15 percent zero coupon bond, Professor Light, then you probably shouldn’t be teaching this class.” James “Longo” Long, who had come to Harvard after a stint at Hewlett-Packard in California, was dismayed by what he considered the lack of concrete “business” experience of many classmates who had worked in investment banking and consulting.

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Security Analysis by Benjamin Graham, David Dodd

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No one had ever heard of a venture capital fund, a private equity fund, an index fund, a quant fund, or an emerging market fund. And, interestingly, “famous investor” was largely an oxymoron—the world hadn’t yet heard of Warren Buffett, for example, and only a small circle recognized his teacher at Columbia, Ben Graham. The world of fixed income bore little resemblance to that of today. There was no way to avoid uncertainty regarding the rate at which interest payments could be reinvested because zero-coupon bonds had not been invented. Bonds rated below investment grade couldn’t be issued as such, and the fallen angels that were outstanding had yet to be labeled “junk” or “high yield” bonds. Of course, there were no leveraged loans, residential mortgage–backed securities (RMBSs), or collateralized bond, debt, and loan obligations. And today’s bond professionals might give some thought to how their predecessors arrived at yields to maturity before the existence of computers, calculators, or Bloomberg terminals.

Should those bonds falter, there may not be any recovery at all. During the 1980s, a significant percentage of the high yield bond market consisted of securities that had never been sold directly to investors but were parts of packages of securities and cash given to selling shareholders in acquisitions. The investment bank Drexel Burnham Lambert perfected this strategy, creating such instruments as zero-coupon bonds (paying no interest for, say, five years) or “pay-in-kind (PIK) preferreds,” which, instead of paying cash interest, just issued more preferred stock. Almost no one thought these securities were worth their nominal value, but selling shareholders generally approved the transactions. As the decade ended, however, the junk bond market collapsed and so did several of Drexel’s deals. These problems, coupled with Drexel’s legal difficulties with the SEC and prosecutors, led to the firm’s bankruptcy filing in 1990.

(Schwed), 6 Whitbread, 719 White Motor Company, 560–562, 565, 579, 586–588 White Motor Securities Corporation, 561 White Rock Mineral Springs Company, 303, 305n, 306 White Sewing Machine Company, 306, 307, 308 Willet and Gray, 95 Williams, John Burr, 18, 364n, 476n Willys-Overland Company, 252, 330 Wilson and Company, 245, 428 Winn–Dixie Stores, 275–276 “With Icahn Agreement, Texaco Emerges from Years of Trying Times” (Potts), 272n Withholding of dividends, 378–381 Woolworth, 86 Working capital: basic rules for, 591–594 requirements for, 245–246 safety of speculative senior issues and, 327–330 World War I, interest rates and bond prices and, 25 WorldCom, 545–547, 717 Wright Aeronautical Corporation, 63, 67, 679 Wright-Hargreaves Mines, Ltd., 522–523 X Xcel Energy, 51–53 Y Yahoo!, 53 Yale University, 630–631, 710n Yield: relationship with risk, 164–168 sacrificing safety for, 164–168 Youngstown Sheet and Tube Company, 244, 430, 461, 462, 476n, 501, 683 Z Zero-coupon bonds, 284 1 Losing money, as Graham noted, can also be psychologically unsettling. Anxiety from the financial damage caused by recently experienced loss or the fear of further loss can significantly impede our ability to take advantage of the next opportunity that comes along. If an undervalued stock falls by half while the fundamentals—after checking and rechecking—are confirmed to be unchanged, we should relish the opportunity to buy significantly more “on sale.”

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

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Structural models price all corporate securities in a common framework, grounded in the pioneering theoretical models of Merton (1974) and Black–Scholes (1973). In the classic “Merton model”, a firm’s capital structure is particularly simple: a single zero-coupon debt and a single equity issue. The firm’s equity can be viewed as a call option on the firm’s assets (struck at the maturity value D of its debt), while the firm’s debt consists of a riskless zero-coupon bond (which guarantees the payment of D) and a short put option on the value of the firm (struck at D). Thus the bondholder is effectively writing a put on the firm’s assets, being long equity but short equity volatility. The value of any option depends crucially on the volatility level of the underlying asset (as well as time horizon and leverage, where leverage is the difference between the current value of the firm’s assets and the value of its debt):• While all corporate stakeholders tend to benefit from rising equity prices, a key difference between the exposures of equity-holders and bondholders is that the former benefit from rising volatility while the latter are hurt by it.

The implications of two hypotheses about yield curve behavior Pure expectations hypothesis Risk premium hypothesis What is the information in forward rates (yield curve steepness)? Market’s rate expectations Required bond risk premia What future events should forward rates forecast? Future interest rate changes Near-term return differentials across bonds What is the best predictor of a 5-year zero-coupon bond’s 1-year return? The 1-year riskless spot rate The 5-year zero’s “rolling yield” (which is also the 1-year forward rate after 4 years) What is the best predictor of next year’s spot yield curve? Implied spot yield curve one year forward Current spot yield curve Roll or slide is another nuanced aspect of carry. The random walk hypothesis assumes that the current yield curve is the best predictor of the future yield curve.

Monte Carlo Simulation and Finance by Don L. McLeish

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SOME BASIC THEORY OF FINANCE backwards Kolmogorov equation 2.27 that if a related process Xt satisfies the stochastic diﬀerential equation dXt = r(Xt , t)Xt dt + σ(Xt , t)dWt then its transition kernel p(t, s, T, z) = ∂ ∂z P [XT (2.47) · z|Xt = s] satisfies a partial diﬀerential equation similar to 2.44; ∂p σ 2 (s, t) ∂ 2 p ∂p = −r(s, t)s − ∂t ∂s 2 ∂s2 (2.48) For a given process Xt this determines one solution. For simplicity, consider the case (natural in finance applications) when the spot interest rate is a function of time, not of the asset price; r(s, t) = r(t). To obtain the solution so that terminal conditions is satisfied, consider a product f (t, s, T, z) = p(t, s, T, z)q(t, T ) where q(t, T ) = exp{− Z (2.49) T r(v)dv} t is the discount function or the price of a zero-coupon bond at time t which pays 1\$ at maturity. Let us try an application of one of the most common methods in solving PDE’s, the “lucky guess” method. Consider a linear combination of terms of the form 2.49 with weight function w(z). i.e. try a solution of the form Z V (s, t) = p(t, s, T, z)q(t, T )w(z)dz (2.50) for suitable weight function w(z). In view of the definition of pas a transition probability density, this integral can be rewritten as a conditional expectation: V (t, s) = E[w(XT )q(t, T )|Xt = s] (2.51) the discounted conditional expectation of the random variable w(XT ) given the current state of the process, where the process is assumed to follow (2.18).

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Tools for Computational Finance by Rüdiger Seydel

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No investment is really free of risks. But bonds can come close to the idealization of being riskless. If the seller of a bond has top ratings, then the return of a bond at maturity can be considered safe, and its value is known today with certainty. Such a bond is regarded as “riskless asset.” The rate earned on a riskless asset is the risk-free interest rate. To avoid the complication of re-investing coupons, zero-coupon bonds are considered. The interest rate, denoted r, depends on the time to maturity T . The interest rate r is the continuously compounded interest which makes an initial investment S0 grow to S0 erT . We shall often assume that r > 0 is constant throughout that time period. A candidate for r is the LIBOR1 , which can be found in the ﬁnancial press. In the mathematical ﬁnance literature, the term “bond” is used as synonym for a risk-free investment.

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The Age of Stagnation by Satyajit Das

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Vindicating Edmund Burke's mistrust of the utopian promises of professors, more extreme policy measures are probable. Extension of the forms of QE can be expected, encompassing purchases of a wider range of assets, including shares, as in Japan. The US is considering canceling Treasury bonds held by the Fed to reduce debt, ignoring the loss to the central bank from the write-down of its holdings. In the UK, one idea considered was for the Bank of England to exchange holdings of government securities for zero coupon bonds, which pay no interest and have no maturity or fixed repayment date, and which were to be valued at face value to avoid loss. The global economy risks becoming trapped in a QE-forever cycle. A weak economy forces policymakers to implement expansionary fiscal measures and QE. If the economy responds, then increased economic activity and the side effects of QE will encourage a withdrawal of the stimulus.

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Valuation: Measuring and Managing the Value of Companies by Tim Koller, McKinsey, Company Inc., Marc Goedhart, David Wessels, Barbara Schwimmer, Franziska Manoury

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In choosing the bond’s duration, the most theoretically sound approach is to discount each year’s cash flow at a cost of equity that matches the maturity of the cash flow. In other words, year 1 cash flows would be discounted at a cost of equity based on a one-year risk-free rate, while year 10 cash flows would be discounted at a cost of equity based on a 10-year discount rate. To do this, use zero-coupon bonds (known as STRIPS)11 rather than Treasury bonds that make interim payments. The interim payments cause their effective maturity to be much shorter than their stated maturity. Using multiple discount rates is quite cumbersome. Therefore, few practitioners discount each cash flow using its matched bond maturity. Instead, most choose a single yield to maturity that best matches the cash flow stream being valued.

Thus, when you measure the cost of debt, estimate what a comparable investment would earn if bought or sold today. Below-Investment-Grade Debt In practice, few financial analysts distinguish between expected and promised returns. But for debt below investment grade, using the yield to maturity as a proxy for the cost of debt can cause significant error. To understand the difference between expected returns and yield to maturity, consider the following example. You have been asked to value a oneyear zero-coupon bond whose face value is \$100. The bond is risky; there is a 25 percent chance the bond will default and you will recover only half the final payment. Finally, the cost of debt (not yield to maturity), estimated using the CAPM, equals 6 percent.29 29 The CAPM applies to any security, not just equities. In practice, the cost of debt is rarely estimated using the CAPM, because infrequent trading makes estimation of beta impossible.

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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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When Mexico devalued at the end of 1994, it came as a complete surprise.A lot of investors were leveraged in Mexican and other Latin American debt, so the devaluation created the fear of further devaluations or the so-called tequila effect.Venezuela, Argentina, and Brazil saw their bonds sell off massively, but the economic reality of the situation in each of those countries was very different. We thought that their Brady bonds were trading at ridiculously low levels. Brady bonds have their principal backed by U.S.Treasuries so when you buy one of those bonds, you basically get two securities:The principal is a U.S. Treasury zero coupon bond, which is easy to price, and the coupon stream, which is represented by local sovereign risk. When the other Latin American Brady bonds sold off in sympathy with Mexico, once you stripped out the zeroes, you were left with sovereign risk for next to nothing. The implied interest rates embedded in these coupons were so high that we felt it was a terrific risk/reward opportunity. The sovereign spread to Treasuries ended up moving from over 1,900 basis points in early 1995 to 400 basis points by early 1997.

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Your Money or Your Life: 9 Steps to Transforming Your Relationship With Money and Achieving Financial Independence: Revised and Updated for the 21st Century by Vicki Robin, Joe Dominguez, Monique Tilford

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So they made what they call a Life Chart for all three of them. For each year from now until they will be eighty-five, they asked themselves, “What needs or desires might come up?” They included all the normal expenses of raising a healthy (but not pampered) child—things like braces, tutoring, summer camp and his first car—and determined how much each might cost. They then bought an investment vehicle called zero-coupon bonds (treasury bonds with no interest, bought at a big discount but repaid at par and especially good for future cash needs), with different bonds coming due in each of the years that their son might need a big-ticket item. And if he doesn’t need braces or want to go to summer camp, they’ll just roll over the money into regular treasury bonds. They also anticipated their own reasonable needs, including housing, health care, education and travel, and calculated how the combination of their cushion and their cache could handle them with ease.

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

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5 In 1987, Howard Rubin, a Merrill Lynch trader, lost \$377 million in mortgage trading. MBSs are split into IO (interest only) and PO (principal only) bonds. IOs pay out only the interest payments on the underlying pool of mortgages. Lower rates mean more prepayments, meaning less interest payments reducing the price of the IOs. Higher interest rates mean lower prepayments and more interest payments, increasing the value of the IOs. POs pay only principal, effectively like zero coupon bonds where you paid \$800 for a bond that at maturity pays \$1,000. POs behave exactly the opposite to IOs. If interest rates go down, then they appreciate in value, as the investor receives the face value of the bond earlier because of higher prepayments. If interest rates go up, then POs decrease in value as you get paid back later. Rubin owned a large amount of POs from Merrill Lynch’s deals that the firm had not managed to sell.

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The Social Life of Money by Nigel Dodd

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Strange, like Minsky, suggests that this problem has deepened the more the financial system has grown. Strange argues that the development of money substitutes encourages overbanking, i.e., “the imprudent expansion of credit with increased profits to the banks but increased risk to the system of financial panic and collapse” (Strange 1994b: 96). A new language had to be invented to describe these devices, she argued, incorporating “money market mutual funds, swaps, options, NOW accounts, zero coupon bonds, off balance-sheet financing, and so on” (Strange 1994b: 110). Overbanking, Strange argued, can lead to the death of money, which, “whether it comes about by inflation or by a political revolution sweeping away the government, inevitably brings trade, investment and economic life generally to a standstill” (Strange 1994b: 95, 99). Finance, by generating a volatile international environment through overbanking, is dangerous for society insofar as it is a threat to its money.

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The Making of Global Capitalism by Leo Panitch, Sam Gindin

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“They plainly did not feel there were equally attractive alternatives in Tokyo.”56 The status of US Treasuries as “the linchpin of the global financial order” was graphically captured in R. Taggart Murphy’s description of what made them so “irresistible” to large Japanese investors: [I]n all the blizzards of financial paper that blew through Tokyo during the 1980s—the Canadian and Australian dollar twofers, the reverse dual currency bonds, the Samurai bonds, the Sushi bonds, the instantly repackaged perpetuals, the zero-coupon bonds, the square trips and double-dip leveraged leases—US Treasury notes bills and bonds held pride of place. These securities . . . backed by the full faith and credit of the US government . . . formed a liquid market of great depth: the securities were traded around the world, and buyers and sellers were thus available twenty-four hours a day. Most other dollar debt securities were priced off Treasuries.

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Den of Thieves by James B. Stewart

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The October "minicrash," as it was quickly dubbed on Wall Street, proved a more long-lasting harbinger of trouble than the dramatic October 1987 crash. Beginning with Integrated and Campeau, and then continuing with alarming regularity, junk-bond issuers began to default on their obligations. Payment terms in highly leveraged deals, especially those completed in the frenzied days prior to the 1987 crash, had managed to disguise the underlying folly of the investments, often through the issuance of so-called "zero-coupon" bonds, "payments in kind," and "re-sets" which required no payments whatsoever for several years. Eventually the piper had to be paid. Like Integrated, the whole junk market began to tumble as companies admitted they couldn't fulfill the promises they had been so eager to make just several years before. By the time the financial data for 1989 were collected and analyzed, a growing suspicion of many participants in the junk-bond market, even of some Milken loyalists, was confirmed: Milken's oft-repeated premise that "investors obtained better returns on low-grade issues than on high-grades" was false.

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The Snowball: Warren Buffett and the Business of Life by Alice Schroeder

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Buffett, who usually dealt with uncomfortable issues by joking about them, ended the 1999 Berkshire annual report (written winter 2000) by saying that he loved running Berkshire, and “if enjoying life promotes longevity, Methuselah’s record is in jeopardy.” 14. This is sort of an inside joke at Berkshire Hathaway. 15. David Henry, “Buffett Still Wary of Tech Stocks—Berkshire Hathaway Chief Happy to Skip ‘Manias,’” USA Today, May 1, 2000. 16. Buffett owned 14 million barrels of oil at the end of 1997, bought 111 million ounces of silver, and owned \$4.6 billion of zero-coupon bonds as well as U.S. Treasuries. The silver represented 20% of the world’s annual mine output and 30% of the above-ground vault inventory (Andrew Kilpatrick, Of Permanent Value: The Story of Warren Buffett: More in ’04, California Edition. Alabama: AKPE, 2004), purchased on terms to avoid disrupting world supply. 17. Interview with Sharon Osberg. The silver was at JP Morgan in London. 18. Buffett measures his performance not by the company’s stock price, which he didn’t control, but by increase in net worth per share, which he did.