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FIXED INCOME FINANCE
A QUANTITATIVE APPROACH
By MARK B. WISE, VINEER BHANSALI The McGraw-Hill Companies, Inc.
Copyright © 2010The McGraw-Hill Companies, Inc.
All rights reserved.
ISBN: 978-0-07-162120-5
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<h2>CHAPTER 1</h2> <p><b>Bond Basics</p> <br> <p>1.1 Treasury Bonds and the Yield Curve</b></p> <p>A zero coupon bond pays its holder an amount <i>P</i> at some time <i>T</i> years in the future. <i>P</i> is called the principal of the bond, and <i>T</i> is the time of maturity. The bonds can be issued, for example, by corporations or by the U.S. Federal Reserve. Suppose an investor purchases a zero coupon bond today; how much should the investor pay? This depends on a number of factors. For example, how sure is the investor that the institution that issued the bond will be able to make the principal payment at the time of maturity <i>T</i>? If the institution is a corporation, it might go bankrupt before that date and not be able to make the full principal payment <i>P</i>. This possibility is a source of risk for the investor; it is called <i>credit risk</i>. If the institution is the U.S. Treasury, the credit risk is nonexistent. Yet even for a Treasury bond, we would not pay $100 today to get $100 at some time in the future because the present value is degraded by inflation and the amount of future inflation is uncertain. The present value, i.e., the amount an investor would pay today for a zero coupon Treasury bond that matures in <i>T</i> years and has principal <i>P</i>, can be written as</p> <p>Price = 1/(1 + <i>Y</i><sub>1</sub>)<i><sup>T</sup> P</i> (1.1.1)</p> <p>We can think of <i>Y</i><sub>1</sub> in <b>Equation 1.1.1</b> as a yearly interest rate by which we are discounting the value of the payment <i>P</i> that occurs at time <i>T</i>. After all, if you had invested that amount today and received yearly interest at a rate <i>Y</i><sub>1</sub> that was compounded annually, then the value of your investment at maturity <i>T</i> would equal the principal <i>P. Y</i><sub>1</sub> is called the <i>yield to maturity T</i>.</p> <p>The yield is often quoted in units of percent or basis points (bp). One hundred basis points equals 1 percent. If <i>T</i> = 10 years and the (annual) yield to this maturity is equal to 5 percent or 500 bp, the price of a zero coupon Treasury bond with principal $100 is $61.39.</p> <p><b>Equation 1.1.1</b> is written in a way that suggests that the principal is discounted annually. However, there is nothing special about discounting annually. Suppose we discount every 1/<i>n</i> years by an amount <i>Y<sub>n</sub>/n</i>, where <i>n</i> is a natural number greater than 1. Then <b>Equation 1.1.1</b> becomes</p> <p>Price = 1/(1 + <i>Y<sub>n</sub>)<sup>nT</sup> P</i> (1.1.2)</p> <p>Equating the prices in <b>Equations 1.1.1</b> and <b>1.1.2</b> gives</p> <p><i>Y<sub>n</sub> = n</i> [(1 + <i>Y</i><sub>1</sub>])<sup>1/<i>n</i></sup>] (1.1.3)</p> <p>Taking the limit of <b>Equation 1.1.2</b>, keeping <i>Y<sub>n</sub></i> fixed at <i>Y</i>, as <i>n</i> -> ∞ gives the formula</p> <p>Price = <i>e<sup>-YT</sup> P</i> (1.1.4)</p> <p>which corresponds to discounting continuously in time by the fixed yield <i>Y</i>. In this limit, the discount factor for each infinitesimal time interval [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where we have neglected quadratic terms in the infinitesimal time interval <i>dt</i>. Repeating this infinitesimal discounting for each successive time interval of length <i>dt</i> gives the price in <b>Equation 1.1.4</b> written as the exponential of an integral,</p> <p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.5)</p> <p>as <i>n</i> -> ∞. The yield for discounting continuously in time <i>Y</i> is related to the yield for yearly discounting <i>Y</i><sub>1</sub> by</p> <p><i>Y</i> = log(1 + <i>Y</i><sub>1</sub>) (1.1.6)</p> <p>In other words, annual discounting and continuous discounting are equivalent. They are just different ways of writing the same price.</p> <p>There is no reason that all maturities should be discounted by the same factor. Suppose that the discounting rate for the time interval [<i>t, t + dt</i>] (i.e., the "short rate" at time <i>t</i>t;) is <i>y(t)</i>; the price then becomes</p> <p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.7)</p> <p>and the yield to maturity <i>T</i> can be written as</p> <p>[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1.8)</p> <p>Multiplying the above by <i>T</i> and differentiating with respect to the maturity, the "short rate" at any time <i>T</i> is given by</p> <p><i>y(T) = d[TY(T)]/dT</i> (1.1.9)</p> <p><i>YY(T)</i> is also called the <i>spot rate</i>. It is a function of the maturity <i>T</i>, and this function is known as the <i>yield curve</i>. Under typical circumstances, we can expect that the spot rate will be an increasing function of <i>T</i>. We may be confident that inflation will be contained over the near term, but as the period of time increases, that confidence diminishes, and the investor who purchases a zero coupon bond should demand compensation for that source of risk. Of course, if investors feel that the economy is about to go into recessiiiion, then we might expect that inflationary pressures and interest rates will fall in the future, and in that case <i>Y(T)</i> could, for some range of <i>T</i>, be
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