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# Fixed Income Finance: A Quantitative Approach

## Overview

A complete guide for professionals with advanced mathematical skills but little or no financial knowledge . . .

You’re smart. Logical. Mathematically adept. One of those people who can make quick work of long, difficult equations.

But when it comes to managing a financial portfolio and managing risk, you wonder if you’re missing out.

*Fixed Income Finance* is the book for you.

It’s the perfect introduction to the concepts,

formulas, applications, and methodology,

all derived from first principles, that you need to succeed in the world of quantitative finance—with a special emphasis on fixed incomes. Written by two of the sharpest analytical minds in their fields, this instructive guide takes you through the basics of fixed income finance, including many new and original results, to help you understand:

- Treasury Bonds and the Yield Curve
- The Macroeconomics behind Term

Structure Models - Structural Models for Corporate

Bonds and Portfolio Diversification - Options
- Fixed Income Derivatives
- Numerical Techniques

Filled with step-by-step equations, clear and concise concepts, and ready-to-use formulas, this essential workbook bridges the gap between basic beginners’ primers and more advanced surveys to provide hands-on tools you can begin to use immediately. It’s all you need to put your math skills to work—

and make the money work for you.

Brilliantly researched, impeccably detailed,

and thoroughly comprehensive, *Fixed IncomeFinance* is applied mathematics at its best and most useful.

## Product Details

ISBN-13: | 9780071621205 |
---|---|

Publisher: | McGraw-Hill Professional Publishing |

Publication date: | 12/23/2009 |

Pages: | 256 |

Product dimensions: | 6.10(w) x 9.10(h) x 1.10(d) |

## About the Author

Mark Wise is the John A. McCone

Professor of High Energy Physics at the

California Institute of Technology. He is the winner of the 2001 J.J. Sakurai Prize of the

American Physical Society and a member of the American Academy of Arts and Sciences and National Academy of Sciences. He is also the coauthor of *Heavy Quark Physics*.

Vineer Bhansali is an executive vice president, portfolio manager, firm-wide head of analytics for portfolio management, and a senior member of PIMCO’s portfolio management team. He is the author of *Pricing and Managing Exotic and Hybrid Options* and currently serves as an associate editor for the *International Journal of Theoretical andApplied Finance*.

## Read an Excerpt

#### 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

All rights reserved.

ISBN: 978-0-07-162120-5

#### Excerpt

<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*(Continues...)*

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## Table of Contents

Section 1: Bond Basics: Treasury bonds and the Yield Curve/Corporate Bonds and Credit Risk/ Derivatives/ Mortgages/ Municipal Bonds/Real Return Bonds

Section 2: Probability Theory and Stochastic Processes:

Normal Random Variables/The Central Limit Theory/ The Probability Distribution for Corporate Bonds Returns/ Correlated Random Variables/ Random Walks/Survival Probabilities/ Correlated Random Walks/Simulation

Section 3: Term Structure Models: One Factor Models and Two Factor Models/Bond Prices, Volatilities/Eurodollar Futures/Futures and Forward Contracts/ Macroeconomics and Two Factor Models

Section 4: Options: Call and Put Options on a Stock/The Merton Model/Options on Interest Rate Sensitive Securities

Section 5: Portfolio Allocation: Utility Functions/The Sharpe Ratio/Beyond Mean and Variance/ Value at Risk/ Examples