## One Factor Measures of Price Sensitivity

The exercises for this chapter are built around a spreadsheet exercise. Set up a column of interest rates from 1.75 to 8.25 in 25 basis point increments. In the next column compute the price of a perpetuity with a face of 100 and a coupon of 5 100x05 y where y is the rate in the first column. In the next column compute the price of a one-year bond with a face of 100 and an annual coupon of 5 105 (1+y). 5.1 Graph the prices of the perpetuity and the one-year bond as a function of the interest...

## Swaptions Caps And Floors

Swaptions i.e., options on swaps are particularly liquid fixed income options. A receiver swaption gives the owner the right to receive fixed in an interest rate swap. For example, a European-style receiver might give its owner the right on May 15, 2002, to receive fixed on a 10-year swap at a fixed rate of 5.75 . A payer swaption gives the owner the right to pay fixed in an interest rate swap. For example, an American-style payer might give its owner the right at any time on or before May 15,...

## Pricing American And Bermudan Bond Options In A Term Structure Model

By way of introducing arbitrage-free pricing, Chapter 9 described the pricing of European bond options. Basically, either 19.2 or 19.4 is used to determine the value of the option at expiration and then the standard tree methodology is used to determine the value of the option on FIGURE 19.3 Value of a 99-18.25 Straddle on the 5s of February 15, 2011, at Expiration FIGURE 19.3 Value of a 99-18.25 Straddle on the 5s of February 15, 2011, at Expiration earlier dates. This section, therefore,...

## The Science of Term Structure Models

Part One described how to price fixed income securities with fixed cash flows relative to other securities with fixed cash flows. While many such securities do trade in fixed income markets, there are also many securities whose cash flows are not fixed. Consider a European call option on 1,000 face value of a particular bond with a strike at par. This security gives the holder the right to buy 1,000 face value of the bond for 1,000 on the maturity date of the option. If the price of the bond on...

## Yieldto Maturity

3.1 On May 15, 2001, the price of the 11.625s of November 15, 2002, was 110-214 4. Verify that the yield-to-maturity was 4.2139 . Explain this yield relative to the spot rates from question 2.3. 3.2 On May 15, 2001, the price of the 6.75s of May 15, 2005, was 106-211 8. Use a calculator or spreadsheet to find the yield of the bond. 3.3 Consider a 10-year par bond yielding 5 . How much of the bond's value comes from principal and how much from coupon payments How does your answer change for a...

## Twovariable Regressionbased Hedging

The change in the 20-year yield is probably better predicted by changes in both 10- and 30-year yields than by changes in 30-year yields alone. Consequently, a market maker hedging a long position in 20-year bonds may very well consider selling a combination of 10- and 30-year bonds rather than 30-year bonds alone. Appropriate risk weights for the 10- and 30-year bonds may be found by estimating the following regression model Ay20 a p10 x Ay10 x Ay e, 8.14 The coefficients P10 and P30 give the...

## The Short Rate Process and the Shape of the Term Structure

Given the initial term structure and assumptions about the true interest rate process for the short-term rate, Chapter 9 showed how to derive the risk-neutral process used to determine arbitrage prices for all fixed income securities. Models that follow this approach and take the initial term structure as given are called arbitrage-free models of the term structure. Another approach, to be described in this and subsequent chapters, is to derive the risk-neutral process from assumptions about...

## Asset Swap Spreads And Asset Swaps

Chapter 17 discussed the convenience of measuring the value of bonds, like agency securities, relative to rate curves that are not contaminated by individual security effects. TED spreads, based on Eurodollar futures rates, serve this function for relatively short-term bonds. For longer-term bonds, market participants rely on asset swap spreads. The asset swap spread of a bond is the spread such that discounting the bond's cash 3Rather than pay fixed on a swap, a corporation may sell Treasuries...

## Special Repo Rates And The Auction Cycle

Current issues tend to be more liquid. This means that their bid-ask spreads are particularly low and that trades of large size can be conducted relatively quickly. This phenomenon is partly self-fulfilling. Since everyone expects a recent issue to be liquid, investors and traders who demand liquidity and who trade frequently flock to that issue and thus endow it with the anticipated liquidity. Also, the dealer community, which trades as part of its business, tends to own a lot of a new issue...

## Info

The constant 0 denotes the long-run value or central tendency of the short-term rate in the risk-neutral process and the positive constant k denotes the speed of mean reversion. Note that in this specification the greater the difference between r and 0 the greater the expected change in the short-term rate toward 0. Because the process 11.8 is the risk-neutral process, the drift combines both interest rate expectations and risk premium. Furthermore, market prices do not depend on how the...

## Reverse Repurchase Agreements And Short Positions

Rather than wanting to sell the 57 8s of November 15, 2005, as in the previous section, assume that the mutual fund wants to buy the bonds from a trading desk. Also assume that the trading desk doesn't happen to have that bond in inventory. The trading desk may very well sell the bonds anyway i.e., go short the bonds , planning to buy them from another client at a later time. When the trade settles and the mutual fund pays for the bonds, the trading desk is obliged to deliver the 57 8s. But...

## Forward Yield And Forward Dv01

In the example of the previous section, the spot yield of the 5.50s of May 15, 2009, is 4.904 and the forward yield is 5.081 . Analogous to the treatment of spot and forward rates in Chapter 2, it is useful to think of the spot yield as a blend of the repo rate and the forward yield. In the example, the spot yield of 4.904 from November 27, 2001, to May 15, 2009, is a mix of the 1.80 repo rate from November 27, 2001, to the forward date March 28, 2002, and of the 5.081 forward yield from March...

## Forward Price Of A Coupon Bond

This section derives the forward price of a coupon bond and begins with the following example Forward contract transaction date November 26, 2001 Underlying security 100 face amount of the 5.50s of May 15, 2009 Forward date March 28, 2002 Price of 5.50s of May 15, 2009, for November 27, 2001, settle 103.6844 Accrued interest for November 27, 2001, settle .1823 Repo rate from November 27, 2001, to March 28, 2002 1.80 Accrued interest for March 28, 2002, settle 2.0207 Number of days from November...

## Volatility And Convexity

2Those who thought the economy would take some time to recover predicted a U-shaped recovery. Those even more pessimistic expected an L-shaped recovery. Note that the expected interest rate on date 1 is .5x8 .5x12 or 10 and that the expected rate on date 2 is .25x14 .5x10 .25x6 or 10 . In the previous section, with no volatility around expectations, flat expectations of 10 imply a flat term structure of spot rates. That is not the case in the presence of volatility. The price of a one-year zero...

## Volatility As A Function Of The Short Rate The Coxingersollross And Lognormal Models

The models in Chapter 11 along with Model 3 assume that the basis point volatility of the short rate is independent of the level of the short rate. This is almost certainly not true at extreme levels of the short rate. Periods of high inflation and high short-term interest rates are inherently unstable and, as a result, the basis point volatility of the short rate tends to be high. Also, when the short-term rate is very low, its basis point volatility is limited by the fact that interest rates...

## Properties Of The Twofactor Model

Figure 13.2 graphs the rate curves generated by the V2 model along with the swap rate data on February 16, 2001. Apart from the three-month rate, the model is flexible enough to fit the shape of the term structure. The long-lived factor's true process and the risk premium give enough flexibility to capture the intermediate terms and long end of the curves while the short-lived factor process gives enough flexibility to capture the shorter to intermediate terms. As mentioned in Part One, the...

## Trading Case Study Trading 2s5s10s in Swaps with a Two Factor Model

This case study uses the V2 model of Chapter 13 to illustrate how a term structure model might be used in a particular trade, namely a butterfly in swaps.5 The V2 model, like any term structure model, has parameters and factors. The parameters of this model are the coefficients of mean reversion, the target levels of the factors, the risk premium, the volatility of the factors, and the correlation of the two factors. These are parameters in the sense that the model assumes that these values are...

## Profit And Loss Pl Attribution

Term structure models are important not only for valuation and hedging but also for analyzing the performance of trades and trading strategies. In particular, it is good practice to compare the actual performance of a trade with its performance predicted by the model being used for valuation and hedging. Aside from occasionally revealing errors in reported market prices, calculated P amp L, or even hedge ratios, this comparison allows an investment or trading operation to assess its valuation...

## General Collateral And Specials

Investors using the repo market to earn interest on cash balances with the security of U.S. Treasury collateral do not usually care about which particular Treasury securities they take as collateral. These investors are said to accept general collateral GC . Other participants in the market, however, do care about the specific issues used as collateral. Commercial and investment banks engaging in repurchase agreements to finance particular security holdings have to deliver those particular...

## Timedependent Volatility Model

In 12.2 the volatility of the short rate starts at the constant o and then exponentially declines to zero. Volatility could have easily been designed to decline to another constant instead of zero, but Model 3 serves its pedagogical purpose well enough. Setting o 126 basis points and a .025, Figure 12.1 graphs the standard deviation of the terminal distribution of the short rate at various horizons.1 Note that the standard deviation rises rapidly with horizon at first but then rises more...

## Repurchase Agreements And Cash Management

Suppose that a corporation has accumulated cash to spend on constructing a new facility. While not wanting to leave the cash in a non-interest-bearing account, the corporation would also not want to risk these earmarked funds on an investment that might turn out poorly. Balancing the goals of revenue and safety, the corporation may very well decide to extend a short-term loan and simultaneously take collateral to protect its cash. Holding collateral makes it less important to keep...

## Lognormal Model With Mean Reversion The Blackkarasinski Model

The Vasicek model, a normal model with mean reversion, was the last model presented in Chapter 11. The last model presented in this chapter is a lognormal model with mean reversion called the Black-Karasinski model. The model allows the mean reverting parameter, the central tendency of the short rate, and volatility to depend on time, firmly placing the model in the arbitrage-free class. A user may, of course, use or remove as much time dependence as desired. The dynamics of the model are...

## Tree For The Original Salomon Brothers Model3

This section shows how to construct a binomial tree to approximate the dynamics for a lognormal model with a deterministic drift. Describe the model as follows By Ito's Lemma which is beyond the mathematical scope of this book , d ln r y - 22 2 dt 12.8 d ln r a t - o2 2 dt adw 12.9 Redefining the notation of the time-dependent drift so that a t a t -a2 2, equation 12.9 becomes d ln r a t dt adw 12.10 Recalling the models of Chapter 11, equation 12.10 says that the natural logarithm of the short...

## Normally Distributed Rates Zero Drift Model

The discussion begins with a particularly simple model to be called Model 1. The continuously compounded, instantaneous rate r t is assumed to evolve in the following way The quantity dr denotes the change in the rate over a small time interval, dt, measured in years 7 denotes the annual basis point volatility of rate changes and dw denotes a normally distributed random variable with a mean of zero and a standard deviation of Idt.1 Say, for example, that the current value of the short-term rate...

## Drift And Risk Premium Model

The term structures implied by Model 1 always look like Figure 11.2 relatively flat for early terms and then downward sloping. Chapter 10 pointed out that the term structure tends to slope upward and that this behavior might be explained by the existence of a risk premium. The model of this section, to be called Model 2, adds a drift to Model 1, interpreted FIGURE 11.3 Par Rate Volatility from Model 1 and Selected Implied Volatilities, February 16, 2001 FIGURE 11.4 Sensitivity of Spot Rates to...

## Mathematical Description Of Expectations Convexity And Risk Premium

This section presents an approach to understanding the components of return in fixed income markets. While the treatment is mathematical, the aim is intuition rather than mathematical rigor. 8See, for example, Homer and Sylla 1996 , pp. 394-409. 9See, for example, Homer and Sylla 1996 , pp. 394-409. Let P y, t c be the price of a bond at time t with a yield y and a continuously paid coupon rate of c. A continuously paid coupon means that over a small time interval, dt, the bond makes a coupon...

## Key Rate and Bucket Exposures

A major weakness of the approach taken in Chapters 5 and 6, as well as in some of the term structure models in Part Three, is the assumption that movements in the entire term structure can be described by one interest rate factor. To put it bluntly, the change in the six-month rate is assumed to predict perfectly the change in the 10-year and 30-year rates. So, for example, a naive DV01 analysis would allow for the hedging of a position in 10- or 30-year bonds with six-month securities....

## Example Pricing A Cmt Swap

Equipped with the tree built in the previous section, this section prices a particular derivative security, namely 1,000,000 face value of a stylized constant maturity Treasury CMT swap. This swap pays every six months until it matures, where yCMT is a semiannually compounded yield, of a predetermined maturity, at the time of payment. The text prices a one-year CMT swap on the six-month yield. Since six-month semiannually compounded yields equal six-month spot rates, rates in the tree of the...

## Arbitrage Pricing Of Derivatives

The text now turns to the pricing of a derivative security. What is the price of a call option, maturing in six months, to purchase 1,000 face value of a then six-month zero at 975 Begin with the price tree for this call option. If on date 1 the six-month rate is 5.50 and a six-month zero sells for 973.23, the right to buy that zero at 975 is worthless. On the other hand, if the six-month rate turns out to be 4.50 and the price of a six-month zero is 978, then the right to buy the zero at 975...

## Key Rate Shifts

In this technique a set of key rates is assumed to describe the movements of the entire term structure. Put another way, the technique assumes that given the key rates any other rate may be determined. The following choices must be made to apply the technique the number of key rates, the type of rate to be used usually spot rates or par yields , the terms of the key rates, and the rule for computing all other rates given the key rates. These choices are examined in the next several sections....

## Key Rate 01s And Key Rate Durations

Table 7.1 summarizes the calculation of key rate 01s, the key rate equivalent of DV01, and of key rate durations, the key rate equivalent of duration. The security used in the example is a nonprepayable mortgage requiring a payment of 3,250 every six months for 30 years. So that the the numbers are easily reproduced, it is assumed that the par yield curve is flat at 5 , but no such restrictive assumption is necessary. Like DV01 and duration, key rate exposures can be calculated assuming any...

## Bucket Shifts And Exposures

A bucket is jargon for a region of some curve, like a term structure of interest rates. Bucket shifts are similar to key rate shifts but differ in two respects. First, bucket analysis usually uses very many buckets while key rate analysis tends to use a relatively small number of key rates. Second, each bucket shift is a parallel shift of forward rates as opposed to the shapes of the key rate shifts described previously. The reasons for these differences can be explained in the context for...

## The Barbell Versus The Bullet

In the asset-liability context, barbelling refers to the use of a portfolio of short- and long-maturity bonds rather than intermediate-maturity bonds. An asset-liability manager might have liabilities each with duration equal to nine years and, as a result, with portfolio duration equal to nine years. The proceeds gained from incurring those liabilities could be used to purchase several assets with duration equal to nine years, or alternatively, to purchase 2- and 30-year securities that, as a...

## Hedging Example Part I Hedging A Call Option

Since it is usual to regard a call option as depending on the price of a bond, rather than the reverse, the call is referred to as the derivative security and the bond as the underlying security. The rightmost columns of Table 5.1 4Were prices available without error, it would be desirable to choose a very small difference between the two rates and estimate DV01 at a particular rate as accu rately as possible. Unfortunately, however, prices are usually not available without error. The models...

## Duration Dv01 Maturity And Coupon A Graphical Analysis

Figure 6.1 uses the equations of this chapter to show how Macaulay duration varies across bonds. For the purposes of this figure all yields are fixed at 5 . At this yield, the Macaulay duration of a perpetuity is 20.5. Since a perpetuity has no maturity, this duration is shown in Figure 6.1 as a horizontal line. Also, since, by equation 6.31 , the Macaulay duration of a perpetuity does not depend on coupon, this line is a benchmark for the duration of any coupon bond with a sufficiently long...

## RJ2 1 fj2 1 023 1 fj2

Equations 3.10 clearly demonstrate that yield-to-maturity is a summary of all the spot rates that enter into the bond pricing equation. Recall from Table 2.1 that the first four spot rates have values of 5.008 , 4.929 , 4.864 , and 4.886 . Thus, the bond's yield of 4.8875 is a blend of these four rates. Furthermore, this blend is closest to the two-year spot rate of 4.886 because most of this bond's value comes from its principal payment to be made in two years. Equations 3.10 can be used to be...

## Convexity

As mentioned in the discussion of Figure 5.3 and as seen in Tables 5.1 and 5.2, interest rate sensitivity changes with the level of rates. To illustrate this point more clearly, Figure 5.4 graphs the DV01 of the option and underlying bond as a function of the level of rates. The DV01 of the bond declines relatively gently as rates rise, while the DV01 of the option declines sometimes gently and sometimes violently depending on the level of rates. Convexity measures how interest rate sensitivity...

## Piecewise Cubics

The first step in building a smooth curve is to assume a functional form for the discount function, for spot rates, or for forward rates. For example, an extremely simple functional form for the discount function might be a cubic polynomial d t 1 at bt2 ct3 4.20 Given constants a, b, and c, equation 4.20 provides the discount factor for any term t. Note that the intercept is one so that d 0 1. The goal, therefore, is to find a set of constants such that the discount function 4.20 reasonably...

## Yieldtomaturity And Realized Return

Yield-to-maturity is sometimes described as a measure of a bond's return if held to maturity. The argument is made as follows. Repeating equation 3.1 , the yield-to-maturity of the 61 4s of February 15, 2003, is defined such that 3.125 3.125 3.125 103.125 ino 1i11f - -- -- -- 102 18.125 32 1 J 2 1 y 2 2 1 y 2 3 1 y 2 4 ' 3.17 Multiplying both sides by 1 y 2 4 gives 3.125 1 3.125 1 2 3.125 1 gt 103.125 102.5664 1 gt 3.18 The interpretation of the left-hand side of equation 3.18 is as follows. On...

## Duration

DV01 measures the dollar change in the value of a security for a basis point change in interest rates. Another measure of interest rate sensitivity, duration, measures the percentage change in the value of a security for a unit change in rates.6 Mathematically, letting D denote duration, 6A unit change means a change of one. In the rate context, a change of one is a change of 10,000 basis points. As in the case of DV01, when an explicit formula for the price-rate function is available, the...

## The Negative Convexity Of Callable Bonds

A callable bond is a bond that the issuer may repurchase or call at some fixed set of prices on some fixed set of dates. Chapter 19 will discuss callable bonds in detail and will demonstrate that the value of a callable bond to an investor equals the value of the underlying noncallable bond minus the value of the issuer's embedded option. Continuing with the example of this chapter, assume for pedagogical reasons that there exists a 5 Treasury bond maturing on February 15, 2011, and callable in...

## Estimating Price Changes And Returns With Dv01 Duration And Convexity

Price changes and returns as a result of changes in rates can be estimated with the measures of price sensitivity used in previous sections. Despite the abundance of calculating machines that, strictly speaking, makes these approximations unnecessary, an understanding of these estimation techniques builds intuition about the behavior of fixed income securities and, with practice, allows for some rapid mental calculations. A second-order Taylor approximation of the price-rate function with...

## Hedging Example Part Ii A Short Convexity Position

In the first section of this hedging example the market maker buys 47.37 million of the 5s of February 15, 2011, against a short of 100 million options. Figure 5.5 shows the profit and loss, or P amp L, of a long position of 47.37 million bonds and of a long position of 100 million options as rates change. Since the market maker is actually short the options, the P amp L of the position at any rate level is the P amp L of the long bond position minus the P amp L of the long option position. By...

## Yieldtomaturity And Relative Value The Coupon Effect

All securities depicted in Figure 3.2 are fairly priced. In other words, their present values are properly computed using a single discount function or term structure of spot or forward rates. Nevertheless, as explained in the previous section, zero coupon bonds, par coupon bonds, and mortgages of the same maturity have different yields to maturity. Therefore, it is incorrect to say, for example, that a 15-year zero is a better investment than a 15-year par bond or a 15-year mortgage because...

## Term Structure Of Dv01

Denote the price-rate function of a fixed income security by P y , where y is an interest rate factor. Despite the usual use of y to denote a yield, this factor might be a yield, a spot rate, a forward rate, or a factor in one of the models of Part Three. In any case, since this chapter describes one-factor measures of price sensitivity, the single number y completely describes the term structure of interest rates and, holding everything but interest rates constant, allows for the unique...

## Bond Prices Discount Factors and Arbitrage

How much are people willing to pay today in order to receive 1,000 one year from today One person might be willing to pay up to 960 because throwing a 960 party today would be as pleasurable as having to wait a year before throwing a 1,000 party. Another person might be willing to pay up to 950 because the enjoyment of a 950 stereo system starting today is worth as much as enjoying a 1,000 stereo system starting one year from today. Finally, a third person might be willing to pay up to 940...

## Bad Days

The phenomenon of bad days is an example of how confusing yields can be when cash flows are not exactly six months apart. On August 31, 2001, the Treasury sold a new two-year note with a coupon of 35 8 and a maturity date of August 31, 2003. The price of the note for settlement on September 10, 2001, was 100-71 4 with accrued interest of .100138 for a full price of 100.32670. According to convention, the cash flow dates of the bond are assumed to be February 28, 2002, August 31, 2002, February...

## Compounding Conventions

Since the previous chapters assumed that cash flows arrive every six months, the text there could focus on semiannually compounded rates. Allowing for the possibility that cash flows arrive at any time requires the consideration of other compounding conventions. After elaborating on this point, this section argues that the choice of convention does not really matter. Discount factors are traded, directly through zero coupon bonds or indirectly through coupon bonds. Therefore, it is really...

## Bond Prices Spot Rates and Forward Rates

While discount factors can be used to describe bond prices, investors often find it more intuitive to quantify the time value of money with rates of interest. This chapter defines spot and forward rates, shows how they can be derived from bond prices, and explains why they are useful to investors. An investment of 100 at an annual rate of 5 earns 5 over the year, but when is the 5 paid The investment is worth less if the 5 is paid at the end of the year than if 2.50 is paid after six months and...

## Securities

Tools for Today's Markets Second Edition Copyright 2002 by Bruce Tuckman. All rights reserved. Published by John Wiley amp Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior...