## C [ yH

Reinvested Future Value (RFV) at Horizon H RFV (1, H) accumulated reinvested value from payments in years 1 through year H (after the Hth year payment has been received). The last equality is intended to relate the RFV back to the earlier PV concepts. We shall try to show these throwback relationships whenever possible. Also, for the special case of a level annuity, Horizon Present Value (HPV) of Tail Cash Flow at Horizon H HPV(H+1, M) PV of flows in years H+1 to M with discounting to end of...

## Reinvestment Volatility

When we move from the PV, which declines with higher discount rates, to the RFV, which increases with higher reinvestment rates, another volatility measure becomes important. The reinvestment volatility (RFV-VOL) is rarely characterized in the same quantitative way as the Duration concept, but doing so leads to some interesting results that should be more widely appreciated and that may be particularly useful for long-term holders of fixed-income exposures such as insurance companies and...

## Reinvested Future Value

10 Annual Payments of 10 8 Discount Rate 9.26 is obtained. And as in the preceding section, this PV(1,1) 9.26 grows to exactly the RFV(1,1) 10 when invested at 8 . Similarly, for H 7, Table 2 shows that the RFV(1,7) 89.23. From Table 1, this 7-year flow has a cumulative PV(1,7) 52.06. A simple computation shows that More generally, as demonstrated in the Technical Appendix, Like the PV, the FV has the appeal of great simplicity. Rather than think through a complex pattern of payments, we can...

## Summary and Investment Implications

In these chapters we have discussed four important types of bond swaps the Substitution Swap, the Intermarket Spread Swap, the Interest Rate Anticipation Swap and the Pure Yield Pickup Swap. The Substitution Swap is very elementary. It merely substitutes an almost identical bond for a pickup in price (a higher yield-to-maturity). The Intermarket Spread Swap depends on a yield spread judgment that one department of the market is overpriced compared with another. The Interest Rate Anticipation...

## Horizon Duration and Horizon Volatility

Lockstep rate changes, it can be shown that the horizon-to-Duration gap relationship holds not only for longer horizons, but indeed for any FV horizon regardless of its placement relative to the investment's span of flows see the Technical Appendix . More precisely, for any FV horizon H, the percentage volatility in the TFV H can be approximated by the extent that the horizon H exceeds the Macaulay Duration D 1, M of the entire flow that is, for any H, TFV-VOL H H-D 1, M . 1 y This formulation...

## Definition of the Yieldto Call

When a bond is callable, i.e., redeemable prior to maturity at the issuer's option, the cash flow implicit in the yield-to-maturity figure is subject to possible early alteration. Most corporate bonds issued today are callable, but with a certain period of call protection before the call option can be exercised. At the expiration of this period the bond may be called at a specified call price which usually involves some premium over par. This call price may decline in steps towards par in...

## The Volatility of Bond Prices

It is sometimes erroneously supposed that the volatility of high-grade bond prices in response to a given change in yield is entirely a function of maturity. This is only partly true. There are two other factors which affect volatility importantly the coupon rate and the general level of yields. All things else being equal, with the same percentage change in yield, the volatility of the price of a bond increases 1 As maturity lengthens, the longer the maturity, the greater the price volatility...

## The Macaulay Duration

One of the concepts just discussed is that a bond's maturity date or more generally, the date of any cash flow's last payment is a poor gauge of the flow's life. As one might suspect, the problem of finding a good measure of a flow's life is closely related to the problem of determining its PV volatility. One natural way is to simply compute an average life by just determining the time to each payment, weighted by the size of the payment. However, a little experimentation quickly reveals a...

## The Future Value of Per Period

As shown in the preceding chapter, the Future Value of 1 received or paid today, and invested for T semiannual periods at an interest rate R per period, will be For example, as we have seen, assuming a 7 rate, 1 today would have a Future Value in 3 4 years of Now suppose that the payment of 1 were to be made six months from now. What would be its IVi-yzar Future Value This 1 would then be subject to 3 years or 6 periods of compounded reinvestment. Consequently, its Future Value 3 4 years from...

## Interest on Interest

The recent high level of bond yields and the uncertainty whether yields will be high in the years ahead emphasizes the importance of interest-on-interest, that is to say, the rate at which receipts from coupons can be reinvested in the future. An original investment compounds automatically at the purchase yield only until the funds are paid back in the form of coupons and finally of principal. However, some investors mistakenly expect that a bond purchased at a given yield will always produce...

## Index

A, 189,196 Accrued Interest compensation, 155 computation, 156 definition, 155 delivery date in bond's coupon payment cycle, 158 distorting effect of coupon payment cycle, 157 Dollar Prices, 155-162 see also Dollar fraction of coupon due as, 156 market values quoted without, 157 mechanical computation for particular obtain in dollars, 156 what it represents, 157 AI, 156,159,191,196 Amortization, 102,104,105,107,152-154 Annualized interest rate, 169 Beyond redemption evaluation, 182-184 coupon,...

## Yield Pickup Swaps and Realized Losses

Probably a majority of institutional bond swaps are done purely for the purpose of achieving an immediate gain in return, either in terms of current coupon income or in terms of yield-to-maturity or both. These swaps can be made and often are made without reference to substitutions or to yield spreads, interest rate trends, or overvaluation or undervaluation of the issues involved. For example, suppose the investor swaps from the 30-year 4's at 67.18 to yield 6.50 into the 30-year 7's at 100 to...

## The Total Future Value Volatility at Longer Horizons

The preceding development of an RFV-VOL 1,H volatility also provides an answer to the question of the volatility TFV-VOL H of a cash flow's TFV H with a horizon H that coincides with the flow's last payment M, that is, where H M. With horizons that match the flow's last payment, there are by definition no tail flows and so the total future value, TFV, consists of just the reinvestment-driven RFV 1,H . Because the reinvestment effect is always positive, TFV-VOL M RFV-VOL 1, M M - D 1, M higher...

## The Volatility of Yields

Heretofore we have used and compared two ways of measuring the volatility of yields 1 by percentage change 3373 and 2 by basis point change 100 basis points . We have found that the valuation of a pure coupon stream exactly follows percentage changes in yield and that the valuation of a pure lump sum payment loosely follows basis point changes in yield see Table 15 . Which standard of yield volatility is true to real life Common sense says that yields fluctuate according to percentages of...

## Evaluating Bond Portfolio Swaps

If all bond swaps were good bond swaps, we would use a more constructive title for this chapter and call it EVALUATING BOND PORTFOLIO IMPROVEMENTS. Alas, however, there are as many bad swaps as good, so we have chosen the conventional and neutral term swap. The purpose of this chapter and the next is to present a comprehensive method of evaluating several types of swaps. This method combines in one figure most of the advantages and disadvantages of each specific swap in a way that permits an...

## The Rate Anticipation Swap

When portfolio managers anticipate an important change in the level or structure of interest rates they often make swaps designed to protect or benefit their portfolios. Most commonly these Rate Anticipation Swaps consist of shortening maturities if higher long yields are expected or lengthening maturities if lower long yields are expected. It should be noted that the decisive factor is the expected long rate the change in the long rate will almost always be the chief determinant of the value...

## Generalizing the PV Model to Equities

The YTM is a well-defined measure because the coupon flows over the maturity of high-grade noncallable bonds are well defined. Therefore, for a given market price, there is only one uniform rate that can discount these fixed flows back to the given price. However, turning to investments with less well-defined flows, such as equities or callable bonds, the problem becomes more complex on a number of counts. At the outset, there needs to be some process for estimating the flows themselves. Such...

## Classification of Bond Swaps

These chapters will limit their consideration to four important types of swaps which depend for their validity on correct mathematical comparisons. These we will call The Substitution Swap is the simplest of all. The investor now holds a particular bond. For conciseness, we shall refer to this bond as the H-bond H for now held . He is offered a bond for proposed purchase which we shall refer to as the P-bond P for purchase . Except for price, the P-bond is essentially identical to the H-bond in...

## Methods for Computing the Yieldto Call

Computation of the Present Value at a 7X Discount Rate of _an 8 3 4Z Bond Called in 5 Years at 107_ Present Value of 1.00 x .709 Due at End of 10 Semiannual Periods 3.555 Discount Rate Per Period _ Present Value of Redemption Payment 758.54 758.54 Present Value of 1.00 Per Period for 10 Periods 0 3.556 Discount Rate Per Period Present Value of Coupon Payments 363785 363.85 Total Present Value of Bond's Cash Flow 0 73 Discount Rate 1,122.39 Present Value as t of 1000 Face Value 112.24 6 Digit...

## Understanding Yieldto Maturity

This concept of the yield-to-maturity has many implications and interpretations, all closely related and, in fact, all mathematically equivalent. First of all, we see that yield-to-maturity can be viewed as simply an alternative method for stating a bond's price. Given the market price, one can always determine the yield-to-maturity. Given the yield-to-maturity to sufficient accuracy , one can always find the market price. A second interpretation is that the yield-to-maturity corresponds to the...

## Horizon Analysis

The preceding discussion assumed that the reinvestment rate and the discount rate are always equal and always move in a lockstep fashion. However, there is an important form of total return analysis for fixed-income portfolios where these assumptions break down. As demonstrated earlier, the total value TFV H of a cash flow at any given horizon H consists of the RFV 1,H plus the HPV H 1, M . Viewing the TFV H in terms of these two basic components has the advantage of allowing for reinvestment...

## References

Sidney Homer, The Bond Buyer's Primer New York Salomon Brothers Hutzler, 1968 . 2. Sidney Homer, Richard Sylla, and Henry Kaufman, A History of Interest Rates New Brunswick, NJ Rutgers University Press, 1996 . 3. J. Peter Williamson, Computerized Approaches to Bond Switching, Financial Analysts Journal July August 1970 . 4. Lawrence Fisher and Roman L. Weil, Coping with the Risk of Interest Rate Fluctuations Returns to Bondholders from Naive and Optimal Strategies, Journal of Business 44,...

## Realized Compound Yield Prior to Maturity

Suppose the investor purchased a 30-year 4 bond at 671.82 to yield 6.50 , but decided to sell this same bond 2 years later when it achieved a market value of 730.34. This market value at sale corresponds to a 6 yield-to-maturity. The consequent Future Value and the realized compound yield over this two year holding period are computed as shown in Table 65. It should be noted that as long as the bond has the indicated market value at time of sale, the investor has achieved the resultant realized...

## Realized Compound Yield and the Recovery of Book Loss

The Future Value and realized compound yield can also be useful in contexts where a book loss must be recovered in order to justify a Pure Yield Pickup Swap. This general problem of correctly computing recovery times for such proposed swaps is described in some detail in Chapter 7. The investor wants to determine the time required for the improvement in his book cash flow from the swap to accumulate to the point of equaling an initial book loss. This accumulation of book cash flow consists of 1...

## Immunization

A common investment problem is to provide a given dollar payment at a specified future point in time. Obviously, with zero-coupon bonds, this problem has a simple solution Just use a zero-coupon bond that matures at the specified horizon. However, back when Inside the Yield Book was written, the zero-coupon bond was only a hypothetical construct although Sidney Homer and I made considerable use of the zero-coupon concept as an analytic tool in several of the Inside the Yield Book chapters . At...

## Realized Compound Yield Beyond Maturity

We can also extend the realized compound yield concept to a future point beyond the bond's maturity. The bond's maturity date is, in a sense, a rather artificial investment horizon. Presumably, an investor has planning periods which are convenient for his portfolio and his objectives. The evaluation of any particular investment instrument should be made within a time framework chosen by the investor, rather than pinned to an arbitrary characteristic like a bond's maturity date. By extending the...

## The Basic Concept of Present Value

Every investment is an exchange of current resources for some future flow of payments. In the broadest sense, the concept of PV is a gauge of the value of those future payments in current terms. One could argue that the PV concept is at work at least implicitly in every investment deci sion, both in the primary and the secondary market. The PV idea is a very old concept, and every investor has some intuitive sense of how it works. In fact, it is such a basic tool, and so widely used and taught,...

## Simple and Compound Interest

Suppose 1.00 were loaned today for one year at simple interest of 7 payable at maturity. One year hence the creditor would receive the following The generalized formula for this total repayment is extremely simple. If P principal is invested today at simple interest rate R expressed as a decimal, e.g., not 7 but .07 , then the payment received T years hence number of interest periods which, in this case, equals the number of years will be P T X R X P which can be usefully simplified to be 1 1 1...

## Realized Compound Yield Over the Bonds Life

In Chapter I, we proposed the concept of total realized compound yield as an important value measure for investors seeking full compounding over the life of a long-term bond. It supplements yield-to-maturity in two ways by including interest-on-interest for the full life of the investment at various assumed rates, and by including, where pertinent, interim price fluctuations. Thus, every bond offering can be related to four interest rates 1 the coupon rate 2 the yield-to-maturity which can...

## The Future Value of Today

Taking the specific example cited at the end of Chapter 8, suppose income can be reinvested at an annual rate of 7 compounded semiannually, i.e., at a rate of 3Vz per semiannual period. Then 1 today, as we saw, would become at the end of six months. Compounding this amount for a second six month period, one obtains at the end of the first year. This figure is equivalent to 1.00 X 1.035 X 1.035 1.00 X 1.035 2, i.e., to our original dollar multiplied by the second power of the growth factor of...

## The Future Value of a Level Cash Flow

Now suppose we again have a series of 7 semiannual payments, but of 20 each instead of 1 as before. The first 20 payment is received in six months. It is immediately reinvested at the assumed 7 interest rate, and compounded for the 6 semiannual periods remaining until the end of the 3Yi years. The Future Value of this first 20 payment will be 20 multiplied by the Future Value of 1 after 6 periods of compounding at 3Y2 per period, i.e., 20 X 1.035 6 20 X 1.229 24.58 In other words, the first 20...

## The Yields of Premium Bonds Par Bonds and Discount Bonds

As a result of the extraordinarily wide two-way fluctuations of bond yields over the last few years and the heavy volume of new corporate bond issues that have come out at different levels of the market, the investor today has available to him a much larger selection of high-grade seasoned corporate bond issues than ever before. Coupons range from 2 up to 9Vi for issues of uniform high quality and long maturity. Offering prices recently ranged simultaneously from 60 up to 115. Yields of...

## The Substitution Swap

The Substitution Swap is simple in concept. Both the H-bond and the P-bond are equivalent in quality, coupon and maturity. The swap is executed at a basis point pickup which is expected to be eradicated by the end of the workout period. It will prove very revealing to determine the value of a concrete Substitution Swap in terms of improved realized compound yield. For this example, take as the H-bond 30-year 7's priced at par to yield 7.00 . The P-bonds are, of course, then also 30-year 7's but...

## The Time Value of Money

In evaluating any investment where money is paid today in order to purchase future returns, the impact of time itself on both the Present Value and the Future Value of expected payments must be considered. One dollar received today is worth more from either point of view than one dollar received one year from now. The dollar received today can be put to work right away. It can be reinvested so as to return more than one dollar one year from now. Even in the case of the investor who spends all...

## Accrued Interest and Dollar Prices

In the preceding chapter, the yield-to-maturity concept was developed in the context of a bond having a remaining life corresponding to whole multiples of semiannual interest periods. In such cases, the bond's Present Value is well defined and consists solely of Principal Value free of any Accrued Interest. Such rounded calculations are adequate for all the theoretical problems discussed in this book, but it seems desirable here to go a step further. In this chapter, therefore, we shall enlarge...

## The Intermarket Spread Swap

The Intermarket Spread Swap is really a swap from one department of the bond market to another. The motivation is the investor's belief that the yield spreads between the two market components are out of line for one reason or another, so that a better value is afforded by the P-bond. The swap is executed so that the anticipated realignment will provide a relative capital advantage or a better yield with at least an equal capital performance. For clarity, we will at first pin the H-bond to a...

## Using the Yield Book to Find Realized Compound Yields

Throughout this book we have emphasized the concept of realized compound yield as a comprehensive guide to bond values. It reveals the fully compounded growth rate of any investment under varying reinvestment rates and thus permits consistent comparisons between alternate investments. We have demonstrated how realized compound yields can be calculated by the use of Compound Interest Tables. However, for those who find it more convenient to use only the Yield Book, there is a straightforward...

## The Price Volatility of Premium Bonds Par Bonds and Discount Bonds

The yield analysis in Chapter 4 will suffice to guide that very special investor who feels the need to check his bond market values only once every few years. However, few investors now cling to this rigidly long-term orientation. In fact, most bond portfolio managers pay close attention to interim movements in interest rates and the corresponding bond price reactions. In the context of this discussion, this means that the intermediate term price potentials of premium, par and discount bonds...

## Preface to the Edition

The Yield Book can appropriately be called the playing field of the game of bond investment. Its structure and dimensions and the basics of price-yield relationships deserve the closest study. Too often the dollars and cents significance of bond yields is taken for granted and sometimes even is misunderstood. Our book will attempt to explore some of the basic but less obvious relationships between coupon, maturity, price and yield with the aim of aiding the investor in judging and comparing...

## Realized Compound Yieldslo Maturity

In Chapter 1, the term realized compound yield was used to describe the total effective compound yield obtained from a bond purchased at a given price when the coupon income is reinvested and thus compounded at a specified reinvestment rate over the entire life of the bond. Only when the reinvestment rate equals the yield-to-maturity at purchase, does the realized compound yield coincide with the Yield Book's yield-to-maturity. However, for reinvestment rates differing from the purchase...

## Coupon Bonds in Real Life

In real life almost all bonds are coupon bonds which combine a stream of coupon payments with a lump sum payment at maturity. Therefore, their volatility is determined by a combination of the volatility characteristics of perpetuals and the very different volatility characteristics of the lump sum payment. In the case of long par or premium bonds, the coupon stream plays a greater role in determining the bonds' Present Value than in the case of discount bonds and, therefore, their volatility...

## The Cross Over Yield

Table 21 shows that high coupon bonds with 5-year call protection can command sizeable price premiums and still give potentially reasonable yields-to-call. At high premiums, however, the yield-to-call, which is the limiting yield, should be well above short or medium-term market yields to compensate for the risk of non-call. While at all prices below call price the yield-to-maturity is always the minimum yield, as the price level rises to and above call price the yield-to-call drops more...

## Horizon Duration and the Generalized TFV Volatility

As noted earlier, one can also have horizon H dates that precede the last payment date, that is, H lt M. Recall that in such cases, the total TFV H will be the sum of the reinvested flows to that date RFV 1,H , together with the going-forward HPV H 1, M of the remaining flows. These two terms react in opposite ways to rate changes, so that the TFV sensitivity will depend on the balance between the two volatility terms. As might be expected, for relatively short horizons, the TFV is dominated by...

## Book

LEIBOWITZ, PH.D. With ,t new foreword by TIENRY KAUFMAN ami two new sections by MAKTIN I.. LlillJOWIT The Classic That Created the Science of Bond Analysis by Sidney Homer and Martin L. Leibowitz, Ph.D. For active fixed-income investors, this book has always been the holy writ, where it all began. Leibowitz s revisions make certain that the book will continue in that role. But for all investors, of any stripe, Inside the Yield Book is the essential work for...