Bond Duration and Immunization

Early Developments and Recent Contributions

Gabriel Hawawini

• Publisher: Routledge
• Year: 2017
• ISBN:
  9781351381109
• Edition: 1
• Topic: Business
• Subtopic: Business General
• Index: Business

First published in 1982, Bond Duration and Immunization is a collection of seminal papers featuring articles from high profile academics such as Frederick McCaulay, John Hicks, and F.M. Redington. This collection also features several articles published in British actuarial journals often unavailable outside of the UK, and a strong collection of articles which contextually offer a significant contribution to the field. This strong collection will appeal to anyone working or researching in the area of bond duration and immunization.

PART I

BOND DURATION

FROM ‘SOME THEORETICAL PROBLEMS SUGGESTED BY THE MOVEMENTS OF INTEREST RATES, BOND YIELD, AND STOCK PRICES IN THE UNITED STATES SINCE 1856’

Frederick R. Macaulay

We have, so far in this chapter, been discussing the subject of long time interest rates without asking the question: how much longer term is one loan than another? For a study of the relations between long and short time interest rates, it would seem highly desirable to have some adequate measure of ‘longness’. Let us use the word ‘duration’ to signify the essence of the time element in a loan. If one loan is essentially a longer term loan than another we shall speak of it as having greater ‘duration’.

Now the promise contained in a loan is either a promise to make one and only one future payment or a promise to make more than one future payment. If two loans are made at the same rate of interest, and if each loan involves a promise to make one future payment only, the loan whose future payment is to be made earlier is clearly a shorter term loan than the other. For example, if $100 be lent for one year at 5 per cent per annum, the only payment to be $105 at the end of the year, and if another $100 be lent for two years at 5 per cent per annum, the only payment to be $110.25 at the end of the two years, the first loan is clearly a shorter term loan than the second. If, on the other hand, either or both loans involve a promise to make more than one future payment, or if the rates of interest ascribed to the two loans are not the same, it may be extremely difficult to decide which is essentially the longer term loan.

It is clear that ‘number of years to maturity’ is a most inadequate measure of ‘duration’. We must remember that the ‘maturity’ of a loan is the date of the last and final payment only. It tells us nothing about the sizes of any other payments or the dates on which they are to be made. It is clearly only one of the factors determining ‘duration’. Sometimes, as in the case of a low coupon, short term bond, it may be overwhelmingly the most important factor. At other times, as in the case of a long term, diminishing annuity, its importance may be so small as to be almost negligible. Because of its nature, length of time to maturity is not an accurate or even a good measure of ‘duration’. ‘Duration’ is a reality of which ‘maturity’ is only one factor.

Whether one bond represents an essentially shorter or an essentially longer term loan than another bond depends not only upon the respective ‘maturities’ of the two bonds but also upon their respective ‘coupon rates’—and, under certain circumstances, on their respective ‘yields’. Only if maturities, coupon rates and yields are identical can we say, without calculation, that the ‘durations’ of two bonds are the same.

If two bonds have the same maturit and the same yield but one has a higher coupon rate than the other, the one having the higher coupon rate represents an essentially shorter term loan than the other. For example, if each bond is selling on a 5 per cent basis, a 6 per cent bond maturing in 25 years necessarily represents an essentially shorter term loan than a 4 per cent bond maturing in 25 years. This may easily be seen by comparing a $400 face value 6 per cent bond maturing in 25 years with a $500 face value 4 per cent bond maturing in 25 years. On both bonds the total of all future payments, both principal and interest, is $1,000. But on the 6 per cent bond the payments are $12 each six months for 24½ years, and then a final payment of $412, while on the 4 per cent bond the payments are $10 each six months for 24½ years, and then a final payment of $510. It is plain that the $1,000 is being paid earlier on the 6 than on the 4 per cent bond. Though both have the same ‘maturity’, the 6 per cent bond represents a loan of shorter ‘duration’ than the 4 per cent bond.

The difference in ‘duration’ of the two bonds is manifest in their prices. As the payments are made earlier on the 6 per cent bond, its price (if the ‘yields’ of the two bonds are the same) is necessarily higher. For example, as each bond ‘yields’ 5 per cent, the price of the $400 face value 6 per cent bond will be $456.72, while the price of the $500 face value 4 per cent bond will be only $429,10.

We see, then, that if two bonds have the same yield and the same maturity but different coupon rates, the bond having the higher coupon rate represents the loan of shorter ‘duration’. Instead of examining in a similar manner the case in which the two bonds have the same coupon rate and the same maturity but different yields, and the case in which they have the same coupon rate and the same yield but different maturities, we shall now consider directly the general problem of how to measure ‘duration’. Let us approach this problem by considering the maturity of a bond as a function of the maturities of the separate loans of which it may be said to consist.

It would seem almost natural to assume that the ‘duration’ of any loan involving more than one future payment should be some sort of a weighted average of the maturities of the individual loans that correspond to each future payment. Two sets of weights immediately present themselves—the present and the future values of the various individual loans.

Future value weighting seems clearly inadmissible. It gives absurdly long ‘durations’. If $2,000 be lent at 5 per cent per annum in the form of two loans, one of $1,000 at 5 per cent per annum¹⁹ payable in one lump sum of $1,050 at the end of one year, and one of $1,000 at 5 per cent per annum payable in one lump sum of $131,501.26 at the end of 100 years, the ‘average maturity’ or ‘duration’ of the two loans, if calculated by taking an arithmetic average of the two maturities, using the present values as weights, is 50½ years. If the future values ($1,050 and $131,501.26) be used as weights, the ‘average maturity’ is found to be more than 99 years.

In this illustration, the present values (or amounts lent) were equal. Let us examine a case in which the future values are equal. If $959.98 be lent at 5 per cent per annum in the form of two loans, one of $952.38 at 5 per cent per annum payable in one lump sum of $1,000 at the end of one year, and one of $7.60 at 5 per cent per annum payable in one lump sum of $1,000 at the end of 100 years, the ‘average maturity’ or ‘duration’ of the two loans, if calculated by taking an arithmetic average of the two maturities, using the present values ($952.38 and $7.60) as weights, is about 21½ months. If the future values be used as weights, the average maturity is 50½ years.

How absurd it seems to think of a loan of $2,000 made up of two loans each of $1,000, one maturing in one year and one in 100 years, as having a ‘duration’ of over 99 years. And how absurd to think of a loan of $1,000 made up of two loans, one of $952.38 maturing in one year and the other of $7.60—less than11 per cent of the larger loan—maturing in 100 years, as having a ‘duration’ of 50½ years.²⁰

But are not the results obtained by using present values as weights also open to criticism? If the ‘durations’ obtained by using future value weighting seem unmistakably too long, does not at least one of the ‘durations’ obtained from present value weighting seem very short?

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