Chapter 7: Duration and Immunization
The book value of a bond converges to its maturity value as the maturity date approaches. That is, as the coupons are paid, the outstanding balance of the amortized bond approaches the redemption value. But also, the market value is affected by the current yield rate dictated by the economy. The market value may not equal the book value for the bondholder. That is, if the yield rate is now higher than when the bond was purchased, then the present value of the remaining payments (the market value) is smaller based on this higher yield rate. Thus, the market value of the bond at a given point in time is directly related to the “yield to maturity” existing in the market at that time.
Since changes to the yield rate affect the bond’s market value, can we measure the sensitivity of the bond’s market value to any change in yield rates. Using the derivative of the market value P(i), as a function of yield rate, we may determine the rate of change in the market value relative to change in the yield rate.
Consider a zero-coupon bond with a maturity value of 1 in n years. The current price, based on a yield rate i, is P = (1 + i)-n and the rate of change is . This rate of change is negative since increasing the yield rate results in a decrease in the present value (or price). To see the percentage increase in price, find the ratio of the derivative to the current price. The magnitude of this ratio is the modified duration, DM:
That is, . For our zero-coupon example above, we have
is the per-dollar rate of increase in price relative to a changing yield rate.
The duration D (also called Macaulay duration) of a bond is given by .
For our zero-coupon example above, we have . Note that here the sensitivity of the market value is measured in years because for a given price the interest rate determines the number of years to reach a given maturity value. In the case of our zero coupon bond, . Using linear approximation (ie, the tangent line) we note that , but we don’t really need this approximation since it’s easy enough to compute the bond price for any given yield rate i.
Clearly, in our example above, the sensitivity increases as time-to-maturity n increases. Thus, a longer term bond is more sensitive to changes in the yield rate and experiences a bigger change in market value.(See example 7.2, page 349.)
Similarly, for a series of annual payments or coupons K1 , K2 , …, Kn with a fixed yield rate i, the market value or price and so the modified duration is
and the Macaulay duration is
In particular, for bonds with coupons Kt = Fr for t = 1, 2, …, n-1 and payment at maturity
Kn = Fr + F, the duration becomes
If we define “weights” then the duration becomes a weighted average of the payment times , where since , we know .
Note each weight is the percentage of the present value associated with that payment time. Thus for bonds, the coupons are relatively small compared to the redemption amount and so the weights associated with coupon payments are small. As a result, the final payment Kn at maturity has the greatest weight and so the duration would be close to n. That is, .
Duration of a Portfolio
Suppose an investment portfolio results in a set of m cashflows where each cashflow consists of n years of annual payments. Let the payments of the kth cashflow be represented by
. Let Pk be the present value of the kth cashflow, .
Likewise, define the duration of the kth cashflow, . Now, consider the total of the m cashflows and define the total present value , so that . It follows the duration for the total of the m cashflows is given by
We may define weights , where , such that the duration is a weighted average of the durations of individual cashflows. Note each weight is the percentage of the overall present value associated with the kth cashflow.
“One of the most important interpretations of duration” (see page 355) is that two cashflows that have the same duration will be affected in the same way by small changes in yield-to-maturity. In particular, if the duration for the total of a set of cashflows is D years, then a zero-coupon bond maturing in D years will have the same duration. If the zero-coupon bond has the same present value as the set of cashflows, then for small changes in the yield-to-maturity, the change in the total present value of the cashflows will be about the same as the change in value of the zero-coupon bond.
Asset Matching and Immunization
A company needs to plan its investments so that resulting income provides the funds required to make any liability payments for which the company is responsible. Let Lt be the net liability due at time t and let At be the asset income available at time t to cover the liability payments. The asset income and liabilities due are said to be exactly matched if At = Lt at each point in time (typically, discrete and equally-spaced times t = 0, 1, 2, …, n ).
But even when plans are made so that asset income and liability due payments are matched and PVA(i0) < PVL(i0) based on the current yield rate i0, if changes in the economy result in changing yield rates, then asset and liability valuations may fall out of balance. We should “attempt to structure the asset cashflows so that small changes in interest rates don’t put the asset-liability relationship into a deficit position” (pg. 361) such that under the new interest rate i the present value for assets is less than the present value for liabilities, PVA(i) < PVL(i).
Redington’s immunization theory says that with careful structuring of asset income in relation to liabilities due, small changes in interest result in PVA(i) > PVL(i). That is, whether the interest rate increases or decreases by a small amount, the balance won’t change to a deficit position.
Suppose there is an asset-liability balance PVA(i0) = PVL(i0) based on current rate i0 , and assume that at i0 we have
.
If we define the difference , then we know and . Thus, i0 is a critical point of h where a relative minimum exists. Hence, in a neighborhood , and equivalently , which represents a surplus position. That is, under these assumptions, a small change in interest results in moving from a matched position to a surplus position. This is the basic theory of immunization.
Consider the immunization assumptions above. The assumptions PVA(i0) = PVL(i0) and imply the modified durations are equal
As a result, changes in the interest rate would have approximately the same affect on each of these present values. The latter assumption is equivalent to saying the asset income flow is more widely spread (greater variance) in time than the liabilities due.
Note the immunization assumptions
may be written in terms of the present value factor as
, and
Example 7.6
Example 7.6 (page 363) uses these equations to verify a portfolio is immunized. Two zero-coupon bonds of par value X and Y mature at times t1 = 2 and t2 = 14, respectively. We’re asked to determine the amounts X and Y such that these bonds will provide the necessary asset income to cover total liabilities with present value $300,000, given i = 0.10. Using the assumptions
,
we set up two equations in terms of the two unknowns X and Y:
and
(where values and times for each Lt are given in example 7.4, pg. 357)
Solving this linear system yields X = $195,407.21 and Y = $525,977.95.
Is this portfolio, consisting of the above two bonds, considered to be immunized? Check to see if the third condition is satisfied. That is, verify that
is true for these values of X and Y . Since
, the condition is satisfied and the portfolio is immunized. Hence, for small changes in the interest rate, the asset-liability relationship won’t change to a deficit position. Keep in mind that for larger changes in the interest rate, it is possible that a net deficit may occur.
The book notes a special situation, called fully immunized, where a net surplus PVA(i) > PVL(i) results (ie., h(i) > 0 ) for any rate i > 0. The concept of full immunization is discussed on pages 365 – 367 but not included here.