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47

= 13.5.

Also, it is important to note that duration for a 10-year mortgage is significantly shorter than for a 10-year bond. This happens because a mortgage repays some principal in every installment but the bond repays principal only at its redemption date. (This statement is not exacdy true with premium and discount bonds, but the effect is generally not large enough to invalidate die above comparison between bonds and mortgages.)

4. Per dollar of dividend, formula (9.13) gives

- (b) 1.08/.08 1/.08

Note that this answer is independent of die dividend rate being paid on the preferred stock. Also note diat duration for preferred stock is longer than for any of die odier investments. This seems reasonable, since preferred stock involves payments being made into perpetuity.

9.9 IMMUNIZATION

Until this point in the book we have largely been analyzing individual transactions. We now shift our focus and consider a whole collection of transactions. More specifically, we consider the interrelationship between assets and liabilities for some financial 1 5 , such as a bank, an insurance company, or a pension fund.

The assets will generate a series of cash inflows. We label the cash inflows as Ai,A2, . . . , which are made at times 1,2, ... ,n. Similarly, die liabilities will generate a series of cash outflows. These are labelled Lj, L2 , . . . , L and occur at times 1,2, . . . , n. The issue is how to achieve an equilibrium or safe balance between these cash inflows and outflows.

Before going further with the development of the theory, let us consider the nature of the problems that can arise if such equilibrium does not exist. The primary problem is the risk of adverse effects created by changes in the level of interest rates.

We can illustrate this problem with an example. Consider a financial institution, such as a bank or an insurance company, which issues a one-year instrument with a guaranteed rate of interest, e.g. a certificate of deposit (CD) or a guaranteed investment contract (GIC). A significant risk to the financial institution exists if the assets backing fliese contracts are invested either "too long" or "too short."

First, consider the case in which die assets are invested "too long," e.g. in investments with duration equal to two. The financial institution is vulnerable to losses if interest rates rise. Under these conditions contract holders are likely to withdraw their funds at the end of the year. The financial institution may

lye to sell assets to pay these departing contract holders. However, the assets t would have to be sold have declined in value due to the rise in interest (ates. Losses may be incurred as a result. Second, consider the case in which die assets are invested "too short," e.g. in very short-term investments with duration close to zero. Now the financial institution is vulnerable to losses if interest rates fall. Since its assets are invested very short-term, its interest earnings will decline quickly and may not fbe sufficient to pay the guaranteed interest on the contracts at the end of the year. Thus, losses may be incurred in this case as well.

Immunization is a technique that has been developed to structure the assets and liabilities in a manner that would reduce or even eliminate the type of ilem illustrated above. In other words, the enterprise would be immunized" against the adverse effects created by changes in the level of iterest rates.

Let the net receipt at time t be denoted by R,, i.e.

R, = A,-L, for r = 1,2.....n. (9.18)

;We will assume that the present value of the cash inflows from the assets is equal to the present value of the cash outflows from the liabilities. Thus, from formula (9.14) we have

P(0 =0. (9.19)

We now institute a small change in the rate of interest from i to i -t- e.

If we expand P(0 as a Taylor series as far as second derivatives, we have

Pii + e) = P(i) + ePiO + L. P"(i + J), where 0 < ? < e .

Now P(i) will have a local minimum at i if two conditions hold. First, we must have

P(i) = 0. (9.20)

Note tiiat formula (9.20) can be interpreted as requiring that modified duration of the net receipts must equal zero. Second, we must have

P"(i) > 0. (9.21)

If P(0 has a local minimum, we obtain a key result in immunization theory; jiiamely, that small changes in the interest rate in either direction will increase the present value of receipts. This certainly seems to be a highly desirable result, if it can be achieved.

The second derivative of P{i) is used to define the property of convexity, denoted by c, as

(9.22)

P(i)



Money market fund: Two-year zero coupon bonds;

V = 0

The weighted average of diese two values reflecting our investment allocation is .J

.5(0) + .5

= .90909.

Pii) = E (1 +)"

so diat

--1

Pii) = -E(i+o

Pit)

When we evaluate diis expression for n = 360 and / = .( 85, we obtain

11,283.80

= 99.85.

(1.0085)(112.0591)

Thus, the modified duration of die payments is just under 100 months of the 360-mondi term of die mortgage.

12. To measure die convexity of die payments we need to take second derivative of Pii)

p"ii) = E /(/ + i)(i +0"" /=1

= vj: it+t)v. 1=1

In order to evaluate diis second derivative, we need to be able to

evaluate E Since this summation has 360 terms in it, direct calculation is

P(.l)=-- + i=0.

This gives us two equations in two unknowns, so that we do not have the flexibility in-i this case to tiy to apply formula (9.21). Solving the two equations in two unbownsi gives us j: = 500 and = 500. "

Although we could not apply formula (9.21) in developing the investment allocationj we need to test our answer for convexity. We have

P"(.l) = £«2- = 826.45 >0. (1-1) (1.1)3

Thus, our strategy satisfies formula (9.21) anyway, which is a happy result.

Let us test die result empirically for changes in / of 1 % in eidier direction. We have P(.IO) = 0

P(.ll) = 500 + li:H)»-il22 = .0406 > 0

(1.11)2 1.11

P(.09) = 500 +<i:i)<M-1122 = 0421 > 0

(1.09)2 1.09

Thus, die value of Pit) increases for movements of / away from / = 10% in er direction. This may seem too good to be true, but it Ulusfrates a result diat immuni is attempting to achieve.

Example 9.8 For the assets in Example 9.7 compute: (1) the modifud\ duration, and (2) the convexity.

1. Using the asset portion of die expressions above, we have Pit) = x + 1.21y(l +0- = 1000.00

Pii) = -2.42y(l + O" = -909.09

when evaluated at jc = 500, = 500, and / = .1. Then applying formula (9.15) wel

have *

V = - Zl® = .90909. >

Pii)

Anodier approach is to use formula (9.16). We have die following modi durations:

As an exercise die reader will be asked to confirm that this value equals the modified duration of die liability.

2. Again, using die asset portion of the expressions above, we have

P"(0 = 7.26y(l +0" = 2479.34

when evaluated at = 500 and i = .1. Then applying formula (9.22), we have

- = "(0 = 2.47934. Pii)

As an exercise die reader will be asked to confirm diat diis value is greater dian die convexity of die liability.

Example 9.9 For a 30-year home mortgage with level payments and an I interest rate of 10.2% convertible monthly, find: (I) the modified duration of I the payments, and (2) the convexity of the payments.

1. The mondily rate of interest is .102/12 = .0085. Then per dollar of mondily payment, we have



This pattern reflects the normal yield curve with positive slope. Thus, \ already see one new feature in this model. Classical immunization theory <Jo not recognize the term structure of interest rates. ,

Let and be the amount of fiinds withdrawn by the holders of the CDg at the ends of years 1 and 2, respectively. The reader should be careful to note tiiat sy and 2 are amounts withdrawn, not rates of withdrawal. For each dollar deposited in the CDs we have the equation of value

1 = (1.08)->Ji + (1.08)-2j2 which can be expressed as

S2 = (1.08)2-1.081.

Explicit recognition of fiiture witiidrawal rates is a second new feahire in tiiis model.

Let /?i and pj equal the proportion of funds the bank invests in one-year: and two-year notes, respectively. Clearly, pi + P2 = I . Let /be the, forward (reinvestment) rate on one-year notes in the second year. Explicit? consideration of reinvestment rates is a third new feature in this model. Finally, let A2 equal the accumulated value of the banks fiinds arising from fliis, transaction at the end of the second year. Then we have

Pl(1.08)-5,1(1 +/) +

/>2 (1.085)2-2 (1 -/7i)(l.085)2 - (1.08)2 + (1.08)J,

(1.08)(1 +/) - (1.085)2 + 5,(.08 -/) + (1.085)2 - (1.08)2

We need to analyze the impact of fluctuations in/on the value of A2 in order to develop a recommendation concerning the selection of pj (and thus P2 also).

First, we consider the case of a decrease in interest rates. Suppose that/ is equal to 7%. This will create a low rate of witiidrawal by the holders of die CDs at die end of Uie first year, since tiiere are incentives to leave fiinds in die CD. Assume the amount withdrawn is only 10 cents per dollar originally invested. We then have

A2 = -.021625/71 + .011825. We want A2 to be greater tiian 0, which will happen if /?i < .5468.

Second, we consider the case of an increase in interest rates. Suppose that /is equal to 9.5%. This will create a high rate of witiidrawal by die holders of the CDs at the end of tiie first year, since high rates are available elsewhere. Assume the amount withdrawn is 90 cents per dollar originally invested. We tiien have

A2 - .005375pi -.002675.

fwe want A2 to be greater than 0, which will happen if > .4977.

Thus, die recommended investment strategy for the bank based on these fassumptions is to choose Pi such that .4977 < p < .5468. Since bp = 1 the investment allocation is determined.

f Application of this technique requires investment managers to make certain key assumptions which are not required in classical immunization. One key assumption is how high and how low to set the forward (reinvestment) rate in working through the two cases. A second key assumption is the level of witiidrawal rates that will develop in the two cases. Unfortunately, the technique is sensitive to these assumptions, which means that relatively modest changes in the assumptions can resuh in significantly different allocation strategies. Nevertheless, despite tiiis sensitivity, the technique has proven to be usefiil in practical applications.

i EXERCISES

9.4 Recognition of inflation

The nominal rate of interest is 8% and the rate of inflation is 5%. A single deposit

is invested for 10 years. Let:

A = value of the investment at the end of 10 years measured in "constant

dollars," i.e. in dollars valued at time 0. = value of die investment at die end of 10 years computed at the real rate of interest. Find die ratio A/B.

2. Rework Exercise 1 assuming equal level deposits at die beginning of each year during die 10-year period instead of a single deposit.

3. Money is invested for five years in a savings account earning 7% effective. If the rate of inflation is 10%, find die percentage of purchasing power lost during die period of investment.

4. A worker invests a sum of money for 20 years for retirement in a ftind earning 8 % effective. Interest income is subject to a 25% income tax rate. On Basis A die money accumulates tax-free and is subject to income tax at the end of the period of investment. On Basis the interest income is subject to income tax each year as it is earned. Find die ratio of the after-tax accumulation on Basis A to die after-tax accumulation on Basis B.

5. An employee has just retired after working exacdy 25 years for one employer. The employee was hired at $10,000 per year and received annual pay raises of 4%. The employee is eligible to receive a pension based on salary computed on one of three bases as specified below.



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