sensible family puts its grocery money (a short-term ob-

in full?

ligation) into common stocks (a long-term asset). Ordi-

We™ll get to that, but the real news here is broader

nary Joes and Janes grasp the principle of “matching”

than GM. According to experts, most pension funds

actually lost ground, even though, as you may recall, it without even thinking about it.

But fund managers”the pros”insist on shorter,

was a rather good year for stocks and bonds.

unmatching bond portfolios for a simple, stupefying

True, pension-fund assets did have a banner year.

reason. They are graded”usually by consultants”

But as is sometimes overlooked, pension funds also

according to how they perform against standard (and

have liabilities (their obligations to retirees). And at

shorter term) bond indexes. Thus, rather than invest to

most funds, liabilities grew at a rate that put asset

keep up with liabilities, managers are investing so as to

growth to shame. At the margin, that means more

avoid lagging behind the popular index in any year.

companies™ pension plans will be “underfunded.” And

down the road, assuming no reversal in the trend, more

companies will have to pony up more cash.

What™s to blame? The decline in interest rates that

SOURCE: Roger Lowenstein, “How Pension Funds Lost in Market

brought joy to everyone else. As rates fall, pension

Boom,” The Wall Street Journal, February 1, 1996. Reprinted by

funds have to set aside more money today to pay off a permission of Dow Jones & Company, Inc. via Copyright Clearance

fixed obligation tomorrow. In accounting-speak, this Center, Inc. © 1996 Dow Jones & Company, Inc. All Rights

“discounted present value” of their liabilities rises. Reserved Worldwide.

Suppose that the insurance company chooses to fund its obligation with $10,000 of 8% an-

nual coupon bonds, selling at par value, with six years to maturity. As long as the market in-

terest rate stays at 8%, the company has fully funded the obligation, as the present value of the

obligation exactly equals the value of the bonds.

Table 10.4A shows that if interest rates remain at 8%, the accumulated funds from the bond

will grow to exactly the $14,693.28 obligation. Over the five-year period, the year-end coupon

income of $800 is reinvested at the prevailing 8% market interest rate. At the end of the pe-

riod, the bonds can be sold for $10,000; they still will sell at par value because the coupon rate

still equals the market interest rate. Total income after five years from reinvested coupons and

the sale of the bond is precisely $14,693.28.

If interest rates change, however, two offsetting influences will affect the ability of the fund

to grow to the targeted value of $14,693.28. If interest rates rise, the fund will suffer a capital

loss, impairing its ability to satisfy the obligation. The bonds will be worth less in five years

than if interest rates had remained at 8%. However, at a higher interest rate, reinvested

coupons will grow at a faster rate, offsetting the capital loss. In other words, fixed-income in-

vestors face two offsetting types of interest rate risk: price risk and reinvestment rate risk. In-

creases in interest rates cause capital losses but at the same time increase the rate at which

reinvested income will grow. If the portfolio duration is chosen appropriately, these two ef-

fects will cancel out exactly. When the portfolio duration is set equal to the investor™s horizon

date, the accumulated value of the investment fund at the horizon date will be unaffected by

351

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Essentials of Investments, Portfolios Companies, 2003

Fifth Edition

352 Part THREE Debt Securities

Years Remaining Accumulated Value of

TA B L E 10.4 Payment Number until Obligation Invested Payment

Terminal value of

A. Rates remain at 8%

a bond portfolio

800 (1.08)4

after five years 1 4 1,088.39

(all proceeds 800 (1.08)3

2 3 1,007.77

reinvested) 800 (1.08)2

3 2 933.12

800 (1.08)1

4 1 864.00

800 (1.08)0

5 0 800.00

Sale of bond 0 10,800/1.08 10,000.00

14,693.28

B. Rates fall to 7%

800 (1.07)4

1 4 1,048.64

800 (1.07)3

2 3 980.03

800 (1.07)2

3 2 915.92

800 (1.07)1

4 1 856.00

800 (1.07)0

5 0 800.00

Sale of bond 0 10,800/1.07 10,093.46

14,694.05

C. Rates increase to 9%

800 (1.09)4

1 4 1,129.27

800 (1.09)3

2 3 1,036.02

800 (1.09)2

3 2 950.48

800 (1.09)1

4 1 872.00

800 (1.09)0

5 0 800.00

Sale of bond 0 10,800/1.09 9,908.26

14,696.02

Note: The sale price of the bond portfolio equals the portfolio™s final payment ($10,800) divided by 1 r, because the time to maturity of

the bonds will be one year at the time of sale.

interest rate fluctuations. For a horizon equal to the portfolio™s duration, price risk and rein-

vestment risk exactly cancel out. The obligation is immunized.

In the example we are discussing, the duration of the six-year maturity bonds used to fund

the GIC is five years. You can confirm this following the procedure in Spreadsheet 10.1. The

duration of the (zero-coupon) GIC is also five years. Because the fully funded plan has equal

duration for its assets and liabilities, the insurance company should be immunized against in-

terest rate fluctuations. To confirm that this is the case, let us now investigate whether the bond

can generate enough income to pay off the obligation five years from now regardless of inter-

est rate movements.

Tables 10.4B and C consider two possible interest rate scenarios: Rates either fall to 7% or

increase to 9%. In both cases, the annual coupon payments from the bond are reinvested at the

new interest rate, which is assumed to change before the first coupon payment, and the bond

is sold in year 5 to help satisfy the obligation of the GIC.

Table 10.4B shows that if interest rates fall to 7%, the total funds will accumulate to

$14,694.05, providing a small surplus of $0.77. If rates increase to 9% as in Table 10.4C, the

fund accumulates to $14,696.02, providing a small surplus of $2.74.

Several points are worth highlighting. First, duration matching balances the difference be-

tween the accumulated value of the coupon payments (reinvestment rate risk) and the sale

value of the bond (price risk). That is, when interest rates fall, the coupons grow less than in

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Essentials of Investments, Portfolios Companies, 2003

Fifth Edition

353

10 Managing Bond Portfolios

Accumulated value of invested funds

Obligation

t

0

t* D = 5 years

F I G U R E 10.4

Growth of invested funds

Note: The solid curve represents the growth of portfolio value at the original interest rate. If interest rates increase at time t* the portfolio value falls but

increases thereafter at the faster rate represented by the broken curve. At time D (duration) the curves cross.

the base case, but the gain on the sale of the bond offsets this. When interest rates rise, the re-

sale value of the bond falls, but the coupons more than make up for this loss because they are

reinvested at the higher rate. Figure 10.4 illustrates this case. The solid curve traces out the ac-

cumulated value of the bonds if interest rates remain at 8%. The dashed curve shows that

value if interest rates happen to increase. The initial impact is a capital loss, but this loss even-

tually is offset by the now-faster growth rate of reinvested funds. At the five-year horizon date,

the two effects just cancel, leaving the company able to satisfy its obligation with the accu-

mulated proceeds from the bond. The nearby box discusses this trade-off between price and

reinvestment rate risk, suggesting how duration can be used to tailor a bond portfolio to the

horizon of the investor.

We can also analyze immunization in terms of present as opposed to future values. Table

10.5A shows the initial balance sheet for the insurance company™s GIC account. Both assets

and the obligation have market values of $10,000, so that the plan is just fully funded. Table

10.5B and C show that whether the interest rate increases or decreases, the value of the bonds

funding the GIC and the present value of the company™s obligation change by virtually iden-

tical amounts. Regardless of the interest rate change, the plan remains fully funded, with the

surplus in Table 10.5B and C just about zero. The duration-matching strategy has ensured that

both assets and liabilities react equally to interest rate fluctuations.

Figure 10.5 is a graph of the present values of the bond and the single-payment obligation

as a function of the interest rate. At the current rate of 8%, the values are equal, and the ob-

ligation is fully funded by the bond. Moreover, the two present value curves are tangent at

Bodie’Kane’Marcus: III. Debt Securities 10. Managing Bond © The McGraw’Hill

Essentials of Investments, Portfolios Companies, 2003

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354 Part THREE Debt Securities

A. Interest rate 8%

TA B L E 10.5

Market value Assets Liabilities

balance sheets

Bonds $10,000 Obligation $10,000

B. Interest rate 7%

Assets Liabilities

Bonds $10,476.65 Obligation $10,476.11

C. Interest rate 9%

Assets Liabilities

Bonds $9,551.41 Obligation $9,549.62

Notes:

Value of bonds 800 Annuity factor(r, 6) 10,000 PV factor(r, 6)

14,693.28

Value of obligation 14,693.28 PV factor(r, 5)

r)5

(1

F I G U R E 10.5 Values ($)

Immunization. The 14,000

coupon bond fully Coupon bond

funds the obligation

at an interest rate of 12,000

Single

8%. Moreover, the

payment

present value curves obligation

10,000

are tangent at 8%,

so the obligation will

remain fully funded

8,000

even if rates change.

Interest

6,000 rate

8% 10%

0 5% 15% 20%

y 8%. As interest rates change, the change in value of both the asset and the obligation are

equal, so the obligation remains fully funded. For greater changes in the interest rate, however,

the present value curves diverge. This reflects the fact that the fund actually shows a small sur-

plus at market interest rates other than 8%.

Why is there any surplus in the fund? After all, we claimed that a duration-matched asset

and liability mix would make the investor indifferent to interest rate shifts. Actually, such a

claim is valid only for small changes in the interest rate, because as bond yields change, so too

does duration. (Recall Rule 4 for duration.) In fact, while the duration of the bond in this

example is equal to five years at a yield to maturity of 8%, the duration rises to 5.02 years

when the bond yield falls to 7% and drops to 4.97 years at y 9%. That is, the bond and the

Bodie’Kane’Marcus: III. Debt Securities 10. Managing Bond © The McGraw’Hill

Essentials of Investments, Portfolios Companies, 2003

Fifth Edition