Finance Short Response 4

W E B E X T E N S I O N 5C A Closer Look at Bond Risk: Duration

T his extension explains how to manage the risk of a bond portfolio using the con-cept of duration. 5.1 BOND RISK In our discussion of bond valuation in Chapter 5, we discussed interest rate and rein- vestment rate risk. Interest rate (price) risk is the risk that the price of a debt secu- rity will fall as a result of increases in interest rates, and reinvestment rate risk is the risk of earning a less than expected return when debt principal or interest payments are reinvested at rates that are lower than the original yield to maturity.

To illustrate how to reduce interest rate and reinvestment rate risks, we will consider a firm that is obligated to pay a worker a lump-sum retirement benefit of $10,000 at the end of 10 years. Assume that the yield curve is horizontal, the current interest rate on all Treasury securities is 9%, and the type of security used to fund the retirement benefit is Treasury bonds. The present value of $10,000, discounted back 10 years at 9%, is $10,000(0.4224) = $4,224. Therefore, the firm could invest $4,224 in Treasury bonds and expect to be able to meet its obligation 10 years hence.1

Suppose, however, that interest rates change from the current 9% rate immedi- ately after the firm has bought the Treasury bonds. How will this affect the situation? The answer is, “It all depends.” If rates fall, then the value of the bonds in the port- folio will rise, but this benefit will be offset to a greater or lesser degree by a decline in the rate at which the coupon payment of 0.09($4,224) = $380.16 can be reinvested. The reverse would hold if interest rates rose above 9%. Here are some examples (for simplicity, we assume annual coupons).

1. The firm buys $4,224 of 9%, 10-year maturity bonds; rates fall to 7% immediately after the purchase and remain at that level:

Portfolio value at the end of 10 years

¼ Future value of

10 interest payments of $380:16 each

compounded at 7%

þ Maturity value

¼ $5; 252 þ $4; 224 ¼ $9; 476

Therefore, the firm cannot meet its $10,000 obligation, and it must contribute additional funds.

1For the sake of simplicity, we assume that the firm can buy a fraction of a bond. 1

2. The firm buys $4,224 of 9%, 40-year bonds; rates fall to 7% immediately after the purchase and remain at that level:

Portfolio value at the end of 10 years

¼ $5; 252 þ Value of 30-year 9% bonds

when rd ¼ 7% ¼ $5; 252 þ $5; 272 ¼ $10; 524

In this situation, the firm has excess capital at the end of the 10-year period.

3. The firm buys $4,224 of 9%, 10-year bonds; rates rise to 12% immediately after the purchase and remain at that level:

Portfolio value at the end of 10 years

¼ Future value of

10 interest payments of $380:16 each

compounded at 12% þ Maturity

value

¼ $6; 671 þ $4; 224 ¼ $10; 895

This situation also produces a funding surplus.

4. The firm buys $4,224 of 9%, 40-year bonds; rates rise to 12% immediately after the purchase and remain at that level:

Portfolio value at the end of 10 years

¼ $6; 671 þ Value of 30-year 9% bonds

when rd ¼ 12% ¼ $6; 671 þ $3; 203 ¼ $9; 874

This time, a shortfall occurs.

Here are some generalizations drawn from the examples.

1. If interest rates fall and the portfolio is invested in relatively short-term bonds, then the reinvestment rate penalty exceeds the capital gains and so a net short- fall occurs. However, if the portfolio had been invested in relatively long-term bonds, then a drop in rates would produce capital gains that would more than offset the shortfall caused by low reinvestment rates.

2. If interest rates rise and the portfolio is invested in relatively short-term bonds, then gains from high reinvestment rates will more than offset capital losses, and the final portfolio value will exceed the required amount. However, if the portfolio had been invested in long-term bonds, then capital losses would more than offset reinvestment gains, and a net shortfall would result.

If a company has many cash obligations expected in the future, then the complexity of estimating the effects of interest rate changes is obviously exac- erbated. Still, methods have been devised to help deal with the risks associated with changing interest rates. Several methods are discussed in the following sections.

2 Web Extension 5C: A Closer Look at Bond Risk: Duration

5.2 IMMUNIZATION Bond portfolios can be immunized against interest rate and reinvestment rate risk, much as people can be immunized against the flu. In brief, immunization involves selecting bonds with coupons and maturities such that the benefits or losses from changes in reinvestment rates are exactly offset by losses or gains in the prices of the bonds. In other words, if a bond’s reinvestment rate risk exactly matches its inter- est rate price risk, then the bond is immunized against the adverse effects of changes in interest rates.

To see what is involved, refer back to our example of a firm that buys $4,224 of 9% Treasury bonds to meet a $10,000 obligation 10 years hence. In the example, we see that if the firm buys bonds with a 10-year maturity and interest rates remain con- stant, then the obligation can be met exactly. However, if interest rates fall from 9% to 7%, then a shortfall will occur because the coupons received will be reinvested at a rate of 7% rather than the 9% reinvestment rate required to reach the $10,000 tar- get. But suppose the firm had bought 40-year rather than 10-year bonds. A decline in interest rates would still have the same effect on the compounded coupon payments, but now the firm would hold 9% coupon, 30-year bonds in a 7% market 10 years hence, so the bonds would have a value greater than par. In this case, the bonds would have risen by more than enough to offset the shortfall in compounded interest. Can we buy bonds with a maturity somewhere between 10 and 40 years such that the net effect of changes in reinvested cash flows and changes in bond values at year 10 will always be positive? The answer is “yes,” as we explain next.

5.3 DURATION The key to immunizing a portfolio is to buy bonds that have a duration equal to the years until the funds will be needed. Duration cannot be defined in simple terms like maturity, but it can be thought of as the weighted average maturity of all the cash flows (coupon payments plus maturity value) provided by a bond, and it is exception- ally useful to help manage the risk inherent in a bond portfolio. The duration for- mula and an example of the calculation are provided below, but first we present some additional points about duration.

1. Duration is similar to the concept of discounted payback in capital budgeting in the sense that, the longer the duration, the longer funds are tied up in the bond.

2. To immunize a bond portfolio, buy bonds that have a duration equal to the number of years until the funds will be needed. In our example, the firm should buy bonds with a duration of 10 years.

3. Duration is a measure of bond volatility. The percentage change in the value of a bond (or bond portfolio) will be approximately equal to the bond’s duration mul- tiplied by the percentage-point change in interest rates.2 Therefore, a 2- percentage-point increase in interest rates will lower the value of a bond with a 10-year duration by about 20%, but the value of a 5-year duration bond will fall by only 10%. As this example shows, if Bond A has twice the duration of Bond B then Bond A also has twice the volatility of Bond B. A corporate treasurer (or any other investor) who is concerned about declines in the market value of his or her

2Actually, duration measures the negative of the percentage change in bond price. Duration is a measure of the bond price’s elasticity with respect to the interest rate, but this elasticity measure is valid only for small changes in the interest rate.

Web Extension 5C: A Closer Look at Bond Risk: Duration 3

portfolio should buy bonds with low durations. (This is important even if the in- vestor buys a bond mutual fund.)

4. The duration of a zero coupon bond is equal to its maturity, but the duration of any coupon bond is less than its maturity. (Remember, the duration is a weighted average maturity of the cash flows, the only cash flow from a zero occurs at ma- turity, and coupon bonds have cash flows prior to maturity.) And with all else held constant, the higher the coupon rate the shorter the duration, because a high-coupon bond provides significant early cash flows even if it has a long maturity.

Duration is calculated using this formula:

Duration ¼ ∑ N

t¼1 tðCFtÞ ð1 þ rdÞt

∑ N

t¼1 CFt

ð1 þ rdÞt

¼ ∑ N

t¼1 tðCFtÞ ð1 þ rdÞt VB

(5C-1)

Here rd is the required return on the bond, N is the bond’s years to maturity, t is the year each cash flow occurs, and CFt is the cash flow in year t (CFt = INT for t < N and CFt = INT + M for t = N, where INT is the interest payment and M is the principal payment). Notice that the denominator of Equation 5C-1 is simply the value of the bond, VB.

To simplify calculations in Excel, we define the present value of cash flow t (PV of CFt ) as

PV of CFt ¼ CFt

ð1 þ rdÞt

We can rewrite Equation 5C-1 as

Duration ¼ ∑ N

t¼1 tðPVofCFtÞ

VB

(5C-2)

To illustrate the duration calculation, consider a 20-year, 9% annual coupon bond bought at its par value of $1,000. It provides cash flows of $90 per year for 19 years plus $1,090 in the 20th year. To calculate duration, we used an Excel model as shown in Figure 5C-1; see the worksheet Web 5C in the file Ch05 Tool Kit.xls for details.

Column 1 of Figure 5C-1 gives the year each cash flow occurs, Column 2 gives the cash flows, Column 3 shows the PV of each cash flow, and Column 4 shows the product of t and the PV of each cash flow. The sum the PVs in Column 3 is the value of the bond, VB. The duration is equal to the value of the bond divided by the sum of Column 4.

This 20-year bond’s 9.95 duration is close to that of our illustrative firm’s 10-year liability. So, if the firm bought a portfolio of these 20-year bonds and then reinvested

4 Web Extension 5C: A Closer Look at Bond Risk: Duration

the coupons as they came in, then the accumulated interest payments, plus the value of the bond after 10 years, would be close to or exceed $10,000 regardless of whether interest rates rose, fell, or remained constant at 9%. See Ch05 Tool Kit.xls for an example. There is also an Excel function for duration; the Tool Kit also de- monstrates its use.

Unfortunately, other complications arise. Our simple example looked at a single interest rate change that occurred immediately after funding. In reality, interest rates change every day, which causes bonds’ durations to change; this, in turn, requires that bond portfolios be rebalanced periodically to remain immunized.

FIGURE 5C-1 Duration

7 A B C D E

Inputs Years to maturity = 20

9.00%

$90.0 $1,080

9.00%

t (1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

$90 $90 $90 $90 $90 $90 $90 $90 $90 $90 $90 $90 $90 $90 $90 $90 $90 $90 $90

$1,090 $194.49

VB =

Duration = Sum of t(PV of CFt)/VB =

$1,000,00 Sum of

t(PV of CFt) =

$17.50 $19.08 $20.80 $22.67 $24.71 $26.93 $29.36 $32.00 $34.88 $38.02 $41.44 $45.17 $49.23 $53.66 $58.49 $63.76 $69.50 $75.75 $82.57 82.57

151.50 208.49 255.03 292.47 321.98 344.63 361.34 372.95 380.17 383.66 383.98 381.63 377.05 370.63 362.69 353.54 343.43 332.58

3,889.79

$9,990.11

CFt (2)

PV of CFt (3)

t(PV of CFt) (4)

Coupon rate =

Annual payment = Par value = FV =

Going rate, r =

8 9

10 11 12 13 14

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

37 38 39 9.95

resource See the worksheet Web 5C in Ch05 Tool Kit.xls on the textbook’s Web site.

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