web extension: bond risk, duration, and zero coupon bonds · web extension: bond risk, duration,...

4C H A P T E R

4E-1

Web Extension: Bond Risk,Duration, and Zero Coupon Bonds

This extension explains how to manage the risk of a bond portfolio using the concept ofduration. It also explains zero coupon bonds.

Bond RiskIn our discussion of bond valuation in Chapter 4, we discussed interest rate andreinvestment rate risk. Interest rate (price) risk is the risk that the price of a debtsecurity will fall as a result of increases in interest rates, and reinvestment rate riskis the risk of earning a return less than expected when debt principal or interestpayments are reinvested at rates less than the original yield to maturity.

To illustrate how to reduce interest rate and reinvestment rate risks, we willconsider 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 cur-rent interest rate on all Treasury securities is 9 percent, and the type of security usedto fund the retirement benefit is Treasury bonds. The present value of $10,000,discounted back 10 years at 9 percent, is $10,000(0.4224) � $4,224. Therefore,the firm could invest $4,224 in Treasury bonds and expect to be able to meet itsobligation 10 years hence.

Suppose, however, that interest rates change from the current 9 percent rateimmediately after the firm has bought the Treasury bonds. How would this affectthe situation? The answer is, “It all depends.” If rates fall, then the value of thebonds in the portfolio will rise, but this benefit will be offset to a greater or lesserdegree by a decline in the rate at which the coupon payment of 0.09($4,224) �

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4E-2 • Chapter 4 Web Extension: Bond Risk, Duration, and Zero Coupon Bonds

$380.16 can be reinvested. The reverse would hold if interest rates rise above 9percent. Here are some examples (for simplicity, we assume annual coupons):

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

Future value of Portfolio value at 10 interest payments Maturity

the end of 10 years�

of $380.16 each �

valuecompounded at 7%

� $5,252 � $4,224 � $9,476

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

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

Value of 30-year,Portfolio value at

� $5,252 � 9% bonds when the end of 10 years

rd � 7%� $5,252 � $5,272 � $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 percent, 10-year bonds; rates rise to 12 percent

immediately after the purchase and remain at that level:

Future value of Portfolio value at 10 interest payments Maturity

the end of 10 years�

of $380.16 each �

valuecompounded at 12%

� $6,671 � $4,224 � $10,895

This situation also produces a funding surplus.4. The firm buys $4,224 of 9 percent, 40-year bonds; rates rise to 12 percent

immediately after the purchase and remain at that level:

Value of 30-year,Portfolio value at

� $6,671 � 9% bonds when the end of 10 years

rd � 12%� $6,671 � $3,203 � $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, so a net shortfalloccurs. However, if the portfolio had been invested in relatively long-termbonds, a drop in rates would produce capital gains that would more than off-set 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

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Chapter 4 Web Extension: Bond Risk, Duration, and Zero Coupon Bonds • 4E-3

portfolio had been invested in long-term bonds, then capital losses wouldmore than offset reinvestment gains, and a net shortfall would result.

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

Immunization Bond portfolios can be immunized against interest rate and rein-vestment rate risk, much as people can be immunized against flu. In brief, immuniza-tion involves selecting bonds with coupons and maturities such that the benefits orlosses from changes in reinvestment rates are exactly offset by losses or gains inthe prices of the bonds. In other words, if a bond’s reinvestment rate risk exactlymatches its interest rate price risk, then the bond is immunized against the adverseeffects of changes in interest rates.

To see what’s involved, refer back to our example of a firm that buys $4,224of 9 percent Treasury bonds to meet a $10,000 obligation 10 years hence. In theexample, we see that if the firm buys bonds with a 10-year maturity and interestrates remain constant, then the obligation can be met exactly. However, if interestrates fall from 9 percent to 7 percent, a shortfall will occur because the couponsreceived will be reinvested at a rate of 7 percent rather than the 9 percent reinvest-ment rate required to reach the $10,000 target. But, suppose the firm had bought40-year rather than 10-year bonds. A decline in interest rates would still have thesame effect on the compounded coupon payments, but now the firm would hold 9percent coupon, 30-year bonds in a 7 percent market 10 years hence, so the bondswould have a value greater than par. In this case, the bonds would have risen bymore than enough to offset the shortfall in compounded interest. Bonds with amaturity somewhere between 10 and 40 years would result in a breakeven situa-tion in which the reinvestment shortfall was exactly offset by capital gains.

Duration The key to immunizing a portfolio is to buy bonds that have a dura-tion equal to the years until the funds will be needed. Duration cannot be definedin simple terms like maturity, but it can be thought of as the weighted averagematurity of all the cash flows (coupon payments plus maturity value) provided bya bond, and it is exceptionally useful to help manage the risk inherent in a bondportfolio. The duration formula and an example of the calculation are providedbelow, but first we present some additional points about duration:

1. Duration is to a bond what payback is to a capital budgeting project, becausethe 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 thenumber of years until the funds will be needed. In our example, the firm shouldbuy bonds with a duration of 10 years.

3. A corporate treasurer (or any other investor) who is terribly concerned aboutdeclines in the market value of his or her portfolio should buy bonds with lowdurations. (This is important even if the investor buys a bond mutual fund.) Thepercentage change in the value of a bond (or bond portfolio) will be approxi-mately equal to the bond’s duration times the percentage point change in inter-est rates. Therefore, a 2 percentage point increase in interest rates will lower thevalue of a bond with a 10-year duration by about 20 percent, but the value of a5-year duration bond will fall by only 10 percent. So, twice the duration, twicethe volatility.

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4. The duration of a zero coupon bond is equal to its maturity, but the durationof any coupon bond is less than its maturity. (Remember, the duration is aweighted average age maturity of the cash flows, the only cash flow from azero occurs at maturity, and coupon bonds have cash flows prior to maturity.)Further, the higher the coupon rate, the shorter the duration, other things heldconstant, because a high-coupon bond provides significant early cash flowseven if it has a long maturity.

Duration is calculated using this formula:

| 4E-1 |Duration � an

t�1a t (PVCFt )

an

t�1 PVCFt

b � an

t�1

t (PVCFt )

Value

See IFM9 Ch04 Tool Kit.xls

for calculations.

DurationTable 4E-1

PVCF/Value �t CF PVCF (9%) PVCF/$1,000 t(PVCF/Value)

(1) (2) (3) (4) (5)

1 $ 90 $ 82.57 0.08257 0.082572 90 75.75 0.07575 0.15150� � � � �

� � � � �

� � � � �

19 90 17.50 0.01750 0.3325820 1,090 194.49 0.19449 3.88979

Value � $1,000.00 Duration � 9.95011

Here n is the bond’s years to maturity, t is the year each cash flow occurs, andPVCFt is the present value of the cash flow at Year t discounted at the current rateof interest. Note that the denominator of the equation is the PV of the cash flows,which is the current market value of the bond.

To illustrate the duration calculation, consider a 20-year, 9 percent annualcoupon bond bought at its par value of $1,000. It provides cash flows of $90 peryear for 19 years, and $1,090 in the 20th year. To calculate duration, we used aspreadsheet set up as shown in Table 4E-1.

Column 1 of Table 4E-1 gives the year each cash flow occurs, Column 2 givesthe cash flows, Column 3 shows the PV of each cash flow, Column 4 shows thepercentage of each PV cash flow to the total PV of cash flows, and Column 5 mul-tiplies each year by its percentage of the PV of total cash flows. The sum of theweighted percentages is the bond’s duration.

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Since a 20-year bond’s 9.95 duration is close to that of our illustrative firm’s10-year liability, if the firm bought a portfolio of 20-year bonds and then rein-vested the coupons as they came in, the accumulated interest payments, plus thevalue of the bond after 10 years, would be close to $10,000 irrespective of whetherinterest rates rose, fell, or remained constant at 9 percent.

Unfortunately, other complications arise. Our simple example looked at a sin-gle interest rate change that occurred immediately after funding. In reality, interestrates change every day, which causes bonds’ durations to change, and this, in turn,requires that bond portfolios be rebalanced periodically to remain immunized.Still, this can be done, and computer programs are available to assist in the rebal-ancing process.

Zero (or Very Low) Coupon BondsSome bonds pay no interest but are offered at a substantial discount below theirpar values and hence provide capital appreciation rather than interest income.These securities are called zero coupon bonds (“zeros”), or original issue discountbonds (OIDs). Some corporations use these bonds to manage their maturity struc-ture. In addition, these bonds provide some desirable tax features for corpora-tions, as we discuss later in this section.

Corporations first used zeros in a major way in 1981. In recent years IBM,Alcoa, JCPenney, ITT, Cities Service, GMAC, Martin-Marietta, and many othercompanies have used them to raise billions of dollars. Municipal governments alsosell “zero munis.” Shortly after corporations began to issue zeros, investment bankersfigured out a way to create zeros from U.S. Treasury bonds, which were issuedonly in coupon form. In 1983 Salomon Brothers bought $1 billion of 7 percent,30-year Treasuries. Each bond had 60 coupons worth $35 each, which repre-sented the interest payments due every six months. Salomon then in effect clippedthe coupons and placed them in 60 piles; the last pile also contained the now“stripped” bond itself, which represented a promise of $1,000 in the year 2013.These 60 piles of U.S. Treasury promises were then placed with the trust depart-ment of a bank and used as collateral for “zero coupon U.S. Treasury Trust Certifi-cates,” which are, in essence, zero coupon Treasury bonds. A pension fund that, in1983, expected to need money in 2007 could have bought 24-year certificatesbacked by the interest the Treasury will pay in 2007.

In 1985 the Treasury Department began allowing investors to strip long-termU.S. Treasury bonds and directly register the newly created zero coupon bonds, calledSTRIPs, with the Treasury Department. This bypasses the role formerly played byinvestment banks. Now virtually all U.S. Treasury zeros are held in the form ofSTRIPs. These STRIPs are, of course, safer than corporate zeros, so they are verypopular with pension fund managers.

To understand how zeros are used and analyzed, consider the zeros of Van-denberg Corporation, a shopping center developer. Vandenberg is developing a newshopping center in Orange County, California, and it needs $50 million. The com-pany does not anticipate major cash flows from the project for about five years.However, Pieter Vandenberg, the president, plans to sell the center once it is fullydeveloped and rented, which should take about five years. Therefore, Vandenbergwants to use a financing vehicle that will not require cash outflows for five years.He has decided on a five-year zero coupon bond, with a maturity value of $1,000.

Vandenberg Corporation is an A-rated company, and A-rated zeros with five-year maturities yield 9 percent at this time (five-year coupon bonds also yield

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9 percent). The company is in the 40 percent federal-plus-state tax bracket. PieterVandenberg wants to know the firm’s after-tax cost of capital if it uses 9 percent,five-year maturity zeros, and he also wants to know what the bond’s cash flowswill be. Table 4E-2 provides an analysis of the situation, and the following num-bered paragraphs explain the table itself:

1. The information in the “Basic Data” section, except the issue price, was givenin the preceding paragraph, and the information in the “Analysis” section wascalculated using the known data. The maturity value of the bond is always setat $1,000 or some multiple thereof.

2. The issue price is the PV of $1,000, discounted back five years at the rate rd �9%. Using a financial calculator, we input N � 5, I � 9, and FV � 1000,then press the PV key to find PV � $649.93. Note that $649.93, compoundedannually for five years at 9 percent, will grow to $1,000 as shown on the timeline in Table 4E-2.

3. The accrued values as shown on Line 1 in the analysis section represent thecompounded value of the bond at the end of each year. The accrued value forYear 0 is the issue price; the accrued value for Year 1 is found as $649.93(1.09)1 � $708.42; the accrued value at the end of Year 2 is $649.93(1.09)2 �$772.18; and, in general, the value at the end of any Year n is

Accrued value at the end of Year n � Issue price � (1 � rd)n

4. The interest deduction as shown on Line 2 represents the increase in accruedvalue during the year. Thus, interest in Year 1 � $708.42 � $649.93 �$58.49. In general,

Interest in Year n � Accrued valuen � Accrued valuen � 1

This method of calculating taxable interest is specified in the Tax Code.

Analysis of Zero Coupon BondTable 4E-2

Basic DataMaturity value $1,000rd 9.00%Maturity 5 yearsCorporate tax rate 40.00%Issue price $649.93

Analysis

(1) Year-end accrued value(2) Interest deduction(3) Tax savings (40%)(4) Cash flow to Vandenberg

After-tax cost of debt � 5.40%.Number of $1,000 zeros the company must issue to raise $50 million � Amount needed/Price per bond

� $50,000,000/$649.93� 76,931 bonds.

Face amount of bonds: (76,931)($1,000) � $76,931,000.

See IFM9 Ch04 Tool Kit.xls

for detailed calculations.

0 2 3

1,000.0082.5733.03

–966.97

917.4375.7530.30

+30.30

841.6869.5027.80

+27.80

772.1863.7625.50

+25.50

708.4258.4923.40

+23.40

649.93

+649.93

4 51

Years

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5. The company can take a tax deduction for interest each year, even though thepayment is not made in cash. This deduction lowers the taxes that would oth-erwise be paid, producing the following savings:

Tax savings � (Interest deduction)(T)� $58.49(0.4)� $23.40 in Year 1

6. Line 4 represents a cash flow time line; it shows the cash flow at the end ofYears 0 through 5. At Year 0, the company receives the $649.93 issue price.The company then has positive cash inflows equal to the tax savings duringYears 1 through 4. Finally, in Year 5, it must pay the $1,000 maturity value,but it gets one more interest tax saving for that year. Therefore, the net cashflow in Year 5 is �$1,000 � $33.03 � �$966.97.

7. We can find the IRR of the cash flows shown on Line 4 using the IRR func-tion of a financial calculator by simply inputting the annual cash flows in thecash flow register. The IRR is 5.4 percent, and it is the after-tax cost of zerocoupon debt to the company. Conceptually, here is the situation:

The value rd(AT) � 0.054 � 5.4%, found with a financial calculator, producesthe equality, and it is the after-tax cost of the zero coupon bond.

8. Note that rd(1 � T) � 9%(0.6) � 5.4%. As we will see in Chapter 10, the costof capital for regular coupon debt is found using the formula rd(1 � T). Thus,there is symmetrical treatment for tax purposes of zero coupon and regularcoupon debt, so both have the same tax implications. This was Congress’s intent,and it is why the Tax Code specifies the treatment set forth in Table 4E-2.1

Not all original issue discount bonds (OIDs) have zero coupons. For example,Vandenberg might have sold an issue of five-year bonds with a 5 percent couponat a time when other bonds with similar ratings and maturities were yielding 9percent. Such a bond would have had a value of $844.41:

If an investor had purchased these bonds at a price of $844.41, the yield to matu-rity would have been 9 percent. The discount of $1,000 � $844.41 � $155.59would have been amortized over the bond’s five-year life, and it would have been

Bond value � a5

t�1

$50

(1.09) t�

$1,000

(1.09) 5� $844.41

$649.93

(1 � rd(AT) )0

�$23.40

(1 � rd(AT) )1

�$25.50

(1 � rd(AT) )2

�$27.80

(1 � rd(AT) )3

�$30.30

(1 � rd(AT) )4

��$966.97

(1 � rd(AT) )5

� 0

an

t�0

CFn

(1 � rd(AT) )n

� 0

1The purchaser of a zero coupon bond must calculate interest income on the bond in the same manner as the issuer cal-culates the interest deduction. Thus, in Year 1, a buyer of a bond would report interest income of $58.49 and would paytaxes in the amount of T(Interest income), even though no cash was received. T, of course, would be the bondholder’spersonal tax rate. Because of the tax situation, most zero coupon bonds are bought by pension funds and other tax-exempt entities. Individuals do, however, buy taxable zeros for their Individual Retirement Accounts (IRAs). Also, state and local governments issue “tax-exempt muni zeros,” which are purchased by individuals in high tax brackets.

Note too that we have analyzed the bond as if the cash flows accrued annually. Generally, to facilitate comparisonswith semiannual payment coupon bonds, the analysis is conducted on a semiannual basis.

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handled by both Vandenberg and the bondholders exactly as the discount on thezeros was handled.

Thus, zero coupon bonds are just one type of original issue discount bond.Any nonconvertible bond whose coupon rate is set below the going market rate atthe time of its issue will sell at a discount, and it will be classified (for tax andother purposes) as an OID bond.

Corporate (and municipal) zeros are generally callable at the option of theissuer, just like coupon bonds, after some stated call protection period. The callprice is set at a premium over the accrued value at the time of the call. StrippedU.S. Treasury bonds (Treasury zeros) are not callable. Thus, Treasury zeros arecompletely protected against reinvestment risk (the risk of having to invest cashflows from a bond at a lower rate because of a decline in interest rates).

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web extension: bond risk, duration, and zero coupon bonds · web extension: bond risk, duration,...

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