Risk and Return Problems and Questions

BOND VALUATION LG7-4

Present Value of Bond Cash Flows

Any bond’s value computation directly applies time value of money concepts. Bondholders know the interest payments that they are scheduled to receive and the repayment of the par value at maturity. The current price of a bond is, therefore, the present value of these future cash flows discounted at the prevailing market interest rate. The prevailing market interest rate will depend on the bond’s term to maturity, credit quality, and tax status.

The simplest type of bond for time value of money calculations is a zero coupon bond. As you might guess from its name, a zero coupon bond makes no interest payments. Instead, the bond pays only the par value payment at its maturity date. So a zero coupon bond sells at a substantial discount from its par value. For example, a bond with a par value of $1,000, maturing in 20 years, and priced to yield 6 percent, might be purchased for about $306.56. At the end of 20 years, the bond investor will receive $1,000. The difference between $1,000 and $306.56 (which is $693.44) represents the interest income received over the 20 years based upon the discount rate of 6 percent. The time line for this zero coupon bond valuation appears as:

discount bond A bond selling for lower than its par value.

zero coupon bond A bond that does not make interest payments but generally sells at a deep discount and then pays the par value at the maturity date.

Page 170We compute the zero’s price by finding the present value of the $1,000 cash flow received in 20 years. However, to be consistent with regular coupon-paying bonds, zero coupon bonds are priced using semiannual compounding. So the formula and calculator valuation would use 40 semiannual periods at a 3 percent interest rate rather than 20 periods at 6 percent. Using the present value equation of Chapter 4 results in

So the zero coupon bond’s price is indeed a steep discount to its par value. This makes sense because investors would only buy a security that pays $1,000 in many years for a price that is much lower to make enough profit to make up for the for-gone semiannual interest payments. For comparison’s sake, instead of the 20-year zero, consider a 20-year bond with a 7 percent coupon. So this 20-year maturity bond pays $35 in interest payments every six months. We can think of these interest payments as an annuity stream. If the market discount rate is 6 percent annually, the time line appears as

The time line shows the 40 semiannual payments (with the accompanying semiannual interest rate at 3 percent) of $35 and the par value payment at the bond’s maturity. Think through this: When bonds pay semiannual payments, the discount rate must be a semiannual rate. Thus, the 6 percent annual rate becomes a 3 percent semiannual rate. So we then compute the price of this bond by adding the present value of the interest payment annuity cash flow to the present value of the future par value. A combination of the present value equations for the annuity cash flows and the value of the par redemption appear in the bond valuation equation 7-1:

where PMT is the interest payment, N is the number of periods until maturity, and i is the market interest rate per period on securities with the same bond characteristics. If this bond paid interest annually, then these variables would take yearly period values. Since this bond pays semiannually, PMT, N, and i are all denoted in semiannual periods. The price of this coupon bond should be:

Of the $1,115.57 bond price, most of the value comes from the semiannual $35 coupon payments ($809.017) and not the value from the future par value payment ($306.557).

EXAMPLE 7-4 Find the Value of a Bond LG7-4

Consider a 15-year bond that has a 5.5 percent coupon, paid semiannually. If the current market interest rate is 6.5 percent, and the bond is priced at $940, should you buy this bond?

SOLUTION:

Compute the value of the bond using equation 7-1. Use semiannual compounding (N = 2 × 15 = 30, I = 6.5 ÷ 2 = 3.25, and PMT = 0.055 × $1,000 ÷ 2 = $27.50) as:

So this bond’s value is $905.09, which is less than the $940 price. The bond is overvalued in the market and you should not buy it.

Similar to Problems 7-21, 7-22, 7-23, 7-24, self-test problem 1

Because equation 7-1 is quite complex, we usually compute bond prices using a financial calculator or computer program. An investor would compute the bond value using a financial calculator by entering N = 40, I = 3, PMT = 35, FV = 1000, and computing the present value (PV). The calculator solution is $1,115.57.1

BOND YIELDS LG7-6

Current Yield

Although we speak about “the prevailing interest rate,” bond relationships reflect many interest rates (also called yields). Some rates are difficult to calculate but accurately reflect the return the bond is offering. Others, like the current yield, are easy to compute but only approximate the bond’s true return. A bond’s current yield is defined as the bond’s annual coupon rate divided by the bond’s current market price. Current yield measures the rate of return a bondholder would earn annually from the coupon interest payments alone if the bond were purchased at a stated price. Current yield does not measure the total expected return because it does not account for any capital gains or losses that will occur from purchasing the bond at a discount or premium to par.

Yield to Maturity

Yield to maturity is a more meaningful equation for investors than the simple current yield calculation. The yield to maturity calculation tells bond investors the total rate of return that they might expect if the bond were bought at a particular price and held to maturity. While the yield to maturity calculation provides more information than the current yield calculation, it’s also more difficult to compute, because we must compute the bond’s cash flows’ internal rate of return. This calculation seeks to equate the bond’s current market price with the value of all anticipated future interest and par value payments. In other words, it is the discount rate that equates the present value of all future cash flows with the current price of the bond. To calculate yield to maturity, investors must solve for the interest rate, i, in equation 7-2, or solve for i in:

current yield Return from interest payments; computed as the annual interest payment divided by the current bond price.

yield to maturity The total return the bond offers if purchased at the current price and held to maturity.

Computing Current Yield and Yield to Maturity LG7-6

You have identified a 3.5 percent Treasury bond with four years left to maturity and a quoted price of 96:09. Calculate the bond’s current yield and yield to maturity.

SOLUTION:

(1) First, identify that the bond’s price is $962.81 (= 96 09/32% × $1,000 = 0.9628125 × $1,000).

(2) The annual $35 in interest payments is paid in two $17.50 semiannual payments. Therefore, the current yield of the bond is 3.64 percent (= $35 ÷ $962.81).

(3) The yield to maturity is computed using equation 7-2 and the financial calculator as N = 8, PV = − 962.81, PMT = 17.50, and FV = 1,000. Computing the interest rate (I) results in 2.263 percent and multiplying by 2 gives the yield to maturity of 4.53 percent.

(4) Note that the current yield is less than the yield to maturity because it does not account for the capital gain to be earned if held to maturity.

Similar to Problems 7–13, 7–14, 7–27, 7–28, self-test problem 2

In this case, N is the number of periods until the bond can be called and i is the prevailing market rate. The prevailing market interest rate will probably differ from the rate for a noncallable bond. The previous section demonstrated via the yield curve that bonds with different maturities have different yields. A bond that matures in 20 years but is likely to be called in five years will carry a yield appropriate for a 5-year bond.

Now, reconsider the 20-year bond with a 7 percent coupon that we discussed previously (see pp. 170–171). If the bond can be called in five years with a call price of $1,070, the appropriate discount rate happens to be 5.75 percent annually at that time (instead of the 6 percent in the original problem). This time line would be

The changes in this time line are only 10 semiannual payments of $35 (rather than 40 such semiannual payments), a 2.875 percent semiannual discount rate, and the call price payment of $1,070. The price of this callable bond would be:

In this example, the callable bond would be priced at $1,106.38, which is slightly lower than an identical bond that was not callable, priced at $1,115.57.

If a bond is likely to be called, then the yield to maturity calculation does not give investors a good estimate of their return. Bondholders can use instead a yield to call calculation, which differs from the yield to maturity only in that its calculation assumes that the investor will receive the par value and call premium at the earliest call date. For example, reconsider the 7 percent coupon bond (paid semiannually) with 8 years to maturity, which we examined previously (see p. 174). The current bond price is $1,130 (which is slightly lower than the yield to maturity bond price of $1,150). If the bond can be called in three years at a specific call price of the par value plus one annual coupon, then what is the yield to call? The yield to call is computed as N = 6, PV = − 1130, PMT = 35, and FV = 1070. The resulting interest rate (I) is 2.26 percent. The yield to call for this bond is thus 4.52 percent (= 2 × 2.26%).

Municipal Bonds and Yield

Municipal bonds (munis) seem to offer low yields to maturity compared to the return that corporate bonds and Treasury securities offer. Munis offer lower rates because the interest income they generate for investors is tax-exempt—at least at the federal level.2

Yield to call The total return that the bond offers if purchased at the current price and held until the bond is called.

Settlement is the bond’s settlement date. This is the purchase date of the bond; typically it is today. Maturity is the bond’s maturity date. Rate is the bond’s annual coupon rate. Pr is the bond’s price per $100 face value. Note that the par value of a bond is typically $1,000, so an adjustment is needed for this input. Redemption is the bond’s redemption value per $100 face value. Frequency is the number of coupon payments per year. For semiannual, frequency = 2. Basis is the type of day count basis to use.

Consider the bond valuation problem of Example 7-4. The spreadsheet solution is the same as the TVM calculator solution and the pricing equation.

Also consider the yield to maturity problem in Example 7-6. This spreadsheet solves for the yield to maturity.

Log in to your Connect course to watch instructional videos on using spreadsheets. Also note that the solutions for all the examples in the book are illustrated using spreadsheets in videos that are also available in Connect.

EXAMPLE 7-7 Which Bond Has a Better After-Tax Yield? LG7-6

Imagine a time when you have a high income, placing you in the 31 percent marginal tax bracket. You are interested in investing some money in a bond issue and have three alternatives. The first is a corporate bond with a 6.4 percent yield to maturity. The second bond is a Treasury that offers a 5.7 percent yield. The third choice is a municipal bond priced at a yield to maturity of 4.0 percent. Which bond gives you the highest after-tax yield?

SOLUTION:

The Treasury and corporate bonds are both taxable, so we can compare them directly with each other. The yield of 6.4 percent on the corporate is clearly higher than the 5.7 percent yield offered by the Treasury bond. To include a comparison with the nontaxable municipal bond, compute its equivalent taxable yield as in equation 7-4:

The municipal bond’s equivalent taxable yield of 5.80 percent is higher than the Treasury but lower than the corporate bond.

Similar to Problems 7-15, 7-16, 7-31, 7-32, 7-37, 7-38, self-test problem 3

Specifically, income from municipal bonds is not subject to taxation by the federal government or the state government where the bonds are issued. As a result, municipal bond investors willingly accept lower yields than those they can obtain from taxable bonds. Generally speaking, investors compare the after-tax interest income earned on taxable bonds against the return earned on municipal bonds. For example, suppose an investor in the 35 percent marginal income tax bracket has $100,000 to invest in either corporate or municipal bonds. The $100,000 investment would earn a taxable $7,000 annually from 7 percent corporate bonds or $5,000 from tax-exempt 5 percent municipal bonds. After taxes, the corporate bond leaves the investor with $4,550 [=(1 − 0.35) × $7,000]. Obviously, this is less than the tax-free income of $5,000 generated by the muni bond.

A common way to compare yields from muni bonds versus those from taxable bonds is to convert the yield to maturity of the muni to a taxable equivalent yield, as shown in equation 7-4.

For high-income investors (in the 35 percent marginal tax bracket) a 5 percent muni bond has an equivalent taxable yield of 7.69 percent [= 0.05 ÷ (1 − 0.35)]. The 5 percent muni is more attractive for this investor than a 7 percent corporate bond. However, for an investor with lower income (in the 28 percent marginal tax bracket) the equivalent taxable yield is only 6.94 percent. The corporate bond provides more after-tax profit than the muni for this investor. It’s easy to see why muni bonds are popular among high-income investors (those with substantial marginal tax rates).

Summarizing Yields

In this section, we have presented several different types of interest rates, or yields, associated with bonds. See a summary in Table 7.4. Many of these yields relate to one another. Consider the bonds and associated yields reported in Table 7.5. Treasury bonds (1) to (3) show how coupon rates, current yield, and yield to maturity relate. When a bond trades at its par value (usually $1,000), then the coupon rate, current yield, and yield to maturity are all the same. When that bond is priced at a premium (bond 2), then both the current yield and the yield to maturity will be lower than the coupon rate. They are both higher than the coupon rate when the bond trades at a discount. Notice that yield to maturity is higher than current yield for discount bonds, and that yield to maturity is lower than current yield for premium bonds. In other words, the current yield always lies between the coupon rate and the yield to maturity. Both the current yield and the yield to maturity move in the opposite direction to the bond’s price.

Bonds (4) to (6) are callable corporate bonds. Recall that all the yields (current yield, yield to maturity, and yield to call) are identical when the bond trades at par value. When interest rates fall and bond prices increase, as with bond (5), the issuing corporation has a strong incentive to call the bond after five years, as allowed in the indenture agreement. So investors should base their purchase decisions on the yield to call. When interest rates increase, bond prices decline (as bond (6) shows). In this case, investors could compute the yield to call (as shown), but the information isn’t useful because the company will not likely call the bond while interest rates are high.

TABLE 7.4 Summary of Interest Rates

taxable equivalent yield Modification of a municipal bond’s yield to maturity used to compare muni bond yields to taxable bond yields.

Bonds issued for municipal projects tend to offer lower yields because their income is tax exempt.

The last three bonds shown in the table are municipal bonds. Recall that these bonds typically offer lower yields because the income from munis is tax exempt. It is easier to compare municipal bonds with Treasuries and corporate bonds if you compute the municipal bond’s taxable equivalent yield first. Here, we use a marginal tax rate of 35 percent in the calculation.

TABLE 7.5 Price, Coupon, and Yield Relationships of a 10-Year Bond

The last column of the table shows that the taxable equivalent yield of the municipal bonds is really quite competitive with corporate bond yields. Any investor with income taxed at the 35 percent marginal tax bracket would prefer the municipal bond over the corporate bond if the muni’s taxable equivalent yield is higher than the yield to maturity (or yield to call) of the corporate bond.

The table also shows that Treasury securities offer lower yields than corporate bonds with similar terms to maturity. The difference (or spread) between Treasury and corporate yields gives rise to a discussion of bond credit risk, which follows.