CHAPTER 7

INTEREST RATES AND BOND VALUATION

Answers to Concepts Review and Critical Thinking Questions

1. No. As interest rates fluctuate, the value of a government security will fluctuate. Long-term

government securities have substantial interest rate risk.

2. All else the same, the government security will have lower coupons because of its lower default risk,

so it will have greater interest rate risk.

3. No. If the bid were higher than the ask, the implication would be that a dealer was willing to sell a

bond and immediately buy it back at a higher price. How many such transactions would you like to

do?

4. Prices and yields move in opposite directions. Since the bid price must be lower, the bid yield must be

higher.

5. There are two benefits. First, the company can take advantage of interest rate declines by calling in an

issue and replacing it with a lower coupon issue. Second, a company might wish to eliminate a

covenant for some reason. Calling the issue does this. The cost to the company is a higher coupon. A

put provision is desirable from an investorâ€™s standpoint, so it helps the company by reducing the

coupon rate on the bond. The cost to the company is that it may have to buy back the bond at an

unattractive price.

6. Bond issuers look at outstanding bonds of similar maturity and risk. The yields on such bonds are

used to establish the coupon rate necessary for a particular issue to initially sell for par value. Bond

issuers also simply ask potential purchasers what coupon rate would be necessary to attract them. The

coupon rate is fixed and simply determines what the bondâ€™s coupon payments will be. The required

return is what investors actually demand on the issue, and it will fluctuate through time. The coupon

rate and required return are equal only if the bond sells for exactly par.

7. Yes. Some investors have obligations that are denominated in dollars; i.e., they are nominal. Their

primary concern is that an investment provide the needed nominal dollar amounts. Pension funds, for

example, often must plan for pension payments many years in the future. If those payments are fixed

in dollar terms, then it is the nominal return on an investment that is important.

8. Companies pay to have their bonds rated simply because unrated bonds can be difficult to sell; many

large investors are prohibited from investing in unrated issues.

9. Treasury bonds have no credit risk, so a rating is not necessary. Junk bonds often are not rated

because there would no point in an issuer paying a rating agency to assign its bonds a low rating (itâ€™s

like paying someone to kick you!).

10. The term structure is based on pure discount bonds. The yield curve is based on coupon-bearing

issues.

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Solutions to Questions and Problems

NOTE: All end of chapter problems were solved using a spreadsheet. Many problems require multiple

steps. Due to space and readability constraints, when these intermediate steps are included in this

solutions manual, rounding may appear to have occurred. However, the final answer for each problem is

found without rounding during any step in the problem.

Basic

1. The yield to maturity is the required rate of return on a bond expressed as a nominal annual interest

rate. For noncallable bonds, the yield to maturity and required rate of return are interchangeable

terms. Unlike YTM and required return, the coupon rate is not a return used as the interest rate in

bond cash flow valuation, but is a fixed percentage of par over the life of the bond used to set the

coupon payment amount. For the example given, the coupon rate on the bond is still 10 percent, and

the YTM is 8 percent.

2. Price and yield move in opposite directions; if interest rates rise, the price of the bond will fall. This is

because the fixed coupon payments determined by the fixed coupon rate are not as valuable when

interest rates riseâ€”hence, the price of the bond decreases.

NOTE: Most problems do not explicitly list a par value for bonds. Even though a bond can have any par

value, in general, corporate bonds in Canada will have a par value of $1,000. We will use this par value in

all problems unless a different par value is explicitly stated.

3. The price of any bond is the PV of the interest payment, plus the PV of the par value. Notice this

problem assumes an annual coupon. The price of the bond will be:

P = $80({1 â€“ [1/(1 + .06)]10 } / .06) + $1,000[1 / (1 + .06)10] = $1,147.20

We would like to introduce shorthand notation here. Rather than write (or type, as the case may be)

the entire equation for the PV of a lump sum, or the PVA equation, it is common to abbreviate the

equations as:

PVIFR,t = 1 / (1 + r)t

which stands for Present Value Interest Factor

PVIFAR,t = ({1 â€“ [1/(1 + r)]t } / r )

which stands for Present Value Interest Factor of an Annuity

These abbreviations are short hand notation for the equations in which the interest rate and the

number of periods are substituted into the equation and solved. We will use this shorthand notation in

remainder of the solutions key.

4. Here we need to find the YTM of a bond. The equation for the bond price is:

P = $884.50 = $90(PVIFAR%,9) + $1,000(PVIFR%,9)

Notice the equation cannot be solved directly for R. Using a spreadsheet, a financial calculator, or trial

and error, we find:

R = YTM = 11.09%

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If you are using trial and error to find the YTM of the bond, you might be wondering how to pick an

interest rate to start the process. First, we know the YTM has to be higher than the coupon rate since

the bond is a discount bond. That still leaves a lot of interest rates to check. One way to get a starting

point is to use the following equation, which will give you an approximation of the YTM:

Approximate YTM = [Annual interest payment + (Price difference from par / Years to maturity)] /

[(Price + Par value) / 2]

Solving for this problem, we get:

Approximate YTM = [$90 + ($115.50 / 9] / [($884.50 + 1,000) / 2] = 10.91%

This is not the exact YTM, but it is close, and it will give you a place to start.

5. Here we need to find the coupon rate of the bond. All we need to do is to set up the bond pricing

equation and solve for the coupon payment as follows:

P = $870 = C(PVIFA6.8%,16) + $1,000(PVIF6.8%,16)

Solving for the coupon payment, we get:

C = $54.42

The coupon payment is the coupon rate times par value. Using this relationship, we get:

Coupon rate = $54.42 / $1,000 = 5.44%

6. To find the price of this bond, we need to realize that the maturity of the bond is 10 years. The bond

was issued one year ago, with 11 years to maturity, so there are 10 years left on the bond. Also, the

coupons are semiannual, so we need to use the semiannual interest rate and the number of semiannual

periods. The price of the bond is:

P = $41.00(PVIFA3.7%,20) + $1,000(PVIF3.7%,20) = $1,055.83

7. Here we are finding the YTM of a semiannual coupon bond. The bond price equation is:

P = $970 = $43(PVIFAR%,20) + $1,000(PVIFR%,20)

Since we cannot solve the equation directly for R, using a spreadsheet, a financial calculator, or trial

and error, we find:

R = 4.531%

Since the coupon payments are semiannual, this is the semiannual interest rate. The YTM is the APR

of the bond, so:

YTM = 2

Ã—

4.531% = 9.06%

8. Here we need to find the coupon rate of the bond. All we need to do is to set up the bond pricing

equation and solve for the coupon payment as follows:

P = $1,145 = C(PVIFA3.75%,29) + $1,000(PVIF3.75%,29)

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