**Quantitative Problems Chapter 5**

**1.** (a) The yield to maturity would be 5% for a one-year bond, 6% for a two-year bond, 6.33% for a three-year bond, 6.5% for a four-year bond, and 6.6% for a five-year bond.

(b) The yield to maturity would be 5% for a one-year bond, 4.5% for a two-year bond, 4.33% for a three-year bond, 4.25% for a four-year bond, and 4.2% for a five-year bond. The upward-sloping yield curve in (a) would be even steeper if people preferred short-term bonds over long-term bonds because long-term bonds would then have a positive risk premium. The downward-sloping yield curve in (b) would be less steep and might even have a slight positive upward slope if the long-term bonds have a positive risk premium.

**2.** Government economists have forecasted one-year T-bill rates for the following five years, as follows:

Year 1-year rate

1 4.25%

2 5.15%

3 5.50%

4 6.25%

5 7.10%

You have liquidity premium 0.25% for the next two years and 0.50% thereafter. Would you be willing to purchase a four-year T-bond at a 5.75% interest rate?

**Solution:** Your required interest rate on a 4-year bond Average interest on 4 one-year bonds Liquidity Premium

(4.25% 5.15% 5.50% 6.25%)/4 0.5%

5.29% 0.50% 5.79%

At a rate of 5.75%, the T-bond is just below your required rate.

**3.** What is the yield on a $1,000,000 municipal bond with a coupon rate of 8%, paying interest annually, versus the yield of a $1,000,000 corporate bond with a coupon rate of 10% paying interest annually? Assume that you are in the 25% tax bracket.

**Solution:** Municipal bond coupon payments equal $80,000 per year. No taxes are deducted; therefore, the yield would equal 8%.

The coupon payments on a corporate bond equal $100,000 per year. But you only

keep $75,000 because you are in the 25% tax bracket. Therefore your after-tax yield

is only 7.5%

**4.** Consider the decision to purchase either a 5-year corporate bond or a 5-year municipal bond. The corporate bond is a 12% annual coupon bond with a par value of $1,000. It is currently yielding 11.5%. The municipal bond has an 8.5% annual coupon and a par value of $1,000. It is currently yielding 7%. Which of the two bonds would be more beneficial to you? Assume that your marginal tax rate is 35%.

**Solution:** Municipal

Purchase Price $1,061.50

After-tax Coupon Payment $85

Par Value $1,000

Calculated YTM 7%

Corporate Bond

Purchase Price $1,018.25

After-tax Coupon Payment $78

Par Value $1,000

Calculated YTM 7.35%

The corporate bond offers a higher yield, and is the better buy.

**5.** Debt issued by Southeastern Corporation currently yields 12%. A municipal bond of equal risk currently yields 8%. At what marginal tax rate would an investor be indifferent between these two bonds?

**Solution:** Corporate Bonds (1 Tax Rate) Municipal Bonds

12% * (1 – Tax Rate) 8%

1 – Tax Rate 0.67, or Tax Rate 0.33

**6.** 1-year T-bill rates are expected to steadily increase by 150 basis points per year over the next 6 years. Determine the required interest rate on a 3-year T-bond and a 6-year T-bond if the current 1-year interest rate is 7.5%. Assume that the Pure Expectations Hypothesis for interest rates holds.

**Solution:** 3 year bond:

year 1 interest rate 7.5%

year 2 interest rate 9.0%

year 3 interest rate 10.5%

number of years 3

(7.5% 9.0% 10.5%)/3 9.0%

6 year bond:

year 1 interest rate 7.5%

year 2 interest rate 9.0%

year 3 interest rate 10.5%

year 4 interest rate 12%

year 5 interest rate 13.5%

year 6 interest rate 15%

(7.5% 9.0% 10.5% 12% 13.5% 15%)/6 11.25%

**7.** The one-year interest rate over the next 10 years will be 3%, 4.5%, 6%, 7.5%, 9%, 10.5%, 13%, 14.5%, 16%, 17.5%. Using the pure expectations theory, what will be the interest rates on a 3 year bond, 6 year bond, and 9 year bond?

**Solution:** 3 year bond [(3 4.5 6)]/(3) 4.5%

6 year bond [(3 4.5 6 7.5 9 10.5)]/(6) 6.75%

9 year bond [(3 4.5 6 7.5 9 10.5 13 14.5 16)]/(9) 9.333%

**8.** Using the information for the previous question, now assume that investor prefer holding short-term bonds. A liquidity premium of 10 basis points is required for each year of a bond’s maturity. What will be the interest rates on a 3 year bond, 6 year bond, and 9 year bond?

**Solution:** To solve this problem, you will need to use the following equation:

3 year bond (0.30) [(3 4.5 6)]/(3) 4.8%

6 year bond (0.60) [(3 4.5 6 7.5 9 10.5)]/(6) 7.35%

9 year bond (0.90) [(3 4.5 6 7.5 9 10.5 13 14.5 16)]/(9) 10.233%

**9.** Which bond would produce a greater return if the pure expectations theory was to hold true, a 2-year bond with an interest rate of 15% or two 1 year bonds with sequential interest payment of 13% and 17%?

**Solution:** Both of the bonds would produce the same return. Two 2 year bonds:

(13% 17%)/2

15%

**10.** Little Monsters Inc. borrowed $1,000,000 for two years from NorthernBank Inc. at an 11.5% interest rate. The current risk-free rate is 2% and Little Monsters’s financial condition warrants a default risk premium of 3% and a liquidity risk premium of 2%. The maturity risk premium for two-year loan is 1% and inflation is expected to be 3% next year. What does this information imply about the rate of inflation in the second year?

**Solution:** If inflation were expected to remain constant at 3% over the life of the loan, the interest rate on the two-year loan would be 11%. Since the actual two-year interest rate is 11.5%, the one-year interest rate in year 2 must be 12%, since 11.5 (11

12)/2.

The required rate of 12% Rf DRP LP MRP Inflation Premium.

2% 3% 2% 1% Inflation Premium.

So, the Inflation Premium in year 2 is 4%. But this is an average premium over two years. Inflation Premium 4% (Year 1 Inflation Year 2 Inflation)/2

(3% x)/2

Or, x 5%

**11.** One year T-bill rates are 2% currently. If interest rates are expected to go up, after 3 years, by 2% every year, what should be the required interest rate on a 10-year bond issued today?

**12.** One year T-bill rates over the next 4 years are expected to be 3%, 4%, 5%, & 5.5%. If 4-year T-bonds are yielding 4.5%, what is the liquidity premium on this bond?

**Solution:** 4.5% (3% 4% 5% 5.5%)/4 LP

4.5% 4.375% LP

0.125% LP

**13.** At your favorite bond store, Bonds-R-Us, you see the following prices:

(a) 1-year $100 zero selling for $90.19

(b) 3-year 10% coupon $1000 par bond selling for $1000

(c) 2-year 10% coupon $1000 par bond selling for $1000

Assume that the pure expectations theory for the term structure of interest rates holds, no liquidity or maturity premium exists, and the bonds are equally risky. What is the implied 1-year rate two years from now?

**Solution:** From (a), you know that the 1-year rate today is 10.877%.

Using this information, (c) tells you that:

1000 100/1.10877 1100/(1 2-year rate)2

So, the 2-year rate today is 9.95%.

Using these two rates, (b) tells you that:

1000 100/1.10877 100/1.09952 1100/(1 3-year rate)3

So, the 3-year rate today is 9.97%

1-year rate 2 years from now (3 9.97% – 2 9.95%) 10.01%

**14.** You observe the following market interest rates, for both borrowing and lending:

one-year rate 5%

two-year rate 6%

one-year rate one year from now 7.25%

How can you take advantage of these rates to earn a riskless profit? Assume that the Pure Expectation Theory for interest rates holds.

**Solution:** Borrow $100 today at the two-year rate. You will be required to payback $100 (1.06)2, or $112.36 in two years.

Simultaneously, lend the $100 you borrowed at the one-year rate of 5%. A year later, you will have $105. Again, lend the $105 at the one-year rate of 7.25%. At the end of that year, you will have $105 (1.0725) $112.6125.

Use the $112.6125 to payoff the $112.36 you owe. This leaves $0.2525 in profit. Recall that this required no cash upfront. As long as the future rate of 7.25% is guaranteed, you can risklessly make a profit.

**15.** The expected one-year interest rate two years from now is [(1 i3t k3t)3/(1 i2t k2t)2] 1

[(1 0.06 0.0035)3/(1 0.05 0.0025)2] 1 0.075 7.5%.