Understanding Interest Rates, Bond Pricing, and Yields

Posted on Mar 9, 2025 in International Business Economics

Interest Rate Calculations and Bond Valuation

Question 8: Nominal Interest Rate Calculation

Assume you just deposited $1,000 into a bank account. The current real interest rate is 2%, and inflation is expected to be 6% over the next year. What nominal interest rate would you require from the bank over the next year? How much money will you have at the end of one year? If you are saving to buy a stereo that currently sells for $1,050, will you have enough to buy it?

Solution: The required nominal rate would be:

At this rate, you would expect to have $1,000 * 1.08 = $1,080 at the end of the year. In theory, the price of the stereo will increase with the rate of inflation. So, one year later, the stereo will cost $1,050 * 1.06 = $1,113. You will be short by $33.

Question 9: Rate of Return on a Bond

A 10-year, 7% coupon bond with a face value of $1,000 is currently selling for $871.65. Compute your rate of return if you sell the bond next year for $880.10.

Solution:

Question 10: Bond Price for a Desired Return

You have paid $980.30 for an 8% coupon bond with a face value of $1,000 that matures in five years. You plan on holding the bond for one year. If you want to earn a 9% rate of return on this investment, what price must you sell the bond for? Is this realistic?

Solution: To find the price, solve:

Although this appears possible, the yield to maturity when you purchased the bond was 8.5%. At that yield, you only expect the price to be $983.62 next year. In fact, the yield would have to drop to 8.35% for the price to be $988.53.

Question 13: Portfolio Duration

The duration of a $100 million portfolio is 10 years. $40 million in new securities are added to the portfolio, increasing the duration of the portfolio to 12.5 years. What is the duration of the $40 million in new securities?

Solution: First, note that the portfolio now has $140 million in it. The duration of a portfolio is the weighted average duration of its individual securities. Let D equal the duration of the $40 million in new securities. Then, this implies:

The new securities have a duration of 18.75 years.

Chapter 4: Bond Pricing and Risk

Question 2: Junk Bond Valuation

Consider a $1,000-par junk bond paying a 12% annual coupon. The issuing company has a 20% chance of defaulting this year; in which case, the bond would not pay anything. If the company survives the first year, paying the annual coupon payment, it then has a 25% chance of defaulting in the second year. If the company defaults in the second year, neither the final coupon payment nor the par value of the bond will be paid. What price must investors pay for this bond to expect a 10% yield to maturity? At that price, what is the expected holding period return? Standard deviation of returns? Assume that periodic cash flows are reinvested at 10%.

Solution: The expected cash flow at t₁ = 0.20 * (0) + 0.80 * (120) = 96

The expected cash flow at t₂ = 0.25 * (0) + 0.75 * (1,120) = 840

The price today should be:

Question 3: Loanable Funds Framework

Last month, corporations supplied $250 billion in bonds to investors at an average market rate of 11.8%. This month, an additional $25 billion in bonds became available, and market rates increased to 12.2%. Assuming a Loanable Funds Framework for interest rates, and that the demand curve remained constant, derive a linear equation for the demand for bonds, using prices instead of interest rates.

Solution: First, translate the interest rates into prices.

We know two points on the demand curve:

So, the slope =

Question 4: Equilibrium Price and Quantity of Bonds

An economist has estimated that, near the point of equilibrium, the demand curve and supply curve for bonds can be estimated using the following equations:

a. What is the expected equilibrium price and quantity of bonds in this market?

b. Given your answer to part (a), what is the expected interest rate in this market?

Solution:

a. Solve the equations simultaneously:

This implies that P = 814.2857.

b.

Chapter 5: Term Structure and Yields

Question 2: T-Bill Rates and Bond Yields

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

Solution: Your required interest rate on a 4-year bond = Average interest on four 1-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.

Question 3: Municipal vs. Corporate Bond Yields

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%.

Question 5: Indifference Tax Rate

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

Question 10: Inflation Expectations

Little Monsters Inc. borrowed $1,000,000 for two years from Northern Bank 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 a 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% = R_f + 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

Question 11: Interest Rate on a 10-Year Bond

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?

Solution:

Question 12: Liquidity Premium Calculation

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

Question 13: Implied One-Year Rate

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%

Question 15: Predicting One-Year Interest Rate

If the interest rates on one- to five-year bonds are currently 4%, 5%, 6%, 7%, and 8%, and the term premiums for one- to five-year bonds are 0%, 0.25%, 0.35%, 0.40%, and 0.50%, predict what the one-year interest rate will be two years from now.

Solution: The expected one-year interest rate two years from now is

= [(1 + i_3t – l_3t)³/(1 + i_2t – l_2t)²] – 1

= [(1 + 0.06 – 0.0035)³/(1 + 0.05 – 0.0025)²] – 1

= 0.075 = 7.5%