Mastering Bond Valuation: Sample Exercises and Solutions

C. Sample Bond Valuation Exercises

In this section, you’ll find a series of sample exercises designed to test and enhance your understanding of bond valuation concepts. These exercises cover key topics such as bond pricing, yield calculations, duration, and convexity. Each exercise is followed by a detailed solution and explanation to help you grasp the underlying principles and techniques. Whether you’re preparing for an exam or seeking to deepen your knowledge, these exercises will provide you with the practical experience needed to master bond valuation.

Exercise 1: Bond Pricing Basics

Problem:
Consider a bond with a face value of $1,000, a coupon rate of 5% paid annually, and a maturity of 10 years. If the market interest rate is 4%, what is the price of the bond?

Solution:
To calculate the bond price, we need to find the present value of the bond’s future cash flows, which include the annual coupon payments and the face value at maturity.

• Annual Coupon Payment (C): $1,000 * 5% = $50
• Number of Periods (n): 10
• Market Interest Rate (r): 4% or 0.04

The bond price (P) is calculated using the formula:

Calculating each component:

• Present Value of Coupons:

  $$ = 50 \times \left(\frac{1 - (1 + 0.04)^{-10}}{0.04}\right) \approx 405.45 $$

• Present Value of Face Value:

  $$ = \frac{1,000}{(1 + 0.04)^{10}} \approx 675.56 $$

• Total Price:

  $$ = 405.45 + 675.56 = 1,081.01 $$

Explanation:
The bond is priced at $1,081.01 because the market interest rate is lower than the coupon rate, making the bond more attractive and increasing its price above par value.

Exercise 2: Yield to Maturity (YTM)

Problem:
A bond with a face value of $1,000 and a coupon rate of 6% paid annually is currently selling for $950. It has 5 years until maturity. What is the bond’s yield to maturity (YTM)?

Solution:
The YTM is the interest rate that equates the present value of the bond’s cash flows to its current price.

Using the formula:

This equation can be solved using trial and error, financial calculator, or spreadsheet software. The approximate YTM is found to be 7.16%.

Explanation:
The YTM of 7.16% reflects the bond’s current price below par, indicating a higher yield to compensate for the lower purchase price.

Exercise 3: Duration and Interest Rate Risk

Problem:
Calculate the Macaulay Duration of a 3-year bond with a face value of $1,000, a coupon rate of 5% paid annually, and a YTM of 4%.

Solution:
The Macaulay Duration (D) is the weighted average time to receive the bond’s cash flows, calculated as:

Where:

• \(C = 50\)
• \(F = 1,000\)
• \(YTM = 0.04\)
• \(n = 3\)

Calculating each component:

• Present Value of Cash Flows:

  $$ = \frac{50}{1.04} + \frac{50}{1.04^2} + \frac{50 + 1,000}{1.04^3} $$

• Total Present Value:

  $$ = 48.08 + 46.23 + 1,039.68 = 1,134.00 $$

• Duration:

  $$ D = \frac{1 \times 48.08 + 2 \times 46.23 + 3 \times 1,039.68}{1,134.00} \approx 2.89 \text{ years} $$

Explanation:
The Macaulay Duration of 2.89 years indicates the bond’s sensitivity to interest rate changes. It measures the weighted average time to receive cash flows, impacting how price changes with interest rate fluctuations.

Exercise 4: Convexity and Price Sensitivity

Problem:
A bond has a duration of 5 years and a convexity of 40. If interest rates decrease by 1%, estimate the percentage change in the bond’s price.

Solution:
The percentage change in the bond’s price can be estimated using the duration and convexity formula:

Where:

• \(D = 5\)
• \(C = 40\)
• \(\Delta y = -0.01\)

Calculating the change:

• Duration Effect:

  $$ = -5 \times (-0.01) = 0.05 \text{ or } 5\% $$

• Convexity Effect:

  $$ = \frac{1}{2} \times 40 \times (0.01)^2 = 0.002 \text{ or } 0.2\% $$

• Total Percentage Change:

  $$ = 5\% + 0.2\% = 5.2\% $$

Explanation:
The bond’s price is expected to increase by approximately 5.2% due to the decrease in interest rates, with convexity providing an additional price sensitivity measure beyond duration.

Exercise 5: Zero-Coupon Bond Valuation

Problem:
Calculate the price of a zero-coupon bond with a face value of $1,000, maturing in 8 years, if the market interest rate is 3%.

Solution:
The price of a zero-coupon bond is calculated as the present value of its face value:

Where:

• \(F = 1,000\)
• \(r = 0.03\)
• \(n = 8\)

Explanation:
The zero-coupon bond is priced at $789.41, reflecting the present value of the $1,000 face value discounted at the market interest rate over 8 years.

Exercise 6: Calculating Yield to Call

Problem:
A callable bond with a face value of $1,000, a coupon rate of 7% paid annually, and a call price of $1,050 is callable in 3 years. If the bond is currently selling for $1,020, what is the yield to call (YTC)?

Solution:
The YTC is calculated by equating the present value of the bond’s cash flows to its current price, assuming it is called at the earliest opportunity:

Solving this equation using trial and error, financial calculator, or spreadsheet software, the approximate YTC is found to be 6.47%.

Explanation:
The YTC of 6.47% considers the bond’s call feature, providing a yield based on the assumption that the issuer will call the bond at the earliest opportunity.

Exercise 7: Current Yield Calculation

Problem:
A bond with a face value of $1,000 and a coupon rate of 8% is currently selling for $1,100. Calculate the current yield.

Solution:
The current yield is calculated as:

Explanation:
The current yield of 7.27% reflects the bond’s coupon income relative to its current market price, providing an indication of income return excluding capital gains or losses.

Exercise 8: Bond Price Change with Interest Rate Shift

Problem:
A bond with a duration of 4 years is currently priced at $950. If interest rates increase by 0.5%, estimate the new price of the bond.

Solution:
The price change can be estimated using the duration formula:

Where:

• \(D = 4\)
• \(\Delta y = 0.005\)
• \(P = 950\)

Calculating the change:

• Price Change:

  $$ = -4 \times 0.005 \times 950 = -19 $$

• New Price:

  $$ = 950 - 19 = 931 $$

Explanation:
The bond’s price is expected to decrease to $931 due to the increase in interest rates, illustrating the inverse relationship between bond prices and interest rates.

Exercise 9: Calculating Effective Duration

Problem:
A bond with a face value of $1,000, a coupon rate of 6%, and a maturity of 5 years is priced at $980. If the bond’s price changes to $970 when interest rates increase by 0.1%, calculate the effective duration.

Solution:
Effective duration is calculated as:

Where:

• \(\Delta P = 970 - 980 = -10\)
• \(P = 980\)
• \(\Delta y = 0.001\)

Explanation:
The effective duration of -10.20 indicates the bond’s sensitivity to interest rate changes, providing a measure of price volatility for small changes in yield.

Exercise 10: Yield Spread Analysis

Problem:
Two bonds, Bond A and Bond B, have the same maturity and face value. Bond A has a yield of 5%, while Bond B has a yield of 7%. Calculate the yield spread between the two bonds.

Solution:
The yield spread is calculated as the difference in yields between the two bonds:

Explanation:
The yield spread of 2% reflects the difference in risk or credit quality between the two bonds, with Bond B offering a higher yield to compensate for perceived higher risk.

Bonds and Fixed Income Securities Quiz: C. Sample Bond Valuation Exercises

This comprehensive set of exercises and solutions will help you solidify your understanding of bond valuation concepts. By practicing these problems, you’ll gain confidence in your ability to analyze and value bonds, preparing you for success in both exams and real-world applications.