Introduction & Learning Objectives
This guide is your interactive companion to understanding **Interest Rates and Bond Valuation**, based on **Chapter 7 of Ross's textbook**. It applies the time value of money concepts to a bond's cash flows to determine its value and explores the key features that affect its yield.
Learning Objectives (Ross, page 201)
- Define important bond features and types of bonds.
- Explain bond values and yields and why they fluctuate.
- Describe bond ratings and what they mean.
- Outline the impact of inflation on interest rates.
- Illustrate the term structure of interest rates and the determinants of bond yields.
1. Bonds and Bond Valuation
Bond Features & Valuation (Ross, page 202)
A **bond** is an interest-only loan. Its value is the present value of its future cash flows, which consist of a stream of **coupon payments** and a final **face value** (or par value) payment at **maturity**.
Where $C$ is the coupon payment, $F$ is the face value, $r$ is the yield to maturity (YTM) per period, and $t$ is the number of periods to maturity.
Key Relationships:
- Bond prices and interest rates move in **opposite directions**.
- If YTM = Coupon Rate, the bond sells at **par value**.
- If YTM > Coupon Rate, the bond sells at a **discount**.
- If YTM < Coupon Rate, the bond sells at a **premium**.
Interest Rate Risk (Ross, page 206)
**Interest rate risk** is the risk to bondholders from fluctuating interest rates. This risk is greater for bonds with a **longer time to maturity** and a **lower coupon rate**.
Exam Hint:
Be able to explain these relationships and provide examples. The concept of YTM and its relationship to price was tested in **Question 1(iv) from the May 2025 exam**.
2. Bond Features and Ratings
Bond Indenture and Security (Ross, page 214)
The **indenture** is the legal document that details the bond's terms, including its **security** (collateral or mortgage), **seniority**, and repayment arrangements (**sinking fund** or **call provision**).
Exam Hint: You may be asked to explain these terms, as seen in **Question 6(a) from the January 2024 exam**.
**Protective covenants** are part of the indenture that limit a company's actions. **Negative covenants** prohibit certain actions (e.g., paying dividends above a certain level), while **positive covenants** require certain actions (e.g., maintaining a specific level of working capital).
Bond Ratings (Ross, page 217)
Bond ratings from agencies like Moody's and S&P assess a bond's **creditworthiness** or its likelihood of default. **Investment-grade bonds** are rated BBB or higher, while those below are considered **junk bonds**.
Exam Hint: The rating only addresses default risk, not interest rate risk. This is a common point of confusion.
3. Inflation and Interest Rates
Real vs. Nominal Rates (Ross, page 230)
The **nominal rate** ($R$) is the percentage change in the number of dollars you have. The **real rate** ($r$) is the percentage change in your buying power, after adjusting for inflation ($h$). The relationship is given by the **Fisher effect**:
Exam Hint: Be able to use this formula to find the missing rate. You must be consistent in your calculations: use nominal cash flows with nominal rates, and real cash flows with real rates.
Determinants of Bond Yields (Ross, page 233)
A bond's yield is composed of six components:
- Real Rate of Interest
- Inflation Premium (compensation for future inflation)
- Interest Rate Risk Premium (compensation for interest rate risk)
- Default Risk Premium (compensation for credit risk)
- Taxability Premium (compensation for unfavorable tax treatment)
- Liquidity Premium (compensation for a lack of liquidity)
4. Problems & Solutions
Problem Statement: A Microgates Industries bond has a 10 percent coupon rate and a $1,000 face value. Interest is paid semiannually, and the bond has 20 years to maturity. If investors require a 12 percent yield, what is the bond's value? What is the effective annual yield on the bond?
Solution:
1. Bond Value:
This is a semiannual bond, so we adjust the coupon rate, YTM, and maturity.
$C = \$100 / 2 = \$50$. $r = 12\% / 2 = 6\%$. $t = 20 \times 2 = 40$.
$\text{Bond Value} = \$50 \times \frac{1 - 1/(1.06)^{40}}{0.06} + \frac{\$1,000}{(1.06)^{40}} = \$50 \times 15.0463 + \$97.22 = \$849.54$
2. Effective Annual Yield:
$EAR = (1 + \frac{\text{Quoted Rate}}{m})^m - 1 = (1 + \frac{0.12}{2})^2 - 1 = (1.06)^2 - 1 = 0.1236$ or 12.36%
Problem Statement: Westco Co. issued 15-year bonds a year ago at a coupon rate of 5.4 percent. The bonds make semiannual payments and have a par value of $1,000. If the YTM on these bonds is 4.5 percent, what is the current price of the bond in dollars?
Solution:
The bond was issued a year ago with a 15-year maturity, so it now has **14 years to maturity**. We adjust for semiannual periods.
Coupon Payment ($C$) = $54 / 2 = \$27$
YTM per period ($r$) = $4.5\% / 2 = 2.25\%$
Number of periods ($t$) = $14 \times 2 = 28$
Face Value ($F$) = $1,000
Bond Price = $27 \times \frac{1 - 1/(1.0225)^{28}}{0.0225} + \frac{1,000}{(1.0225)^{28}} = 27 \times 20.897 + 531.05 = 564.22 + 531.05 = \$1,095.27
Problem Statement: Bond P is a premium bond making semiannual payments. The bond pays a coupon rate of 6.8 percent, has a YTM of 6.2 percent, and has 13 years to maturity. Bond Y is a discount bond making semiannual payments. This bond pays a coupon rate of 6.2 percent, has a YTM of 6.8 percent, and also has 13 years to maturity. The bonds have a par value of $1,000. What is the price of each bond today? If interest rates remain unchanged, what do you expect the price of these bonds to be 1 year from now? In 3 years? In 8 years? In 12 years? In 13 years? What's going on here? Illustrate your answers by graphing bond prices versus time to maturity.
Solution:
Bond Prices Today:
Bond P: $C = 34$, $r = 3.1\%$, $t = 26$. Price = $34 \times PVIFA(3.1\%, 26) + 1000/(1.031)^{26} = 34 \times 17.65 + 448.24 = \$1,048.34$
Bond Y: $C = 31$, $r = 3.4\%$, $t = 26$. Price = $31 \times PVIFA(3.4\%, 26) + 1000/(1.034)^{26} = 31 \times 17.39 + 413.41 = \$951.48$
Prices over time: If interest rates remain unchanged, the price of the premium bond (P) will gradually decrease toward par value. The price of the discount bond (Y) will gradually increase toward par value. At maturity (13 years from now), both will sell for exactly $1,000.
Problem Statement: Bond P is a premium bond with a coupon rate of 9 percent. Bond D has a coupon rate of 5 percent and is currently selling at a discount. Both bonds make annual payments, have a par value of $1,000, a YTM of 7 percent, and 15 years to maturity. What is the current yield for Bond P? For Bond D? If interest rates remain unchanged, what is the expected capital gains yield over the next year for Bond P? For Bond D? Explain your answers and the interrelationships among the various types of yields.
Solution:
1. Bond Prices Today:
Bond P: Coupon = $90, YTM = 7%, t = 15. Price = $90 \times PVIFA(7\%, 15) + 1000/(1.07)^{15} = $90 \times 9.108 + 362.45 = $819.72 + 362.45 = $1,182.17
Bond D: Coupon = $50, YTM = 7%, t = 15. Price = $50 \times PVIFA(7\%, 15) + 362.45 = $455.40 + 362.45 = $817.85
2. Current Yields:
Current Yield P = $90 / 1182.17 = 7.61\%
Current Yield D = $50 / 817.85 = 6.11\%
3. Capital Gains Yield:
Total Return = Current Yield + Capital Gains Yield. Since YTM is 7% for both, this is the total return.
Capital Gains Yield P = 7% - 7.61% = -0.61\%
Capital Gains Yield D = 7% - 6.11% = +0.89\%
Explanation:
The premium bond (P) has a coupon rate greater than its YTM, so it must have a capital loss over time as its price falls to par. The discount bond (D) has a coupon rate less than its YTM, so it must have a capital gain over time as its price rises to par.
4. Minicase: S&S Air's Expansion Plans
S&S Air plans to issue $35 million in new 10-year bonds. As a financial analyst, you must prepare a memo to advise on the effect of various bond features on the coupon rate, as well as the advantages and disadvantages of each.
Textbook Link: Ross, Chapter 7, Minicase (Page 246)
Solution:
Effect on Coupon Rate: A secured bond (with collateral) has lower default risk, so it will have a **lower** coupon rate than an unsecured bond (debenture).
Advantages: Lower borrowing cost for the company. Less risk for the bondholder.
Disadvantages: The company must pledge specific assets, limiting its future financial flexibility.
Solution:
Effect on Coupon Rate: A **sinking fund** reduces the default risk because it provides for the orderly repayment of the bond before maturity. This will lead to a **lower** coupon rate.
Advantages: Reduces default risk, making the bond more attractive to investors and lowering the firm's borrowing cost. It also helps manage liquidity by smoothing out the repayment process.
Disadvantages: The firm must make cash payments to the fund, even if it is experiencing financial difficulty.
Solution:
Effect on Coupon Rate: A **call provision** is beneficial to the issuer but detrimental to the bondholder, who may have their bond redeemed when interest rates fall. To compensate for this risk, the bond must offer a **higher** coupon rate.
Advantages: The company can refinance its debt at a lower rate if interest rates fall, reducing its borrowing cost.
Disadvantages: Increases the cost of the bond (higher coupon rate) and may make it less attractive to investors.
Solution:
Effect on Coupon Rate: A **floating-rate coupon** adjusts to an interest rate index, which reduces the interest rate risk for the bondholder. This will typically result in a **lower** average coupon rate compared to a fixed-rate bond.
Advantages: Reduces interest rate risk for the investor. It can be attractive to firms that want to match their debt payments with variable-rate revenues.
Disadvantages: The firm's interest expense will fluctuate with market rates, making it difficult to predict future cash flows.