For a discount bond that sells below its face value, how does the yield to maturity compare with the coupon rate and the current yield?

Introduction / Context:
Bond valuation concepts such as coupon rate, current yield and yield to maturity are important in finance and investment examinations. A discount bond is one that sells for less than its face (par) value. Understanding the relationship among the three yields for discount and premium bonds helps investors assess return. This question asks you to identify how yield to maturity behaves relative to the coupon rate and the current yield for a discount bond.

Given Data / Assumptions:

• The bond is a discount bond, so its market price is below its face value.
• The bond pays a fixed coupon rate on face value every period.
• Current yield is defined as annual coupon payment divided by current market price.
• Yield to maturity (YTM) is the internal rate of return on the bond if held to maturity, assuming coupons are reinvested at the same rate.

Concept / Approach:
For a discount bond, the investor receives periodic coupon payments and also a capital gain because the bond will repay face value, which is higher than the price paid. The coupon rate is calculated on face value. Current yield considers only coupon income relative to current price, so for a discount bond it is higher than the coupon rate. The yield to maturity accounts for both coupon income and capital gain over the remaining life of the bond, so for a discount bond it is highest among the three. Therefore, YTM is greater than both the coupon rate and the current yield.

Step-by-Step Solution:
Step 1: Note that the bond sells below par, so there will be a capital gain when it matures at face value. Step 2: Recognize that the coupon rate is fixed as annual coupon divided by face value. Step 3: Understand that current yield equals annual coupon divided by market price; for a discount bond, this is higher than the coupon rate because price is lower than face value. Step 4: Recall that yield to maturity includes both coupon income and capital gain, so it is higher than current yield for a discount bond. Step 5: Conclude that option C, stating that YTM is greater than both the coupon rate and current yield, is correct.

Verification / Alternative check:
You can verify with a simple numerical example. Suppose a bond has a face value of 1000, a coupon rate of 8 percent, so the annual coupon is 80, and it currently sells for 900. The coupon rate is 8 percent. The current yield is 80 divided by 900, which is about 8.89 percent, already above the coupon rate. If the bond matures at 1000 after a few years, the investor also gains 100 in capital value, which when spread over the remaining years increases the overall annualized return above 8.89 percent. That total return is the yield to maturity, which is therefore greater than both 8 percent and 8.89 percent.

Why Other Options Are Wrong:
Option A is wrong because YTM cannot be equal to both coupon rate and current yield for a discount bond; equality holds only when a bond sells at par. Option B is wrong because it claims YTM is less than current yield for a discount bond, which ignores the capital gain element. Option D is wrong because it claims YTM equals current yield but only exceeds the coupon rate; this relationship is not correct for a discount bond, since YTM must also exceed current yield.

Common Pitfalls:
Students often confuse current yield with yield to maturity, treating them as identical. Another common error is to remember the relationship incorrectly, perhaps thinking that current yield is always the highest. A helpful rule is to link the bond price relative to par with the ranking of yields: for a discount bond, price is below par and yields rank as coupon rate less than current yield less than yield to maturity; for a premium bond, the order reverses. Remembering this simple pattern can prevent mistakes in exam questions.

Final Answer:
For a discount bond, the yield to maturity is greater than both the coupon rate and the current yield.