“Analyzing Bond Prices and Yields in a Changing Yield Curve Environment: A Comparative Study of 5% and 8% Coupon Bonds”

QUESTION

Consider the zero-yield curve reported in the following table. Consider two bonds, both with 5 years to maturity, but with different coupon rates. Let the two coupon rates be 5% and 8%. (a) Compute the prices and the yields to maturity of these coupon bonds. (b) How do the yields to maturity compare to each other? If they are different, why are they different? Would the difference in yields imply that one is a better ‘buy’ than the other? (c) What happens to the prices of these bonds if (a) the whole yield curve shifts uniformly upward by 50 basis points, (b) the yields for the maturities between 6 and 7.5 move downward by 25 basis point (while the rest of the curve remains unchanged)?

Zero-yield curve (semi-annually compounded)

Maturity Yield(% p.a.)
0.25  6.33
0.50  6.49
0.75  6.62
1.00  6.71
1.25  6.79
1.50  6.84
1.75  6.87
2.00  6.88
2.25  6.89
2.50  6.88
2.75  6.86
3.00  6.83
3.25  6.80
3.50  6.76
3.75  6.72
4.00  6.67
4.25  6.62
4.50  6.57
4.75  6.51
5.00  6.45
5.25  6.39
5.50  6.31
5.75  6.24
6.00  6.15
6.25  6.05
6.50  5.94
6.75  5.81
7.00  5.67
7.25  5.50
7.50  5.31

ANSWER

In this essay, we will analyze two bonds with different coupon rates (5% and 8%) both with a 5-year maturity in the context of the provided zero-yield curve. We will compute their prices and yields to maturity, compare these yields, and discuss the implications of changes in the yield curve on the bond prices.

(a) Computation of Prices and Yields to Maturity

To calculate the prices of the two bonds, we can use the present value formula for bonds:

Bond Price = (C * [1 – (1 + YTM)^(-n)]) / YTM + (FV / (1 + YTM)^n)

Where:

C represents the annual coupon payment,

YTM is the yield to maturity,

n is the number of periods until maturity, and

FV is the face value of the bond.

For both bonds, we have:

Coupon rate (C1) = 5%

Coupon rate (C2) = 8%

Maturity (n) = 5 years

We also need the relevant yield to maturity (YTM) from the zero-yield curve for a 5-year maturity, which is 6.45%.

For the 5% coupon bond:

C1 = 5% of face value

YTM = 6.45%

n = 5 years

FV = 100 (assuming a face value of $100)

Plugging these values into the bond pricing formula:

Bond Price (5% Coupon) = (5 * [1 – (1 + 0.0645)^(-5)]) / 0.0645 + (100 / (1 + 0.0645)^5)

Bond Price (5% Coupon) ≈ $97.18

For the 8% coupon bond:

C2 = 8% of face value

YTM = 6.45%

n = 5 years

FV = 100 (assuming a face value of $100)

Plugging these values into the bond pricing formula:

Bond Price (8% Coupon) = (8 * [1 – (1 + 0.0645)^(-5)]) / 0.0645 + (100 / (1 + 0.0645)^5)

Bond Price (8% Coupon) ≈ $104.57

(b) Comparison of Yields to Maturity:

The yield to maturity (YTM) for the 5% coupon bond is 6.45%, which matches the YTM from the zero-yield curve for a 5-year maturity. However, the YTM for the 8% coupon bond is also 6.45%, despite having a higher coupon rate. This might seem counterintuitive, but it’s because the price of the 8% coupon bond is higher than its face value, which lowers its effective yield.

Comparing the two YTMs, we can see that they are the same despite the different coupon rates. This is because the higher coupon payment on the 8% bond is offset by its higher price. Investors receive more income from the coupon, but they also pay more upfront for the bond.

The difference in coupon rates doesn’t necessarily imply that one bond is a better ‘buy’ than the other. It depends on the investor’s objectives and risk tolerance. The 8% coupon bond provides higher income but also carries a higher initial cost, while the 5% coupon bond offers lower income but comes with a lower purchase price. Investors seeking more immediate income might prefer the 8% bond, while those looking for capital appreciation might favor the 5% bond.

(c) Impact of Yield Curve Shifts:

(i) If the whole yield curve shifts uniformly upward by 50 basis points, both bonds’ prices will decrease because their coupon rates are fixed. When interest rates rise, bond prices fall. However, the 8% coupon bond’s price will likely decrease more than the 5% coupon bond’s price because it has a higher duration due to its higher coupon rate.

(ii) If the yields for maturities between 6 and 7.5 years move downward by 25 basis points while the rest of the curve remains unchanged, both bond prices are likely to increase. Lower interest rates make existing bonds with higher coupon rates more attractive, leading to price appreciation. However, the 8% coupon bond will likely experience a smaller price increase than the 5% coupon bond because its coupon rate is already higher.

In summary, the comparison of yields and the impact of yield curve changes on bond prices demonstrate the complex relationship between coupon rates, bond prices, and interest rate movements. Investors should consider their investment goals and market conditions when choosing between bonds with different coupon rates.