Assume that you are an aggressive bond trader and therefore, want to speculate on interest rate swing. Market interest rates are currently 9 percent, but you expect the interest rates to fall to 7…

Model Answer: Fundamentals of Investment (FIN253) – TU Board 2080

Given:

• Current market interest rate (r): 9%
• Expected interest rate after 1 year (r₁): 7%
• Two bonds to choose from:
  1. Zero-coupon bond (ZCB):
     • Maturity: 25 years
     • Par value: Rs. 1000
     • Coupon rate: 0%
  2. Coupon-paying bond:
     • Maturity: 20 years
     • Coupon rate: 5.5%
     • Par value: Rs. 1000
• Both bonds have the same credit rating and par value.

---

a. Maximizing Capital Gain Income (8 marks)

Objective: Determine which bond will provide the highest capital gain if interest rates fall from 9% to 7%.

Step 1: Calculate Current Price of Both Bonds (at 9%)

Since interest rates are expected to fall, the bonds will appreciate in value. We first find their current market prices using the YTM (Yield to Maturity) formula:

$$P=\frac{C}{(1+r{)}^1}+\frac{C}{(1+r{)}^2}+\cdots +\frac{C+F}{(1+r{)}^n}$$

For zero-coupon bonds, the formula simplifies to:

$$P_{ZCB}=\frac{F}{(1+r{)}^n}$$

For coupon-paying bonds, we use the full formula.

1. Zero-Coupon Bond (25-year, 0% coupon)

$$P_{ZCB}=\frac{1000}{(1+0.09{)}^{25}}=\frac{1000}{(1.09{)}^{25}}$$

Calculating $(1.09{)}^{25}$: $$(1.09{)}^{25}\approx 10.683$$

$$P_{ZCB}=\frac{1000}{10.683}\approx \text{Rs. 93.62}$$

2. Coupon-Paying Bond (20-year, 5.5% coupon)

Using the present value of an annuity + lump sum formula:

$$P_{Coupon}=\sum _{t=1}^{20}\frac{55}{(1.09{)}^t}+\frac{1000}{(1.09{)}^{20}}$$

First, calculate the present value of coupons (annuity): $$P{V}_{Coupons}=55\times \left(\frac{1-(1+0.09{)}^{-20}}{0.09}\right)$$

$$(1.09{)}^{-20}\approx 0.1653$$

$$P{V}_{Coupons}=55\times \left(\frac{1-0.1653}{0.09}\right)=55\times 9.277=\text{Rs. 510.24}$$

Next, calculate the present value of par value: $$P{V}_{Par}=\frac{1000}{(1.09{)}^{20}}=\frac{1000}{4.6609}\approx \text{Rs. 214.53}$$

$$P_{Coupon}=510.24+214.53=\text{Rs. 724.77}$$

Step 2: Calculate Future Price of Both Bonds (at 7%) After 1 Year

Since interest rates fall to 7%, we recalculate the bond prices after 1 year.

1. Zero-Coupon Bond (Remaining Maturity: 24 years)

$$P_{ZCB,future}=\frac{1000}{(1+0.07{)}^{24}}=\frac{1000}{(1.07{)}^{24}}$$

$$(1.07{)}^{24}\approx 4.6016$$

$$P_{ZCB,future}=\frac{1000}{4.6016}\approx \text{Rs. 217.31}$$

2. Coupon-Paying Bond (Remaining Maturity: 19 years)

$$P_{Coupon,future}=\sum _{t=1}^{19}\frac{55}{(1.07{)}^t}+\frac{1000}{(1.07{)}^{19}}$$

First, present value of coupons: $$P{V}_{Coupons,future}=55\times \left(\frac{1-(1.07{)}^{-19}}{0.07}\right)$$

$$(1.07{)}^{-19}\approx 0.2755$$

$$P{V}_{Coupons,future}=55\times \left(\frac{1-0.2755}{0.07}\right)=55\times 10.357=\text{Rs. 569.64}$$

Next, present value of par value: $$P{V}_{Par,future}=\frac{1000}{(1.07{)}^{19}}=\frac{1000}{3.6306}\approx \text{Rs. 275.42}$$

$$P_{Coupon,future}=569.64+275.42=\text{Rs. 845.06}$$

Step 3: Calculate Capital Gains for Both Bonds

Capital gain = Future Price - Current Price

1. Zero-Coupon Bond

$${\text{Capital Gain}}_{ZCB}=217.31-93.62=\text{Rs. 123.69}$$

2. Coupon-Paying Bond

$${\text{Capital Gain}}_{Coupon}=845.06-724.77=\text{Rs. 120.29}$$

Conclusion for Part (a):

The zero-coupon bond provides a higher capital gain (Rs. 123.69) compared to the coupon-paying bond (Rs. 120.29).

Answer:

To maximize capital gain income, the aggressive bond trader should select the 25-year zero-coupon bond.

---

b. Maximizing Total Return (4 marks)

Objective: Determine which bond provides the highest total return (capital gain + coupon income) over the investment horizon.

Step 1: Calculate Total Return for Zero-Coupon Bond

• No coupon payments (since it’s a zero-coupon bond).
• Total return = Capital gain only = Rs. 123.69

Step 2: Calculate Total Return for Coupon-Paying Bond

• Coupon income received in 1 year = Rs. 55 (since it pays annually).
• Capital gain = Rs. 120.29 (from part a).
• Total return = Coupon income + Capital gain = 55 + 120.29 = Rs. 175.29

Comparison:

Conclusion for Part (b):

The coupon-paying bond provides a higher total return (Rs. 175.29) due to the additional coupon income.

Answer:

To maximize total return, the aggressive bond trader should select the 20-year, 5.5% coupon-paying bond.

---

c. Why Did One Bond Provide Better Capital Gains Than the Other? (3 marks)

Key Reasons:

1. Duration & Price Sensitivity:

   • The zero-coupon bond has a longer duration (25 years) compared to the coupon-paying bond (20 years).
   • Longer-duration bonds are more sensitive to interest rate changes (higher price volatility).
   • When interest rates fall, the zero-coupon bond’s price increases more because its entire value is concentrated at maturity.

2. Coupon Payments & Reinvestment Risk:

   • The coupon-paying bond provides periodic cash flows (Rs. 55 annually), which reduces its price appreciation potential compared to a zero-coupon bond.
   • The zero-coupon bond’s price is purely based on the present value of its par value, making it more responsive to interest rate changes.

3. Time to Maturity & Reinvestment:

   • The zero-coupon bond has 25 years left, while the coupon bond has 20 years.
   • A longer time to maturity means greater price appreciation when rates fall.

Mathematical Explanation:

• Price of a zero-coupon bond is inversely proportional to $(1+r{)}^n$.
  • A small change in $r$ has a larger impact when $n$ is large.
• Coupon bonds have multiple cash flows, so their price changes are less extreme than zero-coupon bonds.

Answer:

The zero-coupon bond provided better capital gains because it has a longer duration (25 years vs. 20 years), making it more sensitive to interest rate declines. Since its entire value is concentrated at maturity, its price rises more sharply when rates fall compared to the coupon-paying bond, which has periodic cash flows that moderate its price appreciation.