Bond Price Sensitivity

Bond Price Sensitivity

The yield of the 10-year US Treasury bond is 1.20%. It is the risk-free rate. You work

for investment manager and your boss asks you to calculate the price of a 10-year

corporate bond that yields 3.00% more than its risk-free rate and has a face value of

$1,000. The fixed coupon of this corporate bond is 5.00%. Both bonds pay coupons

annually.

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Bond Price Sensitivity

• What is the current price of the corporate bond?

• Calculate the price of the bond if its yield increased by 1.00%.

• Calculate the price of the bond if its yield decreased by 1.00%.

• Please discuss the risk associated with this change in interest rates?

Given Information:

• Risk-free rate (10-year Treasury yield): 1.20%

• Corporate bond yield spread: 3.00%

• Corporate bond yield = 1.20% + 3.00% = 4.20%

• Coupon rate: 5.00%

• Face value: $1,000

• Time to maturity: 10 years

• Coupons paid annually

 1. Price of the Corporate Bond (Yield = 4.20%)

The bond price is the present value of coupon payments + present value of face value:

Price=∑t=11050(1+0.042)t+1000(1+0.042)10\text{Price} = \sum_{t=1}^{10} \frac{50}{(1 + 0.042)^t} + \frac{1000}{(1 + 0.042)^{10}}Price=t=1∑10​(1+0.042)t50​+(1+0.042)101000​

Let’s compute:

Coupon = $50 annually (5% of $1,000)

Using the present value of annuity formula:

PVcoupons=50×(1−1(1.042)10)÷0.042≈50×8.1109=405.54PV_{\text{coupons}} = 50 \times \left(1 – \frac{1}{(1.042)^{10}}\right) \div 0.042 \approx 50 \times 8.1109 = 405.54PVcoupons​=50×(1−(1.042)101​)÷0.042≈50×8.1109=405.54 PVface=1000(1.042)10≈1000÷1.519=657.23PV_{\text{face}} = \frac{1000}{(1.042)^{10}} \approx 1000 \div 1.519 = 657.23PVface​=(1.042)101000​≈1000÷1.519=657.23 Total Price=405.54+657.23=1062.77\text{Total Price} = 405.54 + 657.23 = \boxed{1062.77}Total Price=405.54+657.23=1062.77​

 2. Price if Yield Increases to 5.20%

New yield = 4.20% + 1.00% = 5.20%

PVcoupons=50×(1−1(1.052)10)÷0.052≈50×7.3807=369.04PV_{\text{coupons}} = 50 \times \left(1 – \frac{1}{(1.052)^{10}}\right) \div 0.052 \approx 50 \times 7.3807 = 369.04PVcoupons​=50×(1−(1.052)101​)÷0.052≈50×7.3807=369.04 PVface=1000(1.052)10≈1000÷1.658=603.12PV_{\text{face}} = \frac{1000}{(1.052)^{10}} \approx 1000 \div 1.658 = 603.12PVface​=(1.052)101000​≈1000÷1.658=603.12 Total Price=369.04+603.12=972.16\text{Total Price} = 369.04 + 603.12 = \boxed{972.16}Total Price=369.04+603.12=972.16

3. Price if Yield Decreases to 3.20%

New yield = 4.20% – 1.00% = 3.20%

PVcoupons=50×(1−1(1.032)10)÷0.032≈50×8.5302=426.51PV_{\text{coupons}} = 50 \times \left(1 – \frac{1}{(1.032)^{10}}\right) \div 0.032 \approx 50 \times 8.5302 = 426.51PVcoupons​=50×(1−(1.032)101​)÷0.032≈50×8.5302=426.51 PVface=1000(1.032)10≈1000÷1.374=727.56PV_{\text{face}} = \frac{1000}{(1.032)^{10}} \approx 1000 \div 1.374 = 727.56PVface​=(1.032)101000​≈1000÷1.374=727.56 Total Price=426.51+727.56=1154.07\text{Total Price} = 426.51 + 727.56 = \boxed{1154.07}Total Price=426.51+727.56=1154.07​

 4. Discussion: Interest Rate Risk

This analysis shows bond prices are inversely related to interest rates:

• When yields rise, bond prices fall.

• When yields drop, bond prices rise.

This is known as interest rate risk. Longer-term bonds are more sensitive to interest rate changes. Also,Investors holding bonds face potential losses if rates increase after purchase. On the other hand, they may enjoy capital gains if rates fall.

The magnitude of this risk depends on:

• Duration of the bond (longer = higher risk)

• Coupon rate (lower coupons = higher price sensitivity)

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