To calculate the price of a bond, we need to discount its future cash flows (coupon payments and face value) back to the present using the bond’s yield to maturity as the discount rate. The formula for the present value of a bond is:

BondPrice=∑t=1N​(1+r)tC​+(1+r)NF​

Where:

• $C$ = Annual coupon payment
• $F$ = Face value of the bond
• $r$ = Yield to maturity (YTM)
• $N$ = Number of years to maturity
• $t$ = Time period

Let’s break down the calculations.

Given Information:

• Risk-free rate (10-year US Treasury bond yield): 1.20%
• Corporate bond yield premium: 3.00%
• Face Value (F): $1,000
• Fixed Coupon Rate: 5.00%
• Maturity (N): 10 years
• Coupon Frequency: Annually

1. Determine the Corporate Bond’s Yield to Maturity (YTM)

The corporate bond’s yield is the risk-free rate plus the yield premium: Corporate Bond YTM (r) = Risk-free rate + Yield premium Corporate Bond YTM (r) = 1.20% + 3.00% = 4.20% = 0.042

2. Calculate the Annual Coupon Payment (C)

Coupon Payment (C) = Face Value * Coupon Rate C = $1,000 * 5.00% = $50

A. What is the current price of the corporate bond?

Using the corporate bond’s YTM of 4.20% and the annual coupon of $50:

We need to calculate the present value of 10 annual coupon payments of $50 and the present value of the $1,000 face value received in 10 years.

BondPrice=(1+0.042)150​+(1+0.042)250​+…+(1+0.042)1050​+(1+0.042)101000​

Using a financial calculator or spreadsheet:

• N = 10
• I/Y = 4.20
• PMT = 50
• FV = 1000

Current Price of the Corporate Bond = $1,064.24

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

If the yield increases by 1.00%, the new YTM will be: New YTM (r) = Original YTM + 1.00% = 4.20% + 1.00% = 5.20% = 0.052

Now, calculate the bond price with the new YTM:

• N = 10
• I/Y = 5.20
• PMT = 50
• FV = 1000

Price of the Bond if Yield Increased by 1.00% = $984.77

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

If the yield decreased by 1.00%, the new YTM will be: New YTM (r) = Original YTM – 1.00% = 4.20% – 1.00% = 3.20% = 0.032

Now, calculate the bond price with the new YTM:

• N = 10
• I/Y = 3.20
• PMT = 50
• FV = 1000

Price of the Bond if Yield Decreased by 1.00% = $1,151.78

D. Discuss the risk associated with this change in interest rates.

The changes in bond prices calculated above illustrate interest rate risk, which is the risk that a bond’s price will fall as interest rates rise. Conversely, if interest rates fall, the bond’s price will increase.

Here’s a breakdown of the observations and associated risk:

1. Inverse Relationship: As demonstrated by the calculations, there is an inverse relationship between bond prices and interest rates (yields). When interest rates (yields) increased from 4.20% to 5.20%, the bond’s price decreased from $1,064.24 to $984.77. When interest rates (yields) decreased from 4.20% to 3.20%, the bond’s price increased from $1,064.24 to $1,151.78.

2. Impact on Bondholders:

   • Rising Interest Rates: If an investor holds this corporate bond and market interest rates rise (meaning newly issued bonds offer higher yields), their existing bond becomes less attractive because its fixed coupon rate is lower than what new bonds offer. To sell their bond in the secondary market, the investor would have to lower its price, resulting in a capital loss if they sell before maturity.
   • Falling Interest Rates: Conversely, if interest rates fall, the investor’s existing bond with its higher fixed coupon becomes more attractive, and its market price will rise. This benefits the investor if they choose to sell the bond, realizing a capital gain.

3. Duration: The sensitivity of a bond’s price to changes in interest rates is measured by its duration. Bonds with longer maturities and lower coupon rates generally have higher durations, meaning they are more sensitive to interest rate changes. Our 10-year bond is relatively long-term, which contributes to its price sensitivity.

4. Reinvestment Risk (Less Relevant in this specific example but good to note): While interest rate risk relates to the bond’s price, another related risk is reinvestment risk. If an investor holds a bond to maturity and interest rates have fallen, they might face difficulty reinvesting their principal and coupon payments at the same high yield as their original bond. This is more relevant for investors who rely on a consistent income stream from bond investments.

In summary, the risk associated with these changes in interest rates is primarily interest rate risk, which can lead to capital gains or losses for bond investors who may need to sell their bonds before maturity. The longer the maturity of the bond, the greater this risk tends to be.