Question

Suppose a risk-free bond has a face value of \(\$ 100,000\) with a maturity date three years from now. The bond also gives coupon payments of \(\$ 5,000\) at the end of each of the next three years. What will this bond sell for if the annual interest rate for risk-free lending in the economy is a. 5 percent? b. 10 percent?

Short answer

The bond price will decrease when the interest rate increases from 5% to 10%. The exact prices can be calculated by adding up the present value of the coupon payments and the face value for each interest rate respectively.

Step-by-step solution

Understand the problem

The bond pays $5,000 each year for three years and then pays its face value of $100,000. These payments are discounted at the economy’s risk-free interest rate to find their present values, which sum to the price of the bond. In this case, the interest rate changes and you are asked to find the bond price in each case.

Calculate the present value of the coupon payments

The present value (PV) of each coupon payment can be expressed as \( PV = C / (1 + r)^n \), where \( C \) is the coupon payment, \( r \) is the interest rate, and \( n \) is the number of years. Calculating the present value of the coupon payments for each of the three years and adding them together, we get: For an interest rate of 5%: \( PV_{coupons} = 5000 / (1+0.05) + 5000 / (1+0.05)^2 + 5000 / (1+0.05)^3 \) For an interest rate of 10%: \( PV_{coupons} = 5000 / (1+0.10) + 5000 / (1+0.10)^2 + 5000 / (1+0.10)^3 \)

Calculate the present value of the face value

The present value of the face value is calculated similar to the coupon payments but does not repeat over the years. It can be calculated as \( PV = FV / (1 + r)^n \) where \( FV \) is the face value. We thus have: For an interest rate of 5%: \( PV_{face value} = 100000 / (1+0.05)^3 \) For an interest rate of 10%: \( PV_{face value} = 100000 / (1+0.10)^3 \)

Calculate the bond price

Finally, the bond price is the sum of the present values of the coupon payments and the face value. Therefore: For an interest rate of 5%: \( Bond Price = PV_{coupons} + PV_{face value} \) For an interest rate of 10%: \( Bond Price = PV_{coupons} + PV_{face value} \)

Key concepts

Present Value Calculation

Understanding the present value calculation is key to grasping bond pricing. This concept is based on the idea that a sum of money today is worth more than the same sum in the future due to its potential earning capacity. To calculate the present value (PV) of future cash flows, like those from a bond's coupon payments and its face value at maturity, we discount them back to their value today.

For example, if a bond pays annual coupons, we calculate the present value of each payment by dividing the coupon amount (\(C\)) by \(1 + r\) raised to the power of the period (\(n\)). This can look like \( PV = \frac{C}{(1 + r)^n} \) where \( r \) is the discount rate or interest rate, and \( n \) is the time in years until the payment will be made. These calculations apply to each individual future payment chronologically until the bond's maturity.

To understand it better, let's take the individual coupon payment PV calculation: If you are to receive a coupon payment of \(5,000\) in one year and the discount rate is 5%, the present value of this single payment is \( PV = \frac{5,000}{(1 + 0.05)^1} \). Using the same formula and discount rate, the present value of the same payment in two years would be \( \frac{5,000}{(1 + 0.05)^2} \) and so on for all future payments. Each present value is then added together to calculate the total present value of all future cash flows from the bond.

Coupon Payments

Coupon payments are regular interest payments that a bond issuer makes to the bondholder. Typically, these payments are a fixed amount or a percentage of the bond's face value and are paid at specified intervals until maturity. When calculating the price of a bond, each coupon must be considered individually as a future cash flow.

Let's break down the concept with an emphasis on the calculation. Suppose each year a bond pays a coupon of \(5,000\) to bondholders. These payments are, in essence, separate mini-loans the bondholder has given to the issuer and, consequently, must be valued individually using the present value formula discussed earlier. It's important to note that a bond's total value includes the sum of all these individual present values.

However, the coupon rate and the frequency of payments impact the bond's valuation. For a semi-annual coupon payment, the formula for present value needs to be adjusted to reflect the frequency by dividing the annual interest rate by two and doubling the number of periods. This adjustment reflects more frequent compounding and results in a different bond price as compared to annual payments.

Risk-Free Interest Rate

The risk-free interest rate plays a crucial role in financial calculations, including bond pricing. It's the return on an investment with no risk of financial loss, often associated with government bonds.

When used as a discount rate in present value calculations, the risk-free rate helps determine the value of future cash flows in today's dollars without considering credit risk. Looking back at our earlier example, the risk-free rate is the \( r \) in the present value formulas. It's a critical aspect because it's used to discount the bond's future cash flows to their present value.

If the risk-free rate increases, the present value of the bond's future cash flows decreases, leading to a lower bond price, and vice versa. In our exercise, we consider two different risk-free rates, 5% and 10%. Each interest rate will give a different bond price, showing the inverse relationship between interest rates and bond prices. Essentially, understanding the risk-free rate gives investors a benchmark for measuring investment performance and helps in assessing the value of future cash flows like those from bonds.