Question

Imagine that a local water company issued \(10,000 ten-year bond at an interest rate of 6%. You are thinking about buying this bond one year before the end of the ten years, but interest rates are now 9%. a. Given the change in interest rates, would you expect to pay more or less than \)10,000 for the bond? b. Calculate what you would actually be willing to pay for this bond.

Short answer

Given the change in interest rates, you would expect to pay less than $10,000 for the bond. The actual price you would be willing to pay for this bond is $8965.55.

Step-by-step solution

Determine the cash flows from the bond

The bond was issued at \(10,000 and has a 6% interest rate with 10 years to maturity. This means that the annual coupon payment (interest payment) will be 6% of \)10,000. Calculate the annual coupon payment: Annual coupon payment = 0.06 * 10,000 = $600 Since we're buying the bond one year before the end of the ten years, there is just one year left for the bond. The cash flows for this bond will be: 1. An annual coupon payment of $600 at the end of year 9 2. The face value of the bond ($10,000) at the end of year 10

Calculate the present value of cash flows

To determine the value of the bond, we need to find the present value of the cash flows using the new interest rate of 9%. We'll use the following formula to calculate the present value of each cash flow: Present Value = Cash Flow / (1 + Interest Rate)^n where n is the number of years until the cash flow occurs. Let's calculate the present value of each cash flow: PV_Coupon_Payment = \(600 / (1 + 0.09)^1 = \)600 / 1.09 = $550.46 PV_Face_Value = \(10,000 / (1 + 0.09)^2 = \)10,000 / 1.1881 = $8415.09 Now, add the present values of each cash flow to find the total present value: Total Present Value = PV_Coupon_Payment + PV_Face_Value = \(550.46 + \)8415.09 = $8965.55

Determine whether we should expect to pay more or less than $10,000

Now that we've calculated the total present value of the bond with the new interest rate, we can compare it with the original issue price of \(10,000. The total present value is \)8965.55, which is less than \(10,000. Therefore, we should expect to pay less than \)10,000 for the bond given the change in interest rates.

Calculate the actual price we would be willing to pay

Since we've already calculated the total present value of the bond with the new interest rate of 9%, the actual price we would be willing to pay for this bond is: Price of the bond = Total Present Value = $8965.55 In conclusion, given the change in interest rates, we would expect to pay less than \(10,000 for the bond. The actual price we would be willing to pay for this bond is \)8965.55.

Key concepts

Interest Rates

Interest rates play a crucial role in bond valuation. They represent the cost of borrowing money. When interest rates rise, the present value of future bond payments falls. This is because new bonds might offer higher returns compared to older bonds with lower rates. In our example, the original bond had a 6% interest rate. However, current rates have risen to 9%. This makes the bond less attractive, meaning we'd pay less than its original value. Changes in interest rates directly affect bond prices. If rates increase, the bond's value typically decreases, and vice versa.

Present Value

The concept of present value is essential in determining how much a bond is worth today. It refers to the current worth of a future sum of money. Essentially, it helps us understand how much future cash flows are worth now, given a specific interest rate. To calculate the present value, we use this formula: \[ \text{Present Value} = \frac{\text{Cash Flow}}{(1 + \text{Interest Rate})^n} \]

• "Cash Flow" is the amount you'll receive in the future (like annual coupon payments or the face value).
• "Interest Rate" is the rate used to discount the future cash flows.
• "n" is the number of years until the cash flow occurs.

In the bond example, we calculated the present value of a \(600 coupon payment and the \)10,000 face value to determine the bond's worth.

Coupon Payment

Coupon payments are regular interest payments made to bondholders. They are calculated based on the bond's coupon rate. In this scenario, the bond had a 6% coupon rate, meaning it paid $600 annually (6% of $10,000). These payments are crucial for investors as they provide a steady income stream. For buyers looking to purchase a bond close to its maturity, it's vital to consider the remaining coupon payments, which are used to assess the bond's current value. Thus, understanding the bond's coupon structure helps calculate potential returns and evaluate whether it's a worthwhile investment.