Problem 14 - Applied Calculus

Question

(a) What is the present value of a \(\$ 1000\) bond which pays \(\$ 50\) a year for 10 years, starting one year from now? Assume the interest rate is \(5 \%\) per year, compounded annually. (b) Since \(\$ 50\) is \(5 \%\) of \(\$ 1000\), this bond is called a \(5 \%\) bond. What does your answer to part (a) tell you about the relationship between the principal and the present value of this bond if the interest rate is \(5 \%\) ? (c) If the interest rate is more than \(5 \%\) per year, compounded annually, which is larger: the principal or the present value of the bond? Why is the bond then described as trading at a discount? (d) If the interest rate is less than \(5 \%\) per year, compounded annually, why is the bond described as trading at a premium?

Step-by-Step Solution

Answer

(a) Present value is $1000. (b) At 5%, present value equals principal. (c) Present value < principal at >5%, bond trades at discount. (d) <5%, bond trades at premium due to more attractive coupons.

Step 1: Understand the Bond Components

This bond pays a fixed coupon of $50 each year for 10 years with a face value of $1000. The coupon payment is the interest payment received by the bondholder. The interest rate used for calculations, compounded annually, is 5% per year.

Step 2: Calculate Present Value of Annuity (Coupon Payments)

To find the present value of the $50 annual coupon payments, we use the present value of an annuity formula: \( PV = C \times \left(1 - (1 + r)^{-n}\right) / r \), where \( C = 50 \), \( r = 0.05 \), and \( n = 10 \). Calculate: \[ PV = 50 \times \left(1 - (1 + 0.05)^{-10}\right) / 0.05 \approx 386.09 \].

Step 3: Calculate Present Value of Face Value

The present value of the face value (paid at maturity) is calculated using \( PV = F / (1 + r)^n \), where \( F = 1000 \), \( r = 0.05 \), and \( n = 10 \). Calculate: \[ PV = \frac{1000}{(1 + 0.05)^{10}} \approx 613.91 \].

Step 4: Add Present Values to Find Total Present Value

Add the present value of the annuity and the present value of the face value: \( 386.09 + 613.91 = 1000 \). This is the total present value of the bond when the interest rate matches the bond's coupon rate.

Step 5: Understand the Principal and Present Value Relationship

In part (b), the principal (face value) and present value are equal due to the interest rate equalling the coupon rate. This indicates the bond is priced at par value.

Step 6: Analyze Higher Interest Rates

For part (c), if the interest rate increases above 5%, the present value becomes less than the principal, causing the bond to trade at a discount. This happens because the fixed coupons are less attractive compared to new bonds issued at higher rates.

Step 7: Consider Lower Interest Rates

For part (d), if the interest rate is less than 5%, the present value exceeds the principal, making the bond trade at a premium. This occurs because the bond's fixed coupons are more attractive relative to the market rate.

Key Concepts

Interest Rates

Interest rates are crucial in determining how much money your investments will make over time. They are expressed as a percentage and represent the cost of borrowing money. Regarding bonds, the interest rate affects how much you earn from the bond's interest payments or coupons. The stated interest rate on a bond is its coupon rate. However, the market interest rate is what influences the bond's present value.
If the market interest rate is equal to the bond's coupon rate, as in our example with a 5% interest rate, the bond is said to be trading at par. This means the bond's present value is the same as its principal value.

• If the market interest rate increases above the bond's coupon rate, the bond's present value falls below its principal, leading it to trade at a discount.
• If the market interest rate is less than the bond's coupon rate, the bond’s present value exceeds its principal, and it trades at a premium.

Hence, understanding interest rates can help you determine whether a bond is a good investment.

Compounded Annually

Compounding annually refers to the process where the interest earned over a year is added to the principal amount, allowing you to earn interest on the new total in the following year. When we say interest is compounded annually, it means that the computation of interest happens once every year.
The formula for compounding looks like this:
\[ A = P imes (1 + r)^n \] Where:

• \( A \) is the amount after n years,
• \( P \) is the principal amount,
• \( r \) is the annual interest rate (expressed as a decimal),
• \( n \) is the number of years.

In our exercise, this annual compounding affects both the coupon payments and the bond's face value, influencing their present values. Understanding annual compounding is essential for evaluating long-term investments effectively.

Annuity Formula

The annuity formula is used to calculate the present value of a series of payments made over time, such as the periodic coupon payments of a bond. This calculation allows you to find out how much these future payments are worth in today's dollars.
The annuity formula is: \[ PV = C \times \left(1 - (1 + r)^{-n}\right) / r \] In this equation:

• \( PV \) is the present value,
• \( C \) is the annual coupon payment,
• \( r \) is the interest rate per period (as a decimal),
• \( n \) is the total number of payments.

Using this formula, you can determine how much the coupon payments you receive over time are worth now. For investors, understanding the annuity formula helps in comparing the value of different bonds or investment options.

Bond Pricing

Bond pricing is about determining a bond's present value based on future cash flows, such as coupon payments and the face value at maturity. The bond's price reflects the current worth of receiving those future payments.
To calculate a bond price, understanding the present value of both the annual coupon payments (using the annuity formula) and the present value of the bond's face value is crucial. Once these two values are calculated, they are added together to determine the bond's total present value.
Here's what affects bond pricing:

• Interest rates - A higher market rate than the bond's coupon rate results in a lower present value, indicating a discount.
• Time to maturity - Longer maturity generally means more interest compounding, affecting the present value.
• Compounding frequency - While our example uses annual compounding, other bonds might use different compounding periods, impacting their present value calculations.

Mastering bond pricing helps investors make informed decisions about buying or selling bonds, ensuring they understand a bond's value in different market conditions.