Question

Suppose Ford Motor Company issues a five year bond with a face value of \(5,000 that pays an annual coupon payment of \)150. a. What is the interest rate Ford is paying on the borrowed funds? b. Suppose the market interest rate rises from 3% to 4% a year after Ford issues the bonds. Will the value of the bond increase or decrease?

Short answer

a. The interest rate Ford is paying on the borrowed funds is 3%. b. The value of the bond decreases by $286.20 when the market interest rate rises from 3% to 4%.

Step-by-step solution

a. Calculate the bond's coupon rate

The coupon rate is the annual coupon payment divided by the face value of the bond. In this case, the annual coupon payment is \(150, and the face value is \)5,000. So, the coupon rate is: Coupon rate = \( \frac{Annual \thinspace Coupon \thinspace Payment}{Face \thinspace Value} \) Calculate the coupon rate: Coupon rate = \( \frac{150}{5000} \) = 0.03 or 3% Therefore, the interest rate Ford is paying on the borrowed funds is 3%.

b. Calculate the bond's initial price

Now, let's calculate the bond's price with a market interest rate of 3%. We will use the formula below to calculate the present value of the bond, which is its price: Bond price = \( \sum_{i=1}^{n} \frac{C}{(1 + r)^{i}} \) + \( \frac{FV}{(1 + r)^{n}} \) where: C = annual coupon payment ($150) r = market interest rate (0.03) n = number of years to maturity (5) FV = face value of the bond ($5,000) Plug in the values and calculate the bond's initial price: Bond price = \(\frac{150}{(1 + 0.03)^1} + \frac{150}{(1 + 0.03)^2} + \frac{150}{(1 + 0.03)^3} + \frac{150}{(1 + 0.03)^4} + \frac{150}{(1 + 0.03)^5} + \frac{5000}{(1 + 0.03)^5} \) ≈ $5,000

b. Calculate the bond's price after the market interest rate rises to 4%

Now, let's recalculate the bond's price with a market interest rate of 4%: r = 0.04 Bond price = \(\frac{150}{(1 + 0.04)^1} + \frac{150}{(1 + 0.04)^2} + \frac{150}{(1 + 0.04)^3} + \frac{150}{(1 + 0.04)^4} + \frac{150}{(1 + 0.04)^5} + \frac{5000}{(1 + 0.04)^5} \) ≈ $4,713.80

b. Determine the change in bond value

Now, let's compare the bond's initial price (\(5,000) and its price after the market interest rate rises to 4% (\)4,713.80): Change in bond value = Initial price - New price Change in bond value = \(5,000 - \)4,713.80 = $286.20 Since the bond's value has decreased by $286.20 when the market interest rate rises from 3% to 4%, the value of the bond decreases when the market interest rate rises.

Key concepts

Coupon Rate

A bond's coupon rate is an essential concept in bond valuation. It's the annual interest payment made by the bond issuer to the bondholder, expressed as a percentage of the bond's face value. In simpler terms, the coupon rate tells us what percentage of the face value the investor will receive each year as interest.

For instance, if a bond has a face value of \(5,000 and an annual coupon payment of \)150, the coupon rate is calculated as follows:

• Coupon Rate = \( \frac{150}{5000} = 0.03 \) or 3%

This means that Ford is paying 3% annually on the money they borrowed through this bond.

The coupon rate is fixed when the bond is issued and does not change over the life of the bond. However, it is crucial in determining the bond's attractiveness to potential investors.

Market Interest Rate

The market interest rate is the prevailing rate of return on investments in the market at the time. It plays a crucial role in bond pricing because it influences how a bond's yield compares to the yields available on new issues.

When the market interest rate changes, it affects the value of existing bonds. If the market interest rate increases, existing bonds with lower coupon rates become less attractive. This is because they offer lower returns compared to new bonds issued at the higher prevailing rates, often resulting in a price drop for these bonds.

In the exercise example, initially, the market interest rate was 3%. When it rose to 4%, the bonds, which initially seemed more valuable due to their competitive coupon rate, lost value. As calculated, the bond's price dropped from approximately $5,000 to $4,713.80. The bondholders now effectively get less return, diminishing the bond's value.

Present Value

Present value is a financial concept used to determine the current worth of a future sum of money or stream of cash flows, given a specific interest rate. In the context of bonds, it helps to establish the bond's price today based on future coupon payments and the face value that will be received at maturity.

The formula for calculating the present value of a bond is:

• Bond price = \[ \sum_{i=1}^{n} \frac{C}{(1 + r)^{i}} + \frac{FV}{(1 + r)^{n}} \]
• where \( C \) is the annual coupon payment
• \( r \) is the market interest rate
• \( n \) is the number of years to maturity
• \( FV \) is the face value of the bond

Calculating the present value involves summing the present values of each individual coupon payment and the present value of the face value of the bond that the investor will receive at the end of its term.

Understanding present value is crucial for investors when deciding whether to buy or sell a bond. It helps them assess if a bond is overvalued or undervalued compared to its potential returns.