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Chapter 7: Problem 1

Callaghan Motors' bonds have 10 years remaining to maturity. Interest is paid annually; they have a \(\$ 1,000\) par value; the coupon interest rate is 8 percent; and the yield to maturity is 9 percent. What is the bond's current market price?

Short Answer

Expert verified
The bond's market price is approximately $938.23.

Step by step solution

01

Identify the Bond Parameters

First, identify and note the parameters given in the exercise. The par value of the bond is \(\$1,000\), the coupon interest rate is 8 percent, the yield to maturity (YTM) is 9 percent, and the bond has 10 years remaining until maturity.
02

Calculate the Annual Coupon Payment

The annual coupon payment is calculated by multiplying the coupon interest rate by the par value of the bond. \[\text{Coupon Payment} = 0.08 \times 1000 = \$80\]
03

Set Up the Present Value Formula

The current market price of the bond can be calculated by summing the present values of all future cash flows, which include the annual coupon payments and the lump sum of the par value at maturity. The formula is:\[P = \sum_{t=1}^{10} \frac{C}{(1 + YTM)^t} + \frac{F}{(1 + YTM)^{10}}\]Where:- \(P\) is the price of the bond,- \(C\) is the annual coupon payment \(\$80\),- \(F\) is the face value \(1000\),- \(YTM\) is the yield to maturity \(0.09\).
04

Calculate Present Values of Coupon Payments

Calculate the present value of each of the 10 annual coupon payments using the YTM:\[PV(C) = \sum_{t=1}^{10} \frac{80}{(1.09)^t}\]This can be summed up directly to a present value of approximately \(\$515.82\).
05

Calculate Present Value of Par Value

Calculate the present value of the \(\\(1,000\) par value received at the end of 10 years:\[PV(F) = \frac{1000}{(1.09)^{10}} \approx \\)422.41\]
06

Sum the Present Values

Add the present value of the coupon payments and the present value of the par value to find the bond's current market price:\[P = 515.82 + 422.41 = \$938.23\]
07

Conclusion

The bond's current market price based on the given yield to maturity is approximately \(\$938.23\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Present Value
The concept of present value is pivotal in bond valuation and financial management. It allows us to understand what a future sum of money is worth today. In the context of bonds, investors are interested in valuing the series of future cash flows (coupon payments) and the face value of the bond. To simplify, when we talk about present value, we're considering the time value of money:
  • It is based on the premise that a dollar today is worth more than a dollar in the future, due to its earning potential.
  • This is calculated by discounting the future cash flows back to the present using a specific interest rate, in this example, the yield to maturity (YTM).
For Callaghan Motors' bonds, we calculate the present value of each coupon payment and the par value at maturity. The present value formula helps in determining how much an investor should pay today for all the future cash flows.
Yield to Maturity
Yield to maturity (YTM) is a critical concept for anyone dealing with bond investments. It essentially measures the return an investor can expect if they hold the bond until it matures. Think of it as the internal rate of return on bond investments. YTM takes into account several factors:
  • The bond's current market price.
  • The par value.
  • The coupon interest rate.
  • The total number of years remaining to maturity.
In our exercise, the YTM is given as 9%, which means if an investor buys the bond at its current market price and holds it for 10 years, they can expect an annual return of approximately 9%. Calculating the market price using this YTM helps investors make informed decisions about the risks and returns of their investments.
Coupon Interest Rate
Another vital component in bond valuation is the coupon interest rate. This rate is the annual interest payment made to the bondholders during the lifespan of the bond. It gives an idea of the bond's periodic income for the investor. In simpler terms:
  • The coupon rate is expressed as a percentage of the bond's par value.
  • It determines the fixed coupon payment made to the investor annually.
For Callaghan Motors' bonds, the coupon rate is 8%, resulting in an $80 annual payment (8% of $1,000 par value). This regular payment is an attractive feature for many investors seeking steady income. The difference between the coupon rate and the current YTM can also indicate whether a bond is trading at a premium or a discount.
Financial Management
Financial management involves effective planning, organizing, directing, and controlling financial activities. In the context of bonds, understanding bond valuation, yield to maturity, and coupon interest rates helps in making informed financial decisions. Here are some key points:
  • Effective financial management ensures that an organization can create value and returns, maximize wealth, and achieve financial health.
  • For investors, understanding and managing bonds is crucial to achieving investment objectives, be it for income, growth, or diversification.
In financial management, bond valuation tools, such as calculating present value and understanding yield to maturity, help managers and investors assess the bonds' fair market value and make strategic investment decisions.

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Most popular questions from this chapter

Lloyd Corporation's 14 percent coupon rate, semiannual payment, \(\$ 1,000\) par value bonds, which mature in 30 years, are callable 5 years from today at \(\$ 1,050 .\) They sell at a price of \(\$ 1,353.54,\) and the yield curve is flat. Assume interest rates are expected to remain at their current level. a. What is the best estimate of these bonds' remaining life? b. If Lloyd plans to raise additional capital and wants to use debt financing, what coupon rate would it have to set in order to issue new bonds at par?

Nungesser Corporation's outstanding bonds have a \(\$ 1,000\) par value, a 9 percent semiannual coupon, 8 years to maturity, and an 8.5 percent YTM. What is the bond's price?

Heymann Company bonds have 4 years left to maturity. Interest is paid annually, and the bonds have a \(\$ 1,000\) par value and a coupon rate of 9 percent. a. What is the yield to maturity at a current market price of (1) \(\$ 829\) or (2) \(\$ 1,104 ?\) b. Would you pay \(\$ 829\) for each bond if you thought that a "fair" market interest rate for such bonds was 12 percent-that is, if \(\mathrm{r}_{\mathrm{d}}=12\) percent? Explain your answer.

A semiannual coupon bond that matures in 7 years sells for \(\$ 1,020\). It has a face value of \(\$ 1,000\) and a yield to maturity of 10.5883 percent. What is its current yield?

Six years ago, the singleton Company issued 20 -year bonds with a 14 percent annual coupon rate at their \(\$ 1,000\) par value. The bonds had a 9 percent call premium, with 5 years of call protection. Today, singleton called the bonds. Compute the realized rate of return for an investor who purchased the bonds when they were issued and held them until they were called. Explain why the investor should or should not be happy that singleton called them.
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