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Chapter 7: Problem 28

Valuing Bonds The Moulon Rouge Corporation has two different bonds currently outstanding. Bond M has a face value of \(\$ 20,000\) and matures in 20 years. The bond makes no payments for the first six years, then pays \(\$ 1,000\) every six months over the subsequent eight years, and finally pays \(\$ 1,750\) every six months over the last six years. Bond \(\mathrm{N}\) also has a face value of \(\$ 20,000\) and a maturity of 20 years; it makes no coupon payments over the life of the bond. If the required return on both these bonds is 12 percent compounded semiannually, What is the furrent nrice nfand M? Pf Bond N?

Short Answer

Expert verified
The present value of Bond M is approximately \(\$12,878.40\), and the present value of Bond N is approximately \(\$1,643.02\).

Step by step solution

01

Calculate the semiannual discount rate for both bonds

We are given an annual required return of 12% compounded semiannually, which means that there are two compounding periods per year. We can calculate the semiannual discount rate (r) using the following formula: \(1 + annual\ required\ return = (1 + r)^{number\ of\ compounding\ periods}\) \(1.12 = (1 + r)^2\) Solving for r: r 鈮� 0.0573
02

Calculate the present value of the cash flows of Bond M

Bond M has three sets of cash flows: no payments for the first 6 years, \(1,000\) every six months for 8 years, and \(1,750\) every six months for 6 years. First, let's find the present value of each set of cash flows individually. - First 6 years: No payments, so the present value of these cash flows is \(0\). - Next 8 years: \(PV_1 = 1000 \times \frac{1 - (1 + 0.0573)^{-16}}{0.0573} 鈮� \$8,273.56\) - Last 6 years: \(PV_2 = 1750 \times \frac{1 - (1 + 0.0573)^{-12}}{0.0573} 鈮� \$10,839.48\) Now, we need to find the present value of the cash flow set for the last 6 years and then add all of the present values to find the total present value of Bond M: - Present value of the 8 year cash flow at the end of the 6th year: \(PV_3 = \frac{PV_1}{(1 + 0.0573)^{12}} 鈮� \$4,867.42\) - Present value of the 6 year cash flow at the end of the 6th year: \(PV_4 = \frac{PV_2}{(1 + 0.0573)^{12}} 鈮� \$6,367.96\) - Total Present value of Bond M: \(PV_M = PV_3 + PV_4 鈮� \$11,235.38\)
03

Determine the present value of the face value of both bonds

For both bonds, the face value (\(\$20,000\)) is paid back at the end of the 20-year term. To find the present value of the face value: \(PV_{FV} = \frac{20000}{(1 + 0.0573)^{40}} 鈮� \$1,643.02\)
04

Calculate the present value of Bond N

Bond N makes no coupon payments over the life of the bond, so the only value to consider is the present value of the face value. \(PV_N = PV_{FV} 鈮� \$1,643.02\)
05

Calculate the final present value of Bond M and Bond N

We now have all the components to calculate the present value of Bond M and Bond N: Bond M: \(PV_M = PV_M + PV_{FV} 鈮� \$11,235.38 + \$1,643.02 鈮� \$12,878.40\) Bond N: \(PV_N 鈮� \$1,643.02\) So, the present value of Bond M is approximately \(12,878.40\), and the present value of Bond N is approximately \(1,643.02\).

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Key Concepts

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

Present Value Calculation
The concept of present value is a cornerstone in finance and is integral when valuing bonds. It helps in determining the current worth of future cash flows, considering a specified rate of return. When applied to bond valuation, it means discounting the bond's expected cash flows, such as coupon payments and face value, back to their present values. This calculation offers insights into what these cash flows are worth today.

The formula for calculating present value is crucial. It is:\[PV = \frac{C}{(1+r)^n}\]where:
  • **PV** stands for present value,
  • **C** is the future cash flow,
  • **r** represents the discount rate (or required rate of return), and
  • **n** is the number of periods until cash flow occurs.
For the terms in our exercise, this means identifying each bond's future payments and discounting them back to present value at the specified rate of return, which is compounded semiannually. Calculating these helps investors understand the bond's value under current market conditions.
Semiannual Compounding
Compounding frequency significantly affects the present value calculations for bonds. Semiannual compounding means interest is calculated twice a year. Thus, the stated annual interest rate must be split into two periods, and the compounding effect will be applied more frequently.

For instance, in the exercise, the annual required return is 12%, but with semiannual compounding, this affects the calculations such that:\[1 + annual\ required\ return = (1 + r)^2\]where "r" is the semiannual discount rate. Solving this formula gives us the effective period interest rate used to discount cash flows every six months. This method produces more accurate results, reflecting real-world calculations more closely. Such understanding is vital for precise financial planning and valuation.
Cash Flow Analysis
Cash flow analysis in bond valuation involves dissecting the expected cash flows from a bond over its lifespan. With bonds like those in our exercise, various cash flows occur at different periods.

Let's break this down. Bond M, for instance, has complex cash flows, which need individual valuation:
  • **First 6 years**: No payments, simplifying the present value here to zero,
  • **Next 8 years**: Regular payments of \(\$1,000\) every six months, creating periodic cash flows requiring a separate present value calculation,
  • **Last 6 years**: Higher payments elevate bond return, further necessitating separate present value assessments.
This detailed cash flow evaluation allows investors to understand how different phases of cash disbursements affect overall bond value. In essence, focusing on each segment of cash flow ensures a robust, accurate bond valuation.

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Most popular questions from this chapter

Bond Prices Mullineaux Co. issued 11-year bonds one year ago at a coupon rate of 8.6 percent. The bonds make semiannual payments. If the YTM on these bonds is 7.5 percent, what is the current bond price?

Interpreting Bond Yields Is the yield to maturity on a bond the same thing as the required return? Is YTM the same thing as the coupon rate? Suppose today a 10 percent coupon bond sells at par. Two years from now, the required return on the same bond is 8 percent. What is the coupon rate on the bond now? The YTM?

Interest Rate Risk Both Bond Bob and Bond Tom have 8 percent coupons, make semiannual payments, and are priced at par value. Bond Bob has 2 years to maturity, whereas Bond Tom has 15 years to maturity. If interest rates suddenly rise by 2 percent, what is the percentage change in the price of Bond Bob? Of Bond Tom? If rates were to suddenly fall by 2 percent instead, what would the percentage change in the price of Bond Bob be then? Of Bond Tom? Illustrate your answers by graphing bond prices versus YTM. What does this problem tell you about the interest rate risk of longer-term bonds?

Zero Coupon Bonds Suppose your company needs to raise \(\$ 10\) million and you want to issue 30 -year bonds for this purpose. Assume the required return on your bond issue will be 9 percent, and you're evaluating two issue alternatives: a 9 percent annual coupon bond and a zero coupon bond. Your company's tax rate is 35 percent. a. How many of the coupon bonds would you need to issue to raise the \(\$ 10 \mathrm{mil}\) lion? How many of the zeroes would you need to issue? b. In 30 years, what will your company's repayment be if you issue the coupon bonds? What if you issue the zeroes? c. Based on your answers in ( \(a\) ) and ( \(b\) ), why would you ever want to issue the zeroes? To answer, calculate the firm's aftertax cash outflows for the first year under the two different scenarios. Assume the IRS amortization rules apply for the zero coupon bonds.

Finding the Bond Maturity Massey Co. has 12 percent coupon bonds making annual payments with a YTM of 9 percent. The current yield on these bonds is 9.80 percent. How many years do these bonds have left until they mature?
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