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Cash Flows

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Homework Solutions Chapter 6

Valuing Bonds + Financing

(For problems that do not give the face value of the bond, assume $1000)

            [pic 1]

  1. A technology firm issues bonds and uses the cash to finance a daring new project. They believe the project might generate a 30% IRR, which would more than cover the 3% YTM they would owe bondholders. Why would anyone purchase their 3% bonds instead of investing directly in the project, with a higher potential return?

Risk aversion. Investors purchase bonds because they don’t want the uncertainty that comes with “owning” part of a company and its projects. In finance, lower-risk investments generate lower returns.  

  1. Find the price of a 4% semi-annual coupon bond, with a face value of $1,000 that will mature in 5 years. Assume the YTM is 6%.

[pic 2]$914.69


[pic 3]$914.69

Calculator: CF0=0, CF1=20, F1=9, CF2=1020, F2=1

  1. Re-do the calculation above, but imagine that you purchased the bond above for only $500, 5 years ago.

Your calculation will not change. The price of the bond today isn’t influenced by how much you paid in the past. The present value of that bond’s future cash flows will still be the same. This is why bond pricing problems usually read, “it will mature in X years,” not “it is an X-year bond.”

  1. Suzie tried to find the price of a semi-annual coupon bond, with a face value of $100, 3 years until maturity, a coupon rate of 10%, and a YTM of 4%. However, she made two errors. Examine her work below and identify the errors:

[pic 4]

Her calculator work: CF0=0, CF1=10,F1=6,CF2=100,F2=1.

First, she calculated an annual coupon instead of a semi-annual coupon. It should be only $5. Second, when entering everything into her calculator, she made an error with cash flow timing. If there are 6 coupon payments followed by a face-value re-payment, she should have entered 5 coupon payments followed by 1 period of “coupon + face value” payment. This is a common mistake, because of the way we write out the bond pricing equation. The final coupon cash flow should come at the same time as face value repayment.

  1. A 5% Treasury note pays semi-annual coupons every February 15 and August 15. What is the price of this bond on September 5, 2001 given the bond matures on August 15, 2004. Assume that the settlement day on September 5 is 20 days from the previous coupon period. Also assume there are 181 days from August 15 to February 15. Finally, the bond’s annual YTM is 14%. Assume the face value is $1000.

[pic 5]$840.49

We’ve found price as of February 15 2002, which is 161 (181-20) days after the purchase date in September. So we need to discount our $840.49 number by 161/181ths of a period.

[pic 6]= $791.42

  1. Describe the difference between “Total return” and “Yield to Maturity”
  • YTM is forward-looking: what do you think your total return will be if you hold the bond until maturity? It’s expected interest + expected interest on interest + expected capital gains
  • TR = backwards looking. It’s interest + interest on interest + capital gains

  1. One bond has an annual coupon rate of 8 percent, another has an annual coupon rate of 12 percent.  Both bonds have a FV=$1000, 3-year maturities, and sell at a yield to maturity of 10 percent.  Assuming the reinvestment rate is 8% and you hold the bond until it matures, find the total return of each bond. Which is higher? Why does this make sense?  

        Recall that Total Return = [pic 7]

Bond 1

Step One: Calculate price

[pic 8] $950.26

Step Two: Calculate the future value of total income

[pic 9] $259.71

        Total income to date earned on the bond as of year 3 = $259.71 + $1,000 =$1,259.71

        Total return =         [pic 10]        = 9.85%

Bond 2

Step One: Calculate price

[pic 11]$1,049.73

Step Two: Calculate FV of total income

[pic 12]$389.58

        Total income earned on the bond in year 4 = $389.58 + $1,000 = $1,389.58

        Total return = [pic 13]9.79%

When YTM exceeds reinvestment rate, lower coupon bonds do better than otherwise identical higher coupon bonds. This is because more of their return is coming from capital gains than coupons or interest on coupons. So when reinvestment rate drops and coupon income drops, are hurt less.

  1. A General Motors bond carries a coupon rate of 8 percent, has 9 years until maturity, and sells at a yield to maturity of 7 percent.  
  1. What interest payments do bondholders receive each year?
  2. At what price does the bond sell? (Assume annual interest payments.)
  3. What will happen to the bond price if the yield to maturity falls to 6 percent?

        a.        With a par value of $1,000 and a coupon rate of 8%, the bondholder receives $80 per year.

b.        [pic 14]

        c.        If the yield to maturity is 6%, the bond will sell for:

      [pic 15]

  1.  A 30-year maturity bond with face value of $1,000 makes semi-annual coupon payments and has a coupon rate of 8 percent.  What is the bond’s yield to maturity if the bond is selling for
  1. $900?
  2. $1,000?
  3. $1,100?
  1. [pic 16]⇒ r = 4.483%

        Using a financial calculator, compute the yield to maturity by entering:

n = 60; PV = (−)900; FV = 1000; PMT = 40, compute i = 4.483%

        Verify the solution as follows:

[pic 17]

(difference due to rounding)

        Therefore, the annualized bond equivalent yield to maturity is:

        4.483% × 2 = 8.966%



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