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CHAPTER 8

VALUATION OF KNOWN CASH FLOWS: BONDS

Objectives

• To show how to value contracts and securities that promise a stream of cash flows that are known with certainty.

• To understand the shape of the yield curve.

• To understand how bond prices and yields change over time.

Outline

8.1 Using Present Value Formulas to Value Known Cash Flows

8.2 The Basic Building Blocks: Pure Discount Bonds

8.3 Coupon Bonds, Current Yield, and Yield to Maturity

8.4 Reading Bond Listings

8.5 Why Yields for the Same Maturity Differ

8.6 The Behavior of Bond Prices over Time

Summary

• A change in market interest rates causes a change in the opposite direction in the market values of all existing contracts promising fixed payments in the future.

• The market prices of $1 to be received at every possible date in the future are the basic building blocks for valuing all other streams of known cash flows. These prices are inferred from the observed market prices of traded bonds and then applied to other streams of known cash flows to value them.

• An equivalent valuation can be carried out by applying a discounted cash flow formula with a different discount rate for each future time period.

• Differences in the prices of fixed-income securities of a given maturity arise from differences in coupon rates, default risk, tax treatment, callability, convertibility, and other features.

• Over time the prices of bonds converge towards their face value. Before maturity, however, bond prices can fluctuate a great deal as a result of changes in market interest rates.

Solutions to Problems at End of Chapter

Bond Valuation with a Flat Term Structure

1. Suppose you want to know the price of a 10-year 7% coupon Treasury bond that pays interest annually.

a. You have been told that the yield to maturity is 8%. What is the price?

b. What is the price if coupons are paid semiannually, and the yield to maturity is 8% per year?

c. Now you have been told that the yield to maturity is 7% per year. What is the price? Could you have guessed the answer without calculating it? What if coupons are paid semiannually?

SOLUTION:

a. With coupons paid once a year:

|n |i |PV |FV |PMT |Result |

|10 |8 |? |100 |7 |PV =93.29 |

Price = 93.29

b. With coupons paid twice a year:

|n |i |PV |FV |PMT |Result |

|20 |4 |? |100 |3.5 |PV =93.20 |

Price = 93.20

c. Price = 100. When the coupon rate and yield to maturity are the same, the bond sells at par value (i.e. the price equals the face value of the bond).

2. Assume six months ago the US Treasury yield curve was flat at a rate of 4% per year (with annual compounding) and you bought a 30-year US Treasury bond. Today it is flat at a rate of 5% per year. What rate of return did you earn on your initial investment:

a. If the bond was a 4% coupon bond?

b. If the bond was a zero coupon bond?

c. How do your answer change if compounding is semiannual?

SOLUTION:

a and b.

Step 1: Find prices of the bonds six months ago:

| |n |i |PV |FV |PMT |Result |

|Zero coupon |30 |4 |? |100 |0 |PV =30.83 |

Step 2: Find prices of the bonds today:

| |n |i |PV |FV |PMT |Result |

|Zero coupon |29.5 |5 |? |100 |0 |23.71 |

Step 3: Find rates of return:

Rate of return = (coupon + change in price)/initial price

4% coupon bond: r = (4 + 84.74 ( 100)/100 = (0.1126 or (11.26%

Zero-coupon bond: r = (0 + 23.71 ( 30.83)/30.83 = (0.2309 or (23.09%. Note that the zero-coupon bond is more sensitive to yield changes than the 4% coupon bond.

c.

Step 1: Find prices of the bonds six months ago:

| |n |i |PV |FV |PMT |Result |

|Zero coupon |60 |2 |? |100 |0 |PV =30.48 |

Step 2: Find prices of the bonds today:

| |n |i |PV |FV |PMT |Result |

|Zero coupon |59 |2.5 |? |100 |0 |23.30 |

Step 3: Find rates of return:

Rate of return = (coupon + change in price) / initial price

4% coupon bond: r = (2 + 84.66 ( 100)/100 = (0.1334 or (13.34%

Zero coupon bond: r = (0 + 23.30 ( 30.48)/30.48 = (0.2356 or (23.56%. Note that the zero-coupon bond is more sensitive to yield changes than the 4% coupon bond.

Bond Valuation With a Non-Flat Term Structure

3. Suppose you observe the following prices for zero-coupon bonds (pure discount bonds) that have no risk of default:

|Maturity |Price per $1 of Face Value |Yield to Maturity |

|1 year |0.97 |3.093% |

|2 years |0.90 | |

a. What should be the price of a 2-year coupon bond that pays a 6% coupon rate, assuming coupon payments are made once a year starting one year from now?

b. Find the missing entry in the table.

c. What should be the yield to maturity of the 2-year coupon bond in Part a?

d. Why are your answers to parts b and c of this question different?

SOLUTION:

a. Present value of first year's cash flow = 6 x .97 = 5.82

Present value of second year's cash flow = 106 x .90 = 95.4

Total present value = 101.22

b. The yield to maturity on a 2-year zero coupon bond with price of 90 and face value of 100 is 5.41%

|n |i |PV |FV |PMT |Result |

|2 |? |-90 |100 |0 |i = 5.41% |

c. The yield to maturity on a 2-year 6% coupon bond with price of 101.22 is

|n |i |PV |FV |PMT |Result |

|2 |? |-101.22 |100 |6 |i = 5.34% |

d. The two bonds are different because they have different coupon rates. Thus they have different yields to maturity.

Coupon Stripping

4. You would like to create a 2-year synthetic zero-coupon bond. Assume you are aware of the following information: 1-year zero- coupon bonds are trading for $0.93 per dollar of face value and 2-year 7% coupon bonds (annual payments) are selling at $985.30 (Face value = $1,000).

a. What are the two cash flows from the 2-year coupon bond?

b. Assume you can purchase the 2-year coupon bond and unbundle the two cash flows and sell them.

i. How much will you receive from the sale of the first payment?

ii. How much do you need to receive from the sale of the 2-year Treasury strip to break even?

SOLUTION:

a. $70 at the end of the first year and $1070 at the end of year 2.

b. i. I would receive .93 x $70 = $65.10 from the sale of the first payment.

ii. To break even, I would need to receive $985.30- $65.10 = $920.20 from the sale of the 2-year strip.

The Law of One price and Bond Pricing

5. Assume that all of the bonds listed in the following table are the same except for their pattern of promised cash flows over time. Prices are quoted per $1 of face value. Use the information in the table and the Law of One Price to infer the values of the missing entries. Assume that coupon payments are annual.

|Coupon rate |Maturity |Price |Yield to maturity |

|6% |2 years | |5.5% |

|0 |2 years | | |

|7% |2 years | | |

|0 |1 year |$0.95 | |

SOLUTION:

Bond 1:

|n |i |PV |PMT |FV |Result |

Bond 4:

|n |i |PV |PMT |FV |Result |

From Bond 1 and Bond 4, we can get the missing entries for the 2-year zero-coupon bond. We know from bond 1 that:

1.0092 = 0.06/1.055 +1.06/(1.055)2. This is also equal to 0.06/(1+z1) + 1.06/(1+z2)2 where z1 and z2 are the yields to maturity on one-year zero-coupon and two-year zero-coupon bonds respectively. From bond 4 , we have z1, we can find z2.

1.0092 – 0.06/1.0526 = 1.06/(1+z2)2, hence z2 = 5.51%.

To get the price P per $1 face value of the 2-year zero-coupon bond, using the same reasoning:

1.0092 – 0.06x0.95 = 1.06xP, hence P = 0.8983

To find the entries for bond 3: first find the price, then the yield to maturity. To find the price, we can use z1 and z2 found earlier:

PV of coupon payment in year 1: 0.07 x 0.95 = 0.0665

PV of coupon + principal payments in year 2: 1.07 x 0.8983 =0.9612

Total present value of bond 3 = 1.0277

|n |i |PMT |PV |FV |Result |

Hence the table becomes:

|Coupon rate |Maturity |Price |Yield to maturity |

|6% |2 years |$1.0092 |5.5% |

|0 |2 years |$0.8983 |5.51% |

|7% |2 years |$1.0277 |5.50% |

|0 |1 year |$0.95 |5.26% |

Bond Features and Bond Valuation

6. What effect would adding the following features have on the market price of a similar bond which does not have this feature?

a. 10-year bond is callable by the company after 5 years (compare to a 10-year non-callable bond);

b. bond is convertible into 10 shares of common stock at any time (compare to a non-convertible bond);

c. 10-year bond can be “put back” to the company after 3 years at par (puttable bond) (compare to a 10-year non-puttable bond)

d. 25-year bond has tax-exempt coupon payments

SOLUTION:

a. The callable bond would have a lower price than the non-callable bond to compensate the bondholders for granting the issuer the right to call the bonds.

b. The convertible bond would have a higher price because it gives the bondholders the right to convert their bonds into shares of stock.

c. The puttable bond would have a higher price because it gives the bondholders the right to sell their bonds back to the issuer at par.

d. The bond with the tax-exempt coupon has a higher price because the bondholder is exempted from paying taxes on the coupons. (Coupons are usually considered and taxed as personal income).

Inferring the Value of a Bond Guarantee

7. Suppose that the yield curve on dollar bonds that are free of the risk of default is flat at 6% per year. A 2-year 10% coupon bond (with annual coupons and $1,000 face value) issued by Dafolto Corporation is rates B, and it is currently trading at a market price of $918. Aside from its risk of default, the Dafolto bond has no other financially significant features. How much should an investor be willing to pay for a guarantee against Dafolto’s defaulting on this bond?

SOLUTION:

If the bond was free of the risk of default, its yield would be 6%.

|n |i |PMT |PV |FV |Result |

The difference between the price of the bond if it were free of default and its actual price (with risk of default) is the value of a guarantee against default: 1073.3-918 = $155.3

The implied Value of a Call Provision and Convertibility

8. Suppose that the yield curve on bonds that are free of the risk of default is flat at 5% per year. A 20-year default-free coupon bond (with annual coupons and $1,000 face value) that becomes callable after 10 years is trading at par and has a coupon rate of 5.5%.

a. What is the implied value of the call provision?

b. A Safeco Corporation bond which is otherwise identical to the callable 5.5% coupon bond described above, is also convertible into 10 shares of Safeco stock at any time up to the bond’s maturity. If its yield to maturity is currently 3.5% per year, what is the implied value of the conversion feature?

SOLUTION:

a. We have to find the price of the bond if it were only free of the risk of default.

|n |i |PMT |PV |FV |Result |

The bond is traded at par value, hence the difference between the value calculated above and the actual traded value is the implied value of the call provision: 1062.3 – 1000 = $62.3

Note that the call provision decreases the value of the bond.

b. We have to find the price of the Safeco Corporation:

|n |i |PMT |PV |FV |Result |

This bond has the same features as the 5.5% default free callable bond described above, plus an additional feature: it is convertible into stocks. Hence the implied value of the conversion feature is the difference between the values of both bonds: 1284.2-1000 = $284.25. Note that the conversion feature increases the value of the bond.

Changes in Interest Rates and Bond Prices

9. All else being equal, if interest rates rise along the entire yield curve, you should expect that:

i. Bond prices will fall

ii. Bond prices will rise

iii. Prices on long-term bonds will fall more than prices on short-term bonds.

iv. Prices on long-term bonds will rise more than prices on short-term bonds

a. ii and iv are correct

b. We can’t be certain that prices will change

c. Only i is correct

d. Only ii is correct

e. i and iii are correct

SOLUTION:

The correct answer is e.

Bond prices are inversely proportional to yields hence when yields increase, bond prices fall.

Long-term bonds are more sensitive to yield changes than short-term bonds.

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