Chapter 06 Inputs are in Blue Answers are in Red NOTE: Some functions used in these spreadsheets may require that the "Analysis ToolPak" or "Solver Add-In" be installed in Excel. To install these, click on the Office button then "Excel Options," "Add-Ins" and select "Go." Check "Analysis ToolPak" and "Solver Add-In," then click "OK."
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Quiz 3 A bond with face value $1,000 has a current yield of 6% and a coupon rate of 8%. What is the bond’s price? Face value Current Yield Coupon Rate
$ 1,000.00 6% 8%
Solution: Bond price
$ 1,333.33
note: current yield =coupon payment/bond price
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Quiz 4 A 6-year Circular File bond pays interest of $80 annually and sells for $950. What are its coupon rate and yield to maturity? Time Interest Price
6.00 years 80.00 950.00
Solution: Coupon rate
8.00%
Using a financial calculator:
Yield to maturity
9.119%
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Quiz 5 A 6-year Circular File bond pays interest of $80 annually and sells for $950. If Circular File wants to issue a new 6-year bond at face value, what coupon rate must the bond offer? Time Interest Price
6.00 years 80.00 950.00
Solution: In order for the bond to sell at par, the coupon rate must equal the yield to maturity. Using a financial calculator: 9.119% Yield to maturity
Hence the coupon rate must be
9.119%
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Quiz 6 A bond has 8 years until maturity, a coupon rate of 8%, and sells for $1,100. a. What is the current yield on the bond? b. What is the yield to maturity? Time Coupon rate Price
8.00 years 8% 1,100.00
Solution: a.
Current yield
b.
Using a financial calculator:
Yield to maturity
=
=
7.27%
recall: cu current yi yield=coupon pa paym put the kursor on the the answe answer, r, klic
6.3662%
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ent/bond price k on f x to see the functionargument =Ränta (perioder; betalning; nuvärde; slutvärde; typ)
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Quiz 7 General Matter’s outstanding bond issue has a coupon rate of 10%, and it sells at a yield to maturity of 9.25%. The firm wishes to issue additional bonds to the public at face value. What coupon rate must the new bonds offer in order to sell at face value?
Coupon rate YTM
10% 9.25%
Solution: Coupon rate on the new bonds
=
9.25%
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Quiz 8 Refer to Table 6.1 (below). What is the current yield of the 8.75% 2020 maturity bond? Why is this more than its yield to maturity? Table 6.1 Maturity
Coupon
2012 May 15 2013 May 15 2014 May 15 2020 May 15 2025 Aug 15 2030 May 15 2040 May 15
1.375 3.625 4.75 8.75 6.875 6.25 4.375
Coupon Rate Asked Price
Bid Price Asked Price Asked Yield, % 101:05 106:31 111:22 144:17 133:07 128:25 100:28
8.75% 144.00
101:06 107:01 111:23 144:19 133:11 128:27 100:29
0.78 1.23 1.7 3.44 3.94 4.12 4.32
19.00
Solution: Current yield
6.05%
The current yield exceeds the yield to maturity on the bond because the bond is selling at a premium. At maturity the holder of the bond will receive only the $1,000 face value, reducing the total return on investment
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Practice Problem 9 One bond has a coupon rate of 8%, another a coupon rate of 12%. Both bonds have 10-year maturities and sell at a yield to maturity of 10%. If their yields to maturity next year are still 10%, what is the rate of return on each bond? Does the higher coupon bond give a higher rate of return? Coupon rate - bond 1 Coupon rate - bond 2 Maturity YTM YTM next year
8.00% 12.00% 10.00 years 10.00% 10.00%
Solution: Using a financial calculator:
Bond 1: Rate of return
=
10.00%
=
10.00%
Bond 2: Rate of return
Both bonds provide the same rate of return.
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Practice Problem 10 A bond has 8 years until maturity, a coupon rate of 8%, and sells for $1,100. a. If this bond has a yield to maturity of 8% 1 year from now, what will its price be? b. What will be the rate of return on the bond? c. If the inflation rate during the year is 3%, what is the real rate of return on the bond? Time Coupon rate Price YTM Inflation
8.00 years 8% 1,100.00 8% 3%
Solution: =
$ 1,000.00
Bond Price Rate of return
=
$ 1,000.00 -1.82%
Rate of return Real return
=
-1.82% -4.68%
a.
Bond Price
b.
c.
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Practice Problem 11 A General Electric bond carries a coupon rate of 8%, has 9 years until maturity, and sells at a yield to maturity of 7%. a. What interest payments do bondholders receive each year? b. At what price does the bond sell? (Assume (Assume annual interest payments.) c. What will happen to the bond price if the yield to maturity falls to 6%? Face value Time Coupon rate YTM YTM falls to
$ 1,000.00 9.00 years 8% 7% 6%
Solution: a.
Annual interest payments
=
$
b.
Annual interest payments Bond price
=
80.00 $ 1,065.15
=
80.00 $ 1,136.03
c.
Annual interest payments Bond price if YTM falls to 6%
80.00
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Practice Problem 12 A 30-year maturity bond with with face value of $1,000 makes annual coupon payments and has a coupon rate of 8%. What is the bond’s yield to maturity if the bond is selling for a. $
900.00 ?
b. $ 1,000.00 ? c. $ 1,100.00 ? Face value Time Coupon rate
$ 1,000.00 30.00 years 8%
Solution: Using a financial calculator:
a.
Yield to maturity
=
8.971%
b.
Yield to maturity
=
8.000%
c.
Yield to maturity
=
7.180%
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Practice Problem 13 A 30-year maturity bond with with face value of $1,000 makes semi-annual coupon payments and has a coupon rate of 8%. What is the bond’s yield to maturity if the bond is selling for a. $
900.00 ?
b. $ 1,000.00 ? c. $ 1,100.00 ? Face value Time Coupon rate
$ 1,000.00 30.00 years 8%
Solution: Using a financial calculator:
a.
Yield to maturity
=
8.966%
b.
Yield to maturity
=
8.000%
c.
Yield to maturity
=
7.184%
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Practice Problem 14 Fill in the table below for the following zero-coupon bonds. The face value of each bond is $1,000. Price $300.00 $300.00 —
Maturity (years) 30.00 — 10.00
Face value
Yield to Maturity — 8% 10% $ 1,000.00
Solution: maturity
Price = $1,000/(1 + y ) Price $300.00 $300.00 $385.54
Maturity (years) 30.00 15.64 10.00
Yield to Maturity 4.095% 8.000% 10.000%
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Practice Problem 15 Perpetual Life Corp. has issued consol bonds with coupon payments of $60. (Consols pay interest forever and never mature. They are perpetuities.) If the required rate of return on these bonds at the time they were issued was 6%, at what price were they sold to the public? If the required return today is 10%, at what price do the consols sell? Coupon Payments $ Required rate of return at the time of issue Current required rate of return
60.00 6% 10%
Solution: PV of perpetuity = coupon payment/rate of return Present value
= $ 1,000.00
If the required rate of return is 10%, the bond sells for: Present value
= $
600.00
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Practice Problem 16 Sure Tea Co. has issued 9% annual coupon bonds that are now selling at a yield to maturity of 10% and current yield of 9.8375%. What is the remaining maturity of these bonds? Face value Time Coupon rate Yield to maturity Current yield
$ 1,000.00 30.00 years 9% 10% 9.8375%
Solution: Using a financial calculator:
Remaining maturity
=
20.00 years
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Practice Problem 17 Large Industries bonds sell for $1,065.15. The bond life is 9 years, and the yield to maturity is 7%. What must be the coupon rate on the bonds? Face value Bond price Time Yield to maturity
$ 1,000.00 $ 1,065.15 9.00 years 7%
Solution: Using a financial calculator:
Coupon rate
=
8.00%
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Practice Problem 18 a. Several years ago, Castles in the Sand, Inc., issued bonds at face value at a yield to maturity of 7%. Now, with 8 years left until the maturity of the bonds, the company has run into hard times and the yield to maturity on the bonds has increased to 15%. What has happened to the price of the bond? b. Suppose that investors believe that Castles can make good on the promised coupon payments, but that the company will go bankrupt when the bond matures and the principal comes due. The expectation is that investors will receive only 80% of face value at maturity. If they buy the bond today, what yield to maturity do they expect to receive? Face value Yield to maturity Years left Yield to maturity increased to Face value received by the investors
$ 1,000.00 7% 8.00 15% 80%
Solution: =
a.
Price of bond
b.
Price of bond Using a financial calculator: Yield to maturity
=
$
641.01
$
641.01 12.87%
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Practice Problem 19 You buy an 8% coupon, 10-year maturity bond for $980. A year later, the bond price is $1,200. a. What is the new yield to maturity on the bond? b. What is your rate of return over the year? Face value Coupon rate Time Bond price Bond price a year later
$ 1,000.00 8% 10.00 years 980.00 1,200.00
Solution: Using a financial calculator:
a.
Yield to maturity
=
5.165%
b.
Rate of return
=
30.61%
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Practice Problem 20 You buy an 8% coupon, 20-year maturity bond when its yield to maturity is 9%. A year later, the yield to maturity is 10%. What is your rate of return over the year? Face value $ 1,000.00 Coupon rate 8% Time 20.00 years Yield to maturity 9% Yield to maturity next y 10% Solution: Using a financial calculator:
Rate of return
=
bond price at t0 908.71 kr bond price at t1 832.70 kr total return on investment on bond (coupon payment + P1-P0)/ P0 0.4388%
0.44%
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:
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Practice Problem 21 Consider three bonds with 8% coupon rates, all selling at face value. The short-term bond has a maturity of 4 years, the intermediate-term bond has maturity 8 years, and the long-term bond has maturity 30 years. a. What will happen to the price of each bond if their yields increase to 9%? b. What will happen to the price of each bond if their yields decrease to 7%? Face value Coupon rate Short term bond maturity Intermediate-term bond maturity Long-term bond maturity Yield (a) Yield (b)
$ 1,000.00 8% 4.00 years 8.00 years 30.00 years 9% 7%
Solution: a., b. Price of Each Bond at Different Yields to Maturity Maturity of Bond Yield 7% 8% 9%
4 Years $ 1,033.87 $ 1,000.00 $ 967.60
8 Years $ 1,059.71 $ 1,000.00 $ 944.65
30 Years $ 1,124.09 $ 1,000.00 $ 897.26
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Practice Problem 22 A 2-year maturity bond with with face value of $1,000 makes annual coupon payments of $80 and is selling at face value. What will be the rate of return on the bond if its yield to maturity at the end of the year is a.
6% ?
b.
8% ?
c.
10% ? Face value Annual coupon Time
$ 1,000.00 $ 80.00 2.00 years
Solution: a.
Rate of return
=
9.89%
b.
Rate of return
=
8.00%
c.
Rate of return
=
6.18%
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Practice Problem 23 A bond that pays coupons annually is issued issued with a coupon rate of 4%, maturity of 30 years, and a yield to maturity of 7%. What rate of return will be earned by an investor who purchases the bond and holds it for 1 year if the bond’s yield to maturity at the end of the year is 8%? Face value Coupon rate Time Yield to maturity Yield to maturity by end of 1st year
$ 1,000.00 4% 30.00 years 7% 8%
Solution: Rate of return
=
-5.43%
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Practice Problem 24 A bond’s credit credit rating provides a guide to its risk. Long-term Long-term bonds bonds rated Aa currently currently offer yields yields to maturity of 7.5%. A-rated bonds sell at yields of 7.8%. If a 10-year bond with a coupon rate of 7% is downgraded by Moody’s from Aa to A rating, what is the likely effect on the bond price?
Face value Coupon rate Time Current yield to maturity Yield of A rated bonds
$ 1,000.00 7% 10.00 years 7.5% 8%
Solution: The bond’s yield to maturity will increase from 7.5% to 7.8% when the perceived default risk Increases. The bond price will fall: Initial price
=
$965.68
New price
=
$945.83
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Practice Problem 25 Suppose that you buy a 1-year maturity bond for $1,000 that will pay you back $1,000 plus a coupon payment of $70 at the end of the year. What real rate of return will you earn if the inflation rate is a.
2% ?
b.
4% ?
c.
6% ?
d.
8% ? Face value Annual coupon Time
$ 1,000.00 $ 70.00 1.00 year
Solution: a.
Real rate of return
=
4.902%
b.
Real rate of return
=
2.885%
c.
Real rate of return
=
0.9434%
d.
Real rate of return
=
-0.926%
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Practice Problem 26 Suppose that you buy a 1-year maturity bond for $1,000 that will pay you back $1,000 plus a coupon payment of $70 at the end of the year and the inflation rate is a.
2% ?
b.
4% ?
c.
6% ?
d.
8% ?
Now suppose that the bond is a TIPS (inflation-indexed) bond with a coupon rate of 4%. What will the cash flow provided by the bond be for each of the four inflation rates? What will be the real and nominal rates of return on the bond in each scenario? Face value Time Coupon rate
$ 1,000.00 1.00 year 4%
Solution: The principal value of the bond will increase by the inflation rate, and since the coupon is 4% of the principal, the coupon will also increase along with the general level of prices. The total cash flow provided by the bond will be: 1,000 x (1 + inflation rate) + coupon rate x 1,000 x (1 + inflation rate) Since the bond is purchased for face value, or $1,000, total dollar nominal return is therefore the increase in the principal due to the inflation indexing, plus coupon income: Income = ($1,000 x inflation rate) + [coupon rate x $1,000 x (1 + inflation rate)] Finally: Nominal rate of return = income/$1,000 a.
b.
c.
d.
Nominal rate of return
=
6.080%
Real rate of return
=
4.00%
Nominal rate of return
=
8.160%
Real rate of return
=
4.00%
Nominal rate of return
=
10.240%
Real rate of return
=
4.00%
Nominal rate of return
=
12.320%
Real rate of return
=
4.00%
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Practice Problem 27 Suppose that you buy a 1-year maturity bond for $1,000 that will pay you back $1,000 plus a coupon payment of $70 at the end of the year and the inflation rate is a. 2% ? b. 4% ? c. 6% ? d. 8% ? Now suppose the TIPS bond in the previous problem is a 2-year maturity bond. What will be the bondholder’s cash flows in each year in each of the inflation scenarios? Face value Time Coupon rate
$
1,000.00 2.00 years 4%
Solution:
a. b. c. d.
First-Year Cash Flow
Second-Year Cash Flow
$ $ $ $
$ $ $ $
40.80 41.60 42.40 43.20
1,082.02 1,124.86 1,168.54 1,213.06
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Challenge Problem 28 Suppose interest rates increase from 8% to 9%. Which bond will suffer the greater percentage decline in price: a 30-year bond paying annual coupons of 8% or a 30-year zero-coupon bond? Face value Interest rate 1 Interest rate 2 Time Coupon
$ 1,000.00 8.00% 9.00% 30.00 years 8.00%
Solution: Coupon bond Initial price New price Price decline %
= $ 1,000.00 = $ 897.26 = 10.27%
Zero-coupon bond Initial price New price Price decline %
= $ = $ =
99.38 75.37 24.16%
Hence zero-coupon bond will suffer greater price decline.
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Challenge Problem 29 Consider two 30-year maturity bonds. Bond A has a coupon rate of 4%, while bond B has a coup rate of 12%. Both bonds pay their coupons semiannually a. Construct an Excel spreadsheet showing the prices of each of these bonds for yields to matu ranging from 2% to 15% at intervals of 1%. Column A should show the yield to maturity (ranging from 2% to 15%), and columns B and C should compute the prices of the two bonds (using Excel’s bond price function) at each interest rate. b. In columns D and E, compute the percentage difference between the bond price and its valu when yield to maturity is 8%. c. Plot the values in columns D and E as a function of the interest rate. Which bond’s price is proportionally more sensitive to interest rate changes? Face value Coupon rate bond A Coupon rate bond B Time Yield to maturity
$
100.00 4.00% 12.00% 30.00 years 8.00%
-144.96 kr -119.69 kr -100.00 kr
Solution: c. a., b. Yield 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15%
Price A 144.96 119.69 100.00 84.55 72.32 62.58 54.75 48.40 43.21 38.93 35.35 32.35 29.80 27.62
Price B 324.78 277.21 239.04 208.18 183.03 162.36 145.25 130.96 118.93 108.72 100.00 92.48 85.96 80.26
% Diff. Diff. (8%) (8%) A % Diff. Diff. (8%) (8%) B 1 65% 16 124% 119% 11 91% 83% 83 65% 54% 54 43% 32% 32 26% 14% 14 12% 0% 0% 0% -12% -1 -10% -21% -2 -18% -29% -2 -25% -35% -3 -31% -41% -4 -36% -46% -4 -41% -50% -5 -45%
200%
150%
100%
50%
0%
-50%
-100% 350.00
300.00
The price of
250.00
200.00 Price A 150.00
Price B
100.00
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50.00
0%
5%
10%
15%
20%
here you can see the higher the yield to maturity (IRR), the lower the bond price.
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on
rity
% Diff. (8%) A
% Diff. (8%) B
ond A is more sensitive to interest rate changes as reflected in the steeper curve.
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Challenge Problem 30 In Figure 6.7, we saw a plot of the yield the yield curve on bonds and pointed out that curve on stripped Treasury bonds and bonds of different maturities may sell at different yields to maturity. In principle, when we are valuing a stream of cash flows, each cash flow should be discounted by the yield appropriate to its particular maturity. Suppose the yield curve on (zero-coupon) Treasury strips is as follows: Time to Maturity 1 year 2 3–5 6–10
YTM 4.00% 5% 5.50% 6%
You wish to value a 10-year bond with a coupon rate of 10%, paid annually. a. Set up an Excel spreadsheet to value each of the bond’s annual cash flows using this table of yields. Add up the present values of the bond’s 10 cash flows to obtain the bond price. recall:: Yield recall Yield to maturity maturity is internal internal rate rate of retur
b. What is the bond’s yield to maturity? Face value Coupon rate Time to maturity
$
1,000.00 10.00% 10.00 years
Solution: a., b. Cash Flow from Bond 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 100.00 1,100.00
PV of cash flow 96.15384615 90.70294785 85.16136642 80.72167433 76.51343538 70.49605404 66.50571136 62.74123713 59.18984635 614.2342546
Bond price (PV)
=
1302.420374
YTM (RATE) (RATE)
=
5.91%
Year
YTM
1 2 3 4 5 6 7 8 9 10
4.00% 5.00% 5.50% 5.50% 5.50% 6.00% 6.00% 6.00% 6.00% 6.00%
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stripped Treasury bonds:every coupon payment can be seen as a zero coupon bond with time to maturity of the coupon payment.
n (IRR, overall interest rate) earned by an investor who buys the bond today at the market price.
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