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Solutions to Chapter 5

Valuing Bonds

Note: Unless otherwise stated, assume all bonds have $1,000 face (par) value.

1. a. The coupon payments are fixed at $60 per year.

Coupon rate = coupon payment/par value = 60/1000 = 6%, which remains unchanged.

b. When the market yield increases, the bond price will fall. The cash flows are discounted at a higher rate.

c. At a lower price, the bond’s yield to maturity will be higher. The higher

yield to maturity on the bond is commensurate with the higher yields

available in the rest of the bond market.

d. Current yield = coupon payment/bond price. As coupon payment remains the same and the bond price decreases, the current yield increases.

2. When the bond is selling at a discount, $970 in this case, the yield to maturity is greater than 8%. We know that if the discount rate were 8%, the bond would sell at par. At a price below par, the YTM must exceed the coupon rate.

Current yield equals coupon payment/bond price, in this case, 80/970. So current yield is also greater than 8%.

3. Coupon payment = .08 x 1000 = $80

Current yield = 80/bond price = .075

Therefore, bond price = 80/.075 = $1,066.67

4. Par value is $1000 by assumption.

Coupon rate = $80/$1000 = .080 = 8.0%

Current yield = $80/$950 = .0842 = 8.42%

Yield to maturity = 9.12% [n = 6; PV= (-)950; FV = 1000; PMT = 80)

5. To sell at par, the coupon rate must equal yield to maturity. Since Circular bonds

yield 9.12%, this must be the coupon rate.

6. a. Current yield = annual coupon/price = $80/1050 = .0762 = 7.62%.

b. YTM = 7.2789%. On the calculator, enter PV = (-)1050,

FV = 1000, n = 10, PMT = 80, compute i.

7. When the bond is selling at par, its yield to maturity equals its coupon rate. This firm’s bonds are selling at a yield to maturity of 9.25%. So the coupon rate on the new bonds must be 9.25% if they are to sell at par.

8. The current bid yield on the bond was 4.43%. To buy the bond, investors pay the ask price. The investor would pay 105.66 percent of par value. With $1,000 par value, this means paying $1,056.6 to buy a bond.

9. Coupon payment = interest = .05 × 1000 = 50

Capital gain = 1100 – 1000 = 100

Rate of return = = = .15 = 15%

10. Tax on interest received = tax rate × interest = .3 × 50 = 15

After-tax interest received = interest – tax = 50 – 15 = 35

Fast way to calculate:

After-tax interest received = (1 – tax rate) × interest = (1 – .3)× 50 = 35

Tax on capital gain = .5 × .3 × 100 = 15

After-tax capital gain = 100 – 15 = 85

Fast way to calculate:

After-tax capital gain = (1 – tax rate) × capital gain = (1 – .5×.3)×100 = 85

After-tax rate of return =

= = .12 = 12%

11. Bond 1

year 1: PMT = 80, FV = 1000, i = 10%, n = 10; Compute PV0 = $877.11

year 2: PMT = 80, FV = l000, i = 10%, n = 9; Compute PV1 = $884.82

Rate of return = = .10 = 10%

Bond 2

year 1: PMT = 120, FV = 1000, i = 10%, n = 10; Compute PV0 = $1122.89

year 2: PMT = 120, FV = l000, i = 10%, n = 9; Compute PV1 =$1115.18

Rate of return = = .10 = 10%

Both bonds provide the same rate of return.

12. a. If YTM = 8%, price will be $1000.

b. Rate of return =

= = .0286 = 2.86%

c. Real return = – 1

= [pic] – 1 = –.001359, or about – .136%

13. a. With a par value of $1000 and a coupon rate of 8%, the bondholder receives 2 payments of $40 per year, for a total of $80 per year.

b. Assume it is 9%, compounded semi-annually. Per period rate is 9%/2, or 4.5%

Price = 40 × annuity factor(4.5%, 18 years) + 1000/1.04518 = $939.20

c. If the yield to maturity is 7%, compounded semi-annually, the bond will sell above par, specifically for $1,065.95:

Per period rate is 7%/2 = 3.5%

Price = 40 × annuity factor(3.5%, 18 years) + 1000/1.03518 = $1,065.95

14. On your calculator, set n = 30, FV =1000, PMT = 80.

a. Set PV = (-)900 and compute the interest rate to find that YTM = 8.971%

b. Set PV = (-)1000 and compute the interest rate to find that YTM = 8.000%.

c. Set PV = (-)1100 and compute the interest rate to find that YTM = 7.180%

15. On your calculator, set n=60, FV=1000, PMT=40.

a. Set PV = (-)900 and compute the interest rate to find that the (semiannual) YTM =4.483%. The bond equivalent yield to maturity is therefore 4.483 × 2 = 8.966%.

b. Set PV = (-)1000 and compute the interest rate to find that YTM = 4%. The annualized bond equivalent yield to maturity is therefore 4 × 2= 8%.

c. Set PV = (-)1100 and compute the interest rate to find that YTM = 3.592%. The annualized bond equivalent yield to maturity is therefore 3.592 × 2 = 7.184%.

16. In each case we solve this equation for the missing variable:

Price= 1000/(1 + YTM)maturity

|Price |Maturity (years) |YTM |

|300 |30.0 |4.095% |

|300 |15.64 |8.0% |

|385.54 |10.0 |10.0% |

Alternatively the problem can be solved using a financial calculator:

Solving the first question: PV = (-)300, PMT = 0, n = 30, FV = 1000, and compute i.

17. PV of perpetuity = coupon payment/rate of return.

PV = C/r = 60/.06 = $1000

If the required rate of return is 10%, the bond sells for:

PV = C/r = 60/.1 = $600

18. Because current yield = .098375, bond price can be solved from: 90/Price = .098375, which implies that price = $914.87. On your calculator, you can now enter: i = 10;

PV = (-)914.87; FV = 1000; PMT = 90, and solve for n to find that n =20 years.

19. Assume that the yield to maturity is a stated rate. Thus the per period rate is 7%/2 or 3.5%. We must solve the following equation:

PMT × annuity factor(3.5%, 18 periods) + 1000/(1.035)18 = $1065.95

To solve, use a calculator to find the PMT that makes the PV of the bond cash flows equal to $1065.95. You should find PMT = $40. The coupon rate is 2×40/1000 = 8%.

20. a. The coupon rate must be 8% because the bonds were issued at par value

with a yield to maturity of 8%. Now, the price is

40 × Annuity factor(7%, 16 periods) + 1000/1.0716 = $716.60

b. The investors pay $716.60 for the bond. They expect to receive the promised coupons plus $800 at maturity. We calculate the yield to maturity based on these expectations:

40 × Annuity factor(i, 16 periods) + 800/(1 + i)16 = $716.60

which can be solved on the calculator to show that i =6.03%. On an annual basis, this 2×6.03% or 12.06% [n = 16; PV = (-)716.60; FV = 800; PMT = 40]

21. a. Today, at a price of 980 and maturity of 10 years, the bond’s yield to maturity is 8.3% (n = 10, PV = (-) 980, PMT = 80, FV = 1000).

In one year, at a price of 1050 and remaining maturity of 9 years, the bond’s yield to maturity is 7.23% (n = 9, PV = (-) 1050, PMT = 80, FV = 1000).

b. Rate of return = = 15.31%

22. Assume the bond pays an annual coupon. The answer is:

PV0 = $935.82 (n = 10, PMT = 80, FV = 1000, i = 9)

PV1 = $884.82 (n = 9, PMT = 80, FV = 1000, i = 10)

Rate of return = [pic] = 3.10%

If the bond pays coupons semi-annually, the solution becomes more complex. First, decide if the yields are effective annual rates or APRs. Second, make an assumption regarding the rate at which the first (mid-year) coupon payment is reinvested for the second half of the year. Your assumptions will affect the calculated rate of return on the investment. Here is one possible solution:

Assume that the yields are APR and the yield changes from 9% to 10% at the end of the year. The bond prices today and one year from today are:

PV0 = $934.96 (n = 2 × 10 = 20, PMT = 80/2 = 40, FV = 1000, i = 9/2 = 4.5)

PV1 = $883.10 (n = 2 × 9 = 18, PMT = 80/2 = 40, FV = 1000, i = 10/2 = 5)

Assuming that the yield doesn’t increase to 10% until the end of year, the $40 mid-year coupon payment is reinvested for half a year at 9%, compounded monthly. Its future value at the end of the year is: $40 × (1.045) = $41.80 and the rate of return on the bond investment is:

Rate of return = = 3.20%

23. The price of the bond at the end of the year depends on the interest rate at that time. With one year until maturity, the bond price will be $ 1080/(1 + r).

a. Price = 1080/1.06 = $1018.87

Return = [80 + (1018.87 – 1000)]/1000 = .09887 = 9.887%

b. Price = 1080/1.08 = $1000.00

Return = [80 + (1000 – 1000)]/1000 = .0800 = 8.00%

c. Price = 1080/1.10 = $981.82

Return = [80 + (981.82 – 1000)]/1000 = .06182 = 6.182%

24. The bond price is originally $549.69. (On your calculator, input n = 30, PMT =

40, FV =1000, and i = 8%.) After one year, the maturity of the bond will be 29 years and its price will be $490.09. (On your calculator, input n = 29, PMT = 40, FV = 1000, and i = 9%.) The rate of return is therefore [40 + (490.09 – 549.69)]/549.69 = –.0357 = –3.57%.

25. a. Annual coupon = .08 × 1000 = $80.

Total coupons received after 5 years = 5 × 80 = $400

Total cash flows, after 5 years = 400 + 1000 = $1400

Rate of return = ()1/5 – 1 = .075 = 7.5%

b. Future value of coupons after 5 years

= 80 × future value factor(1%, 5 years) = 408.08

Total cash flows, after 5 years = 408.08 + 1000 = $1408.8

Rate of return = ()1/5 – 1 = .0763 = 7.63%

c. Future value of coupons after 5 years

= 80 × future value factor(8.64%, 5 years) = 475.35

Total cash flows, after 5 years = 475.35 + 1000 = $1475.35

Rate of return = ()1/5 – 1 = .0864 = 8.64%

26. To solve for the rate of return using the YTM method, find the discount rate that makes the original price equal to the present value of the bond’s cash flows:

975 = 80 × annuity factor( YTM, 5 years ) + 1000/(1 + YTM)5

Using the calculator, enter PV = (-)975, n = 5, PMT = 80, FV = 1000 and compute i. You will find i = 8.64%, the same answer we found in 26 (c).

27. a. False. Since a bond's coupon payments and principal are fixed, as interest rates

rise, the present value of the bond's future cash flow falls. Hence, the bond price falls.

Example: Two-year bond 3% coupon, paid annual. Current YTM = 6%

Price = 30 × annuity factor(6%, 2) + 1000/(1 + .06)2 = 945

If rate rises to 7%, the new price is:

Price = 30 × annuity factor(7%, 2) + 1000/(1 + .07)2 = 927.68

b. False. If the bond's YMT is greater than its coupon rate, the bond must sell at a discount to make up for the lower coupon rate. For an example, see the bond in a. In both cases, the bond's coupon rate of 3% is less than its YTM and the bond sells for less than its $1,000 par value.

c. False. With a higher coupon rate, everything else equal, the bond pays more future cash flow and will sell for a higher price. Consider a bond identical to the one in a. but with a 6% coupon rate. With the YTM equal to 6%, the bond will sell for par value, $1,000. This is greater the $945 price of the otherwise identical bond with a 3% coupon rate.

d. False. Compare the 3% coupon bond in a with the 6% coupon bond in c. When YTM rises from 6% to 7%, the 3% coupon bond's price falls from $945 to $927.68, a -1.8328% decrease (= (927.68 - 945)/945). The otherwise identical 6% bonds price falls to 981.92 (= 60 × annuity factor(7%, 2) + 1000/(1 + .07)2) when the YTM increases to 7%. This is a -1.808% decrease (= 981.92 - 1000/1000), which is slightly smaller. The prices of bonds with lower coupon rates are more sensitivity to changes in interest rates than bonds with higher coupon rates.

e. False. As interest rates rise, the value of bonds fall. A 10 percent, 5 year Canada bond pays $50 of interest semi-annually (= .10/2 × $1,000). If the interest rate is assumed to be compounded semi-annually, the per period rate of 2% (= 4%/2) rises to 2.5% (=5%/2). The bond price changes from:

Price = 50 × annuity factor(2%, 2×5) + 1000/(1 + .02)10 = $1,269.48

to:

Price = 50 × annuity factor(2.5%, 2×5) + 1000/(1 + .025)10 = $1,218.80

The wealth of the investor falls 4% (=$1,218.80 - $1,269.48/$1,269.48).

28. Internet: Using historical yield-to-maturity data from Bank of Canada

Tips: Students will need to read the instructions on how to put the data into a spreadsheet. They will want to save the data in CSV format so that it will be easily moved into the spreadsheet. The data will be automatically put into Excel if you access the website with Internet Explorer. Watch that the headings for the columns of data in your spreadsheet aren’t out of line (we found that the Government of Canada bond yield heading took two columns, displacing the other two headings – the data itself were in the correct columns).

Expected results: Long-term Government of Canada bonds have the lowest yield, followed by the yields for the provincial long bonds and then for the corporate bonds. The graph of the yields clearly shows the consistent spreads but also how the level of interest rates varies over time. For an even clearer picture, have the students pick data from 1990 onward.

29. a. Strips pay no interest, only principal. Assume each bond pays $100 principal on the maturity date

|Bond |Time to Maturity (Years) |YTM = (100/Price)1/time to maturity - 1 |

|June 2006 |1.5 |= (100/94.81)1/1.5 - 1 = .03617 |

|June 2008 |3.5 |= (100/86.58)1/3.5 - 1 = .04203 |

|June 2012 |7.5 |= (100/69.56)1/7.5 - 1 = .04959 |

|June 2017 |12.5 |= (100/52.61)1/12.5 - 1 = .05272 |

|June 2022 |17.5 |= (100/38.92)1/17.5 - 1 = .05540 |

b. The term structure (yield curve) is upward sloping.

30. Price of bond today

= 40 × PVIFA(5%, 3) + 50 × PVIFA(5%,3) × PVIF(5%,3)

+ 60 × PVIFA(5%,3)×PVIF(5%,6) + 1000 × PVIF(5%, 9)

= 108.93 + 117.62 + 121.93 + 644.61 = $993.09

31. a., b. Price of each bond at different yields to maturity

Maturity of bond

4 years 8 years 30 years

Yield (%)

7 1033.87 1059.71 1124.09

8 1000.00 1000.00 1000.00

9 967.60 944.65 897.26

Difference between prices

(YTM=7% vs YTM=9%) 66.27 115.06 226.83

c. The table shows that prices of longer-term bonds respond with more sensitivity to changes in interest rates. This can be illustrated in a variety of ways. In the table we compare the prices of the bonds at 7 percent and 9 percent yields. When the yield falls from 9 to 7%, the price of the 30-year bond increases $226.83 but the price of the 4-year bond only increases $66.27. Another way to compare the bonds’ sensitivity to changes in the yield is to look at the percentage change in the prices. For example, with an increase in the yield from 8 to 9%, the price of the 4-year bond falls (967.6/1000) –1, or 3.24% but the 30-year bond price falls (897.26/1000) – 1, or 10.27%.

32. The bond’s yield to maturity will increase from 8.5%, effective annual interest (EAR) to 8.8%, EAR, when the perceived default risk increases.

6 month interest rate equivalent to 8.5% EAR = (1.085)1/2 – 1 = .04163

6 month interest rate equivalent to 8.8% EAR = (1.088)1/2 – 1 = .04307

Price at AA rating = $978.2 (n = 2×10 = 20, PMT = 80/2 = 40, FV =1000, i = 4.163)

Price at A rating = $959.4 (n = 2×10 = 20, PMT = 80/2 = 40, FV =1000, i = 4.307)

The price falls by $18.8 dollars due to the drop in the bond rating and the increase in the required rate of return.

33. Internet: Credit spreads on corporate bonds

INTERNET UPDATE: Since going to press, the provision of free current credit spread data at this site has been stopped. They do provide an example of credit spreads as of June 30, 2004 at http://www.bonds-online.com/Search_Quote__Center/Corporate_Agency_Bonds/Spreads/.

Expected results: Generally, the credit spread should be higher for bonds of increasing risk (lower credit rating) and also increase with the term to maturity. However, these data are averages of many different bonds and weird things can happen (for example, a bond’s yield may change before its credit rating is updated).

34. Internet: Canadian corporate bond yields

Tips: The Monday issue of the Globe and Mail has the most complete list of corporate bond prices and yields. Warn students that not all bonds have ratings at both DBRS and S&P. They might have to check both sources for a bond rating. An alternative approach would be to start with the bond rating services, select 5 bonds and then hope to find them listed in the newspaper. However, since many corporate bonds do not trade frequently, it is generally less frustrating to start with bonds with available price data and then hunt for their bond rating.

Expected results: If the bond rating is current (not out-of-date), you would expect to find that bonds with higher yields will have lower bond ratings. However, bonds have conversion options, call provisions and other bells that affect their price which may distort the relationship between their yield and the yield on equivalent maturity, Government of Canada bonds. You may want to ask your students to research the features of the bonds to be sure that it is not convertible or callable.

35. YTM = 4%

Real interest rate = 1 + nominal interest rate = 1.04 - 1 = .0196, or 1.96%

1 + expected rate of inflation 1.02

Real interest rate ≈ nominal interest rate - expected inflation rate = 4% - 2% = 2%

36. The nominal return is 1060/1000, or 6%. The real return is 1.06/(1 + inflation) – 1.

a. 1.06/1.02 – 1 = .0392 = 3.92%

b. 1.06/1.04 – 1 = .0192 = 1.92%

c. 1.06/1.06 – 1 = 0%

d. 1.06/1.08 – 1 = – .0185 = –1.85%

37. The principal value of the bond will increase by the inflation rate, and since the coupon is 4% of the principal, it too will rise along with the general level of prices. The total cash flow provided by the bond will be

1000 × (1 + inflation rate) + coupon rate × 1000 × (1 + inflation rate).

Since the bond is purchased for par value, or $1000, total dollar nominal return is therefore the increase in the principal due to the inflation indexing, plus coupon income:

Income = 1000 × inflation rate + coupon rate × 1000 × (1 + inflation rate)

Finally, the nominal rate of return = income/1000.

a. Nominal return = = .0608 Real return = – 1 = .04

b. Nominal return = = .0816 Real return = – 1 = .04

c. Nominal return = = .1024 Real return = – 1 = .04

d. Nominal return = = .1232 Real return = – 1 = .04

38. First year income Second year income

a. 40x1.02=$40.80 1040 x 1.022 = $1082.02

b. 40x1.04=$41.60 1040 x 1.042 = $1124.86

c. 40x1.06=$42.40 1040 x 1.062 = $1168.54

d. 40x1.08=$43.20 1040 x 1.082 = $1213.06

39. a. YTM = 5.76% (n=15, PV = (-)1048, PMT=62.5, FV=1000)

b. YTC = 6.33% (n=10, PV = (-)1048, PMT=62.5, FV=1100)

40. a. Current price = 1,112.38 (n=6, i=4.8%, PMT=70, FV=1000)

b. Current call price = 1,137.35 (n=6, i=4.35%, PMT=70, FV=1000)

41. a. YTM on ABC bond at issue = 5.5% (since sold at par, coupon rate = required rate of

return)

10-year Gov't of Canada bond yield at issue

= ABC bond YTM - credit spread = 5.5% - .25% = 5.25%

Required yield to meet Canada call:

= 10-year Gov't of Canada bond yield + .15% = 5.25 + .15% = 5.4%

Call price at issue = 1,007.57 (n=10, i=5.4%, PMT=55, FV=1000)

b. Required yield to call bond = 4.9% + .15% = 5.05%

Call price now, 5 years later = 1,019.46 (n=5, i=5.05%, PMT=55, FV=1000)

c. Based on new interest rates, the bond price is:

Price now, 5 years later = 1,021.65 (n=5, i=5%, PMT=55, FV=1000)

Now the current price is greater than the call price. The company can call bonds and

reduce its cost of debt.

42. The coupon bond will fall from an initial price of $1000 (when yield to maturity = 8%) to a new price of $897.26 when YTM immediately rises to 9%. This is a 10.27% decline in the bond price.

The zero coupon bond will fall from an initial price of = $99.38 to a new

price of = $75.37. This is a price decline of 24.16%, far greater than that of

the coupon bond.

The price of the coupon bond is much less sensitive to the change in yield. It seems to act like a shorter maturity bond. This makes sense: the 8% bond makes many coupon payments, most of which come years before the bond’s maturity date. Each payment may be considered to have its own “maturity date” which suggests that the effective maturity of the bond should be measured as some sort of average of the maturities of all the cash flows paid out by the bond. The zero–coupon bond, by contrast, makes only one payment at the final maturity date.

43. a. Annual after-tax coupon = (1 - .35) × .08 × 1000 = $52.

Total coupons received after 5 years = 5 × 52 = $260

Capital gains tax = .5 × .35 × (1000 – 975) = 4.375

After-tax capital gains = 1000 – 975 – 4.375 = 20.625

Total cash flows, after 5 years = 260 + 1000 – 4.375 = $ 1255.625

Rate of return = ()1/5 – 1 = .05189, or 5.189%

Note: This can also be answered by first calculating the five-year rate of return and then converting it into a one-year rate of return. This way students can continue to use the coupons + capital gains/original investment approach:

Five-year rate of return =

= = .28782

The one-year rate of return equivalent to the five-year rate of return is:

(1 + .28782) 1/5 – 1 = .05189, or 5.189%.

b. Future value of coupons after 5 years

= (1 – .35) × 80 × future value factor((1–.35)×1%, 5 years) = 263.4

Total cash flows, after 5 years = 263.4 + 1000 – 4.375 = $1259.025

Rate of return = ()1/5 – 1 = .0525 = 5.246%

c. Future value of coupons after 5 years

= (1 – .35) × 80 × future value factor((1–.35)×8.64%, 5 years) = 290.89

Total cash flows, after 5 years = 290.89 + 1000 – 4.375 = $1286.5

Rate of return = ()1/5 – 1 = .057 = 5.7%

What is the formula to calculate interest? Interest Formula. The Formula for simple interest enables us to find out the interest amount if the principal amount, rate of interest and time duration is given. That being said, the simple interest formula to calculate interest rate is. SI = (P × R ×T) / 100. In which, SI = simple interest. P = principal amount or the original amount being borrowed