FIN303 - California State University, Northridge



FIN303

Answers to the recommended problems

Chapter 2

2-1 0 1 2 3 4 5

| | | | | |

PV = 10,000 FV5 = ?

FV5 = $10,000(1.10)5

= $10,000(1.61051) = $16,105.10.

Alternatively, with a financial calculator enter the following: N = 5, I/YR = 10, PV = -10000, and PMT = 0. Solve for FV = $16,105.10.

2-2 0 5 10 15 20

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PV = ? FV20 = 5,000

With a financial calculator enter the following: N = 20, I/YR = 7, PMT = 0, and FV = 5000. Solve for PV = $1,292.10.

2-3 0 18

| |

PV = 250,000 FV18 = 1,000,000

With a financial calculator enter the following: N = 18, PV = -250000, PMT = 0, and FV = 1000000. Solve for I/YR = 8.01% ≈ 8%.

2-4 0 N = ?

| |

PV = 1 FVN = 2

$2 = $1(1.065)N.

With a financial calculator enter the following: I/YR = 6.5, PV = -1, PMT = 0, and FV = 2. Solve for N = 11.01 ≈ 11 years.

2-5 0 1 2 N – 2 N – 1 N

| | | ( ( ( | | |

PV = 42,180.53 5,000 5,000 5,000 5,000 FV = 250,000

Using your financial calculator, enter the following data: I/YR = 12; PV = -42180.53; PMT = -5000; FV = 250000; N = ? Solve for N = 11. It will take 11 years to accumulate $250,000.

2-6 Ordinary annuity:

0 1 2 3 4 5

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300 300 300 300 300

FVA5 = ?

With a financial calculator enter the following: N = 5, I/YR = 7, PV = 0, and PMT = 300. Solve for FV = $1,725.22.

Annuity due:

0 1 2 3 4 5

| | | | | |

300 300 300 300 300

With a financial calculator, switch to “BEG” and enter the following: N = 5, I/YR = 7, PV = 0, and PMT = 300. Solve for FV = $1,845.99. Don’t forget to switch back to “END” mode.

2-7 0 1 2 3 4 5 6

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100 100 100 200 300 500

PV = ? FV = ?

Using a financial calculator, enter the following: CF0 = 0; CF1 = 100; Nj = 3; CF4 = 200 (Note calculator will show CF2 on screen.); CF5 = 300 (Note calculator will show CF3 on screen.); CF6 = 500 (Note calculator will show CF4 on screen.); and I/YR = 8. Solve for NPV = $923.98.

To solve for the FV of the cash flow stream with a calculator that doesn’t have the NFV key, do the following: Enter N = 6, I/YR = 8, PV = -923.98, and PMT = 0. Solve for FV = $1,466.24. You can check this as follows:

0 1 2 3 4 5 6

| | | | | | |

100 100 100 200 300 500

324.00

233.28

125.97

136.05

146.93

$1,466.23

2-8 Using a financial calculator, enter the following: N = 60, I/YR = 1, PV = -20000, and FV = 0. Solve for PMT = $444.89.

EAR = [pic] – 1.0

= (1.01)12 – 1.0

= 12.68%.

Alternatively, using a financial calculator, enter the following: NOM% = 12 and P/YR = 12. Solve for EFF% = 12.6825%. Remember to change back to P/YR = 1 on your calculator.

2-9 a. 0 1

| | $500(1.06) = $530.00.

-500 FV = ?

Using a financial calculator, enter N = 1, I/YR = 6, PV = -500, PMT = 0, and FV = ? Solve for FV = $530.00.

b. 0 1 2

| | | $500(1.06)2 = $561.80.

-500 FV = ?

Using a financial calculator, enter N = 2, I/YR = 6, PV = -500, PMT = 0, and FV = ? Solve for FV = $561.80.

c. 0 1

| | $500(1/1.06) = $471.70.

PV = ? 500

Using a financial calculator, enter N = 1, I/YR = 6, PMT = 0, and FV = 500, and PV = ? Solve for PV = $471.70.

d. 0 1 2

| | | $500(1/1.06)2 = $445.00.

PV = ? 500

Using a financial calculator, enter N = 2, I/YR = 6, PMT = 0, FV = 500, and PV = ? Solve for PV = $445.00.

2-10 a. 0 1 2 3 4 5 6 7 8 9 10

| | | | | | | | | | | $500(1.06)10 = $895.42.

-500 FV = ?

Using a financial calculator, enter N = 10, I/YR = 6, PV = -500, PMT = 0, and FV = ? Solve for FV = $895.42.

b. 0 1 2 3 4 5 6 7 8 9 10

| | | | | | | | | | | $500(1.12)10 = $1,552.92.

-500 FV = ?

Using a financial calculator, enter N = 10, I/YR = 12, PV = -500, PMT = 0, and FV = ? Solve for FV = $1,552.92.

c. 0 1 2 3 4 5 6 7 8 9 10

| | | | | | | | | | | $500/(1.06)10 = $279.20.

PV = ? 500

Using a financial calculator, enter N = 10, I/YR = 6, PMT = 0, FV = 500, and PV = ? Solve for PV = $279.20.

d. 0 1 2 3 4 5 6 7 8 9 10

| | | | | | | | | | |

PV = ? 1,552.90

$1,552.90/(1.12)10 = $499.99.

Using a financial calculator, enter N = 10, I/YR = 12, PMT = 0, FV = 1552.90, and PV = ? Solve for PV = $499.99.

$1,552.90/(1.06)10 = $867.13.

Using a financial calculator, enter N = 10, I/YR = 6, PMT = 0, FV = 1552.90, and PV = ? Solve for PV = $867.13.

e. The present value is the value today of a sum of money to be received in the future. For example, the value today of $1,552.90 to be received 10 years in the future is about $500 at an interest rate of 12%, but it is approximately $867 if the interest rate is 6%. Therefore, if you had $500 today and invested it at 12%, you would end up with $1,552.90 in 10 years. The present value depends on the interest rate because the interest rate determines the amount of interest you forgo by not having the money today.

2-11 a. 2000 2001 2002 2003 2004 2005

| | | | | |

-6 12 (in millions)

With a calculator, enter N = 5, PV = -6, PMT = 0, FV = 12, and then solve for I/YR = 14.87%.

b. The calculation described in the quotation fails to consider the compounding effect. It can be demonstrated to be incorrect as follows:

$6,000,000(1.20)5 = $6,000,000(2.48832) = $14,929,920,

which is greater than $12 million. Thus, the annual growth rate is less than 20%; in fact, it is about 15%, as shown in part a.

2-12 These problems can all be solved using a financial calculator by entering the known values shown on the time lines and then pressing the I/YR button.

a. 0 1

| |

+700 -749

With a financial calculator, enter: N = 1, PV = 700, PMT = 0, and FV = -749. I/YR = 7%.

b. 0 1

| |

-700 +749

With a financial calculator, enter: N = 1, PV = -700, PMT = 0, and FV = 749. I/YR = 7%.

c. 0 10

| |

+85,000 -201,229

With a financial calculator, enter: N = 10, PV = 85000, PMT = 0, and FV = -201229. I/YR = 9%.

d. 0 1 2 3 4 5

| | | | | |

+9,000 -2,684.80 -2,684.80 -2,684.80 -2,684.80 -2,684.80

With a financial calculator, enter: N = 5, PV = 9000, PMT = -2684.80, and FV = 0. I/YR = 15%.

2-13 a. ?

| |

-200 400

With a financial calculator, enter I/YR = 7, PV = -200, PMT = 0, and FV = 400. Then press the N key to find N = 10.24. Override I/YR with the other values to find N = 7.27, 4.19, and 1.00.

b. ?

| | Enter: I/YR = 10, PV = -200, PMT = 0, and FV = 400.

-200 400 N = 7.27.

c. ?

| | Enter: I/YR = 18, PV = -200, PMT = 0, and FV = 400.

-200 400 N = 4.19.

d. ?

| | Enter: I/YR = 100, PV = -200, PMT = 0, and FV = 400.

-200 400 N = 1.00.

2-14 a. 0 1 2 3 4 5 6 7 8 9 10

| | | | | | | | | | |

400 400 400 400 400 400 400 400 400 400

FV = ?

With a financial calculator, enter N = 10, I/YR = 10, PV = 0, and PMT = -400. Then press the FV key to find FV = $6,374.97.

b. 0 1 2 3 4 5

| | | | | |

200 200 200 200 200

FV = ?

With a financial calculator, enter N = 5, I/YR = 5, PV = 0, and PMT = -200. Then press the FV key to find FV = $1,105.13.

c. 0 1 2 3 4 5

| | | | | |

400 400 400 400 400

FV = ?

With a financial calculator, enter N = 5, I/YR = 0, PV = 0, and PMT = -400. Then press the FV key to find FV = $2,000.

d. To solve part d using a financial calculator, repeat the procedures discussed in parts a, b, and c, but first switch the calculator to “BEG” mode. Make sure you switch the calculator back to “END” mode after working the problem.

1. 0 1 2 3 4 5 6 7 8 9 10

| | | | | | | | | | |

400 400 400 400 400 400 400 400 400 400 FV = ?

With a financial calculator on BEG, enter: N = 10, I/YR = 10, PV = 0, and PMT = -400. FV = $7,012.47.

2. 0 1 2 3 4 5

| | | | | |

200 200 200 200 200 FV = ?

With a financial calculator on BEG, enter: N = 5, I/YR = 5, PV = 0, and PMT = -200. FV = $1,160.38.

3. 0 1 2 3 4 5

| | | | | |

400 400 400 400 400 FV = ?

With a financial calculator on BEG, enter: N = 5, I/YR = 0, PV = 0, and PMT = -400. FV = $2,000.

2-15 a. 0 1 2 3 4 5 6 7 8 9 10

| | | | | | | | | | |

PV = ? 400 400 400 400 400 400 400 400 400 400

With a financial calculator, simply enter the known values and then press the key for the unknown. Enter: N = 10, I/YR = 10, PMT = -400, and FV = 0. PV = $2,457.83.

b. 0 1 2 3 4 5

| | | | | |

PV = ? 200 200 200 200 200

With a financial calculator, enter: N = 5, I/YR = 5, PMT = -200, and FV = 0. PV = $865.90.

c. 0 1 2 3 4 5

| | | | | |

PV = ? 400 400 400 400 400

With a financial calculator, enter: N = 5, I/YR = 0, PMT = -400, and FV = 0. PV = $2,000.00.

d. 1. 0 1 2 3 4 5 6 7 8 9 10

| | | | | | | | | | |

400 400 400 400 400 400 400 400 400 400

PV = ?

With a financial calculator on BEG, enter: N = 10, I/YR = 10, PMT = -400, and FV = 0. PV = $2,703.61.

2. 0 1 2 3 4 5

| | | | | |

200 200 200 200 200

PV = ?

With a financial calculator on BEG, enter: N = 5, I/YR = 5, PMT = -200, and FV = 0. PV = $909.19.

3. 0 1 2 3 4 5

| | | | | |

400 400 400 400 400

PV = ?

With a financial calculator on BEG, enter: N = 5, I/YR = 0, PMT = -400, and FV = 0. PV = $2,000.00.

Chapter 7

Look at these questions too:

7-4 The price of the bond will fall and its YTM will rise if interest rates rise. If the bond still has a long term to maturity, its YTM will reflect long-term rates. Of course, the bond’s price will be less affected by a change in interest rates if it has been outstanding a long time and matures shortly. While this is true, it should be noted that the YTM will increase only for buyers who purchase the bond after the change in interest rates and not for buyers who purchased previous to the change. If the bond is purchased and held to maturity, the bondholder’s YTM will not change, regardless of what happens to interest rates. For example, consider two bonds with an 8% annual coupon and a $1,000 par value. One bond has a 5-year maturity, while the other has a 20-year maturity. If interest rates rise to 15% immediately after issue the value of the 5-year bond would be $765.35, while the value of the 20-year bond would be $561.85.

7-6 As an investor with a short investment horizon, I would view the 20-year Treasury security as being more risky than the 1-year Treasury security. If I bought the 20-year security, I would bear a considerable amount of interest rate risk. Since my investment horizon is only one year, I would have to sell the 20-year security one year from now, and the price I would receive for it would depend on what happened to interest rates during that year. However, if I purchased the 1-year security I would be assured of receiving my principal at the end of that one year, which is the 1-year Treasury’s maturity date.

7-7 a. If a bond’s price increases, its YTM decreases.

b. If a company’s bonds are downgraded by the rating agencies, its YTM increases.

c. If a change in the bankruptcy code made it more difficult for bondholders to receive payments in the event a firm declared bankruptcy, then the bond’s YTM would increase.

d. If the economy entered a recession, then the possibility of a firm defaulting on its bond would increase; consequently, its YTM would increase.

e. If a bond were to become subordinated to another debt issue, then the bond’s YTM would increase.

Answers to the problems

7-1 With your financial calculator, enter the following:

N = 10; I/YR = YTM = 9%; PMT = 0.08 ( 1,000 = 80; FV = 1000; PV = VB = ?

PV = $935.82.

7-2 VB = $985; M = $1,000; Int = 0.07 ( $1,000 = $70.

a. Current yield = Annual interest/Current price of bond

= $70/$985.00

= 7.11%.

b. N = 10; PV = -985; PMT = 70; FV = 1000; YTM = ?

Solve for I/YR = YTM = 7.2157% ( 7.22%.

c. N = 7; I/YR = 7.2157; PMT = 70; FV = 1000; PV = ?

Solve for VB = PV = $988.46.

7-3 The problem asks you to find the price of a bond, given the following facts: N = 2 ( 8 = 16; I/YR = 8.5/2 = 4.25; PMT = 45; FV = 1000.

With a financial calculator, solve for PV = $1,028.60.

7-4 With your financial calculator, enter the following to find YTM:

N = 10 ( 2 = 20; PV = -1100; PMT = 0.08/2 ( 1,000 = 40; FV = 1000; I/YR = YTM = ?

YTM = 3.31% ( 2 = 6.62%.

With your financial calculator, enter the following to find YTC:

N = 5 ( 2 = 10; PV = -1100; PMT = 0.08/2 ( 1,000 = 40; FV = 1050; I/YR = YTC = ?

YTC = 3.24% ( 2 = 6.49%.

Since the YTC is less than the YTM, investors would expect the bonds to be called and to earn the YTC.

7-5 a. 1. 5%: Bond L: Input N = 15, I/YR = 5, PMT = 100, FV = 1000, PV = ?, PV = $1,518.98.

Bond S: Change N = 1, PV = ? PV = $1,047.62.

2. 8%: Bond L: From Bond S inputs, change N = 15 and I/YR = 8, PV = ?, PV = $1,171.19.

Bond S: Change N = 1, PV = ? PV = $1,018.52.

3. 12%: Bond L: From Bond S inputs, change N = 15 and I/YR = 12, PV = ?, PV = $863.78.

Bond S: Change N = 1, PV = ? PV = $982.14.

b. Think about a bond that matures in one month. Its present value is influenced primarily by the maturity value, which will be received in only one month. Even if interest rates double, the price of the bond will still be close to $1,000. A 1-year bond’s value would fluctuate more than the one-month bond’s value because of the difference in the timing of receipts. However, its value would still be fairly close to $1,000 even if interest rates doubled. A long-term bond paying semiannual coupons, on the other hand, will be dominated by distant receipts, receipts that are multiplied by 1/(1 + rd/2)t, and if rd increases, these multipliers will decrease significantly. Another way to view this problem is from an opportunity point of view. A 1-month bond can be reinvested at the new rate very quickly, and hence the opportunity to invest at this new rate is not lost; however, the long-term bond locks in subnormal returns for a long period of time.

7-7 Percentage

Price at 8% Price at 7% Change   

10-year, 10% annual coupon $1,134.20 $1,210.71 6.75%

10-year zero 463.19 508.35 9.75

5-year zero 680.58 712.99 4.76

30-year zero 99.38 131.37 32.19

$100 perpetuity 1,250.00 1,428.57 14.29

7-8 The rate of return is approximately 15.03%, found with a calculator using the following inputs:

N = 6; PV = -1000; PMT = 140; FV = 1090; I/YR = ? Solve for I/YR = 15.03%.

Despite a 15% return on the bonds, investors are not likely to be happy that they were called. Because if the bonds have been called, this indicates that interest rates have fallen sufficiently that the YTC is less than the YTM. (Since they were originally sold at par, the YTM at issuance= 14%.) Rates are sufficiently low to justify the call. Now investors must reinvest their funds in a much lower interest rate environment.

7-9 a. VB = [pic]

M = $1,000. PMT = 0.09($1,000) = $90.

1. VB = $829: Input N = 4, PV = -829, PMT = 90, FV = 1000, YTM = I/YR = ? I/YR = 14.99%.

2. VB = $1,104: Change PV = -1104, YTM = I/YR = ? I/YR = 6.00%.

b. Yes. At a price of $829, the yield to maturity, 15%, is greater than your required rate of return of 12%. If your required rate of return were 12%, you should be willing to buy the bond at any price below $908.88.

7-10 a. Solving for YTM:

N = 9, PV = -901.40, PMT = 80, FV = 1000

I/YR = YTM = 9.6911%.

b. The current yield is defined as the annual coupon payment divided by the current price.

CY = $80/$901.40 = 8.875%.

Expected capital gains yield can be found as the difference between YTM and the current yield.

CGY = YTM – CY = 9.691% – 8.875% = 0.816%.

Alternatively, you can solve for the capital gains yield by first finding the expected price next year.

N = 8, I/YR = 9.6911, PMT = 80, FV = 1000

PV = -$908.76. VB = $908.76.

Hence, the capital gains yield is the percent price appreciation over the next year.

CGY = (P1 – P0)/P0 = ($908.76 – $901.40)/$901.40 = 0.816%.

c. As long as promised coupon payments are made, the current yield will not change as a result of changing interest rates. However, as rates change they will cause the end-of-year price to change and thus the realized capital gains yield to change. As a result, the realized return to investors will differ from the YTM.

7-16 First, we must find the amount of money we can expect to sell this bond for in 5 years. This is found using the fact that in five years, there will be 15 years remaining until the bond matures and that the expected YTM for this bond at that time will be 8.5%.

N = 15, I/YR = 8.5, PMT = 90, FV = 1000

PV = -$1,041.52. VB = $1,041.52.

This is the value of the bond in 5 years. Therefore, we can solve for the maximum price we would be willing to pay for this bond today, subject to our required rate of return of 10%.

N = 5, I/YR = 10, PMT = 90, FV = 1041.52

PV = -$987.87. VB = $987.87.

You would be willing to pay up to $987.87 for this bond today.

7-18 First, we must find the price Joan paid for this bond.

N = 10, I/YR = 9.79, PMT = 110, FV = 1000

PV = -$1,075.02. VB = $1,075.02.

Then to find the one-period return, we must find the sum of the change in price and the coupon received divided by the starting price.

One-period return = [pic]

One-period return = ($1,060.49 – $1,075.02 + $110)/$1,075.02

One-period return = 8.88%.

Chapter 8

4. Yes, if the portfolio’s beta is equal to zero. In practice, however, it may be impossible to find individual stocks that have a nonpositive beta. In this case it would also be impossible to have a stock portfolio with a zero beta. Even if such a portfolio could be constructed, investors would probably be better off just purchasing Treasury bills, or other zero beta investments.

8-7 The risk premium on a high-beta stock would increase more than that on a low-beta stock.

RPj = Risk Premium for Stock j = (rM – rRF)bj.

If risk aversion increases, the slope of the SML will increase, and so will the market risk premium (rM – rRF). The product (rM – rRF)bj is the risk premium of the jth stock. If bj is low (say, 0.5), then the product will be small; RPj will increase by only half the increase in RPM. However, if bj is large (say, 2.0), then its risk premium will rise by twice the increase in RPM.

8-8 According to the Security Market Line (SML) equation, an increase in beta will increase a company’s expected return by an amount equal to the market risk premium times the change in beta. For example, assume that the risk-free rate is 6%, and the market risk premium is 5%. If the company’s beta doubles from 0.8 to 1.6 its expected return increases from 10% to 14%. Therefore, in general, a company’s expected return will not double when its beta doubles.

Problems

8-1 [pic] = (0.1)(-50%) + (0.2)(-5%) + (0.4)(16%) + (0.2)(25%) + (0.1)(60%)

= 11.40%.

(2 = (-50% – 11.40%)2(0.1) + (-5% – 11.40%)2(0.2) + (16% – 11.40%)2(0.4)

+ (25% – 11.40%)2(0.2) + (60% – 11.40%)2(0.1)

(2 = 712.44; ( = 26.69%.

CV = [pic] = 2.34.

8-3 rRF = 6%; rM = 13%; b = 0.7; r = ?

r = rRF + (rM – rRF)b

= 6% + (13% – 6%)0.7

= 10.9%.

8-4 rRF = 5%; RPM = 6%; rM = ?

rM = 5% + (6%)1 = 11%.

r when b = 1.2 = ?

r = 5% + 6%(1.2) = 12.2%.

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0%

5%

10%

0%

5%

10%

10%

0%

5%

10%

100%

18%

10%

7%

I/YR = ?

I/YR = ?

I/YR = ?

I/YR = ?

?

12%

6%

12%

6%

6%

6%

6%

6%

( (1.08)5

( (1.08)4

( (1.08)3

( (1.08)2

( (1.08)

8%

8%

7%

7%

12%

6.5%

I/YR = ?

7%

10%

0%

5%

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