Tutorial Five
Problems
1. Periodic Interest Rates. In the following table, fill in the periodic rates and the effective annual rates.
Semiannual / 10% / 2Quarterly / 12% / 4
Monthly / 9% / 12
Daily / 5.25% / 365
2. Periodic Interest Rates. You have a savings account in which you leave the funds for one year without adding or withdrawing from the account. What would you rather have: a daily compounded rate of 0.045%, a weekly compounded rate of 0.305%, a monthly compounded rate of 1.5%, a quarterly compounded rate of 4.5%, a semiannually compounded rate of 9.75%, or an annually compounded rate of 19%?
3. EAR. What is the effective annual rate of a mortgage rate that is advertised at 7.25% (APR) over the next twenty years and paid with quarterly payments?
4. EAR. What is the effective rate of a monthly car loan that is advertised at 10% (APR)?
5. Present value with periodic rates. Let’s follow up with Sam Hinds, the dentist, from Chapter 4 and his remodeling project (Problem 12). The cost of the equipment for the project is $18,000, and the purchase will be financed with a 7.5% loan over six years. Originally, the loan called for annual payments. Redo the payments based on quarterly payments (four per year) and monthly payments (twelve per year). Compare the annual cash outflow of the two payments. Why does the monthly payment plan have less total cash outflow each year?
6. Present value with periodic rates. Cooley Landscaping needs to borrow $30,000 for a new front-end dirt loader. The bank is willing to loan the money at 8.5% interest for the next ten years with annual, semiannual, quarterly, or monthly payments. What are the different payments that Cooley Landscaping could choose for these different payment plans?
7. Future Value with Periodic Rates. Matt Johnson delivers newspapers and is putting away $20.00 every month from his paper route collections. Matt is thirteen years old and will use the money when he goes to college in five years. What will be the value of Matt’s account in five years with his monthly payments if he is earning 6% (APR), 8% (APR), or 12% (APR)?
8. Future value with periodic rates. We return to Denise, our hopeful millionaire from Chapter 4 (Example 4.3) and this chapter (Example 5.2). In Chapter 4, Denise was putting away $5,000 per year at the end of each year at 6% interest, with the expectation that in forty-four years she would be a millionaire. If Denise switches to a monthly savings plan and puts one-twelfth of the $5,000 away each month ($416.66), how much will she have in forty-four years at the 6% APR? Why is it more than the $1,000,000 goal? In this chapter, Denise was putting away $546.23 for thirty years at 9% to become a millionaire. Why does it take more per month when she is putting money away at 9% than when she was earning a lower rate of 6% over the forty-four years? Hint: what interest rate would she need for the 30 years putting away $416.66 to match the future value when she started fourteen years earlier at 6%?
So Denise would have to find an investment rate of 10.62% for the next 30 years if she wanted to match the same future value. If she can only get 9%, she will need to increase her monthly contributions to reach the same $1,076,759.95 future value.
9. Payments with periodic rates. What payment does Denise (from problem 8) need to make at the end of each month over the coming forty-four years at 6% to reach her retirement goal of $1,000,000?
10. Savings with Periodic Rates. What investment per month does Patrick need to make at the end of each month into his savings account over the coming eighteen months to reach his vacation goal of $8,000 if he is getting 5% APR on his account?
Tutorial Six
Problems
Bond Prices: Use the following table for problems 1 through 4.
Par Value / Coupon Rate / Years to Maturity / Yield to Maturity / Price$1,000.00 / 8% / 10 / 6% / ?
$1,000.00 / 6% / 10 / 8% / ?
$5,000.00 / 9% / 20 / 7% / ?
$5,000.00 / 12% / 30 / 5% / ?
1. Price the bonds from the above table with annual coupon payments.
2. Price the bonds from the above table with semiannual coupon payments.
ANSWER
Price = $1,000.00 × 1/(1.03)20 + $40.00 (1 – 1/(1.03)20)/ 0.03
Price = $1,000.00 × 0.5537 + $40.00 × 14.8775
Price = $553.67 + $595.10 = $1,148.77
Price = $1,000.00 × 1/(1.04)20 + $30.00 (1 – 1/(1.04)20)/ 0.04
Price = $1,000.00 × 0.4564 + $30.00 × 13.5903
Price = $456.39 + $407.71 = $864.10
Price = $5,000.00 × 1/(1.035)40 + $225.00 (1 – 1/(1.035)40)/ 0.035
Price = $5,000.00 × 0.2526 + $225.00 × 21.3551
Price = $1,262.86 + $4,804.90 = $6,067.75
Price = $5,000.00 × 1/(1.025)60 + $300.00 (1 – 1/(1.025)60)/ 0.025
Price = $5,000.00 × 0.2273 + $300.00 × 30.9087
Price = $1,136.41 + $9,272.60 = $10,409.01
3. Price the bonds from the above table with quarterly coupon payments.
ANSWER
Price = $1,000.00 × 1/(1.015)40 + $20.00 (1 – 1/(1.015)40)/ 0.015
Price = $1,000.00 × 0.5584 + $20.00 × 7.3601
Price = $558.39 + $588.81 = $1,085.84
Price = $1,000.00 × 1/(1.02)40 + $15.00 (1 – 1/(1.02)40)/ 0.02
Price = $1,000.00 × 0.4632 + $15.00 × 6.7101
Price = $463.19 + $402.60 = $863.22
Price = $5,000.00 × 1/(1.0175)80 + $75.00 (1 – 1/(1.0175)80)/ 0.0175
Price = $5,000.00 × 0.2584 + $75.00 × 10.5940
Price = $1,292.10 + $3,178.20 = $4,464
Price = $5,000.00 × 1/(1.0125)120 + $150.00 (1 – 1/(1.0125)120)/ 0.0125
Price = $5,000.00 × 0.2314 + $150.00 × 15.3725
Price = $1,156.89 + $9,223.47 = $10,423.50
4. Price the bonds from the above table with monthly coupon payments.
ANSWER
Price = $1,000.00 × 1/(1.005)120 + $6.67 (1 – 1/(1.005)120)/ 0.005
Price = $1,000.00 × 0.5584 + $6.67 × 7.3601
Price = $558.39 + $588.81 = $1,150.42
Price = $1,000.00 × 1/(1.0067)120 + $5.00 (1 – 1/(1.0067)120)/ 0.0067
Price = $1,000.00 × 0.4632 + $5.00 × 6.7101
Price = $463.19 + $402.60 = $860.13
Price = $5,000.00 × 1/(1.0058)240 + $37.50 (1 – 1/(1.0058)240)/ 0.0058
Price = $5,000.00 × 0.2476 + $37.50 × 128.98
Price = $1,238.01 + $4,836.84 = $6,074.85
Price = $5,000.00 × 1/(1.0042)360 + $50.00 (1 – 1/(1.0042)360)/ 0.0042
Price = $5,000.00 × 0.2314 + $50.00 × 15.3725
Price = $1,156.89 + $9,223.47 = 10,377.65
Yield-to-Maturity: Use the following table for problems 5 through 8.
Par Value / Coupon Rate / Years to Maturity / Yield to Maturity / Price$1,000.00 / 8% / 10 / ? / $1000.00
$1,000.00 / 6% / 10 / ? / $850.00
$5,000.00 / 9% / 20 / ? / $5,400.00
$5,000.00 / 12% / 30 / ? / $4,300.00
5. What is the yield of the above bonds if interest (coupon) is paid annually?
ANSWER
(TVM Keys) Set Calculator to P/Y = 1 and C/Y = 1
INPUT 10 ? -1000.00 80.00 1000.00
KEYS N I/Y PV PMT FV
CPT 8.0
(TVM Keys) Set Calculator to P/Y = 1 and C/Y = 1
INPUT 10 ? -850.00 60.00 1000.00
KEYS N I/Y PV PMT FV
CPT 8.2619
(TVM Keys) Set Calculator to P/Y = 1 and C/Y = 1
INPUT 20 ? -5400.00 450.00 5000.00
KEYS N I/Y PV PMT FV
CPT 8.1746
(TVM Keys) Set Calculator to P/Y = 1 and C/Y = 1
INPUT 30 ? -4300.00 600.00 5000.00
KEYS N I/Y PV PMT FV
CPT 13.9991
6. What is the yield of the above bonds if interest (coupon) is paid semiannually?
ANSWER
(TVM Keys) Set Calculator to P/Y = 2 and C/Y = 2
INPUT 20 ? -1000.00 40.00 1000.00
KEYS N I/Y PV PMT FV
CPT 8.0
(TVM Keys) Set Calculator to P/Y = 2 and C/Y = 2
INPUT 20 ? -850.00 30.00 1000.00
KEYS N I/Y PV PMT FV
CPT 8.2300
(TVM Keys) Set Calculator to P/Y = 2 and C/Y = 2
INPUT 40 ? -5400.00 225.00 5000.00
KEYS N I/Y PV PMT FV
CPT 8.1807
(TVM Keys) Set Calculator to P/Y = 2 and C/Y = 2
INPUT 60 ? -4300.00 300.00 5000.00
KEYS N I/Y PV PMT FV
CPT 13.9936
7. What is the yield of the above bonds if interest (coupon) is paid quarterly?
ANSWER
(TVM Keys) Set Calculator to P/Y = 4 and C/Y = 4
INPUT 40 ? -1000.00 20.00 1000.00
KEYS N I/Y PV PMT FV
CPT 8.0
(TVM Keys) Set Calculator to P/Y = 4 and C/Y = 4
INPUT 40 ? -850.00 15.00 1000.00
KEYS N I/Y PV PMT FV
CPT 8.2140
(TVM Keys) Set Calculator to P/Y = 4 and C/Y = 4
INPUT 80 ? -5400.00 112.50 5000.00
KEYS N I/Y PV PMT FV
CPT 8.1838
(TVM Keys) Set Calculator to P/Y = 4 and C/Y = 4
INPUT 120 ? -4300.00 150.00 5000.00
KEYS N I/Y PV PMT FV
CPT 13.9909
8. What is the yield of the above bonds if interest (coupon) is paid monthly?
ANSWER
(TVM Keys) Set Calculator to P/Y = 12 and C/Y = 12
INPUT 120 ? -1000.00 6.67 1000.00
KEYS N I/Y PV PMT FV
CPT 8.0
(TVM Keys) Set Calculator to P/Y = 12 and C/Y = 12
INPUT 120 ? -850.00 30.00 1000.00
KEYS N I/Y PV PMT FV
CPT 8.2033
(TVM Keys) Set Calculator to P/Y = 12 and C/Y = 12
INPUT 240 ? -5400.00 37.50 5000.00
KEYS N I/Y PV PMT FV
CPT 8.1859
(TVM Keys) Set Calculator to P/Y = 12 and C/Y = 12
INPUT 360 ? -4300.00 50.00 5000.00
KEYS N I/Y PV PMT FV
CPT 13.9891
Tutorial Seven
1. Anderson Motors Inc. has just set the company dividend policy at $0.80 per year. The company plans to be in business forever. What is the price of this stock if
a. An investor wants a 5% return?
b. An investor wants an 8% return?
c. An investor wants a 10% return?
d. An investor wants a 12.5% return?
e. An investor wants a 20% return?
ANSWER
Use the constant dividend infinite dividend stream model:
Price Dividend/r
a. Price $0.80/0.05 $16.00
b. Price $0.80/0.08 $10.00
c. Price $0.80/0.10 $8.00
d. Price $0.80/0.125 $6.40
e. Price $0.80/0.20 $4.00
2. Diettreich Electronics wants its shareholders to earn a 16% return on their investment in the company. At what price would the stock need to be priced today if Diettreich had a
a. $0.40 constant annual dividend forever?
b. $1.00 constant annual dividend forever?
c. $1.80 constant annual dividend forever?
d. $2.40 constant annual dividend forever?
ANSWER
Use the constant dividend with infinite horizon model:
Price Dividend/r
a. Price $0.40/0.16 $2.50
b. Price $1.00/0.16 $6.25
c. Price $1.80/0.16 $11.25
d. Price $2.40/0.16 $15.00
3. Singing Fish Fine Foods has a current annual cash dividend policy of $2.25. The price of the stock is set to yield a 12% return. What is the price of this stock if the dividend will be paid
a. for 10 years?
b. for 15 years?
c. for 40 years?
d. for 60 years?
e. for 100 years?
f. forever?
ANSWER
Use the finite constant dividend model except with f (use infinite constant dividend model)
Price = Dividend × (1 – 1/(1+r)n) / r
a. Price = $2.25 × (1 – 1/(1.12)10 / 0.12 = $2.25 × 5.6502 = $12.71
b. Price = $2.25 × (1 – 1/(1.12)15 / 0.12 = $2.25 × 6.8109 = $15.32
c. Price = $2.25 × (1 – 1/(1.12)40 / 0.12 = $2.25 × 8.2438 = $18.54
d. Price = $2.25 × (1 – 1/(1.12)60 / 0.12 = $2.25 × 8.3240 = $18.73
e. Price = $2.25 × (1 – 1/(1.12)100 / 0.12 = $2.25 × 8.3332 = $18.75
f. Price = $2.25 / 0.12 = $18.75
4. Pfender Guitars has a current annual cash dividend policy of $4.00. The price of the stock is set to yield an 8% return. What is the price of this stock if the dividend will be paid
a. for 10 years and then a liquidating or final dividend of $25.00?
b. for 15 years and then a liquidating or final dividend of $25.00?
c. for 40 years and then a liquidating or final dividend of $25.00?
d. for 60 years and then a liquidating or final dividend of $25.00?
e. for 100 years and then a liquidating or final dividend of $25.00?
f. forever with no liquidating dividend?
ANSWER
Use the finite constant dividend model liquidating dividend except with f (use infinite constant dividend model)
Price = Dividend × (1 – 1/(1 + r)n) / r + Liquidating Dividend × (1/(1 + r)n)
a. Price = $4.00 × (1 – 1/(1.08)10 / 0.08 + $25.00 × 1/1.0810
= $4.00 × 6.7101 + $25 × 0.4632 = $26.84 + $11.58 = $38.42
b. Price = $4.00 × (1 – 1/(1.08)15 / 0.08 + $25.00 × 1/1.0815
= $4.00 × 8.5595 + $25 × 0.3152 = $34.24 + $7.88 = $42.12
c. Price = $4.00 × (1 – 1/(1.08)40 / 0.08 + $25.00 × 1/1.0840
= $4.00 × 11.9246 + $25 × 0.0460 = $47.70 + $1.15 = $48.85
d. Price = $4.00 × (1 – 1/(1.08)60 / 0.08 + $25.00 × 1/1.0860
= $4.00 × 12.3766 + $25 × 0.0099 = $49.51 + $0.24 = $49.75
e. Price = $4.00 × (1 – 1/(1.08)100 / 0.08 + $25.00 × 1/1.08100
= $4.00 × 12.4943 + $25 × 0.0005 = $49.98 + $0.01 = $49.99
f. Price = $4.00 / 0.08 = $50.00
5. King Waterbeds has an annual cash dividend policy that raises the dividend each year by 4%.
Last year’s dividend was $0.50 per share. What is the price of this stock if
a. an investor wants a 6% return?
b. an investor wants an 9% return?
c. an investor wants a 10% return?
d. an investor wants a 14% return?
e. an investor wants a 20% return?
ANSWER
Use the constant growth dividend model with infinite horizon:
Price Last Dividend X(1 g)/(r g)
a. Price $0.50 X(1.04)/(0.06 0.04) $0.52/0.02 $26.00
b. Price $0.50 X(1.04)/(0.09 0.04) $0. 52/0.05 $10.40
c. Price $0.50 X(1.04)/(0.10 0.04) $0. 52/0.06 $8.67
d. Price $0.50 X(1.04)/(0.14 0.04) $0. 52/0.10 $5.20