Problems
Problem Instructions These problems are designed to test your understanding of the material and ability to apply what you have learned to situations that arise in practice – both personal and professional. I have tried to retain the spirit of what you will encounter in practice while recognizing that your knowledge to this point may be limited. As such, you may see similar problems in future modules that expand on these or incorporate important institutional features. Know that all of the problems can be solved with what you have learned in the current and preceding modules. Good luck!
Problem 1. Notation Which of the following present value notations denotes the value as of period 4 of a cash flow received in period 12? a) b) c) d) e)
PV0 (CF) PV0 (CF 12 ) PV4 (CF) PV4 (CF 12 ) PV12 (CF 4 )
Solution: d) 0
1
2
3
4
10
11
12 CF12
PV4(CF12)
Problem 2. Notation Which of the following present value notations denotes the value as of today of a cash flow received in period 6? a) b) c) d) e)
PV0 (CF 6 ) PV6 (CF 0 ) PV4 (CF) PV4 (CF 12 ) PV4 (CF 4 )
Solution: a) 0
1
2
3
4
5
6 CF6
PV0(CF6)
Problem 3. Inheritance You will receive an inheritance of $500,000 in 20 years on your 40th birthday. What is the value of the inheritance today if the discount rate is 10%? Solution: 0 ?
1
2
19
20 $500,000
Problem 4. Inheritance Your brother offers you $150,000 today for a claim to your future inheritance. Should you accept his offer? Solution:
Yes. The present value of your inheritance, $74,321, is substantially less than your brother’s offer, $150,000. Your brother should take finance.
Problem 5. Bond Price What is the present value (i.e., price) today of a bond that will pay its owner $1,000,000 five years from today if the discount rate is 4% per annum? (This is called a zero-coupon or pure discount bond) Solution: 0 1 ?
2
3
4
5 $1 mil
Problem 6. Bond Price The price today of a bond that will pay its owner $1,000,000 in five years is $747,258.17. What is the annual rate of return on this bond? (This rate is also called a bond yield or yield-to-maturity.) Solution: 0
1
2
3
4
$1 mil
$747,258.17 Price PV0 CF 5 747, 258.17
5
1,000,000
1 R 1/5
5
R 1 0.0600 747, 258.1729 1,000,000
Problem 7. Bond Price What is the price two years from today of a bond that will pay its owner $1,000,000 five years from today if the discount rate is fixed at 4% per annum? Solution: 0
1
2 ?
3
4
5 $1 mil
Problem 8. Education Some studies estimate that private college will cost $130,428 per year in 2030 (http://www.cnbc.com/id/47565202). Assuming your child will attend college for four years at a constant cost of $130,428 per year, how much money do you need at the start of their first year – when the first bill is due – to finance all of their college years if you can earn a risk-free return of 5%?
Problem 8. Education (Cont.) Solution: College 1 Year Period 0 130,428 ?
2
3
4
1
2
3
130,428
130,428
130,428
Problem 9. Education Continuing the previous problem, assume that you put the money into a savings account earning an annual risk-free return of 5% per annum. How much money will be in the account at the end of the first year after you make the second payment of $130,428? Solution:
Problem 10. Stock Return If you invested $100 in a portfolio of small stocks in 1925 and reinvested all dividends, your portfolio would be worth $2,655,590 in 2011. (This is true.) What is the typical annual rate of return on your investment? Solution:
0
1
2
85
100
PV0 CF86
86 $2,655,590
CF 86
1 R
86
100
2, 655, 590 86
1 R
1/86
2, 655, 590 R 100
1 0.1258
Problem 11. Company Value Your candy store generates enough after-tax profit to pay dividends of $50,000 per year. You plan on closing the store and liquidating all of the assets for $200,000 three years from today immediately after receiving the last dividend payment. What is the value of your store today if the discount rate is 12%, you do not owe any money (i.e., no debt), and the next dividend will be received one year from today?
Problem 11. Company Value (Cont) Solution: 0
1
2
3
?
50,000
50,000
250,000
