End of Chapter 18 Questions and Answers
1. What is the present value of an offer of $15,000 one year from now if the opportunity
cost of capital (discount rate) is 12% per year simple interest?
Answer: 15000/1.12 = $13,393
2. What is the present value of an offer of $14,000 one year from now if the opportunity
cost of capital (discount rate) is 11% per year simple interest?
Answer: 14000/1.11=$12,613.3
3. What is the present value of an offer of $15,000 two years from now if the opportunity
cost of capital (discount rate) is 12% per year compounded annually?
Answer: 15000/1.122 = $11,958
4. What is the present value of an offer of $14,000 two years from now if the opportunity
cost of capital (discount rate) is 11% per year compounded annually?
Answer: 14,000/1.112 = $11,362.7
5. What is the future value of $20,000 that grows at an annual interest rate of 12% per
year for two years?
Answer: 20000*1.122 = $25,088
6. What is the future value of $25,000 which grows at an annual interest rate of 11% per
year for two years?
Answer: 25000*1.112 = $30,802.5
7. What is the present value of an offer of $15,000 one year from now if the opportunity
cost of capital (discount rate) is 12% per year nominal annual rate compounded monthly?
Answer: 15000/((1+(.12/12))12)=15000/1.0112=15000/1.126825 = $13,312
8. What is the present value of an offer of $14,000 one year from now if the opportunity
cost of capital (discount rate) is 11% per year nominal annual rate compounded monthly?
Answer: 14000/((1+(.11/12))12)=14000/1.00912=14000/1.1157 = $12,548.18
9. What is the future value of $20,000 which grows at a nominal annual interest rate of
12% per year, compounded monthly, for two years?
Answer: 20000*((1+(.12/12))(2*12)) = $25,395
10. What is the future value of $25,000 which grows at a nominal annual interest rate of
11% per year, compounded monthly, for two years?
Answer: 25000*((1+(.11/12))(2*12)) = $31,120.7
11. What is the effective annual rate (EAR) of 8% simple nominal annual rate when
compounded monthly?
Answer: (1+(.08/12))12 - 1 = .0830 = 8.30%
12. What is the effective annual rate (EAR) of 6.5% simple nominal annual rate
compounded monthly?
Answer: (1+(.065/12))12 - 1 = 6.697%
13. What is the monthly loan constant for a 10%, 25-year mortgage?
Answer: MMC = .009087008
14. What is the annualized mortgage constant in Problem #13 above?
Answer: Annualized Mortgage Constant = .109044096
15. If you invested $15,000 at one point in time and received back $30,000 five years
later, what annual interest (or growth) rate (compounded annually) would you have
obtained?
Answer: (30000/15000)(1/5) - 1 = 14.87%
16. If you invested $40,000 at one point in time and received back $100,000 seven years
later, what annual interest (or growth) rate (compounded annually) would you have
obtained?
Answer: (100000/40000) (1/7) – 1 = 13.98%
17. In question 15, what nominal annual rate compounded monthly would you have
obtained?
Answer: 12*((30000/15000)(1/(5*12) - 1) = 13.94%
18. In question 16, what nominal annual rate compounded monthly would you have
obtained?
Answer: 12*((100000/40000)(1/(7*12) - 1) = 13.156%
19. What is the mortgage balance on a $75,000 loan after 5 years with an original term of
30 years a 10% contract rate, and monthly payments?
Answer: $72,430.73
20. How much interest would be paid on problem #19 if paid over the full term?
Answer: $161,944.80 = $681.52 times 360 less $75,000
21. Use the "Rule of 72" to estimate how long it will take $15,000 to grow to $30,000 at
an annual growth rate of 10%.
Answer: DT = 72/(100*.10) = 72/10 = 7.2 years
22. A real estate investor feels that the cash flow from a property will enable her to pay a
lender $15,000 per year, at the end of every year, for 10 years. How much should the
lender be willing to loan her if he requires a 9% annual interest rate (annually
compounded, assuming the first of the 10 equal payments arrives one year from the date
the loan is disbursed)?
Answer: 15000*(1 - (1/1.09)10) / .09 = $96,265
23. You are borrowing $80,000 for 25 years at 10% nominal annual interest compounded
monthly. How much must your monthly payments be if you will completely retire the
loan over the 25-year period (i.e., what is the level payment annuity with a present value
of $80,000)?
Answer: 80000*(.10/12) / (1 - (1+(.10/12)))(25*12)) = $726.96
24. A tenant offers to sign a lease paying a rent of $1,000 per month, in advance (i.e., the
rent will be paid at the beginning of each month), for five years. At 10% nominal annual
interest compounded monthly, what is the present value of this lease?
Answer: 1000*(1+. 10/12)*(1-(1/(1+(.10/12)))(5*12))/(.10/12) = $47,458
25. A lender makes a $30,000 loan at 9.0% for 31 years with monthly payments of
$239.89. To bring the APR up to 9.5% how many points must be charged? Assume no
prepayment.
Answer: 4.369 points based on $1,310.64 over $30,000
26. What is the mortgage balance after 10 years on a 30-year loan at 6.0% with the
original loan at $400,000 and monthly payments assumed?
Answer: PMTs will be $2,398.20 and the PV of the remaining PMTs will be $334,774.90
after 10 years with 240 months left.
27. What is the present value of a 30-year mortgage at a contract rate of 9.0% with
monthly payments. The original loan is $90,000 and prepayment is assumed at the end of
the 12th year. The market yield required by the lender is 12.0%.
Answer: $73,588.84 based on the balance of $77,330.81 and the PMTs of $724.16 all
discounted over 12 years at 12%.
28. How much loan can a household afford if their gross income is $66,000 with current
mortgage rates at 8.125% over 30 years, monthly payments, ignoring the loan to value
constraint? Hint: Use 25% of household income as the limit for payments.
Answer: $185,185.88
29. How much loan could a commercial property support with NOI of $50,000 a debt
coverage ratio requirement of 1.25 and current rates at 9.0% for 25 years, with monthly
payments? Ignore loan to value constraints.
Answer: $50,000/1.25/12 is the payment. The PV is $397,205.41 based on 9.0% for 25
years.