Multiple Choice Questions 11. Section: 5.4 Annuities and Perpetuities B. Chapter 5: Time Value of Money 1 1 n (1 + k) 1 (1.15) PMT $,,(6.5933) $1, 519 k.15 N, I/Y15, PMT,, FV, CPT 1,519 14. Section: 5.7 Loan or Mortgage Arrangements Learning Objective: 5.7 B of annuity of 1 remaining payments at 1% per month. 1 1 n (1 + k) 1 1 (1.1) PMT $3,33.6 3,33.6(69.75) $3,38. 95 k.1 Using a financial calculator (TI BAII Plus), N 1, I/Y 1, PMT -3,33.6, FV, CPT 3,38.95 15. Section: 5.4 Annuities and Perpetuities A. The future value of a perpetuity cannot be computed as it is infinite. Practice Problems Basic 9. Section: 5.4 Annuities and Perpetuities Solutions Manual 1 Chapter 5
Level of difficulty: Basic The value of any perpetual stream of payments can be valued as a perpetuity: PMT $ $16.67 k.1 Each share is worth $16.67. Intermediate 43. Section: 5.4 Annuities and Perpetuities We have two separate annuities to consider: the tuition payments, and the savings amounts. First, find the present value of the four annual tuition payments (at time 8, when Felix is due to begin university studies): 1 (1.7) $1, +.7 4 8 $33,87.11 This is the amount of savings required at time 8. From the perspective of time, this is a future value amount (replaces the $4, in Problem 45.) Next, find the annual savings amount: 8 (1 +.7) 1 $33,87.11 PMT PMT $3,31.44.7 48. Section: 5.4 Annuities and Perpetuities; 5.6 Quoted versus Effective Rates ; 5.6 a. There will be 5 x 1 6 monthly payments. The calculations are most easily done with a financial calculator (TI BAII Plus), 5,, PMT556.11, N 6, CPT I/Y 1.% Because we used monthly payments, and months as the time period, 1.% is the effective monthly rate. b. The compounding period matches the payment frequency, so the nominal rate, or quoted rate, is: Solutions Manual Chapter 5
QR m kmonthly 1 1.% 1.% per year. 49. Section: 5.6 Quoted versus Effective Rates Learning Objective: 5.6 a. Scott will pay interest of ($8 $75) $5 after one week. This implies a nominal interest rate of $5/$75 6.67% per week. With 5 weeks in the year, the nominal rate per year is then 5 x 6.67% 346.84%. 5 b. The effective annual interest rate is k (1 +.667) 1 7.71,77.1% 51. Section: 5.7 Loan or Mortgage Arrangements Learning Objective: 5.7 With semi-annual compounding (the norm in Canada) and monthly payments, m and f1.the effective monthly rate is: k m f monthly QR 1 + m.39 1 1 + 1 1.34% The present value of the mortgage payments over the amortization period (5 years x 1 3 months) is: 1 (1 +.34) $1,95..34 3 $374,553.7 N3, I/Y.34, PMT-1,95, FV, CPT $374,553.7 In addition, Charlie has $13, available as a down payment; the most he can pay for the house is, therefore, $374,553.7 + $13, $54,553.7. 53. Section: 5.8 Comprehensive Examples Learning Objective: 5.8 Solutions Manual 3 Chapter 5
a. 1 st Calculate their yearly income available for investment Monthly income available $9, $3, $85 $1,45 $3,7 Yearly available $(3,7)(1) $44,4 nd Calculate the FV of their investment when they retire: 3 (1 +.1) 1 FV 44,4 $7,33,535 3.1 N3, I/Y1,, PMT- 44,4, CPT FV7,33,535 3 rd Calculate the amount they will have when they retire: $7,33,535 + $5, $7,353,535 b. This is an annuity due problem. 7,353,535, k1%, n3 1 (1 +.1) 7,353,535 PMT.1 3 (1 +.1) So, PMT$79,143 Hit [ nd ] [BGN] [ nd ] [Set] N3, I/Y1, - 7,353,535, FV, CPT PMT79,143 Challenging 54. Section: 5.1 Opportunity Cost; 5.3 Compound Interest Learning Objective: 5.1; 5.3 Find the present value of the money paid back to Veda by each investment, using the interest rate on the alternative (the bank account) as the discount rate. $5 $8 For Investment A: + $453.51 + $691.7 $1144. 58 3 (1 +.5) (1 +.5) For Investment B: $ $4 $7 + + $19.48 + $36.81 + $64.69 $1157.98 1 3 (1 +.5) (1 +.5) (1 +.5) Veda would prefer Investment B, because it has the higher present value. 55. Section: 5.4 Annuities and Perpetuities Solutions Manual 4 Chapter 5
The dividends for the first five years form an ordinary annuity. Starting in year 6, the reduced dividend stream can be thought of as a perpetuity. However, the value of this perpetuity, as determined by our formula, occurs at year 5 (one year before the first $ dividend), and must be discounted to the present: 1 (1.1) 3. +.1 $. 1 +.1 (1 +.1) 5 $ 5 [ $1.81] + [ $16.67.5674] $. 7 63. Section: 5.3 Compound Interest; 5.6 Quoted versus Effective Rates Learning Objective: 5.3; 5.6 Let s assume the present value of the investment is $1. The future value, after doubling, is then $. a. Annually: With annual compounding, the effective rate is the same as the quoted rate, 9%. Using a financial calculator (TI BAII Plus), 1, FV, I/Y 9, PMT, CPT N 8.4 So the investment will double in just over 8 years. b. Quarterly: With quarterly compounding, the effective annual rate is,.9 k 1 + 4 4 1 9.383%, and a financial calculator allows us to find: -1, FV, I/Y 9.383, PMT, CPT N 7.79 The higher effective rate means that only 7.79 years are needed to double the value of the investment. 66. Section: 5.7 Loan or Mortgage Arrangements Learning Objective: 5.7 a. The effective monthly interest rate is, Solutions Manual 5 Chapter 5
k monthly.51 1 + 1 1.46% The amount of the mortgage loan will be ($8, $5,) $3,, and there will be 1 x 5 3 monthly payments, the value of which can be found with a financial calculator, (TI BAII Plus), N3, 3,, I/Y.46, FV, CPT PMT 1,35.89. Alysha s two friends will be paying x $475 $95 in rent, so she will need an additional $1,35.89 $95 $4.89 to make the mortgage payments. b. In two years, Alysha will have made 4 payments, leaving 76. The present value of these payments is the outstanding value of the mortgage loan. Use the calculator again: N76, I/Y.46, PMT 135.89, FV, CPT,336.58.To pay off the loan, and recoup her down payment, Alysha would have to sell the house for at least $,336.58 + $5, $7,336.58. 71. Section: 5.8 Comprehensive Examples Learning Objective: 5.8 Investor A: ke.15 116.18344%. 1 st, consider an ordinary annuity and the present value of the investment when A turns 5 years old is: 1 8 (1 +.1618344) 5 $5,5 $3,749.19.1618344 N8, I/Y16.18344, PMT5,5, FV, CPT - 3,749.19 nd, discount this amount for five years back to today when she is. 1 (1 + k) 1 $3,749.19 (1.1618344) FV5 5 5 $11,18.331 Or, N5, I/Y16.18344, PMT, FV- 3,749.19, CPT 11,18.331 Investor B:.16 k (1 + ) 4 116.985856% 4 Solutions Manual 6 Chapter 5
1 (1.16985856) $11,18.331 + PMT.16985856 1 (1.16985856) PMT$,57.38 Hit [ nd ] [BGN] [ nd ] [Set] N1, I/Y16.985856, 11,18.331, FV, CPT PMT -,57.38 Therefore, Investor B has to make a yearly payment of $,57.38 so that the present value of the two investments is the same. Solutions Manual 7 Chapter 5