8 compound interest: future value and present...
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96 Fundamentals of Business Mathematics in Canada, 1/e
8 Compound Interest: Future Value and Present Value Checkpoint Questions (Section 8.1) 1. We compound interest when we convert it to principal and calculate subsequent interest
on both the principal and the converted interest.
3. The “periodic rate of interest” is the percent interest earned in a single compounding period. The “nominal rate of interest” is the annual interest rate you obtain if you extend the periodic interest rate to a full year. This extension is done by multiplying the periodic rate of interest by the number of compounding periods in a year.
Exercise 8.1 Basic Problems
1. a. i = mj =
12%0.6 = 0.5% per month
b. i = mj =
4%0.6 = 1.5% per quarter
c. i = mj =
2%0.6 = 3.0% per half year
3. a. m = ij =
%65.1%6.6 = 4 (that is, quarterly compounding)
b. m = ij =
%3.3%6.6 = 2 (that is, semiannual compounding)
c. m = ij =
%55.0%6.6 = 12 (that is, monthly compounding)
5. a. j = mi = 2(3.6%) = 7.2% compounded semiannually b. j = mi = 4(1.8%) = 7.2% compounded quarterly c. j = mi = 12(0.6%) = 7.2% compounded monthly
Math App (Section 8.2) The “Magic” of Compound Interest
1. Growth ratio = ( )( )
( )( ) $100$215.89
$100$466.10$1001.08$100$1001.08$100
11
10
20
1
2
−−
=−
−=
−+
−+
PViPVPViPV
n
n= 3.16
That is, the growth over 20 years will be 3.16 times the growth over 10 years.
3. A higher rate of return causes growth to accelerate disproportionately as time passes. Therefore, we expect that the growth ratio will be larger at a 10% rate of return than at an 8% rate of return. Checking our intuition,
Growth ratio = ( )( )
( )( ) $100$259.37
$100$672.75$1001.10$100$1001.10$100
11
10
20
1
2
−−
=−
−=
−+
−+
PViPVPViPV
n
n= 3.59
Chapter 8: Compound Interest: Future Value and Present Value 97
Checkpoint Questions (Section 8.2) 1. The future value of an investment at a future date is the combined value of the
investment’s principal and interest on that date.
3. The more frequent the compounding of the 6% nominal rate, the more interest will be earned by the investment. Therefore, 6% compounded quarterly is the preferred rate. (The other two rates both earn 3% interest in the 6-month term.)
(1) 6% compounded quarterly; (2) 6% compounded semiannually and 6% simple interest (tied)
Exercise 8.2 Basic Problems
1. i = mj =
26% = 3% (per half year)
n = m(Term in years) = 2(7) = 14 compounding periods Maturity value, ( )ni+PVFV 1= = $5000 ( )141.03 = $7562.95
3. i =12
%4.5 = 0.45% (per month); n = 12(6.75) = 81 periods
Maturity value = ( )ni+PV 1 = $4400 ( )811.0045 = $6329.92
5. a. i =2%0.6 = 3.0% (per six months); n = 2(25) = 50 periods
Maturity value, ( )ni+PVFV 1= = $10,000 ( )501.03 = $43,839.06
b. i =2%0.7 = 3.5% (per six months); n = 2(25) = 50 periods
Maturity value, ( )ni+PVFV 1= = $10,000 ( )501.035 = $55,849.27
c. i =2%0.8 = 4.0% (per six months); n = 2(25) = 50 periods
Maturity value, ( )ni+PVFV 1= = $10,000 ( )501.04 = $71,066.83
7. i n Maturity amount
a. 60128 .% = % 12 $100 ( )126001. = $108.30
b. %.%. 02524
18 = 4 $100 ( )4020251. = $108.35
c. %.%. 14228 = 2 $100 ( )20411. = $108.37
d. 8.3% 1 $100 ( )10831. = $108.30
The investor would prefer 8.2% compounded semiannually since it produces the highest maturity value.
98 Fundamentals of Business Mathematics in Canada, 1/e
Exercise 8.2 (continued) Intermediate Problems 9. PV = $12,000; i = 4
%2.7 = 1.8%; n = 4(1.5) = 6
Maturity value, ( )ni+PVFV 1= = $12,000 ( )6018.1 = $13,355.74 Interest charged = FV – PV = $13,355.74 – $12,000 = $1355.74
11. Maturity value at 9% = ( )ni+PV 1 = $10,000 ( )2509.1 = $86,230.81
Maturity value at 8% = ( )ni+PV 1 = $10,000 ( )2508.1 = $68,484.75 Difference: $17,746.06
The difference is 75.484,68$06.746,17$ ×100% = 25.91% of the maturity value at 8% compounded
annually.
13. Interest rate 20 years 25 years 30 years 8% $4660.96 $6848.48 $10,062.66 10% $6727.50 $10,834.71 $17,449.40
15. For the first 2.5 years, i = 1248 %. = 0.7%; n = 12(2.5) = 30
The amount owed after 2.5 years was ( )ni+PVFV 1= = $5000 ( )300071. = $6163.879
For the remaining 2 years, i = 257 %. = 3.75%; n = 2(2) = 4
The total amount required to repay the loan at the end of 4.5 years was FV = $6163.879 ( )403751. = $7141.78
17. Equivalent amount = ( )ni+PVFV 1= = $10,000 ( )48004583333.1 = $12,454.51
19. PV = $2300; i = 2%25.6 = 3.125%; n = 2(1.5 + 2) = 7
Equivalent amount = ( )ni+PVFV 1= = $2300 ( )703125.1 = $2852.83
21. For the first 2 years, PV = $12,000; i = 26% = 3.0%; n = 2(2) = 4
Amount owed after 2 years = ( )ni+PV 1 = $12,000 ( )41.03 = $13,506.106 Amount owed after $3000 payment = $10,506.106 Amount owed after 4 years = ( )ni+PV 1 = $10,506.106 ( )41.03 = $11,824.715 Amount owed after $3000 payment = $8824.715 Amount owed after 6 years = ( )ni+PV 1 = $8824.715 ( )41.03 = $9932.29
23. For the first payment, PV = $1300, i = 4%6 = 1.5%; n =4(4) = 16
For the second payment, PV = $1800, i = 1.5%; n = 4(2.25) = 9 The combined equivalent value 4 years from now is
( ) ( )916 1.015$18001.015$1300 + = $1649.681 + $2058.102 = $3707.78
Chapter 8: Compound Interest: Future Value and Present Value 99
Exercise 8.2 (continued) Advanced Problems
25. Principal i n Maturity amount $3000 2.5% 6 $3000 ( )60251. = $ 3479.08 $3500 2.5% 4 $3500 ( )40251. = $ 3863.35 $4000 2.5% 2 $4000 ( )20251. = $ 4202.50 Total amount owed = $11,544.93
27. For the first 1 2
1 years, PV = $1900; i = 27% = 3.5%; n = 2(1.5) = 3
Amount owed after 1 21 years = ( )ni+PV 1 = $1900 ( )31.035 = $2106.564
Amount owed after $1000 payment = $1106.564 Amount owed today = $1106.564 ( )41.035 = $1269.81
Checkpoint Questions (Section 8.3) 1. The “discount rate” is the interest rate used in a present value calculation.
3. Discounting is the opposite of compounding.
Exercise 8.3 Basic Problems 1. Given: i = 1
6.5% = 6.5% (per year); FV = $10,000 a. n = 1(10) = 10 periods
Principal, ( ) ni+FVPV −= 1 = ( ) 101.065$10,000 − = $5327.26
b. n = 1(20) = 20 periods Principal, ( ) ni+FVPV −= 1 = ( ) 201.065$10,000 − = $2837.97
c. n = 1(30) = 30 periods Principal, ( ) ni+FVPV −= 1 = ( ) 301.065$10,000 − = $1511.86
3. Given: FV = $10,000 a. i = 1
8% = 8% (per year); n = 1(25) = 25 periods
Principal, ( ) ni+FVPV −= 1 = ( ) 251.08$10,000 − = $1460.18
b. i = 28% = 4% (per half year); n = 2(25) = 50 periods
Principal, ( ) ni+FVPV −= 1 = ( ) 501.04$10,000 − = $1407.13
c. i = 48% = 2% (per quarter);n = 4(25) = 100 periods
Principal, ( ) ni+FVPV −= 1 = ( ) 1001.02$10,000 − = $1380.33
d. i = 128% = 6.0 % (per month); n = 12(25) = 300 periods
Principal, ( ) ni+FVPV −= 1 = ( ) 30061.00$10,000 − = $1362.37
100 Fundamentals of Business Mathematics in Canada, 1/e
Exercise 8.3 (continued) 5. Given: i = 4
%5.8 = 2.125% (per quarter); n = 4( 1227 ) = 9; FV = $5437.52
( ) ni+FVPV −= 1 = ( ) 902125152.5437 −.$ = $4500.00
7. Given: i = 12%5 = %641.0 (per month); n = 12( 12
18 ) = 18; FV = $8000
( ) ni+FVPV −= 1 = ( ) 18604118000 −.$ = $7423.10
9. Given: i = 2%5 = 2.5% (per half year); n = 2(3) = 6; FV = $6500
( ) ni+FVPV −= 1 = ( ) 602516500 −.$ = $5604.93
Intermediate Problems
11. Equivalent amount = ( ) ni+FVPV −= 1 = $2600 ( ) 331.0045 − = $2241.95
13. Given: i = 2%2.6 = 3.1%; n = 2(6.5) = 13; FV = $7000
Equivalent amount = ( ) ni+FVPV −= 1 = $7000 ( ) 13031.1 − = $4706.90
15. The economic value of an offer is its present value. PV of second offer = $49,000 + $49,000 ( ) 2011.1 − + $49,000 ( ) 4011.1 −
= $49,000 + $47,939.53 + $46,902.01 = $143,841.54
The $145,000-cash offer is worth $1158.46 more.
17. Equivalent value 3 years from now of the $1400 payment is ( )ni+PVFV 1= = $1400 ( )120151. = $1673.87 Equivalent value 3 years from now of the $1800 payment is ( ) ni+FVPV −= 1 = $1800 ( ) 80151 −. = $1597.88 Combined equivalent value = $1673.87 + $1597.88 = $3271.75
19. Sum of the equivalent values of the payments, 1 year from now, = $1000 ( ) 503875.1 − + $2000 ( ) 9038751 −. = $826.88 + $1420.47 = $2247.35
21. a. Sum of the equivalent values of the payments, 3 years from now, = $6500 ( )41.015 + $8500 ( ) 12015.1 − = $6898.86 + $7109.29 = $14,008.16
(continued)
Chapter 8: Compound Interest: Future Value and Present Value 101
Exercise 8.3 (continued) 21. b. If Johann receives $14,008.16 in 3 years from now, he can invest it at 6%
compounded quarterly for 3 years. Therefore, in 6 years from now he will have ( )ni+PVFV 1= = $14,008.16 ( )12015.1 = $16,748.41.
If Maria had made the originally scheduled payments on time, Johann would have invested the $6500 payment for 4 years. At the end of 6 years, that payment would be worth
( )ni+PVFV 1= = $6500 ( )16015.1 = $8248.41. Then, Maria would provide the $8500 payment, giving Johann a total at the end of
the 6 years of $8248.41 + $8500.00 = $16,748.41.
Therefore, the payment of $14,008.16 in three years from now places Johann in exactly the same financial position as if Maria had made the originally scheduled payments. In either case, Johann will have $16,748.41 at the end of six years.
23. Equivalent payment 1 year from now = $2500 ( )31.0175 + $2500 ( ) 401751 −. = $2633.56 + $2332.40 = $4965.96
25.
Economic value of Joe’s contract = Present value of payments
= $4.8M + 025.1
M10$ + ( )3025.1
M2.17$ + ( )5025.1
M5.17$ + ( )7025.1
M5.18$
= $4,800,000 + $9,756,097 + $15,971,910 + $15,467,450 + $15,563,407 = $61,559,000 rounded to the nearest $1000
Advanced Problems 27. The creditor should accept the economic value, 2¼ years from now, of the three scheduled payments. Allowing for a time value of money of 6% compounded quarterly, the economic value is $2700 ( )51.015 + $1900 ( )31.015 + $1100 ( ) 30151 −. = $2908.667 + $1986.789 + $1051.949 = $5947.41 The creditor should accept $5947.41.
29. Let x represent the size of the loan payments. $4000 = Present value of the payments
= x ( ) 1030081 −. + x ( ) 1530081 −. = 0.9203622x + 0.8829541x
x = $2218.14 Each of the loan payments is $2218.14.
Time: 0 yr 0.5 yr 1.5 yr 2.5 yr 3.5 yr Salary: $4.8 M $10 M $17.2 M $17.5 M $18.5 M
102 Fundamentals of Business Mathematics in Canada, 1/e
Exercise 8.4 Financial Calculator Solutions to Odd-Numbered Problems in Exercise 8.2 Basic Problems 1. Given: PV = $5000; j = 6% ; m = 2 n = m(Term in years) = 2(7) = 14 compounding periods Maturity value, FV = $7562.95
3. Given: PV = $4400; j = 5.4%; m = 12 n = m(Term in years) = 12(6.75) = 81 compounding periods Maturity value, FV = $6329.92
5. Given: PV = $10,000; n = 2(25) = 50; m = 2 for all parts a. j = 6% b. j = 7% Maturity amount, FV = $55,849.27
Maturity amount, FV = $43,839.06
6 I/Y
P/Y 2 ENTER (making C/Y = P/Y = 2)
14 N
5000 + / – PV
0 PMT
CPT FV
Ans: 7562.95
5.4 I/Y
P/Y 12 ENTER (making C/Y = P/Y = 12)
81 N
4400 + / – PV
0 PMT
CPT FV
Ans: 6329.92
6 I/Y
P/Y 2 ENTER (making C/Y = P/Y = 2)
50 N
10000 + / – PV
0 PMT
CPT FV
Ans: 43,839.06
Same P/Y, C/Y, N, PV, PMT 7 I/Y
CPT FV
Ans: 55,849.27
Chapter 8: Compound Interest: Future Value and Present Value 103
Exercise 8.4 (Financial Calculator Solutions to Exercise 8.2 continued) 5. c. j = 8% Maturity amount, FV = $71,066.83
7. Given: PV = $100; Term = 1 year for all parts a. j = 8.0%; m = 12; n = 12(1) = 12 b. j = 8.1%; m = 4; n = 4(1) = 4 c. j = 8.2%; m = 2; n = 2(1) = 2 d. j = 8.3%; m = 1; n = 1(1) = 1 Choose 8.2% compounded semiannually since it produces the highest maturity value.
Intermediate Problems 9. Given: PV = $12,000; Term = 18 months = 1.5 years;
j = 7.2%; m = 4 n = m(Term) = 4(1.5) = 6 compounding periods Maturity value, FV = $13,355.74 Interest charged = FV – PV
= $13,355.74 – $12,000 = $1355.74
Same P/Y, C/Y, N, PV, PMT 8 I/Y
CPT FV
Ans: 71,066.83
8 I/Y
P/Y 12 ENTER (making C/Y = P/Y = 12)
12 N
100 + / – PV
0 PMT
CPT FV
Ans: 108.30
Same PV, PMT 8.1 I/Y
P/Y 4 ENTER (making C/Y = P/Y = 4)
4 N
CPT FV
Ans: 108.35
Same PV, PMT 8.2 I/Y
P/Y 2 ENTER (making C/Y = P/Y = 2)
2 N
CPT FV
Ans: 108.37
Same PV, PMT 8.3 I/Y
P/Y 1 ENTER (making C/Y = P/Y = 1)
1 N
CPT FV
Ans: 108.30
7.2 I/Y
P/Y 4 ENTER (making C/Y = P/Y = 4)
6 N
12000 PV
0 PMT
CPT FV
Ans: -13,355.74
104 Fundamentals of Business Mathematics in Canada, 1/e
Exercise 8.4 (Financial Calculator Solutions to Exercise 8.2 continued) 11. Given: PV = $10,000; Term = 25 years Maturity value at j = 9%, m = 1: Maturity value at j = 8%, m = 1:
13. The solution for the case PV = $1000, j = 8%, m = 1, and n = 1(20) = 20 is presented in the box to the right. Interest rate 20 years 25 years 30 years 8% $4660.96 $6848.48 $10,062.66 10% $6727.50 $10,834.71 $17,449.40 15. For the initial 2.5 years of the loan: Given: PV = $5000; j = 8.4%; m = 12
Term = 2.5 years n = 12(2.5) = 30 compounding periods
After the initial 2.5 years, the balance owing on the loan is FV = $6163.879
For the remaining 2 years of the loan: Given: PV = $6163.879; j = 7.5%; m = 2
Term = 2 years n = 2(2) = 4 compounding periods
After the remaining two years, the balance owing on the loan is FV = $7141.78
The difference is $86,230.81 − $68,484.75 = $17,746.06
which is 75.484,68$06.746,17$ ×100% = 25.91% of the maturity
value at 8% compounded annually.
9 I/Y
P/Y 1 ENTER (making C/Y = P/Y = 1)
25 N
10000 + / – PV
0 PMT
CPT FV
Ans: 86,230.81
Same P/Y, C/Y Same N, PV, PMT,
8 I/Y
CPT FV
Ans: 68,484.75
8 I/Y
P/Y 1 ENTER (making C/Y = P/Y = 1)
20 N
1000 + / – PV
0 PMT
CPT FV
Ans: 4660.96
8.4 I/Y
P/Y 12 ENTER (making C/Y = P/Y = 12)
30 N
5000 PV
0 PMT
CPT FV
Ans: -6163.879
7.5 I/Y
P/Y 2 ENTER (making C/Y = P/Y = 2)
4 N
6163.879 PV
0 PMT
CPT FV
Ans: -7141.78
Chapter 8: Compound Interest: Future Value and Present Value 105
Exercise 8.4 (Financial Calculator Solutions to Exercise 8.2 continued)
17. Given: PV = $10,000; j = 5.5%; m = 12 Term = 4 years n = 12(4) = 48 compounding periods
Equivalent value, FV = $12,454.51
19. Given: PV = $2300; j = 6.25%; m = 2 Term = 3.5 years n = 2(3.5) = 7 compounding periods
Equivalent value, FV = $2852.83
21. Given: PV = $12,000; j = 6%; m = 2
Term = 2 years n = 2(2) = 4 compounding periods
After 2 years, the amount owing on the loan is FV = $13,506.106 After the payment of $3000, the balance owing is $10,506.106.
For the next 2 years of the loan: Given: PV = $10,506.106; j = 6%; m = 2
Term = 2 years n = 2(2) = 4 compounding periods
At the end of four years, the balance owing on the loan is FV = $11,824.715. After the payment of $3000, the balance owing is $8824.715
(continued)
5.5 I/Y
P/Y 12 ENTER (making C/Y = P/Y = 12)
48 N
10000 + / – PV
0 PMT
CPT FV
Ans: 12,454.51
6.25 I/Y
P/Y 2 ENTER (making C/Y = P/Y = 2)
7 N
2300 PV
0 PMT
CPT FV
Ans: -2852.83
Same I/Y, P/Y, C/Y, N, PMT 10506.106 PV
CPT FV
Ans: -11,824.715
6 I/Y
P/Y 2 ENTER (making C/Y = P/Y = 2)
4 N
12000 PV
0 PMT
CPT FV
Ans: -13,506.106
106 Fundamentals of Business Mathematics in Canada, 1/e
Exercise 8.4 (Financial Calculator Solutions to Exercise 8.2 continued) For the next 2 years of the loan: Given: PV = $8824.715; j = 6%; m = 2
Term = 2 years n = 2(2) = 4 compounding periods
On the sixth anniversary of the loan, a payment of FV = $9932.29 will extinguish the debt.
23. The amount owed will be the combined future value. For each amount borrowed, j = 6%, m = 4 PV = $1300; n = 4(4) = 16: PV = $1800; n = 4(2.25) = 9
Total amount owed = $1649.68 + $2058.10 = $3707.78
Advanced Problems 25. The amount owed will be the combined future value. For each amount borrowed, j = 5%, m = 2 PV = $3000; n = 3(2) = 6: PV = $3500; n = 4: PV = $4000; n = 2: Total amount owed = $3479.08 + $3863.35 + $4202.50
= $11,544.93
6 I/Y
P/Y 4 ENTER (making C/Y = P/Y = 4)
16 N
1300 PV
0 PMT
CPT FV
Ans: –1649.68
Same I/Y, P/Y, C/Y, PMT 9 N
1800 PV
CPT FV
Ans: –2058.10
Same I/Y, P/Y, C/Y, N, PMT 8824.715 PV
CPT FV
Ans: -9,932.294
5 I/Y
P/Y 2 ENTER (making C/Y = P/Y = 2)
6 N
3000 PV
0 PMT
CPT FV
Ans: –3479.08
Same I/Y, P/Y, C/Y, PMT 4 N
3500 PV
CPT FV
Ans: –3863.35
Same I/Y, P/Y, C/Y, PMT 2 N
4000 PV
CPT FV
Ans: –4202.50
Chapter 8: Compound Interest: Future Value and Present Value 107
Exercise 8.4 (Financial Calculator Solutions to Exercise 8.2 continued) 27. For the first 1.5 years, j = 7% For the next 2 years, m = 2, PV = $1900, n = 2(1.5) = 3 PV = $2106.564 - $1000 = $1106.564, n = 2(2) = 4 AAt
The amount required today to pay off the remaining principal and accrued interest is $1269.81.
Financial Calculator Solutions to Odd-Numbered Problems in Exercise 8.3 Basic Problems 1. For each part, j = 6.5%, m = 1, P/Y = 1, FV = $10,000 a. n = 1(10) = 10 b. n = 1(20) = 20 c. n = 1(30) = 30
b. Therefore $2837.97 c. Therefore $1511.86 must be invested today. must be invested today.
a. Therefore $5327.26 must be invested today. 3. For each part, j = 8%, FV = $10,000 a. m = 1, n = 1(25) = 25 b. m = 2, n = 2(25) = 50
a. Therefore $1460.18 must be b. Therefore $1407.13 must be invested today. Invested today. (continued)
7 I/Y
P/Y 2 ENTER (making C/Y = P/Y = 2)
3 N
1900 PV
0 PMT
CPT FV Ans: -2106.564
Same I/Y, P/Y, C/Y, PMT 1106.564 PV
4 N
CPT FV
Ans: -1269.81
6.5 I/Y
P/Y 1 ENTER (making C/Y = P/Y = 1)
10 N
10000 FV
0 PMT
CPT PV Ans: -5327.26
Same I/Y, P/Y, C/Y, PMT, FV 20 N
CPT PV
Ans: -2837.97
Same I/Y, P/Y, C/Y, PMT, FV 30 N
CPT PV
Ans: -1511.86
8 I/Y
P/Y 1 ENTER (making C/Y = P/Y = 1)
25 N
10000 FV
0 PMT
CPT PV Ans: -1460.18
Same I/Y, PMT, FV P/Y 2 ENTER
(making C/Y = P/Y = 2) 50 N
CPT PV Ans: -1407.13
108 Fundamentals of Business Mathematics in Canada, 1/e
Exercise 8.4 (Financial Calculator Solutions to Exercise 8.3 continued) 3. c. m = 4, n = 4(25) = 100 d. m = 12, n = 12(25) = 300
c. Therefore $1380.33 must be d. Therefore $1362.37 must be invested today. invested today. 5. Given: FV = $5437.52; j = 8.5%; m = 4 n = m(Term in years)
= 4 ⎟⎠⎞
⎜⎝⎛1227
= 9 compounding periods Principal = PV = $4500.00
7. Given: FV = $8000; j = 5%; m = 12 n = m(Term in years) = 12 ( )12
18 = 18 compounding periods Principal = PV = $7423.10
9. Given: FV = $6500; j = 5%; m = 2 n = m(Term in years) = 2(3) = 6 compounding periods Principal = PV = $5604.93
8.5 I/Y
P/Y 4 ENTER (making C/Y = P/Y = 4)
9 N
0 PMT
5437.52 FV
CPT PV
Ans: –4500.00
5 I/Y
P/Y 12 ENTER (making C/Y = P/Y = 12)
18 N
0 PMT
8000 FV
CPT PV
Ans: –7423.10
5 I/Y
P/Y 2 ENTER (making C/Y = P/Y = 2)
6 N
0 PMT
6500 FV
CPT PV
Ans: –5604.93
Same I/Y, PMT, FV P/Y 4 ENTER
(making C/Y = P/Y = 4) 100 N
CPT PV Ans: -1380.33
Same I/Y, PMT, FV P/Y 12 ENTER
(making C/Y = P/Y = 12) 300 N
CPT PV Ans: -1362.37
Chapter 8: Compound Interest: Future Value and Present Value 109
Exercise 8.4 (Financial Calculator Solutions to Exercise 8.3 continued) Intermediate Problems 11. Given: FV = $2600; j = 5.4%; m = 12 n = m(Term in years) = 12 ( )5.112
15 + = 33 compounding periods Equivalent amount today = PV = $2241.95
13. Given: FV = $7000; j = 6.2%; m = 2 n = m(Term in years) = 2( 5.18 − ) = 13 compounding periods Equivalent amount today = PV = $4706.90
15. The economic value today of an offer is the present value of the payments. Given: j = 4.4%; m = 4 PV of payment due in 6 months PV of payment due in 12 months 17. The economic equivalents of the two originally scheduled payments must be found at the
given focal date, three years from now. Given: j = 6%; m = 4 FV of payment due today PV of payment due in 5 years
A single payment of $1673.865 + $1597.880 = $3271.75 in
three years from now is economically equivalent to the scheduled payments
5.4 I/Y
P/Y 12 ENTER (making C/Y = P/Y = 12)
33 N
0 PMT
2600 FV
CPT PV
Ans: –2241.95
4.4 I/Y
P/Y 4 ENTER (making C/Y = P/Y = 4)
2 N
0 PMT
49000 FV
CPT PV
Ans: –47,939.53
Same I/Y, P/Y, C/Y Same PMT, FV
4 N
CPT PV
Ans: –46,902.01
The economic value of the second offer is $49,000 + $47,939.53 + $46,902.01 = $143,841.54 The $145,000-cash offer is worth $1158.46 more
6.2 I/Y
P/Y 2 ENTER (making C/Y = P/Y = 2)
13 N
0 PMT
7000 FV
CPT PV
Ans: –4706.90
6 I/Y
P/Y 4 ENTER (making C/Y = P/Y = 4)
12 N
0 PMT
1400 + / – PV
CPT FV
Ans: 1673.865
Same I/Y, P/Y, C/Y, PMT 8 N
1800 + / – FV
CPT PV
Ans: 1597.880
110 Fundamentals of Business Mathematics in Canada, 1/e
Exercise 8.4 (Financial Calculator Solutions to Exercise 8.3 continued) 19. The payee should be willing to accept the sum of the economic equivalents of the two
originally scheduled payments at the given focal date, one year from now. Given: j = 7.75%; m = 2 PV of $1000 payment due in 3.5 years PV of $2000 payment due in 5.5 years n = m(Term in years) n = m(Term in years) = 2(3.5 – 1) = 2(5.5 – 1) = 5 compounding periods = 9 compounding periods
The payee should accept a single payment of
$826.884 + $1420.465 = $2247.35 in one year from now in place of the scheduled payments
21. a. Johann should be willing to accept the sum of the economic equivalents of the two originally scheduled payments at the given focal date, three years from now.
Given: j = 6%; m = 4 FV of $6500 payment due in 2 yrs. PV of $8500 payment due in 6 yrs. n = m(Term in years) n = m(Term in years) = 4(1) = 4(3) = 4 compounding periods = 12 compounding periods
Johann should accept single payment of $6898.863 +
$7109.293 = $14,008.16 in three years from now instead of the two scheduled payments.
b. If Johann receives $14,008.16 in 3 years from now, he can invest it at 6%
compounded quarterly for 3 years.
So at the end of the 6 year period, Johann will have a total of $16,748.41, which is the FV of the single equivalent payment that he received three years previously.
7.75 I/Y
P/Y 2 ENTER (making C/Y = P/Y = 2)
5 N
0 PMT
1000 + / – FV
CPT PV
Ans: 826.884
Same I/Y, P/Y, C/Y, PMT 9 N
2000 + / – FV
CPT PV
Ans: 1420.465
6 I/Y
P/Y 4 ENTER (making C/Y = P/Y = 4)
4 N
0 PMT
6500 + / – PV
CPT FV
Ans: 6898.863
Same I/Y, P/Y, C/Y, PMT 12 N
8500 + / – FV
CPT PV
Ans: 7109.293
6 I/Y
P/Y 4 ENTER (making C/Y = P/Y = 4)
12 N
0 PMT
14008.16 + / – PV
CPT FV
Ans: 16748.41
Chapter 8: Compound Interest: Future Value and Present Value 111
Equivalent payment one year from now = $2633.560 + $2332.396 = $4965.96
Exercise 8.4 (Financial Calculator Solutions to Exercise 8.3 continued) If Maria had made the originally scheduled payments on time, Johann would have
invested the $6500 payment for 4 years. At the end of 6 years, that payment would be worth
Then, Maria would provide the $8500 payment, giving Johann a total at the end of the 6 years of $8248.41 + $8500.00 = $16,748.41.
Therefore, the payment of $14,008.16 in three years from now places Johann in exactly the same financial position as if Maria had made the originally scheduled payments. In either case, Johann will have $16,748.41 at the end of six years.
23. First payment: Second payment: PV = $2500, j = 7%, FV = $2500, j = 7%, m = 4, n = 4(1 − 0.25) = 3 m = 4, n = 4(2 –1) = 4 25. The economic value of Joe’s contract is the sum of the present values of the payments. Given: j = 5%; m = 2
Sample calculation: PV of third payment is calculated in the box at the right. Rounded to the nearest $1000, the sum of the present values is $61,559,000.
Time: 0 yr 0.5 yr 1.5 yr 2.5 yr 3.5 yr Salary: $4.8 M $10 M $17.2 M $17.5 M $18.5 M n = 0 1 3 5 7 PV = $4,800,000 $9,756,097 $15,971,910 $15,467,450 $15,563,407
5 I/Y
P/Y 2 ENTER (making C/Y = P/Y = 2)
3 N
0 PMT
17200000 FV
CPT PV
Ans: –15,971,910
Same I/Y, P/Y, C/Y, PMT 16 N
6500 + / – PV
CPT FV
Ans: 8248.41
7 I/Y
P/Y 4 ENTER (making C/Y = P/Y = 4)
3 N
2500 PV
0 PMT
CPT FV
Ans: –2633.560
Same I/Y, P/Y, C/Y, PMT 4 N
2500 FV
CPT PV
Ans: –2332.396
112 Fundamentals of Business Mathematics in Canada, 1/e
Equivalent payment 2¼ years from now = $2908.667 + $1986.789 + $1051.949 = $5947.41
10 I/Y P/Y 12 ENTER
(making C/Y = P/Y = 12) 10 N
0 PMT 1 + / – FV CPT PV
Ans: 0.92036217 Hence, $4000 = 0.92036217x + 0.88295412x
$4000 = 1.80331629x x = $2218.14
The amount of each loan payment is $2218.14.
Exercise 8.4 (Financial Calculator Solutions to Exercise 8.3 continued) Advanced Problems 27. The creditor should accept the economic value, 2¼ years from now, of the three scheduled payments. Allowing for a time value of money of 6% compounded quarterly, the economic value is: First payment: Second payment: Third payment: PV = $2700, j = 6%, PV = $1900, j = 6%, PV = $1100, j = 6%, m = 4, n = 4(2¼ − 1) = 5 m = 4, n = 4(2¼ – 1½) = 3 m = 4, n = 4(3 – 1½) = 3 29. Let x represent the size of each loan payment. PV of first payment: PV of second payment: Checkpoint Questions (Section 8.5) 1. The future value will be the same in both cases. Mathematically, this happens because the
order of multiplication in a product does not matter. In this particular case, (1.04)(1.05)(1.06) = (1.06)(1.05)(1.04)
3. We should not reach this conclusion. It is true that the strip bond owner does not receive any payments from year to year. However, the market value of the bond increases as the time remaining until maturity decreases. This year-to-year change in market value is the owner’s return on investment. The investor can choose to capture the increased value at any time by selling the bond.
6 I/Y
P/Y 4 ENTER (making C/Y = P/Y = 4)
5 N
0 PMT
2700 + / – PV
CPT FV
Ans: 2908.667
Same I/Y, P/Y, C/Y, PMT 3 N
1900 + / – PV
CPT FV
Ans: 1986.789
Same I/Y, P/Y, C/Y, PMT 3 N
1100 + / – FV
CPT PV
Ans: 1051.949
Same I/Y, P/Y, C/Y Same PMT, FV
15 N CPT PV
Ans: 0.88295412
Chapter 8: Compound Interest: Future Value and Present Value 113
Checkpoint Questions (Section 8.5) (continued) 5. The overall decrease will be less than 2x% because the second x% percent decrease acts
on a smaller base than the first x% decrease. Consider, for example, two successive 50% decreases. These do not make the final value zero (corresponding to a decrease of 2 × 50% = 100%). Rather, we reduce the original amount by half twice leaving 4
121
21 =×
or 25% of the initial value. Exercise 8.5 Basic Problems
1. Semiannual interest payment = i(PV) = 2
0.042 × $18,000 = $378.00
3. i = 25.25% = 2.625%; n = 2(7) = 14; PV = $30,000
Maturity value = ( )niPVFV += 1 = $30,000 ( )14026251. = $43,118.70
5. Suppose that $1000 is invested for 3 years at each rate. The maturity value at 4.8% compounded monthly is ( )FV PV i n= +1
= $1000 ( )361.004 = $1154.55
The maturity value at 4.9% compounded semiannually is ( )FV PV i n= +1
= $1000 ( )61.0245 = $1156.30
An investor should choose 4.9% compounded semiannually since it will produce the larger maturity value. 7. Value of $10,000 face-value Sunlife GIC 5 years after its date of issue will be FV = PV(1+ 1i )(1+ 2i )(1+ 3i )(1+ 4i )(1+ 5i )
= $10,000(1.025)(1.03)(1.035)(1.0425)(1.05) = $11,960.98
Value of $10,000 face-value Manulife GIC 5 years after its date of issue will be FV = $10,000(1.0275)(1.0325)(1.035)(1.04)(1.0425)
= $11,904.79 The Sunlife GIC will earn $56.19 more.
4.8 I/Y
P/Y 12 ENTER (making C/Y = P/Y = 12)
36 N
1000 + / – PV
0 PMT
CPT FV
Ans: 1154.55
Same PV, PMT 4.9 I/Y
P/Y 2 ENTER (making C/Y = P/Y = 2)
6 N
CPT FV
Ans: 1156.30
114 Fundamentals of Business Mathematics in Canada, 1/e
Exercise 8.5 (continued) Intermediate Problems 9. Series S102 Bonds were issued on November 1, 2006. a. Value of a $5000 S102 bond on November 1, 2010 was FV = PV(1+ 1i )(1+ 2i )(1+ 3i )(1+ 4i )
= $5000(1.03)(1.0325)(1.02)(1.004) = $5445.417
Therefore the owner of the CSB received $5445.42 when she redeemed the bond on November 1, 2010.
b. The bondholder received all accrued interest up to August 1, 2011. To the amount in part a, we need to add 9 months’ interest at 0.65% per year. I = Prt = $5445.417(0.0065) ( )12
9 = $26.546 The owner received $5445.417 + $26.546 = $5471.96 when she redeemed a $5000 denomination S102 bond on August 21, 2011.
11. The bondholder received interest from the date of issue (November 1, 2005) to March 1, 2011. This period was 5 years plus 4 months long. After 5 years, FV = PV ( ) ( )51 11 ii ++ L
= $300(1.025)(1.03)(1.0325)(1.02)(1.004) = $334.893
The redemption value includes an additional 4 months’ interest at 0.65% per year. Redemption value = $334.893 + $334.893(0.0065) ( )12
4 = $335.62
13. Maturity value of $10,000 invested in the Escalating Rate GIC = PV ( ) ( )51 11 ii ++ L = $10,000(1.02)(1.025)(1.03)(1.0325)(1.07) = $11,896.94 Maturity value of $10,000 invested in the fixed-rate GIC = $10,000 ( )51.0275 = $11,452.73 The Escalating Rate GIC will be worth $11,896.94 – $11,452.73 = $444.21 more at maturity.
15. Initial investment, PV = ( ) ( )51 11 iiFV
++ L
For the Escalating Rate GIC,
PV = 07)1.0325)(1.25)(1.03)((1.02)(1.0
$20,000 = $16,811.05
For the fixed rate GIC, PV = $20,000 ( ) 51.0275 − = $17,463.08
17. Price = Present value of the $5000 face value discounted at the buyer’s required yield of 6.1% compounded semiannually
= ( ) niFV −+1 = $5000 ( ) 3903051 −. = $1549.17
Chapter 8: Compound Interest: Future Value and Present Value 115
2 I/Y
P/Y 1 ENTER (making C/Y = P/Y = 1)
15 N
35000 + / – PV
0 PMT
CPT FV
Ans: 47,105.39
Exercise 8.5 (continued)
19. Market price of one strip bond = FV ( ) ni −+1
= $1000 ( ) 2602625.1 − = $509.82
The number of strip bonds that may be purchased with $12,830 is the integer portion of
82.509$
830,12$ = 25.17
That is, 25 $1000 face value strip bonds can be purchased. 21. A "basket" of goods costing $1000 today would have cost ( )PV FV i n= + −1 = ( ) 15i11000$ −+ 15 years ago if the annual rate of inflation was i. a. If i = 0.02, PV = ( ) 151.02$1000 − = $743.01
b. If i = 0.04, PV = ( ) 151.04$1000 − = $555.26
23. The retirement income goal is $35,000 adjusted for 15 years' nominal growth at the projected annual rate of inflation. a. j = 2%, m = 1: b. j = 3%, m = 1: c. j = 5%, m = 1: ( )FV PV i n= +1 ( )FV PV i n= +1 ( )FV PV i n= +1
= $35,000 ( )1502.1 = $35,000 ( )1503.1 = $35,000 ( )1505.1 = $47,105.39 = $54,528.86 = $72,762.49
25. The Year 2012 employment of 80,000 follows 10 years of 3% annual decline. That is, 80,000 = PV ( )[ ]100301 .−+
= PV ( )10970. = 0.7374241PV
PV = 108,486 108,486 were employed in the forest industry in 2002.
5.25 I/Y
P/Y 2 ENTER (making C/Y = P/Y = 2)
26 N
0 PMT
1000 FV
CPT PV
Ans: –509.82
Same P/Y, C/Y Same N, PV, PMT
3 I/Y
CPT FV
Ans: 54,528.86
Same P/Y, C/Y Same N, PV, PMT
5 I/Y
CPT FV
Ans: 72,762.49
116 Fundamentals of Business Mathematics in Canada, 1/e
Exercise 8.5 (continued) Advanced Problems
27. Proceeds = Present value of the maturity value For the maturity value calculation, PV = $8000, j = 13.5%, m = 12, and i = 12
513 %. = 1.125%, n = 12(4) = 48
( )niPVFV += 1
= $8000 ( )481.01125 = $13,686.73
For the present value (discounting) calculation, FV = $13,686.73, j = 12%, m = 4, i = 4
12% = 3%, and n = 4 ( )122148− = 9
Proceeds = ( ) niFV −+1
= $13,686.73 ( ) 9031 −. = $10,489.74
29. Sales for Year 3 = ( )niPVFV += 1 = $28,600,000 ( )[ ]30401 .−+ = $25,303,450
Sales for Year 7 = $25,303,450 ( )4081. = $34,425,064
Checkpoint Questions (Section 8.6) 1. No. Think of the simplest case where one payment stream consists of a single payment
and the other payment stream consists of a single (different) payment on a different date. These two payments are equivalent at only one discount rate.
3. Compare the future values of the two streams at a focal date 5 (or more) years from now (assuming that money is invested at 7% compounded semiannually as soon as it is received). The two future values will be equal.
13.5 I/Y
P/Y 12 ENTER (making C/Y = P/Y = 12)
48 N
8000 PV
0 PMT
CPT FV
Ans: –13,686.73
Same PMT, FV 12 I/Y
P/Y 4 ENTER (making C/Y = P/Y = 4)
9 N
CPT PV
Ans: 10,489.74
Chapter 8: Compound Interest: Future Value and Present Value 117
Exercise 8.6 Intermediate Problems 1. Let x represent the replacement payment due in 24 months. The equivalent value of the scheduled payments at a focal date 24 months from today is $3000 ( )81.015 + $2000 ( )31.015 = $3379.48 + $2091.36 = $5470.84 The equivalent value of the replacement payments on the same date is $1500 ( )31.015 + x = $1568.52 + x For equivalence of the two payment streams, $1568.52 + x = $5470.84 x = $3902.32 The second payment due in 24 months is $3902.32.
3. Let x represent the amount of the third replacement payment. Calculate x so that the future values of the two payment streams at a focal date 7 months from now are equal. FV of original stream
= $1000 ( )190035.1 + $1500 ( )130035.1 = $1068.637 + $1569.702 = $2638.339 FV of replacement stream
= $800 ( )60035.1 + $900 ( )30035.1 + x = $816.948 + $909.483 = $1726.431 + x Solve for x in
$1726.431 + x = $2638.339 x = $911.91
The third payment should be $911.91.
6 I/Y
P/Y 4 ENTER (making C/Y = P/Y = 4)
8 N
3000 PV
0 PMT
CPT FV Ans: –3379.48
Same I/Y, P/Y, C/Y, PMT 3 N
2000 PV
CPT FV
Ans: −2091.36
Same I/Y, P/Y, C/Y Same N, PMT 1500 PV
CPT FV
Ans: −1568.52
4.2 I/Y
P/Y 12 ENTER (making C/Y = P/Y = 12)
19 N
1000 PV
0 PMT
CPT FV
Ans: –1068.637
Same I/Y, P/Y, C/Y, PMT 13 N
1500 PV
CPT FV
Ans: –1569.702
Same I/Y, P/Y, C/Y, PMT 3 N
900 PV
CPT FV
Ans: –909.483
Same I/Y, P/Y, C/Y, PMT 6 N
800 PV
CPT FV
Ans: –816.948
118 Fundamentals of Business Mathematics in Canada, 1/e
Exercise 8.6 (continued) 5. Let x represent the amount of the second payment. For equivalence at a focal date 5 months from now, $19,000 ( )91.01 + $14,000 ( )31.01 = $10,000 ( )51.01 + x
$20,780.02 + $14,424.21 = $10,510.10 + x x = $24,694.13
The second payment will be $24,694.13.
Advanced Problems 7. The winner should choose the alternative having the larger current economic value. a. Economic value of (1) = $10,000 + $10,000 ( ) 51.06 −
= $17,472.58 The economic value of Alternative (2) is
$6700 + $6700 ( ) 51.06 − + $6700 ( ) 101.06 − + $6700 ( ) 151.06 − = $18,243.55 Alternative (2) should be chosen since it is worth $770.97 more.
b. Economic value of (1) = $10,000 + $10,000 ( ) 51.085 − = $15,934.51 The economic value of alternative (2) is $6700 + $6700 ( ) 51.085 − + $6700 ( ) 101.085 − + $6700 ( ) 151.085 − = $16,089.85 Alternative (1) should be chosen since it is worth $560.60 more.
9. a. Maturity value of loan = $15,000 ( )181.025 = $23,394.88 Let x represent the value of the second replacement payment. For equivalence at a focal date 1 year from now, $23,394.88 ( ) 2031 −. = $5000 ( )2031. + x $22,051.92 = $5304.50 + x
x = $16,747.42 The second payment should be $16,747.42. b. The combined future value, 2 years from now, of the two replacement payments is $5000 ( )41.03 + $16,747.42 ( )21.03 = $5627.54 + $17,767.34= $23,394.88 This is the same amount as the maturity value of the original loan on this date. Therefore, the lender ends up in the same financial position under either alternative.
6 I/Y
P/Y 1 ENTER (making C/Y = P/Y = 1)
5 N
0 PMT
10000 FV
CPT PV
Ans: −7472.58
Same I/Y, P/Y, C/Y, PMT 5 N
6700 FV
CPT PV
Ans: −5006.630
Same I/Y, P/Y, C/Y Same PMT, FV
10 N
CPT PV
Ans: −3741.245
Same I/Y, P/Y, C/Y Same PMT, FV
15 N
CPT PV
Ans: −2795.676
Chapter 8: Compound Interest: Future Value and Present Value 119
Review Problems 1. Given PV = $100. At 8.9% compounded annually, j = 8.9%, m = 1, and n=1(3) = 3 compounding periods. Hence:
i = mj =
18.9% = 8.89% (per year)
( )FV PV i n= +1 = $100 ( )30891. = $129.15 At 8.65% compounded quarterly, j = 8.65%, m = 4, and n=4(3) = 12 compounding periods. Hence:
i = mj =
48.65% = 2.1625% (per quarter)
( )FV PV i n= +1 = $100 ( )120216251. = $129.27 Therefore, 8.65% compounded quarterly is the preferred rate for a 3 year investment. 3. Given: FV = $35,000, j = 8%, m = 4, and n=4(8) = 32 compounding periods. Hence:
i = mj =
48% = 2% (per quarter)
( ) niFVPV −+= 1 = $35,000 ( ) 32021 −. = $18,572.17 5. Maturity value of the compound-interest GIC = PV ( ) ( )51 11 ii ++ L = $10,000(1.03)(1.04)(1.05)(1.06)(1.07) = $12,757.03 7. Current price: PV = FV ( ) ni −+1
= $1000 ( ) 381.0295 − = $331.28
Intermediate Problems 9. Original principal, PV = FV ( ) ni −+1 = $2297.78 ( ) 191.007 − = $2012.56 Interest portion = $2297.78 – $2012.56 = $285.22
8.9 I/Y
P/Y 1 ENTER (making C/Y = P/Y = 1)
3 N
0 PMT
100 + / – PV
CPT FV
Ans: 129.15
Same PMT, PV 8.65 I/Y
P/Y 4 ENTER (making C/Y = P/Y = 4)
12 N
CPT FV
Ans: 129.27
8 I/Y
P/Y 4 ENTER (making C/Y = P/Y = 4)
32 N
0 PMT
35000 FV
CPT PV
Ans: -18,572.17
5.9 I/Y
P/Y 2 ENTER (making C/Y = P/Y = 2)
38 N
0 PMT
1000 FV
CPT PV
Ans: −331.28
120 Fundamentals of Business Mathematics in Canada, 1/e
Review Problems (continued) 11. Amount owed after the $1200 payment 1 year ago = $3000 ( )101.02625 – $1200 = $2687.34 Balance owed today = $2687.34 ( )41.02625 = $2980.82
13. a. Maturity value = PV ( )211 i+ ( )221 i+ ( )231 i+
= $12,000 ( )21.02 ( )21.025 ( )21.03 = $13,915.66
b. Interest earned in the second year = Value after 2 years – Value after 1 year = $12,000 ( )21.02 ( )21.025 – $12,000 ( )21.02 = $13,116.84 – $12,484.80 = $632.04
15. Today’s economic value of Offer 1 = $10,000 + $30,000 ( ) 31.05125 − = $10,000 + $25,822.79 = $35,822.79
Today’s economic value of Offer 2 = $8000 + $33,000 ( ) 41.05125 − = $8000 + $27,020.28 = $35,020.28
Donnelly should accept Offer 1. Its current economic value is $802.51more than that of Offer 2.
17. Given: PV = 4800, j = -12%, m = 1, and n=1(5) = 5 compounding periods. Hence:
i = mj =
15%- = -5% (per year)
( )FV PV i n= +1 = 4800 ( )512.01− = 2533 ads 19. The S96 CSBs were issued on November 1, 2005. a. Redemption value of a $500 face value bond on November 1, 2010 = PV ( ) ( )51 11 ii ++ L = $500(1.025)(1.03)(1.0325)(1.02)(1.004) = $558.16 b. In addition to the amount in Part a, the bond owner is entitled to interest from November 1, 2010 to April 1, 2011 at 0.65% per year. Hence, Redemption value = PV(1 + rt) = $558.16 ( )[ ]12
5006501 .+ = $559.67
- 12 I/Y
P/Y 1 ENTER (making C/Y = P/Y = 1)
5 N
0 PMT
4800 PV
CPT FV
Ans: -2533.11
Chapter 8: Compound Interest: Future Value and Present Value 121
Review Problems (continued) 21. Let x represent the replacement payment due in 4 years. The equivalent value of the scheduled payments at a focal date 4 years from now is $2300 ( )11048751. + $3100 ( )21.04875 = $3882.572 + $3409.617 = $7292.189 The equivalent value of the replacement payments on the same date is x + $2000 ( )81.04875 = x + $2926.886 For equivalence of the two payment streams, x + $2926.886 = $7292.189 x = $4365.30 The replacement payment due in years is $4365.30.
23. Mike should be willing to accept the sum of the economic equivalents of the two originally scheduled payments at the given focal date, six months from now.
Given: j = 6%; m = 4 PV of $1200 payment due in 18 months n = m(Term in years) = 4(1) = 4 compounding periods ( ) ni1FVPV −+= = $1200( ( ) 4015.01 −+ = $1130.621 PV of $3000 payment due in 33 months n = m(Term in years)
= 4 ⎟⎠⎞
⎜⎝⎛ −
126
1233
= 9 compounding periods ( ) ni1FVPV −+= = $3000 ( ) 9015.01 −+ = $2623.777 Therefore Mike should be willing to accept the sum of $1130.621 + $2623.777 = $3754.40 as a single equivalent payment in six months from now. Advanced Problems 25. Value after 1 year: Value after 2 years: Value after 3 years: ( )FV PV i n= +1 Initial investment Initial investment
= $3000 ( )121.0075 = $3281.42 + $4000 = $7920.35 + $3500
= $3281.42 FV = $7281.42 ( )41.02125 FV = $11,420.35 ( )21.03875 = $7920.35 = $12,322.58
6 I/Y
P/Y 4 ENTER (making C/Y = P/Y = 4)
4 N
0 PMT
1200 + / – FV
CPT PV
Ans: 1130.621
Same I/Y, C/Y, P/Y, PMT 9 N
3000 + / – FV
CPT PV
Ans: 2623.777
9 I/Y
P/Y 12 ENTER (making C/Y = P/Y = 12)
12 N
3000 + / – PV
0 PMT
CPT FV
Ans: 3281.42
8.5 I/Y
P/Y 4 ENTER (making C/Y = P/Y = 4)
4 N
7281.42 + / – PV
0 PMT
CPT FV
Ans: 7920.35
7.75 I/Y
P/Y 2 ENTER (making C/Y = P/Y = 2)
2 N
11420.35 + / – PV
0 PMT
CPT FV
Ans: 12322.58
122 Fundamentals of Business Mathematics in Canada, 1/e
Review Problems (continued) 27. Choose any value, say 10,000, for the number employed in 1995. The forecast number of employees 5 years later was: ( )FV PV i n= +1
= 10,000 = 10,000 50.035)] ( [1 −+ = 10,000(0.965)5 = 8368
Number of jobs lost = 10,000 – 8368 = 1632
That is, %,
10000010
1632× = 16.32%
of base metal mining jobs were expected to be lost.
29. Find the equivalent value of each payment, using the focal date of the day the loan was advanced. The sum of those present values will be equivalent to the original loan, $12,500. Thus:
$12,500 = $2000 ( ) 6045.1 − + ( ) 14045.1 − x + ( ) 16045.1 − x $12,500 = $1535.791 + 0.539973 x + 0.494469 x $10,964.209 = 1.034442 x x = $10,599.15 Loralyn will make payments of $10,599.15 seven and eight years after the initial loan.
3.5 + / – I/Y
P/Y 1 ENTER (making C/Y = P/Y = 1)
5 N
10000 PV
0 PMT
CPT FV
Ans: −8368