8 Compound Interest: Future Value and Present...

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


c. i =2%0.8 = 4.0% (per six months); n = 2(25) = 50 periods


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.


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


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


b. i = 28% = 4% (per half year); n = 2(25) = 50 periods


c. i = 48% = 2% (per quarter);n = 4(25) = 100 periods


d. i = 128% = 6.0 % (per month); n = 12(25) = 300 periods



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)


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


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


81 N

4400 + / – PV

0 PMT

CPT FV

Ans: 6329.92

6 I/Y


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


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


12 N

100 + / – PV

0 PMT

CPT FV

Ans: 108.30

Same PV, PMT 8.1 I/Y


4 N

CPT FV

Ans: 108.35



2 N

CPT FV

Ans: 108.37



1 N

CPT FV

Ans: 108.30

7.2 I/Y


6 N

12000 PV

0 PMT

CPT FV

Ans: -13,355.74


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


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


20 N

1000 + / – PV

0 PMT

CPT FV

Ans: 4660.96

8.4 I/Y


30 N

5000 PV

0 PMT

CPT FV

Ans: -6163.879

7.5 I/Y


4 N

6163.879 PV

0 PMT

CPT FV

Ans: -7141.78


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


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


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


48 N

10000 + / – PV

0 PMT

CPT FV

Ans: 12,454.51

6.25 I/Y


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


4 N

12000 PV

0 PMT

CPT FV

Ans: -13,506.106


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


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


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


6 N

3000 PV

0 PMT

CPT FV

Ans: –3479.08


3500 PV

CPT FV

Ans: –3863.35


4000 PV

CPT FV

Ans: –4202.50


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


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


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


25 N

10000 FV

0 PMT


Same I/Y, PMT, FV P/Y 2 ENTER

(making C/Y = P/Y = 2) 50 N



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


9 N

0 PMT

5437.52 FV

CPT PV

Ans: –4500.00

5 I/Y


18 N

0 PMT

8000 FV

CPT PV

Ans: –7423.10

5 I/Y


6 N

0 PMT

6500 FV

CPT PV

Ans: –5604.93


(making C/Y = P/Y = 4) 100 N



(making C/Y = P/Y = 12) 300 N



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


33 N

0 PMT

2600 FV

CPT PV

Ans: –2241.95

4.4 I/Y


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


13 N

0 PMT

7000 FV

CPT PV

Ans: –4706.90

6 I/Y


12 N

0 PMT

1400 + / – PV

CPT FV

Ans: 1673.865


1800 + / – FV

CPT PV

Ans: 1597.880


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


5 N

0 PMT

1000 + / – FV

CPT PV

Ans: 826.884


2000 + / – FV

CPT PV

Ans: 1420.465

6 I/Y


4 N

0 PMT

6500 + / – PV

CPT FV

Ans: 6898.863


8500 + / – FV

CPT PV

Ans: 7109.293

6 I/Y


12 N

0 PMT

14008.16 + / – PV

CPT FV

Ans: 16748.41


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


3 N

0 PMT

17200000 FV

CPT PV

Ans: –15,971,910


6500 + / – PV

CPT FV

Ans: 8248.41

7 I/Y


3 N

2500 PV

0 PMT

CPT FV

Ans: –2633.560


2500 FV

CPT PV

Ans: –2332.396


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


5 N

0 PMT

2700 + / – PV

CPT FV

Ans: 2908.667


1900 + / – PV

CPT FV

Ans: 1986.789


1100 + / – FV

CPT PV

Ans: 1051.949


15 N CPT PV

Ans: 0.88295412


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


36 N

1000 + / – PV

0 PMT

CPT FV

Ans: 1154.55



6 N

CPT FV

Ans: 1156.30


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


2 I/Y


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


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


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


48 N

8000 PV

0 PMT

CPT FV

Ans: –13,686.73

Same PMT, FV 12 I/Y


9 N

CPT PV

Ans: 10,489.74


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


8 N

3000 PV

0 PMT

CPT FV Ans: –3379.48


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


19 N

1000 PV

0 PMT

CPT FV

Ans: –1068.637


1500 PV

CPT FV

Ans: –1569.702


900 PV

CPT FV

Ans: –909.483


800 PV

CPT FV

Ans: –816.948


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


5 N

0 PMT

10000 FV

CPT PV

Ans: −7472.58


6700 FV

CPT PV

Ans: −5006.630


10 N

CPT PV

Ans: −3741.245


15 N

CPT PV

Ans: −2795.676


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


3 N

0 PMT

100 + / – PV

CPT FV

Ans: 129.15

Same PMT, PV 8.65 I/Y


12 N

CPT FV

Ans: 129.27

8 I/Y


32 N

0 PMT

35000 FV

CPT PV

Ans: -18,572.17

5.9 I/Y


38 N

0 PMT

1000 FV

CPT PV

Ans: −331.28


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


5 N

0 PMT

4800 PV

CPT FV

Ans: -2533.11


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


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


12 N

3000 + / – PV

0 PMT

CPT FV

Ans: 3281.42

8.5 I/Y


4 N

7281.42 + / – PV

0 PMT

CPT FV

Ans: 7920.35

7.75 I/Y


2 N

11420.35 + / – PV

0 PMT

CPT FV

Ans: 12322.58


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


5 N

10000 PV

0 PMT

CPT FV

Ans: −8368

8 Compound Interest: Future Value and Present...

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