Appendix B
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Applying
Present and Future Values
QUICK STUDIES
Quick
Study B-1 (10 minutes)
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1.
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2%
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2.
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12%
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3.
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3%
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4.
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1%
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Quick
Study B-2 (10 minutes)
In
Table B.1, where n = 15 and p = $2,745/$10,000 = 0.2745, the i = 9%.
Quick
Study B-3 (10 minutes)
In
Table B.1, where i = 6% and p = $6,651/$10,000 = 0.6651, the n = 7.
Quick
Study B-4 (10 minutes)
In
Table B.1, where n = 5 and i = 9%, the p = 0.6499.
Amount
willing to pay today: 0.6499 x $140,000 = $90,986
Quick
Study B-5 (10 minutes)
In
Table B.2, where n = 10 and i = 12%, the f = 3.1058.
Cash
proceeds at liquidation: 3.1058 x
$630,000 = $1,956,654
Quick
Study B-6 (10 minutes)
In
Table B.3, where n = 6 and i = 7%, the p = 4.7665.
Amount
willing to pay for the project: 4.7665 x
$150,000 = $714,975
Quick
Study B-7 (10 minutes)
In
Table B.4, where n = 30 and i = 10%, the f = 164.494.
Ending
value of the investment program: 164.494
x $1,500 = $246,741
EXERCISES
Exercise
B-1 (10 minutes)
In
Table B.2, where i = 12% and f = $96,463/$10,000 = 9.6463, the n = 20 (implies the investor must wait 20 years before
payment).
Exercise
B-2 (10 minutes)
In
Table B.2, where n = 25 and f = $108,347/$10,000 = 10.8347, the i = 10% (investor must earn 10% interest to
achieve investment goal).
Exercise
B-3 (10 minutes)
In
Table B.3, where n = 8 and p = $57,466/$10,000 = 5.7466, the i = 8% (investor must earn 8% interest to achieve
investment goal).
Exercise
B-4 (10 minutes)
In
Table B.3, where i = 10% and p = $82,014/$10,000 = 8.2014, the n = 18 (investor expects 18 annual payments to be
received).
Exercise
B-5 (10 minutes)
In
Table B.4, where n = 40 and f = $154,762/$1,000 = 154.762, the i = 6% (investor must earn a 6% rate of
interest).
Exercise
B-6 (10 minutes)
In
Table B.4, where i = 8% and f = $303,243/$10,000 = 30.3243, the n = 16 (investor must make 16 annual payments to
achieve investment goal).
Exercise
B-7 (10 minutes)
Interest
rate per period = 12% annual / 12 months per year = 1% per month
Using
Table B.3, where n = 40 and i = 1%, the p = 32.8347. This means:
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Loan balance......
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$16,417.35
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(present value of loan = 32.8347 x $500)
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Down payment...
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6,500.00
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(cash)
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Total cost...........
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$22,917.35
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Exercise
B-8 (15 minutes)
Semiannual
interest payment = $500,000 x 10% x 1/2 = $25,000
Using
Table B.1, where n = 30 and i = 4%, the p = 0.3083 (Principal payment)
Using
Table B.3, where n = 30 and i = 4%, the p = 17.2920 (Interest payments)
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0.3083 x $500,000 =
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$154,150
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present value of maturity amount
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17.2920 x $ 25,000 =
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432,300
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present value of interest payments
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$586,450
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cash proceeds
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Exercise
B-9 (15 minutes)
In
Table B.1, where n = 6 and i = 10%, the p = 0.5645.
Present
value of investment = $606,773 x .5645 = $342,523
Exercise
B-10 (15 minutes)
1. $90,000 x 0.6651 (using Table B.1, i = 6%, n
= 7) = $59,859.
2. $20,000 x 2.4869 (using Table B.3, i = 10%, n
= 3) = $49,738.
Exercise
B-11 (15 minutes)
Amount borrowed =
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present value of $20,000 at 10% for 3 years
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=
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$20,000 x 0.7513 (using Table B.1, i = 10%,
n = 3)
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=
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$15,026
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Exercise
B-12 (10 minutes)
a. p = present value of $60,000 at 9% for 4
years
p
= $60,000 x 0.7084
p
= $42,504
b. p
= present value of $15,000 at 8% for 2 years
p
= $15,000 x 0.8573
p
= $12,859.50
c. There are at least two ways to solve this
problem. (1) We can take the $463 today, compute its future value, and then
compare it to the future value amount of $1,000. (2) We can discount the $1,000 back to the
present and compare it to the $463 today.
àThe same answer results: choose $463 today
f = future value of $463 at 9% for 10
years
f
= $463 / 0.4224
f
= $1096.12 (which implies we’d prefer the $463 today)
p
= present value of $1,000 at 9% for 10 years
p
= $1,000 x 0.4224
p
= $422.40 (which is less than $463 today)
d. f
= future value of $90 at 5% for 8 years
Formula:
$90 = f x 0.6768; then solve for f
f
= $90 / 0.6768
f
= $132.98
e. f
= future value of $158,500 at 10% for 8 years
Formula:
$158,500 = f x 0.4665; then solve for f
f
= $339,764.20
Exercise
B-12 (concluded)
f. There are two aspects to this problem: a
present value of a lump sum part and a present value of an annuity part.
Part 1: p
= present value of $10,000 at 6% for 10 years
p
= $10,000 x 0.5584
p
= $5,584
Part 2: p
= present value of $400 annuity at 6% for 10 years
p
= $400 x 7.3601
p
= $2,944
The answer is the sum of the present
values from parts 1 and 2:
$8,528 = $5,584 + $2,944 (we are
willing to pay $8,528 for this investment)
g. p = present value of $500,000 at 6% for 20
years
p = $500,000 x 11.4699
p = $5,734,950 (present
value of real amount won)
Instructor note: It can be useful to extend this problem and
assume a 30% tax rate. In this case the annuity after-tax declines to $350,000.
Accordingly, the present value of the after-tax amount is $4,014,465. Again,
nothing near the $10 million winnings advertised.
Exercise
B-13 (25 minutes)
1.
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First Annuity
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Future
Payment |
Number of Periods
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Interest Rate
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Table B.1 Value
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Amount Borrowed
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|||||
First payment.......
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$5,000
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1
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6%
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0.9434
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$ 4,717
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||||
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Second payment..
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5,000
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2
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6
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0.8900
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4,450
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||||
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Third payment.....
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5,000
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3
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6
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0.8396
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4,198
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||||
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Fourth payment...
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5,000
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4
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6
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0.7921
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3,961
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||||
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Fifth payment.......
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5,000
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5
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6
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0.7473
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3,737
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Sixth payment......
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5,000
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6
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6
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0.7050
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3,525
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||||
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Total borrowed....
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$24,588
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||||
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Second Annuity
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Future
Payment |
Number of Periods
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Interest Rate
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Table B.1 Value
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Amount Borrowed
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|||||
First payment.......
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$7,500
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1
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6%
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0.9434
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$ 7,076
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||||
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Second payment..
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7,500
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2
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6
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0.8900
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6,675
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||||
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Third payment.....
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7,500
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3
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6
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0.8396
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6,297
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Fourth payment...
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7,500
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4
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6
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0.7921
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5,941
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||||
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Total borrowed....
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$25,989
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||||
Exercise
B-13 (Continued)
2.
First
Annuity
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Payment size...............................
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$
5,000
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|||
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Number of payments...................
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6
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|||
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Interest rate................................
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6%
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Value from Table B.3..................
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4.9173
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Present value of the annuity.......
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$24,587
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(difference
from part (1) due to rounding)
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Second
Annuity
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Payment size...............................
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$
7,500
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|||
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Number of payments...................
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4
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|||
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Interest rate................................
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6%
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|||
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Value from Table B.3..................
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3.4651
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|||
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Present value of the annuity.......
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$25,988
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(difference
from part (1) due to rounding)
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|||
Exercise
B-14 (30 minutes)
1. Present value of the annuity
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Payment size...............................
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$13,000
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||
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Number of payments...................
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4
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||
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Interest rate................................
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4%
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(semiannual)
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Value from Table B.3..................
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3.6299
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||
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||
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Present value of the annuity.......
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$47,189
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||
2. Present value of the annuity
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Payment size...............................
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$13,000
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||
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Number of payments...................
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4
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||
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Interest rate................................
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6%
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(semiannual)
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Value from Table B.3..................
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3.4651
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||
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Present value of the annuity.......
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$45,046
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3. Present value of the annuity
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Payment size...............................
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$13,000
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||
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Number of payments...................
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4
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||
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Interest rate................................
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8%
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(semiannual)
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Value from Table B.3..................
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3.3121
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||
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||
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Present value of the annuity.......
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$43,057
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Exercise
B-15 (15 minutes)
10
years x 4 quarters = 40 interest periods
8%
annual / 4 quarters per year = 2% per quarter
In
Table B.2, where n = 40 and i = 2%, the f = 2.2080.
Total
accumulation = 2.2080 x $7,200 = $15,897.60
Exercise
B-16 (15 minutes)
12%
annual / 12 months per year = 1% per month
2.5
years x 12 months per year = 30 total months
In
Table B.4, where n = 30 and i = 1%, the f = 34.7849.
Total
accumulation = 34.7849 x $50 = $1,739.25
Exercise
B-17 (15 minutes)
10
years x 4 quarters per year = 40 total quarters
12%
annual / 4 quarters per year = 3% per quarter
In
Table B.2, where n = 40 and i = 3%, the f = 3.2620.
In
Table B.4, where n = 40 and i = 3%, the f = 75.4013.
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3.2620 x $100,000 =
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$ 326,200
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future value of initial investment
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75.4013 x $50,000 =
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3,770,065
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future value of periodic investments
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$4,096,265
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future value of fund
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Exercise
B-18 (15 minutes)
In
Table B.2, where n = 9 and i = 7%, the f = 1.8385.
Future
value of investment = $163,170 x 1.8385 =
$299,988
Exercise
B-19 (20 minutes)
a. (1) Present
Value of a single amount.
(2) Multiply $10,000 by p from Table B.1.
(3) Use Table B.1, periods = 8 and interest
rate = 4%.
OR
(1) Future Value of a single amount.
(2) Divide $10,000 by f from Table B.2.
(3) Use Table B.2, periods = 8 and interest
rate = 4%.
b. (1) Future Value of an Annuity.
(2) Divide $10,000 by f from Table B.4.
(3)
Use Table B.4, periods = 8 and
interest rate = 4%.
OR
(1) Present Value of an Annuity.
(2) Multiply $10,000 by p from Table B.1 and
then divide by p from
Table B.3.
(3) Use Tables B.1 and B.3, periods = 8 and interest rate = 4%.
c. (1) Future Value of an Annuity.
(2) Multiply $4,000 by f from Table B.4.
(3) Use Table B.4, periods = 40 and interest =
8%.
d. (1) Present Value of an Annuity.
(2) Multiply $30,000 by p from Table B.3.
(3) Use Table B.3, periods = 20 and interest =
10%.
[Note: Students
must recognize the present value of $225,000
received
today is $225,000.]
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