Chapter TwelveCOMPOUND INTEREST AND PRESENT VALUE
Copyright © 2014 by The McGraw-Hill Companies, Inc. All rights reserved.McGraw-Hill/Irwin
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1. Compare simple interest with compound interest.2. Calculate the compound amount and interest
manually and by table lookup.3. Explain and compute the effective rate (APY).
LU 12-1 Compound Interest (Future Value) – The Big Picture
LEARNING UNIT OBJECTIVES
LU 12-2 Present Value -- The Big Picture
1. Compare present value (PV) with compound interest (FV).
2. Compute present value by table lookup.3. Check the present value answer by compounding.
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COMPOUND INTEREST
• So far we have discussed only simple interest, which is interest on the principal alone. Simple interest is either paid at the end of the loan period or deducted in advance.
• In this chapter we look at the power of compounding—interest paid on earned interest.
• Compounding involves the calculation of interest periodically over the life of the loan (or investment). After each calculation, the interest is added to the principal. Future calculations are on the adjusted principal (old principal plus interest). Compound interest, then, is the interest on the principal plus the interest of prior periods. Future value (FV), or the compound amount, is the final amount of the loan or investment at the end of the last period.
• Before you learn how to calculate compound interest and compare it to simple interest, you must understand the terms that follow.
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COMPOUNDING INTEREST (FUTURE VALUE)
Compound Interest –
The interest on the principal plus the interest of prior
periods
Compounding –
Involves the calculation of interest periodically over the life
of the loan or investment
Present Value –
The value of a loan or investment today
Future Value (compound amount) –
The final amount of the loan or investment at the end of the
last period
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COMPOUNDING TERMS
Compounding Periods Interest Calculated
Compounding Annually
Compounding Semiannually
Compounding Quarterly
Compounding Monthly
Compounding Daily
Once a year
Every 6 months
Every 3 months
Every month
Every day
• Compounding changes the interest rate.• Rate for each period is annual interest rate divided by
the number of times the interest is compounded per year.
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FUTURE VALUE OF $1 AT 8% FOR FOUR PERIODS (FIGURE 12.1)
$0.00$0.50$1.00$1.50
$2.00$2.50$3.00$3.50$4.00$4.50$5.00
0 1 2 3 4Number of periods
Compounding goes from present value to future value
Present
value
After 1
period, $1 is worth $1.08
After 2 periods,
$1 is worth $1.17
After 3 periods,
$1 is worth $1.26
Future
Value
After 4 periods, $1 is worth $1.36
$1.00
$1.08 $1.1664 $1.2597 $1.3605
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FUTURE VALUE OF $1 AT 8% FOR FOUR PERIODS (FIGURE
12.1)
Year 1 Year 2 Year 3 Year 41.00$ 1.08$ 1.17$ 1.26$ 0.08 x .10 x .10 x .10
Interest 0.08$ 0.09$ 0.09$ 0.10$ Beg. Bal 1.00 1.08 1.17 1.26 End of year 1.08$ 1.17$ 1.26$ 1.36$
Manual Calculation
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TOOLS FOR CALCULATING COMPOUND INTEREST
Number of periods (N) Number of years
multiplied by the number of times the interest is compounded per year
Rate for each period (R) Annual interest rate
divided by the number of times the interest is
compounded per year
If you compounded $100 for 4 years at 8% annually,
semiannually, or quarterly, what is N and R?
Annually:
Semiannually:
Quarterly:
Periods Rate
4 x 1 = 4
4 x 4 = 16
4 x 2 = 8
8% / 1 = 8%Annually:
Semiannually:
Quarterly: 8% / 4 = 2%
8% / 2 = 4%
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SIMPLE VERSUS COMPOUND INTEREST
Bill Smith deposited $80 in a savings account for 4 years at an annual interest rate of 8%. What is Bill’s simple interest and maturity value?
I = P x R x T
I = $80 x .08 x 4
I = $25.60
MV = $80 + $25.60
MV = $105.60
I = P x R x T
I = $80 x .08 x 4
I = $25.60
MV = $80 + $25.60
MV = $105.60
Bill Smith deposited $80 in a savings account for 4 years at an annual interest rate of 8%. What is Bill’s interest and compounded amount?
Year 1 Year 2 Year 3 Year 480.00$ 86.40$ 93.31$ 100.77$ x .08 x .08 x .08 x .08
Interest 6.40$ 6.91$ 7.46$ 8.06$ Beg. bal 80.00 86.40 93.31 100.77 End of year 86.40$ 93.31$ 100.77$ 108.83$
Interest: $108.83 -- $80.00 = $28.83
Simple Compounded
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CALCULATING COMPOUND AMOUNT
BY TABLE LOOKUPStep 1. Find the periods: Years multiplied by number of
times interest is compounded in 1 year.
Step 2. Find the rate: Annual rate divided by number of times interest is compounded in 1 year.
Step 3. Go down the period column of the table to the number desired; look across the row to find the rate. At the intersection is the table factor for the compound amount of $1.
Step 4. Multiply the table factor by the amount of the loan. This gives the compound amount.
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Period 1% 1.50% 2% 3% 4% 5% 6% 7% 8% 9% 10%
1 1.0100 1.0150 1.0200 1.0300 1.0400 1.0500 1.0600 1.0700 1.0800 1.0900 1.1000
2 1.0201 1.0302 1.0404 1.0609 1.0816 1.1025 1.1236 1.1449 1.1664 1.1881 1.2100
3 1.0300 1.0457 1.0612 1.0927 1.1249 1.1576 1.1910 1.2250 1.2597 1.2950 1.3310
4 1.0406 1.0614 1.0824 1.1255 1.1699 1.2155 1.2625 1.3108 1.3605 1.4116 1.4641
5 1.0510 1.0773 1.1041 1.1593 1.2167 1.2763 1.3382 1.4026 1.4693 1.5386 1.6105
6 1.0615 1.0934 1.1262 1.1941 1.2653 1.3401 1.4185 1.5007 1.5869 1.6771 1.7716
7 1.0721 1.1098 1.1487 1.2299 1.3159 1.4071 1.5036 1.6058 1.7138 1.8280 1.9487
8 1.0829 1.1265 1.1717 1.2668 1.3686 1.4775 1.5938 1.7182 1.8509 1.9926 2.1436
9 1.0937 1.1434 1.1951 1.3048 1.4233 1.5513 1.6895 1.8385 1.9990 2.1719 2.3579
10 1.1046 1.1605 1.2190 1.3439 1.4802 1.6289 1.7908 1.9672 2.1589 2.3674 2.5937
11 1.1157 1.1780 1.2434 1.3842 1.5395 1.7103 1.8983 2.1049 2.3316 2.5804 2.8531
12 1.1260 1.1960 1.2682 1.4258 1.6010 1.7959 2.0122 2.2522 2.5182 2.8127 3.1384
13 1.1381 1.2135 1.2936 1.4685 1.6651 1.8856 2.1329 2.4098 2.7196 3.0658 3.4523
14 1.1495 1.2318 1.3195 1.5126 1.7317 1.9799 2.2609 2.5785 2.9372 3.3417 3.7975
15 1.1610 1.2502 1.3459 1.5580 1.8009 2.0789 2.3966 2.7590 3.1722 3.6425 4.1772
Future value of $1 at compound interest (Partial)
FUTURE VALUE OF $1 AT COMPOUND INTEREST (TABLE 12.1)
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CALCULATING COMPOUND AMOUNT BY TABLE LOOKUP
Pam Donahue deposits $8,000 in her savings account that pays 6% interest compounded quarterly. What will be the
balance of her account at the end of 5 years?
Periods (N) = 4 x 5 = 20
Rate (R) = 6%/4 = 1.5%Table Factor = 1.3469Compounded Amount:$8,000 x 1.3469 = $10,775.20
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NOMINAL AND EFFECTIVE RATES (APY) OF INTEREST
Truth in
Savings Law
Annual
PercentageYield
Effective rate (APY) = Interest for 1 year
Principal
Nominal Rate (stated rate) –
The rate on which the bank calculates interest
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CALCULATING EFFECTIVE RATE APY
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NOMINAL AND EFFECTIVE RATES (APY) OF INTEREST COMPARED
(FIGURE 12.3)
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Period 6.00% 6.50% 7.00% 7.50% 8.00% 8.50% 9.00% 9.50% 10.00%
1 1.0618 1.0672 1.0725 1.0779 1.0833 1.0887 1.0942 1.0996 1.1052
2 1.1275 1.1388 1.1503 1.1618 1.1735 1.1853 1.1972 1.2092 1.2214
3 1.1972 1.2153 1.2337 1.2523 1.2712 1.2904 1.3099 1.3297 1.3498
4 1.2712 1.2969 1.3231 1.3498 1.3771 1.4049 1.4333 1.4622 1.4917
5 1.3498 1.3840 1.4190 1.4549 1.4917 1.5295 1.5862 1.6079 1.6486
6 1.4333 1.4769 1.5219 1.5682 1.6160 1.6652 1.7159 1.7681 1.8220
7 1.5219 1.5761 1.6322 1.6904 1.7506 1.8129 1.8775 1.9443 2.0136
8 1.6160 1.6819 1.7506 1.8220 1.8963 1.9737 2.0543 2.1381 2.2253
9 1.7159 1.7949 1.8775 1.9639 2.0543 2.1488 2.2477 2.3511 2.4593
10 1.8220 1.9154 2.0136 2.1168 2.2253 2.3394 2.4593 2.5854 2.7179
15 2.4594 2.6509 2.8574 3.0799 3.3197 3.5782 3.8568 4.1571 4.4808
20 3.3198 3.6689 4.0546 4.4810 4.9522 5.4728 6.0482 6.6842 7.3870
25 4.4811 5.0777 5.7536 6.5195 7.3874 8.3708 9.4851 10.7477 12.1782
30 6.0487 7.0275 8.1645 9.4855 11.0202 12.8032 14.8747 17.2813 20.0772
Interest on a 1% deposit compounded daily--360 day basis
COMPOUNDING INTEREST DAILY (TABLE 12.2)
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COMPOUNDING INTEREST DAILY
Use Table 12.2 to calculate what $1,500 compounded daily for
5 years will grow to at 7%.
N = 5
R = 7%
Factor, 1.4190
$1,500 x 1.4190 = $2,128.50
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PRESENT VALUE OF $1 AT 8% FOR FOUR PERIODS (FIGURE
12.4)
$0.00$0.10$0.20$0.30$0.40$0.50$0.60$0.70$0.80$0.90$1.00$1.10$1.20
0 1 2 3 4
Number of periods
Present value goes from the future value to the present value
Present value
$.7350$.7938
$.8573
$.9259
$1.0000Future Value
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CALCULATING PRESENT VALUE BY TABLE LOOKUP
Step 1. Find the periods: Years multiplied by number of times interest is compounded in 1 year.
Step 2. Find the rate: Annual rate divided by number of times interest is compounded in 1 year.
Step 3. Go down the Period column of the table to the number desired; look across the row to find the rate. At the intersection of the two columns is the table factor for the compound value of $1.
Step 4. Multiply the table factor by the future value. This is the present value.
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Period 1% 1.50% 2% 3% 4% 5% 6% 7% 8% 9% 10%
1 0.9901 0.9852 0.9804 0.9709 0.9615 0.9524 0.9434 0.9346 0.9259 0.9174 0.9091
2 0.9803 0.9707 0.9612 0.9426 0.9246 0.9070 0.8900 0.8734 0.8573 0.8417 0.8264
3 0.9706 0.9563 0.9423 0.9151 0.8890 0.8638 0.8396 0.8163 0.7938 0.7722 0.7513
4 0.9610 0.9422 0.9238 0.8885 0.8548 0.8227 0.7921 0.7629 0.7350 0.7084 0.6830
5 0.9515 0.9283 0.9057 0.8626 0.8219 0.7835 0.7473 0.7130 0.6806 0.6499 0.6209
6 0.9420 0.9145 0.8880 0.8375 0.7903 0.7462 0.7050 0.6663 0.6302 0.5963 0.5645
7 0.9327 0.9010 0.8706 0.8131 0.7599 0.7107 0.6651 0.6227 0.5835 0.5470 0.5132
8 0.9235 0.8877 0.8535 0.7894 0.7307 0.6768 0.6274 0.5820 0.5403 0.5019 0.4665
9 0.9143 0.8746 0.8368 0.7664 0.7026 0.6446 0.5919 0.5439 0.5002 0.4604 0.4241
10 0.9053 0.8617 0.8203 0.7441 0.6756 0.6139 0.5584 0.5083 0.4632 0.4224 0.3855
11 0.8963 0.8489 0.8043 0.7224 0.6496 0.5847 0.5268 0.4751 0.4289 0.3875 0.3505
12 0.8874 0.8364 0.7885 0.7014 0.6246 0.5568 0.4970 0.4440 0.3971 0.3555 0.3186
13 0.8787 0.8240 0.7730 0.6810 0.6006 0.5303 0.4688 0.4150 0.3677 0.3262 0.2897
14 0.8700 0.8119 0.7579 0.6611 0.5775 0.5051 0.4423 0.3878 0.3405 0.2992 0.2633
15 0.8613 0.7999 0.7430 0.6419 0.5553 0.4810 0.4173 0.3624 0.3152 0.2745 0.2394
Present value of $1 at end period (partial)
PRESENT VALUE OF $1 AT END PERIOD (TABLE 12.3)
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COMPARING COMPOUND INTEREST (FV)
(TABLE 12.1) WITH PRESENT VALUE (PV) (TABLE 12.3)
Compound value Table 12.1Table Present Future12.1 Value Value1.3605 x $80 = $108.84
(N = 4, R = 8%)
We know the present dollar
amount and find what the dollar amount is worth
in the future.
We know the future dollar amount and find what the
dollar amount is worth in the
present.
Present value Table 12.3Table Future Present12.3 Value Value
.7350 x $108.84 = $80.00
(N = 4, R = 8%)
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CALCULATING PRESENT VALUE AMOUNT
BY TABLE LOOKUPRene Weaver needs $20,000 for college in 4 years. She can earn 8% compounded quarterly at her bank. How much must Rene deposit at
the beginning of the year to have $20,000 in 4 years?
Periods (N) = 4 x 4 = 16Rate (R) = 8%/4 = 2%Table Factor = .7284
Compounded Amount:$20,000 x .7284 = $14,568
Invest Today
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PRACTICE QUIZ-1
1-Complete the following without a table (round each calculation to the nearest cent as needed):
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PRACTICE QUIZ-1
2-Solve the previous problem by using compound value (FV) in Table 12.1.3-Lionel Rodgers deposits $6,000 in Victory Bank, which pays 3% interest compounded semiannually. How much will Lionel have in his account at the end of 8 years?4-Find the effective rate (APY) for the year: principal, $7,000; interest rate, 12%; and compounded quarterly.5-Calculate by Table 12.2 what $1,500 compounded daily for 5 years will grow to at 7%.
For step by step solution watch the video for LU 12-1 ( Go to: McGraw-Hill’s Connect; Assignment # 4; Question 1; Click the eBook & resources options drop down menu; scroll down to LU12-1 and click
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PRACTICE QUIZ-2
Use the present value Table 12.3 to complete:
3-Bill Blum needs $20,000 6 years from today to attend V.P.R. Tech. How much must Bill put in the bank today (12% quarterly) to reach his goal?4-Bob Fry wants to buy his grandson a Ford Taurus in 4 years. The cost of a car will be $24,000. Assuming a bank rate of 8% compounded quarterly, how much must Bob put in the bank today?
For step by step solution watch the video for LU 12-1 ( Go to: McGraw-Hill’s Connect; Assignment # 4; Question 5; Click the eBook & resources options drop down menu; scroll down to LU12-2 and click
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PROBLEM 12-13
Solution:
8 years x 2 = 16 periods 6% 2 = 3% $40,000 x 1.6047 = $64,188
Lynn Ally, owner of a local Subway shop, loaned $40,000 to Pete Hall to help him open a Subway franchise. Pete plans to repay Lynn at the end of 8 years with 6% interest compounded semiannually. How much will Lynn receive at the end of 8 years? LU 12-1(2)
12-27
PROBLEM 12-15
Solution:
Mystic
4 years x 2 = 8 periods
10% 2
= 5%
$10,000 x 1.4775 = $14,775 -- 10,000 $ 4,775
Four Rivers
4 years x 4 = 16 periods
8% 4
= 2%
$10,000 X 1.3728 = $13,728 -- 10,000 $ 3,728
Melvin Indecision has difficulty deciding whether to put his savings in Mystic Bank or Four Rivers Bank. Mystic offers 10% interest compounded semiannually. Four Rivers offers 8% interest compounded quarterly. Melvin has $10,000 to invest. He expects to withdraw the money at the end of 4 years. Which bank gives Melvin the better deal? Check your answer. LU 12-1(3)
12-28
PROBLEM 12-16
Solution:
3 years x 2 = 6 periods9% 2 = 4.5%
$59,534.50 x 1.3023 = $77,531.78
Lee Holmes deposited $15,000 in a new savings account at 9% interest compounded semiannually. At the beginning of year 4, Lee deposits an additional $40,000 at 9% interest compounded semiannually. At the end of 6 years, what is the balance in Lee’s account? LU 12-1(2)
$15,000 x 1.3023 = $19,534.50 + 40,000.00 $ 59,534.50
12-29
PROBLEM 12-27
Solution:
8 years x 2 = 16 periods
6% 2 = 3%
$6,000 x .6232 = $3,739.20
Paul Havlik promised his grandson Jamie that he would give him $6,000 8 years from today for graduating from high school. Assume money is worth 6% interest compounded semiannually. What is the present value of this $6,000?LU 12-2(2)
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DRILL PROBLEMS
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