chapter 12 compound interest and present value mcgraw-hill/irwin copyright © 2011 by the...
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Chapter 12
Compound Interest and Compound Interest and Present ValuePresent Value
McGraw-Hill/Irwin Copyright © 2011 by the McGraw-Hill Companies, Inc. All rights reserved.
12-2
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
Compound Interest and Present Value#12#12Learning Unit ObjectivesCompound Interest (Future Value) – The Big Picture
LU12.1LU12.1
12-3
1. Compare present value (PV) with compound interest (FV)
2. Compute present value by table lookup
3. Check the present value answer by compounding
Compound Interest and Present Value#12#12Learning Unit ObjectivesPresent Value -- The Big PictureLU12.2LU12.2
12-4
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) - is the final amount of the loan or investment at the end of the last
period
12-5
Compounding Terms
Compounding Periods Interested Calculated
Compounding Annually Once a year
Compounding Semiannually Every 6 months
Compounding Quarterly Every 3 months
Compounding Monthly Every month
Compounding Daily Every day
12-6
Figure 12.1 Future Value of $1 at 8% for Four Periods
$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 4
Number 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
12-7
Figure 12.1 Future Value of $1 at 8% for Four Periods
Year 1 Year 2 Year 3 Year 41.00$ 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 1.26End of year 1.08$ 1.17$ 1.26$ 1.36$
Manual Calculation
12-8
Tools for Calculating Compound Interest
Number of periods (N) Number of years
multiplied 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: 4 x 1 = 4Semiannually: 4 x 2 = 8Quarterly: 4 x 4 = 16
Annually: 8% / 1 = 8%Semiannually: 8% / 2 = 4%Quarterly: 8% / 4 = 2%
Periods Rate
12-9
Simple Versus Compound Interest
Bill Smith deposited $80 in a savings account for 4 years at an annual interestrate 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 $80 in a savings account for 4 years at an annual interestrate of 8%. What is Bill’s interest and compounded Amount?
Simple CompoundedCompounded
Year 1 Year 2 Year 3 Year 480.00$ 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 93.31 100.77End of year 86.40$ 93.31$ 100.77$ 108.83$
Interest: $108.83 - $80.00 = $28.83
SimpleSimple
12-10
Calculating Compound Amount 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 is the table factor
Step 4. Multiply the table factor by the amount of the loan.
12-11
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)
Table 12.1 - Future Value of $1 at Compound Interest
12-12
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?
N = 4 x 5 = 20
R = 6% = 1.5% 1
Table Factor = 1.3469
Compounded Amount:
$8,000 x 1.3469 = $10,775.20
12-13
Nominal and Effective Rates (APY) of Interest
Truth in
Savings
Law
Annual
Percentage
Yield Effective Rate = Interest for 1 year (APY) Principal
Nominal Rate (Stated Rate) - The rate on which the bank calculates interest.
12-14
Calculating Effective Rate APY
Blue, 8% compounded quarterlyPeriods = 4 (4 x 1)Percent = 8% = 2% 4Principal = $8,000Table 12.1 lookup: 4 periods, 2%
1.0824 x $8,000Less $8,659.20
$8,000.00 659.20
APY 659.20 = .0824 $8,000
= 8.24%
Sun, 8% compounded semiannuallyPeriods = 2 (2 x 1)Percent = 8% = 4% 2Principal = $8,000Table 12.1 lookup: 2 periods, 4%
1.0816 x $8,000Less $8,652.80
$8,000.00 652.80
APY 652.80 = .0816 $8,000
= 8.16%
12-15
Figure 12.3 - Nominal and Effective Rates (APY) of Interest Compared
Annual
Semiannual
Quarterly
Daily
$1,060.00
$1,060.90
$1,061.40
$1,061.80
6.00
6.09%
6.14%
6.18%
$1,000 + 6%
Beginning Nominal rate Compounding End Effective rate balance of interest period balance (APY) of interest
12-16
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
Table 12.2 - Compounding Interest Daily
12-17
Compounding Interest Daily
Calculate by Table 12.2 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
12-18
Figure 12.4 Present Value of $1 at 8% for Four Periods
$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
$1.0000
Future Value
12-19
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 is the table factor.
Step 4. Multiply the table factor by the future value. This is the present value.
12-20
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)
Table 12.3 - Present Value of $1 at End Period
12-21
Comparing Compound Interest (FV) Table 12.1 with Present Value (PV) Table 12.3
Compound value Table 12.1 Present value Table 12.3Table Present Future Table Future Present12.1 Value Value 12.3 Value Value
1.3605 x $80 = $108.84 .7350 x $108.84 = $80.00
(N = 4, R = 8) (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
12-22
Calculating Present Value Amount by Table Lookup
Rene 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?
N = 4 x 4 = 16
R = 8% = 2% 4
Table Factor = .7284
Compounded Amount:
$20,000 x .7284 = $14,568
Invest Today
12-23
Problem 12-13:
Solution:
8 years x 2 = 16 periods 6% 2
= 3%
$40,000.00 x 1.6047 = $64,188.
12-24
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
12-25
Problem 12-16:
Solution:
3 years x 2 = 6 periods $15,000 x 1.3023 = $19,534.50 +40,000.00 $59,534.50
9% 2
= 4.5%
$59,534.50 x 1.3023 = $77,531.78
12-26
Problem 12-27:
Solution:
8 years x 2 = 16 periods 6% 2
= 3%
$6,000 x .6232 = $3,739.20