Translating Today’s Benefits to the Future Suppose you want to know how much
money you would have in 5 years if you placed $5,000 in the bank today at an interest rate of 6% compounded annually.
future value of a one-time investment.• The future value is the accumulated amount of
your investment fund at the end of a specified period.
This is an exercise that involves the use of compound interest.• Compound interest - Situation where you earn
interest on the original investment and any interest that has been generated by that investment previously.
• Earn interest on your interest• First year: $5,000(1+.06) = $5,300• Second year: $5,300(1+.06) = $5,618• Third year: $5,618(1+.06) = $5,955.08• Fourth year: $5,955.08(1+.06) = $6,312.38• Fifth year: $6,312.38(1+.06) = $6,691.13
Effect of Compound InterestSimple Interest Compound Interest
Year Principal Rate TimeInterestEarned
NewBalance Principal Rate Time
InterestEarned
NewBalance
1 100.00 10% 1 10.00 110.00 100.00 10% 1 10.00 110.00 2 100.00 10% 1 10.00 120.00 110.00 10% 1 11.00 121.00 3 100.00 10% 1 10.00 130.00 121.00 10% 1 12.10 133.10 4 100.00 10% 1 10.00 140.00 133.10 10% 1 13.31 146.41 5 100.00 10% 1 10.00 150.00 146.41 10% 1 14.64 161.05 6 100.00 10% 1 10.00 160.00 161.05 10% 1 16.11 177.16 7 100.00 10% 1 10.00 170.00 177.16 10% 1 17.72 194.87 8 100.00 10% 1 10.00 180.00 194.87 10% 1 19.49 214.36 9 100.00 10% 1 10.00 190.00 214.36 10% 1 21.44 235.79
10 100.00 10% 1 10.00 200.00 235.79 10% 1 23.58 259.37 11 100.00 10% 1 10.00 210.00 259.37 10% 1 25.94 285.31 12 100.00 10% 1 10.00 220.00 285.31 10% 1 28.53 313.84 13 100.00 10% 1 10.00 230.00 313.84 10% 1 31.38 345.23 14 100.00 10% 1 10.00 240.00 345.23 10% 1 34.52 379.75 15 100.00 10% 1 10.00 250.00 379.75 10% 1 37.97 417.72 16 100.00 10% 1 10.00 260.00 417.72 10% 1 41.77 459.50 17 100.00 10% 1 10.00 270.00 459.50 10% 1 45.95 505.45 18 100.00 10% 1 10.00 280.00 505.45 10% 1 50.54 555.99
180.00 455.99
Formula:• FV = PV(1 + r)n
• r = interest rate divided by the compounding factor– (yearly r / compounding factor)
• n = number of compounding periods – (yearly n * compounding factor)
• PV = Present Value of your investment
• Compounding Factors:• Yearly = 1
• Quarterly = 4
• Monthly = 12
• Daily = 365
• Please note that I will always report r’s and n’s as yearly numbers
• You will need to determine the compounding factor
• All of your terms must agree as to time. • If you are taking an action monthly (like investing
every month), then r and n must automatically be converted to monthly compounding.
• If you are rounding in time value of money formulas, you need AT LEAST four (4) numbers after the zeros (0)
• r = .08/12• r=0.006667 (not 0.0067 or 0.007 or etc.)
• Yearly compounding• PV = 5000• r = .06• n = 5• FV = $5,000(1.06)5 • = $6,691.13• Monthly compounding• PV = 5000• r = (.06/12) = .005• n = 5(12) = 60• FV = $5,000(1+.005)60 • = $6,744.25
How do the calculations change if the investment is repeated periodically?
Suppose you want to know how much money you would have in 24 years if you placed $500 in the bank each year for twenty-four years at an annual interest rate of 8%.
future value of a periodic investment or future value of an annuity (stream of payments over time) = FVA
The formula is...
• where PV = the Present Value of the payment in each period
• r = interest rate divided by the compounding factor
• n = number of compounding periods
r
rPVFVA
n 11
Let’s try it… $500/year, 8% interest, 24 years, yearly
compounding• PV = 500
• r = .08
• n = 24
• = 500 (66.7648)
= $33,382.38
08.
108.1500
24
FVA
08.
13412.6500FVA
Let’s try it again… $50/month, 8% interest, 5 years, monthly
compounding• PV = 50
• r = (.08/12) = .006667
• n = 5(12) = 60
• = 50 (73.4769) = $3673.84
Try again with n=120 FVA=$9147.30
006667.
1006667.150
60
FVA
006667.
14898.150FVA
More Practice
You have a really cool grandma who gave you $1,000 for your high school graduation. You invested it in a 5-year CD, earning 5% interest. How much will you have when you cash it out if it is compounded yearly?
How much will you have if it is compounded monthly?
How much will you have if it is compounded daily?
Yearly Compounding 1000(1+.05)5
=$1276.28 Monthly Compounding r = (.05/12) = .004167 n = 5(12) = 60 1000(1+.004167)60 =$1283.36 Daily Compounding r = (.05/365) = .000136986 n = 5(365) = 1825 1000(1+.000136986)1825
=$1284.00
Some more practice...
You have decided to be proactive for the future, and will save $25 a month. At the end of 10 years, how much will you have saved, if you earn 8% interest annually?
Monthly Compounding FVA = PV = $25 a month r = (.08/12) = .006667 n = (10)(12) = 120 FVA = $4573.65
Do I have the money now?
Determining when to use Future Value vs. Present Value Calculation/Tables
Yes No
Is it a lump sum? Is it a lump sum?
Yes No Yes No
Use FV of a single
payment
Use PV of a single
payment
Use FV of an annuity
Use PV of an annuity
Use FV calculation/table
Use PV calculation/table
Future Value of $1 (single amount)
Year 5% 6% 7% 8% 9%
1 1.050 1.060 1.070 1.080 1.090
2 1.103 1.124 1.145 1.166 1.188
3 1.158 1.191 1.225 1.260 1.295
4 1.216 1.262 1.311 1.360 1.412
5 1.276 1.338 1.403 1.469 1.539
6 1.340 1.419 1.501 1.587 1.677
7 1.407 1.504 1.606 1.714 1.828
8 1.477 1.594 1.718 1.851 1.993
9 1.551 1.689 1.838 1.999 2.172
10 1.629 1.791 1.967 2.159 2.367
11 1.710 1.898 2.105 2.332 2.580
12 1.796 2.012 2.252 2.518 2.813
13 1.886 2.133 2.410 2.720 3.066
14 1.980 2.261 2.579 2.937 3.342
15 2.079 2.397 2.759 3.172 3.642
16 2.183 2.540 2.952 3.426 3.970
17 2.292 2.693 3.159 3.700 4.328
18 2.407 2.854 3.380 3.996 4.717
19 2.527 3.026 3.617 4.316 5.142
20 2.653 3.207 3.870 4.661 5.604
Year 5% 6% 7% 8% 9%
1 1.000 1.000 1.000 1.000 1.000
2 2.050 2.060 2.070 2.080 2.090
3 3.153 3.184 3.215 3.246 3.278
4 4.310 4.375 4.440 4.506 4.573
5 5.526 5.637 5.751 5.867 5.985
6 6.802 6.975 7.153 7.336 7.523
7 8.142 8.394 8.654 8.923 9.200
8 9.549 9.897 10.260 10.637 11.028
9 11.027 11.491 11.978 12.488 13.021
10 12.578 13.181 13.816 14.487 15.193
11 14.207 14.972 15.784 16.645 17.560
12 15.917 16.870 17.888 18.977 20.141
13 17.713 18.882 20.141 21.495 22.953
14 19.599 21.015 22.550 24.215 26.019
15 21.579 23.276 25.129 27.152 29.361
16 23.657 25.673 27.888 30.324 33.003
17 25.840 20.213 30.840 33.750 36.974
18 28.132 30.906 33.999 37.450 41.301
19 30.539 33.760 47.379 41.446 46.018
20 33.066 36.786 40.995 45.762 51.160
Future Value of a Series of Annual Deposits (annuity)
Year 5% 6% 7% 8% 9%
1 0.952 0.943 0.935 0.926 0.917
2 0.907 0.890 0.873 0.857 0.842
3 0.864 0.840 0.816 0.794 0.772
4 0.823 0.792 0.763 0.735 0.708
5 0.784 0.747 0.713 0.681 0.650
6 0.746 0.705 0.666 0.630 0.596
7 0.711 0.665 0.623 0.583 0.547
8 0.677 0.627 0.582 0.540 0.502
9 0.645 0.592 0.544 0.500 0.460
10 0.614 0.558 0.508 0.463 0.422
11 0.585 0.527 0.475 0.429 0.388
12 0.557 0.497 0.444 0.397 0.356
13 0.530 0.469 0.415 0.368 0.326
14 0.505 0.442 0.388 0.340 0.299
15 0.481 0.417 0.362 0.315 0.275
16 0.458 0.394 0.339 0.292 0.252
17 0.436 0.371 0.317 0.270 0.231
18 0.416 0.350 0.296 0.250 0.212
19 0.396 0.331 0.277 0.232 0.194
20 0.377 0.312 0.258 0.215 0.178
Present Value of $1 (single amount)
Year 5% 6% 7% 8% 9%
1 0.952 0.943 0.935 0.926 0.917
2 1.859 1.833 1.808 1.783 1.759
3 2.723 2.673 2.624 2.577 2.531
4 3.546 3.465 3.387 3.312 3.240
5 4.329 4.212 4.100 3.993 3.890
6 5.076 4.917 4.767 4.623 4.486
7 5.786 5.582 5.389 5.206 5.033
8 6.463 6.210 5.971 5.747 5.535
9 7.108 6.802 6.515 6.247 5.995
10 7.722 7.360 7.024 6.710 6.418
11 8.306 7.887 7.499 7.139 6.805
12 8.863 8.384 7.943 7.536 7.161
13 9.394 8.853 8.358 7.904 7.487
14 9.899 9.295 8.745 8.244 7.786
15 10.380 9.712 9.108 8.559 8.061
16 10.838 10.106 9.447 8.851 8.313
17 11.274 10.477 9.763 9.122 8.544
18 11.690 10.828 10.059 9.372 8.756
19 12.085 11.158 10.336 9.604 8.950
20 12.462 11.470 10.594 9.818 9.129
Present Value of a Series of Annual Deposits (annuity)