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)