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

_____ frequency of compounding = ___ FV

_____ length of investment = ____ FV

_____ interest rate = _____ FV

Implications…

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

The Time Value of Money in Decision Making

Assume you have the option of two different investment options. First, a friend wanted to borrow $5,000 for three years and pay you back $6,000 in a lump sum. Second, you could invest the same $5,000 for three years in a government bond paying 7 percent annual interest. Which investment would be the best financial decision?

FV = (PV)(1 + r) n

= (5,000)(1+.07)3

= 5,000 x 1.225043

= $6,125.22

You would earn $125.22 more by investing in the government bonds.

Future Value of a Dollar (Single Payment)

Future Value of a Series of Annual Deposits (Annuity)

Present Value of a Dollar (Single Payment)

Present Value of a Series of Annual Deposits (Annuity)