## Time Value of Money: Highlights

This note explains briefly two concepts concerning the time-value-of-money, namely future and present value. Careful application of these concepts will help you evaluate investment/financing situations such as real estate, life insurance, monthly payments on a car, and many others.

### Future Value

Future value is simply the sum to which a dollar amount invested today will grow given some appreciation rate.

To compute the future value of a sum invested today, the formula for interest that is compounded monthly is:

` fv = principal * (1 + rrate/12) ** (12 * termy)`

where
principal = dollar value you have now
termy = term, in years
rrate = annual rate of return in decimal (i.e., use .05 for 5%)

For interest that is compounded annually, use the formula:

` fv = principal * (1 + rrate) ** (termy)`

Example:
I invest 1,000 today at 10% for 10 years compounded monthly. The future value of this amount is 2707.04.

Note that the formula for future value is the formula from Case 1 of present value (below), but solved for the future-sum rather than the present value.

### Present Value

Present value is the value in today's dollars assigned to an amount of money in the future, based on some estimate rate-of-return over the long-term. In this analysis, rate-of-return is calculated based on monthly compounding.

Two cases of present value are discussed next. Case 1 involves a single sum that stays invested over time. Case 2 involves a cash stream that is paid regularly over time (e.g., rent payments), and requires that you also calculate the effects of inflation.

Case 1a: Present value of money invested over time. This tells you what a future sum is worth today, given some rate of return over the time between now and the future. Another way to read this is that you must invest the present value today at the rate-of-return to have some future sum in some years from now (but this only considers the raw dollars, not the purchasing power).

To compute the present value of an invested sum, the formula for interest that is compounded annually is:

`            future-sum pv = -----------------------      (1 + rrate) ** (termy)`

where
future-sum = dollar value you want in termy years
termy = term, in years
rrate = annual rate of return that you can expect, in decimal

Example:
I need to have 10,000 in 5 years. The present value of 10,000 assuming an 8% annually compounded rate-of-return is 6,580. I.e., 6,580 will grow to 10,000 in 5 years at 8%.

Case 1b: This formulation can also be used to estimate the effects of inflation; i.e., compute real purchasing power of present and future sums. Simply use an estimated rate of inflation instead of a rate of return for the rrate variable in the equation.

Example:
In 30 years I will receive 1,000,000 (a megabuck). What is that amount of money worth today (what is the buying power), assuming a rate of inflation of 4.5%? The answer is 267,000.

Case 2: Present value of a cash stream. This tells you the cost in today's dollars of money that you pay over time. Basically, the money you pay in 10 years is worth less than that which you pay tomorrow, and this equation lets you compute just how much.

To compute the present value of a cash stream of monthly payments, the formula is:

`      month=12*termy               paymt   pv = SUM               --------------------------------      month=1            (1 + rrate/12) ** (month - 1)`

where
month = month number
termy = term, in years
paymt = monthly payment, in dollars
rrate = rate of return on money that you can expect, in decimal

Example:
You pay \$500/month in mortgage payments on your mobile home over 10 years. The lender's rate is 6% per annum. Present value is \$45,036...
Or is it? Work it out for yourself.

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