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Present Values

Money can be invested to earn interest. If you are offered the choice between $100,000 now and $100,000 at the end of the year, you naturally take the money now to get a year`s interest. Financial managers make the same point when they say that money in hand today has a time value or when they quote perhaps the most basic financial principle:

A dollar today is worth more than a dollar tomorrow.

We have seen that $100 invested for 1 year at 6 percent will grow to a future value of 100 1.06 = $106. Let`s turn this around: How much do we need to invest now in order to produce $106 at the end of the year? Financial managers refer to this as the present value (PV) of the $106 payoff.

Future value is calculated by multiplying the present investment by 1 plus the interest rate, .06, or 1.06. To calculate present value, we simply reverse the process and divide the future value by 1.06:

Present value = PV = future value = $106 = $100 1.06 1.06

What is the present value of, say, $112.36 to be received 2 years from now? Again we ask, ¬How much would we need to invest now to produce $112.36 after 2 years? ­ The answer is obviously $100; we`ve already calculated that at 6 percent $100 grows to $112.36:

$100 (1.06)2 = $112.36

However, if we don`t know, or forgot the answer, we just divide future value by (1.06)2:

Present value = PV = $112.36 = $100 (1.06)2

In general, for a future value or payment t periods away, present value is Present value = future value after t periods (1 + r)t

In this context the interest rate r is known as the discount rate and the present value is often called the discounted value of the future payment. To calculate present value, we discounted the future value at the interest r.

DISCOUNT RATE Interest rate used to compute present values of future cash flows.

Saving to Buy a New Computer

Suppose you need $3,000 next year to buy a new computer. The interest rate is 8 percent per year. How much money should you set aside now in order to pay for the purchase? Just calculate the present value at an 8 percent interest rate of a $3,000 payment at the end of one year. This value is

PV = $3,000 = $2,778 1.08

Notice that $2,778 invested for 1 year at 8 percent will prove just enough to buy your computer:

Future value = $2,778 1.08 = $3,000

The longer the time before you must make a payment, the less you need to invest today. For example, suppose that you can postpone buying that computer until the end of 2 years. In this case we calculate the present value of the future payment by dividing

$3,000 by (1.08)2: PV = $3,000 = $2,572 (1.08)2

Thus you need to invest $2,778 today to provide $3,000 in 1 year but only $2,572 to provide the same $3,000 in 2 years.