# How to calculate present value

Present value is the concept that cash received today is more valuable than cash received at some point in the future. The reason is that someone who agrees to receive payment at a later date foregoes the ability to invest that cash right now. The only way for someone to agree to a delayed payment is to pay them for the privilege, which is known as interest income.

For example, if a person owns \$10,000 now and invests it at an interest rate of 10%, then she will have earned \$1,000 by having use of the money for one year. If she were instead to not have access to that cash for one year, then she would lose the \$1,000 of interest income. The interest income in this example represents the time value of money.

To extend the example, what is the current payout of cash at which the person would be indifferent to receiving cash now or in one year? In essence, what is the amount that, when invested at 10%, will equal \$10,000 in one year? The general formula used to answer this question, known as the present value of 1 due in N periods, is:

 1 -------------------------------------- (1 + Interest rate) Number of years

The calculation for the example is:

 \$10,000 -------------------- (1 + 10%) 1 year

In essence, if the person receives \$9,090.91 now and invests it at a 10% interest rate, her cash balance will have increased to \$10,000 in one year.

The effect of the present value formula becomes more pronounced if the receipt of cash is delayed to a date even further in the future, because the period during which the recipient of the cash cannot invest the cash is prolonged.

The concept of the time value of money also works in reverse, for expenditures. There is a monetary value associated with delaying the payment of cash, which is known as the future amount of 1 due in N periods. The general formula used to address this situation is:

Amount deferred × (1 + Interest rate) Number of years

For example, if a person could delay the expenditure of \$10,000 for one year and could invest the funds during that year at a 10% interest rate, the value of the deferred expenditure would be \$11,000 in one year.

One of the common uses of the time value of money is to derive the present value of an annuity. An annuity is a series of payments that occur in the same amounts and at the same intervals over a period of time. An annuity is a common feature of a capital budgeting analysis, where a consistent stream of cash flows is expected for multiple years if a fixed asset is purchased. For example, a company is contemplating the purchase of a production line for \$3,000,000, which will generate net positive cash flows of \$1,000,000 per year for the next five years. This stream of incoming cash flows is an annuity. The formula used to derive the present value of an ordinary annuity of 1 per period is:

 1 1 - ---------------------------------------- (1 + Interest rate) Number of years Interest rate

The preceding formula is for an ordinary annuity, which is an annuity where payments are made at the end of each period. If cash were instead received at the beginning of each period, the annuity would be called an annuity due, and would be formulated somewhat differently.

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