How to Calculate PV From PMT

Present value (PV) is one of the most fundamental financial concepts for calculating loan payments and investments. It is the value of future cash payments (PMTs) expressed as if you received or spent the money today in a single lump sum.

You can use a financial calculator or spreadsheet to calculate the present value of a discounted cash flow, such as an annuity, based on its future value or the timing and number of future payments. One common use of PV is to calculate the current price of a bond based on its future payments of interest and the return of principal.

As NYU Stern explains, at the heart of the present value calculation is the discount rate, which expresses the time value of money (TVM). Specifically, TVM is based on the notion that a dollar in your pocket today is worth more than its future value would be if you received it at a future date. The reasons for this are:

• You can spend the money today and not experience the effects of inflation, which robs money of its value over time.
• You can invest the money today and earn a return such as interest or dividends. In fact, interest is an expression of money’s time value, subject to various conditions.

The discount rate is the rate at which you deflate the value of future cash flows to equate to their present value. Another way to look at the discount rate is as the amount of money you would need today to fund a series of future payments (usually monthly payments). A special discount rate is the risk-free rate, commonly equal to the interest rate on U.S. Treasury debt.

The present value of an ordinary annuity (i.e., an annuity that pays interest at the end of each specified period) is as follows:

‌PV = PMT x [(1 – (1/(1+r)ⁿ)) / r]‌

where:

‌PV‌ = present value of an annuity cash flow stream

‌PMT‌ = dollar amount of each annuity payment

‌r‌ = discount rate

‌n‌ = number of periods over which payments will be made

For example, suppose you were offered an annuity that paid $25,000 per year, with the number of years being 20, and which would cost you a total amount of $300,000 to buy today. Is it a good deal? Assume that the risk-free annual interest rate, five percent, is used as the discount rate.

Plugging in the numbers, you get:

‌PV = $25,000 * [(1 – (1 / (1 + .05) ²⁰)) / .05] = $311,555‌

In today’s dollars, the annuity is worth $11,555 more than the cost to buy it, making it the better choice.

You can quickly calculate the present value of an annuity using a business calculator, such as the Hewlett-Packard 12C, which has the required financial functions.

The calculator has special buttons for the PV formula: To enter the variables, include “PMT” for payment, “i” for the discount rate and “n” for the number of payments. After entering the variables, press the PV button to get the annuity’s present value.

Harvard Business School Online notes that you can calculate present value in an Excel spreadsheet or worksheet by using the relevant Excel functions such as the PV function. The Excel formula variables you would enter are Rate (the discount rate), NPER (number of periods) and the PMT function (payment amount).

The spreadsheet accepts an optional parameter: Type. If Type is set to 0 (the default), payments are due at the end of the period. You can set it to 1 for periodic payments due at the beginning of the period.

Keep in mind that when you calculate the PV for an annuity, the answers are never positive numbers. Instead, they are given as negative numbers. The reasoning behind this is that the present value is the amount you would have to spend today to break even with the return on the annuity. Spending is a negative cash flow (i.e., the money is leaving your pocket), so the result is negative.