Present value

Present value, also known as present discounted value, is the value on a given date of a payment or series of payments made at other times.

If the payments are in the future, they are discounted to reflect the time value of money and other factors such as investment risk. If they are in the past, their value is correspondingly enhanced to reflect that those payments have been (or could have been) earning interest in the intervening time. Present value calculations are widely used in business and economics to provide a means to compare cash flows at different times on a meaningful "like to like" basis.

Background

If offered a choice between 100 today or 100 in one year and there is a positive real interest rate throughout the year ceteris paribus, a rational person will choose 100 today. This is described by economists as time preference. Time preference can be measured by auctioning off a risk free security - like a US Treasury bill. If a 100 note, payable in one year, sells for 80, then the present value of 100 one year in the future is 80. This is because money can be put in a bank account or any other (safe) investment that will return interest in the future.

An investor who has some money has two options: to spend it right now or to save it. But the financial compensation for saving it (and not spending it) is that the money value will accrue through the compound interest that he will receive from a borrower (the bank account on which he has the money deposited).

Therefore, to evaluate the real value of an amount of money today after a given period of time, economic agents compound the amount of money at a given (interest) rate. Most actuarial calculations use the risk-free interest rate which corresponds the minimum guaranteed rate provided by a bank's saving account for example. To compare the change in purchasing power, the real interest rate (nominal interest rate minus inflation rate) should be used.

The operation of evaluating a present value into the future value is called a capitalization (how much will 100 today be worth in 5 years?). The reverse operation—evaluating the present value of a future amount of money—is called a discounting (how much will 100 received in 5 years—at a lottery for example—be worth today?).

It follows that if one has to choose between receiving 100 today and 100 in one year, the rational decision is to choose the 100 today. If the money is to be received in one year and assuming the savings account interest rate is 5%, the person has to be offered at least 105 in one year so that two options are equivalent (either receiving 100 today or receiving 105 in one year). This is because if 100 is deposited in a savings account, the value will be 105 after one year.

Calculation

The most commonly applied model of the time value of money is compound interest. To someone who can lend or borrow for $\,t\,$ years at an interest rate $\,i\,$ per year (where interest of "5 percent" is expressed fully as 0.05), the present value of the receiving $\,C\,$ monetary units $\,t\,$ years in the future is:

$C_t = C(1 + i)^{-t}\, = \frac{C}{(1+i)^t} \,$

This is also found from the formula for the future value with negative time.

The purchasing power in today's money of an amount $\,C\,$ of money, $\,t\,$ years into the future, can be computed with the same formula, where in this case $\,i\,$ is an assumed future inflation rate.

The expression $\,(1 + i)^{-t}$ enters almost all calculations of present value. Where the interest rate is expected to be different over the term of the investment, different values for $\,i\,$ may be included; an investment over a two year period would then have PV of:

$\mathrm{PV} = \frac{C}{(1+i_1)(1+i_2)} \,$

Spreadsheets commonly offer functions to compute present value. In Microsoft Excel, there are present value functions for single payments (=NPV) and series of equal, periodic payments (=PV). Programs will calculate present value flexibly for any cash flow and interest rate, or for a schedule of different interest rates at different times.

Technical details

Present value is additive. The present value of a bundle of cash flows is the sum of each one's present value.

In fact, the present value of a cashflow at a constant interest rate is mathematically one point in the Laplace transform of that cashflow, evaluated with the transform variable (usually denoted "s") equal to the interest rate. The full Laplace transform is the curve of all present values, plotted as a function of interest rate. For discrete time, where payments are separated by large time periods, the transform reduces to a sum, but when payments are ongoing on an almost continual basis, the mathematics of continuous functions can be used as an approximation.

Variants/Approaches

There are mainly two flavors of PV. Whenever there will be uncertainties in both timing and amount of the cash flows, the expected present value approach will often be the appropriate technique.

• Traditional Present Value Approach - in this approach a single set of estimated cash flows and a single interest rate (commensurate with the risk, typically a weighted average of cost components) will be used to estimate the fair value.

• Expected Present Value Approach - in this approach multiple cash flows scenarios with different/expected probabilities and a credit-adjusted risk free rate are used to estimate the fair value.

Choice of interest rate

The interest rate used is the risk-free interest rate if there are no risks involved in the project. The rate of return from the project must equal or exceed this rate of return or it would be better to invest the capital in these risk free assets. If there are risks involved in an investment this can be reflected through the use of a risk premium. The risk premium required can be found by comparing the project with the rate of return required from other projects with similar risks. Thus it is possible for investors to take account of any uncertainty involved in various investments.

Annuities, perpetuities and other common forms

Many financial arrangements (including bonds, other loans, leases, salaries, membership dues, annuities, straight-line depreciation charges) stipulate structured payment schedules, which is to say payment of the same amount at regular time intervals. The term "annuity" is often used to refer to any such arrangement when discussing calculation of present value. The expressions for the present value of such payments are summations of geometric series.

A cash flow stream with a limited number (n) of periodic payments (C), receivable at times 1 through n, is an annuity. Future payments are discounted by the periodic rate of interest (i). The present value of this ordinary annuity is determined with this formula:

$PV \,=\,\frac{C}{i}\cdot[1-\frac{1}{\left(1+i\right)^n}]$ $\mathrm = {C}\frac{1-(1+i)^{-n}}{i} \,$

where:

$\,n\,$ = number of years

$\,C\,$ = Amount of cash flows

This formula is usable when the cash flows are spread over the different but in equal intervals and also the amount of these flows is same say $100 at the end of each year from year one to ten. the $\,i\,$ is defined as interest rate / required rate of return

A periodic amount receivable indefinitely is called a perpetuity, although few such instruments exist. The present value of a perpetuity can be calculated by taking the limit of the above formula as n approaches infinity. The bracketed term reduces to one leaving:

$PV\,=\,\frac{C}{i}$

The first formula is found from subtracting from the latter result the present value of a perpetuity delayed n periods.

These calculations must be applied carefully, as there are underlying assumptions:

• That it is not necessary to account for price inflation, or alternatively, that the cost of inflation is incorporated into the interest rate.
• That the likelihood of receiving the payments is high — or, alternatively, that the default risk is incorporated into the interest rate.

See time value of money for further discussion.

See also

• Capital budgeting
• Lifetime value
• Liquidation
• Net present value