Present value

<span lang="en" dir="ltr">In</span> economics and finance, present value (PV), also known as present discounted value (PDV), is the value of an expected income stream determined as of the date of valuation. The present value is usually less than the future value because money has interest-earning potential, a characteristic referred to as the time value of money, except during times of negative interest rates, when the present value will be equal or more than the future value.

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.

Present value of a lump sum

The most commonly applied model of present valuation uses compound interest. The standard formula is:

:<math>PV = \frac{C}{(1+i)^n} \,</math>

Where <math>\,C\,</math> is the future amount of money that must be discounted, <math>\,n\,</math> is the number of compounding periods between the present date and the date where the sum is worth <math>\,C\,</math>, <math>\,i\,</math> is the interest rate for one compounding period (the end of a compounding period is when interest is applied, for example, annually, semiannually, quarterly, monthly, daily). The interest rate, <math>\,i\,</math>, is given as a percentage, but expressed as a decimal in this formula.

Often, <math>v^{n} = \,(1 + i)^{-n}</math> is referred to as the Present Value Factor

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

For example, if you are to receive $1000 in five years, and the effective annual interest rate during this period is 10% (or 0.10), then the present value of this amount is

:<math>PV = \frac{\$1000}{(1+0.10)^{5 = \$620.92 \, </math>

The interpretation is that for an effective annual interest rate of 10%, an individual would be indifferent to receiving $1000 in five years, or $620.92 today.

For example, if a stream of cash flows consists of +$100 at the end of period one, -$50 at the end of period two, and +$35 at the end of period three, and the interest rate per compounding period is 5% (0.05) then the present value of these three Cash Flows are:

:<math>PV_{1} = \frac{\$100}{(1.05)^{1 = \$95.24 \, </math>

:<math>PV_{2} = \frac{-\$50}{(1.05)^{2 = -\$45.35 \, </math>

:<math>PV_{3} = \frac{\$35}{(1.05)^{3 = \$30.23 \, </math> respectively

Thus the net present value would be:

:<math>NPV = PV_{1}+PV_{2}+PV_{3} = \frac{100}{(1.05)^{1 + \frac{-50}{(1.05)^{2 + \frac{35}{(1.05)^{3 = 95.24 - 45.35 + 30.23 = 80.12, </math>

There are a few considerations to be made.

The periods might not be consecutive. If this is the case, the exponents will change to reflect the appropriate number of periods
The interest rates per period might not be the same. The cash flow must be discounted using the interest rate for the appropriate period: if the interest rate changes, the sum must be discounted to the period where the change occurs using the second interest rate, then discounted back to the present using the first interest rate.

:: <math>C \approx PV \left( \frac {1}{n} + \frac {2}{3} i \right) </math>

Where, as above, C is annuity payment, PV is principal, n is number of payments, starting at end of first period, and i is interest rate per period. Equivalently C is the periodic loan repayment for a loan of PV extending over n periods at interest rate, i. The formula is valid (for positive n, i) for ni≤3. For completeness, for ni≥3 the approximation is <math> C \approx PV i</math>.

The formula can, under some circumstances, reduce the calculation to one of mental arithmetic alone. For example, what are the (approximate) loan repayments for a loan of PV = $10,000 repaid annually for n = ten years at 15% interest (i = 0.15)? The applicable approximate formula is C ≈ 10,000*(1/10 + (2/3) 0.15) = 10,000*(0.1+0.1) = 10,000*0.2 = $2000 pa by mental arithmetic alone. The true answer is $1993, very close.

The overall approximation is accurate to within ±6% (for all n≥1) for interest rates 0≤i≤0.20 and within ±10% for interest rates 0.20≤i≤0.40. It is, however, intended only for "rough" calculations.

Present value of a perpetuity

A perpetuity refers to periodic payments, receivable indefinitely, 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.

:<math>PV\,=\,\frac{C}{i}. \qquad (2)</math>

Formula (2) can also be found by subtracting from (1) the present value of a perpetuity delayed n periods, or directly by summing the present value of the payments

:<math>PV = \sum_{k=1}^\infty \frac{C}{(1+i)^{k = \frac{C}{i}, \qquad i > 0,</math>

which form a geometric series.

Again there is a distinction between a perpetuity immediate – when payments received at the end of the period – and a perpetuity due – payment received at the beginning of a period. And similarly to annuity calculations, a perpetuity due and a perpetuity immediate differ by a factor of <math>(1+i) </math>:

:<math> PV_\text{perpetuity due} = PV_\text{perpetuity immediate}(1+i) \,\!</math>

Present value of a lump sum

Present value of a perpetuity

See also

References

Further reading