Present value

 ( Income ) 

There are several types and terms associated with interest rates:

• Compound interest, interest that increases exponentially over subsequent periods,
• Simple interest, additive interest that does not increase
• Effective interest rate, the effective equivalent compared to multiple compound interest periods
• Nominal annual interest, the simple annual interest rate of multiple interest periods
• Discount window, an inverse interest rate when performing calculations in reverse
• Continuously compounded interest, the mathematical limit of an interest rate with a period of zero time.
• Real interest rate, which accounts for inflation.

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.

$PV\; =\; \backslash frac\{C\}\{(1+i)^n\}\; \backslash ,$

Where $\backslash ,C\backslash ,$ is the future amount of money that must be discounted, $\backslash ,n\backslash ,$ is the number of compounding periods between the present date and the date where the sum is worth $\backslash ,C\backslash ,$, $\backslash ,i\backslash ,$ 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, $\backslash ,i\backslash ,$, is given as a percentage, but expressed as a decimal in this formula.

Often, $v^\{n\}\; =\; \backslash ,(1\; +\; i)^\{-n\}$ is referred to as the Present Value Factor

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:

$PV\_\{1\}\; =\; \backslash frac\{\backslash \$100\}\{(1.05)^\{1\}\}\; =\; \backslash \$95.24\; \backslash ,$

$PV\_\{2\}\; =\; \backslash frac\{-\backslash \$50\}\{(1.05)^\{2\}\}\; =\; -\backslash \$45.35\; \backslash ,$

$PV\_\{3\}\; =\; \backslash frac\{\backslash \$35\}\{(1.05)^\{3\}\}\; =\; \backslash \$30.23\; \backslash ,$ respectively

Thus the net present value would be:

$NPV\; =\; PV\_\{1\}+PV\_\{2\}+PV\_\{3\}\; =\; \backslash frac\{100\}\{(1.05)^\{1\}\}\; +\; \backslash frac\{-50\}\{(1.05)^\{2\}\}\; +\; \backslash frac\{35\}\{(1.05)^\{3\}\}\; =\; 95.24\; -\; 45.35\; +\; 30.23\; =\; 80.12,$

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. For example, if the cash flow for period one is $100, and $200 for period two, and the interest rate for the first period is 5%, and 10% for the second, then the net present value would be:

$NPV\; =\; 100\backslash ,(1.05)^\{-1\}\; +\; 200\backslash ,(1.10)^\{-1\}\backslash ,(1.05)^\{-1\}\; =\; \backslash frac\{100\}\{(1.05)^\{1\}\}\; +\; \backslash frac\{200\}\{(1.10)^\{1\}(1.05)^\{1\}\}\; =\; \backslash \$95.24\; +\; \backslash \$173.16\; =\; \backslash \$268.40$

• The interest rate must necessarily coincide with the payment period. If not, either the payment period or the interest rate must be modified. For example, if the interest rate given is the effective annual interest rate, but cash flows are received (and/or paid) quarterly, the interest rate per quarter must be computed. This can be done by converting effective annual interest rate, $\backslash ,\; i\; \backslash ,$, to nominal annual interest rate compounded quarterly:

$(1+i)\; =\; \backslash left(1+\backslash frac\{i^\{4\}\}\{4\}\backslash right)^4$

Here, $i^\{4\}$ is the nominal annual interest rate, compounded quarterly, and the interest rate per quarter is $\backslash frac\{i^\{4\}\}\{4\}$

There are two types of annuities: an annuity-immediate and annuity-due. For an annuity immediate, $\backslash ,\; n\; \backslash ,$ payments are received (or paid) at the end of each period, at times 1 through $\backslash ,\; n\; \backslash ,$, while for an annuity due, $\backslash ,\; n\; \backslash ,$ payments are received (or paid) at the beginning of each period, at times 0 through $\backslash ,\; n-1\; \backslash ,$. This subtle difference must be accounted for when calculating the present value.

An annuity due is an annuity immediate with one more interest-earning period. Thus, the two present values differ by a factor of $(1+i)$:

$PV\_\backslash text\{annuity\; due\}\; =\; PV\_\backslash text\{annuity\; immediate\}(1+i)\; \backslash ,\backslash !$

The present value of an annuity immediate is the value at time 0 of the stream of cash flows:

$PV\; =\; \backslash sum\_\{k=1\}^\{n\}\; \backslash frac\{C\}\{(1+i)^\{k\}\}\; =\; C\backslash left\backslash frac\{1-(1+i)^\{-n\}\}\{i\}\backslash right,\; \backslash qquad\; (1)$

where:

$\backslash ,\; n\; \backslash ,$ = number of periods,

$\backslash ,\; C\; \backslash ,$ = amount of cash flows,

$\backslash ,\; i\; \backslash ,$ = effective periodic interest rate or rate of return.

: $C\; \backslash approx\; PV\; \backslash left(\; \backslash frac\; \{1\}\{n\}\; +\; \backslash frac\; \{2\}\{3\}\; i\; \backslash right)$

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 $C\; \backslash approx\; PV\; i$.

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.

$PV\backslash ,=\backslash ,\backslash frac\{C\}\{i\}.\; \backslash qquad\; (2)$

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

$PV\; =\; \backslash sum\_\{k=1\}^\backslash infty\; \backslash frac\{C\}\{(1+i)^\{k\}\}\; =\; \backslash frac\{C\}\{i\},\; \backslash qquad\; i\; >\; 0,$

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 $(1+i)$:

$PV\_\backslash text\{perpetuity\; due\}\; =\; PV\_\backslash text\{perpetuity\; immediate\}(1+i)\; \backslash ,\backslash !$

See: Bond valuation#Present value approach