How do you adjust the calculation for an annuity due?

An annuity due assumes payments occur at the beginning of the period. Calculate the present value as an ordinary annuity, then multiply the final result by $(1 + r)$.

What if the annuity payments are monthly but the rate is quoted annually?

Divide the annual nominal percentage rate by 12 to find the periodic rate $r$. Multiply the total number of years by 12 to find the total periods $n$.

Can this formula handle a growing annuity?

No. If the payments grow at a constant rate $g$, you must use the growing annuity formula: $$PV = \frac{PMT}{r - g} \left[ 1 - \left(\frac{1+g}{1+r}\right)^n \right]$$

How to calculate the present value of an annuity

The present value of an annuity is found by discounting a series of equal, periodic cash flows back to the present using a constant interest rate. This method applies strictly to ordinary annuities where cash flows are fixed in amount, occur at regular intervals at the end of each period, and the discount rate remains unchanged over the entire duration.

The setup

Define the known parameters of the annuity. You need the periodic cash flow amount ( $PMT$ ), the discount rate per period ( $r$ ), and the total number of periods ( $n$ ). Ensure that the frequency of $r$ and $n$ explicitly match the payment frequency of $PMT$ .

The steps

Identify the values for $PMT$ , $r$ , and $n$ . 2. Convert annual rates or terms to the corresponding periodic rates if payments are non-annual (e.g., divide an annual rate by 12 for monthly payments). 3. Apply the present value of an ordinary annuity formula: $PV = PMT imes \left[ \frac{1 - (1 + r)^{-n}}{r} \right]$ 4. Calculate the term $(1 + r)^{-n}$ . 5. Subtract that result from 1. 6. Divide by $r$ . 7. Multiply the resulting annuity factor by $PMT$ .

Checking the result

The computed $PV$ must strictly be less than the arithmetic sum of all cash flows ( $PMT imes n$ ). This reflects the time value of money, as future cash flows are discounted to a smaller present value.

Common errors

The most frequent error is a period mismatch, such as using an annual discount rate with monthly payments. Another common error is using the ordinary annuity formula for an annuity due (where payments occur at the beginning of the period) without applying the necessary adjustment factor of $(1+r)$ .

Worked example

Calculate the present value of an annuity that pays $1,000 at the end of each year for 5 years, assuming an annual discount rate of 6%.

Identify variables: $PMT = 1000$ , $r = 0.06$ , $n = 5$ . Set up the formula: $PV = 1000 imes \left[ \frac{1 - (1 + 0.06)^{-5}}{0.06} \right]$ Calculate the exponent term: $(1.06)^{-5} = 0.747258$ . Subtract from 1: $1 - 0.747258 = 0.252742$ . Divide by $r$ : $0.252742 / 0.06 = 4.212367$ . Multiply by $PMT$ : $PV = 1000 imes 4.212367 = 4212.37$ . The present value is $4,212.37.

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References: Corporate Finance by Ross, Westerfield, Jaffe · OpenStax Principles of Finance, Chapter 9