How to Figure Out the Present Value of an Annuity at Different Interest Rates

The present value of an annuity is the value of all future payments after backing out, or removing, the interest each earns between the present time and the time of payment. The present-value formula that backs out these interest rates is PMT/(1+i)+PMT/(1+i)^2 +...+PMT/(1+i)^n, where "n" payments of size PMT earning interest at the rate "i" remain. This entails calculating and adding "n" terms, laborious task. This formula can be rewritten in the easier-to-calculate form PMT*[1-1/(1+i)^n]/i. (The asterisk [*] indicates multiplication, and the caret [^] indicates that "n" is an exponent.)

Things You'll Need

• Calculator

Instructions

1. Divide the annually compounded interest rate by the number of payments per year (if you're not given the interest accumulated between consecutive payments). Either one gives you your "i" value. If the interest is compounded at the time of every payment, "i" is found using the formula (1+annual rate)^(1/m)=1+i, if "m" is the number of payments per year.

2. Set PMT to the periodic payout of the annuity. Set "n" to the number of payments left. The formula above does not include payments less than a pay period away in calculating the present value.

3. Plug your numbers into the formula PMT*[1-1/(1+i)^n]/i and solve to find the present value for the annuity. Again, the asterisk [*] indicates multiplication, and the caret [^] indicates that "n" is an exponent.

4. Calculate different present values at different interest rates by repeating Step 3 with different values for "i."

Tips & Warnings

• A derivation of the formula can be found at the link below (see Resources).

References

• Stephen Kellison; Theory of Interest; 1997
• How to Calculate Amortization Tables by Hand

Resources

• What Is the Equation to Determine a Mortgage Payment?