The equivalent annual annuity formula allows you to compare payments or returns, which vary in duration. As an example, you might want to know which is the better deal: A $15,000 car that lasts five years with an average annual operating cost of $8,000, or a $20,000 car that lasts six years with an average annual operating cost of $7,000. The first step in determining the equivalent annuity cost is determining the net present value (NPV) of the payments. Combined with the interest rate and term, the equivalent annuity formula calculates the equivalent cash flow.
Add 1 to the interest rate in decimal format. In the example, say the interest rate is 8 percent for both options. Adding 1 to 0.08 gives you 1.08.
Raise this value to the number of years of payments, expressed as a negative value. In the example, this would be the annual operating costs, so you raise 1.08 to the power of negative 5 for the first option and negative 6 for the second option. This gives you values of 0.680583 and 0.630163, respectively.
Subtract this value from 1. Remember this number, because you'll use it again later. In the example, you get 0.319417 and 0.369830, respectively.
Divide this value by the interest rate. This gives you 3.99271 and 4.62288, respectively.
Multiply this figure by the annual payments. This calculates the net present value of the payments. In the example, you would use the annual operating costs to get $31,942 and $32,360.
Add any upfront costs to calculate the NPV. In the example, you have to add the purchase price of the cars to get a NPV of $46,942 and $52,360, respectively. Looking only at the NPV may mislead you into thinking the first car is the better deal, but the real comparison is the equivalent annuity cost.
Divide the interest rate by the number calculated in Step 3. In the example, divide the interest rate into 0.319417 and 0.369830. This gives you 0.250456 and 0.216315, respectively.
Multiply this figure by the NPV to get the equivalent annuity cost. In the example, this gives you $11,757 and $11,326, respectively. The equivalent annuity cost shows that the second car is actually the cheaper option.
Items you will need
- Business or scientific calculator