Week 7 Solving for the Rate of Return of an Annuity

Solving for the Rate of Return of an Annuity

Recall the rate of return formula you learned about last week:

This formula works well for lump sum problems; that is, problems in which there is one beginning value and one ending value with no cash flows in between. However, it will not work for a problem such as the following:

Joe's Dockyard is financing a new boat with an amortizing loan of $24,000 which is to be repaid in 10 annual installments of $4,247.62 each. What interest rate is Joe paying on the loan?

The cash flows on a timeline:

This is an annuity problem, so we turn to one of the annuity formulas to solve it. The one to use is the PV of an annuity formula because in the problem before us we know all the terms in the equation but one, which is r, the term we are looking for. Also, we will use the interest factor version of the PV of an annuity formula:

The task is to solve for r% in this equation. We do so as follows:

1. Now turn to Table A.4 in Appendix A in the back of your McGraw-Hill Create text and look down the left side until you find the 10 year row.
2. Look out in the 10 year row until you find the PVIFA factor that most closely matches the 5.6502 in the equation above. Note that it occurs in the 12% column. Therefore, r = 12%, which means the interest rate on Joe’s loan is 12%.

To solve for "r" exactly using algebraic techniques is exceedingly difficult. However, it is relatively easy with a financial calculator or in Excel. In Excel enter all the input variables and use Excel’s “RATE” function to produce the answer.