The annual membership fee at your health club is $750 a year and is expected to increase at 5% per year. A life membership is $7,500 and the discount rate is 12%. In order to justify taking out the life membership, what would be your minimum life expectancy?

• 16
• 17
• 18
• 19
• 20

References

Lo, Andrew W.; Wang, Jiang. "MIT Sloan Finance Problems and Solutions Collection Finance Theory I" (PDF). alo.mit.edu. Retrieved November 30, 2023.

Solution: A

Use growing annuity formula assuming that the payments are made at the begining of each year and you pay in full for the year that you die (unfortunately). The value if you have an expected life of T years is:

[[math]]\begin{aligned} P V & = 750+ 750 *\left(\frac{1.05}{1.12}\right)+\cdots+ 750 *\left(\frac{1.05}{1.12}\right)^T \\ & = 750+ 750\left[\frac{1}{0.12-0.05}-\frac{1}{0.12-0.05} \times\left(\frac{1.05}{1.12}\right)^T\right]\end{aligned} [[/math]]

Solve for T . The breakeven point is T >= 16

References

Lo, Andrew W.; Wang, Jiang. "MIT Sloan Finance Problems and Solutions Collection Finance Theory I" (PDF). alo.mit.edu. Retrieved November 30, 2023.