Mental Exercises

After running about 25 miles a week for my last few months in Germany with only minor physical problems, my left knee gave up the ghost on just my second American jog in late July. Since then, and much to my consternation, I've only been jogging a few times and even then just for two or three miles before my knee began hurting and gave me grief for the rest of the day. Since jogging is not an option until my knee is rehabilitated, I decided to join a gym for my fitness jollies, but only got around to it today when I started a free week-long trial. At the end of the week, I must decide on a pricing plan, and am confronted with several options (Yes, the gym is pricey, but it's my only choice given my preferences):

Plan 1- A 12-month contract consisting of a single $99 program fee and a monthly payment of $59 Plan 2 - A 3-month contract consisting of a $225 payment ($75 per month) Plan 3 - Pay a month at a time for $85

Plan 1 can be immediately eliminated since I won't be in Columbia for another year. I will most likely be here for at least three months, however, so Plan 2 seems a good choice as it is $30 cheaper than Plan 3 (85*3=255). Problem solved?


Though the calculation above is where most folks would stop, my finance professors would be insulted if I ignored the structure of the payment! A lump sum payment of, say, $100 today is only equivalent to 10 monthly payments of $10 in a world with no interest, and this is not the world in which we live; money has a time value.

Thus, I must reevaluate Plans 2 and 3. While Plan 2 is cheaper on a monthly basis, it requires me to pay out the money up front. Plan 3, on the other hand, affords me the ability to make smaller monthly payments, which in turn allows me to earn interest on the cash I would have already spent if I had opted for the lump sum. How much interest I expect to earn thus become the key to making a wise decision.

Assuming I could earn 5 percent annually (somewhat heroic, given current conditions), the sums would be as follows (I’m also assuming monthly compounding):

Plan 2 - Since I’m spending all the money up front, there’s no time component here; the plan will cost $225 Plan 3- 85 + 85/(1+(.05/12)² + 85/(1+(.05/12)³ = $ 253.25

Yikes. With a five percent annual rate, the interest I earn on Plan 3 would only defray $1.75 of the cost. Under these circumstances, Plan 2 remains the best bet.

An interesting question remains: How much interest would I have to earn to defray the extra cost of Plan 3? I’ll spare you the calculation, and simply tell you the answer is about 97 percent, or 8 percent per month. If I could earn that rate, the interest earned would defray exactly the extra costs of Plan 3, and I’d be indifferent between it and Plan 2.

Of course, if I were earning a 97 percent annual return, I probably wouldn’t be poring over the minutiae of gym membership plans.