Annualizing and linking returns: a case study

I got an e-mail question on Friday with the following request: We have a discount stock plan at our firm. You can buy discounted stock at the end of each quarter and can sell it immediately for a 15% profit.  The question arises about what your annual return would be.

"Many of my colleagues argue that 15% a quarter sums to  60% annually (ignoring geometric linking which cannot apply since there is no compounding).  I have a problem with this because the money invested is four distinct unrelated transactions and if you divide the total gain by the sum invested you would get 15%"

When I first considered this problem I focused only on a single execution, to determine the annual equivalent (recognizing that it's generally inappropriate to annualize for periods less than a year). If we treat this as an event for a quarter, then the process is pretty simple:

• Add 1 to the return (1.15)
• Raise it to the inverse of the period expressed in years (1/0.25)
• Subtract 1.

And our answer: 74.90 percent. This is the annual equivalent of that single quarterly event.

If the ability to generate a return quarterly is 15%, then we could geometrically link four quarterly returns of 15% each:

• Add 1 to each return (1.15)
• Multiply them together
• Subtract 1.

And we get ... 74.90 percent return!

p.s., The same day I got this e-mail and did the analysis, a technician came to my home to winterize the sprinkler system. The cost for the service? $74.90. Kind of weird, right?

p.p.s., If you think this is the last word on this subject, you'll be pleased to learn that we will address this at much greater length in the November newsletter. As you'll see, depending on one's perspective, we can get different answers!