## Friday, December 23, 2005

### Marginal Discount Rates

For a private institution, setting tuition is a difficult decision. Sooner or later the conversation gets around to the number on everyone's mind: how much of the new money do we get to keep?

Suppose tuition is \$10,000 and our discount rate is 40%. That means we nominally charge each student 10K, but in actuality 4K of that is institutional aid, or funny money, which is excluded from reported revenue in the annual audit. Essentially, it's a way to customize the price per student according to: 1. how much the school wants them to enroll, and 2. how much money they have. If policies are altruistic, the institution may award institutional aid to students with low incomes. A future post on that subject...

Suppose after meeting for hours we decide to raise tuition \$1,000 to 11K. How much do we get to keep? We might assume that since our discount rate is 40%, we can keep 60% of the tuition hike, or \$600 per student. Wrong!

Huh?
The original discount rate is just the ratio Rold = A/T, where A is institutional aid awarded, and T is tuition. So if Suzy gets \$2,000 institutional aid toward the 10K tuition, that's 20% off. Simple. But there's no reason to assume that when tuition increases by 10% that average awards will also increase by 10%, which works out to
Rnew = [A*(1.10)]/[T*(1.10)] = A/T = Rold, our implicit assumption.
This troublesome enthymeme can easily creep in to an unsophisticated analysis. Let's see what it implies.

Typically these calculations are done for new students. It may be assumed that returning students generally do not have their aid packages adjusted for the new tuition rate. Our assumption that we should discount new tuition revenue by the old 40% rate leaves us with the conclusion that these new students will pay \$600 more for tuition in real dollars. We can enforce this with financial aid policies, but we may not want to if it means sacrificing recruiting targets. If your institution has to turn away lots of qualified applicants this probably won't be a problem. But if you are trying to balance an aid budget against enrollment targets, you may very well end up with less net revenue.

The cost of a Coke
If the soda company wants to raise revenue, it's not as simple as just raising the cost of a can of the stuff from .55 to .60. Since revenue is (price)*(number sold), we have to consider the fact that we may lose more revenue through lost customers than we gain through increased prices. Trying to figure out the right balance is the holy grail of setting tuition, and the topic of a future post. The bottom line is, if we have no reason to believe that students can or will pay more tuition, we shouldn't assume that they will (and hence keep the discount rate the same).

We may try to strike a happy medium between higher prices and fewer enrollees by allowing some flexibility in the awarding policies. Set the goals higher, but be willing to back off if admitted students can't or won't pay the sticker price. Rather like buying a car. In this way we can let the market decide how much an education at The College is worth. After the dust settles from Fall enrollment, we can then calculate the real discount rate.

Calculating Marginal Discount
Suppose, after raising tuition 10% to 11K, our new enrollees averaged 5K in institutional aid. Even with tuition leveraging methods, you can't completely predict what kind of applicants you're getting--rich, poor, talented, not--so complete control of your destiny is unattainable. To compute the marginal discount, calculate