In my last post, I linked to an Op-Ed by Amherst President Tony Marx in the LA Times.  In the piece, he argues against a proposed minimum spending rate on colleges’ endowment funds.  His points are especially compelling given the state of the market.  For the record, I stand in basic agreement with his reasoning.

However, an anonymous reader of this blog read the Op-Ed with far more attention to detail than I did.  This reader alerted me to a very significant quantitative error by President Marx.  Upon further review, I found not one - but two - quantitative mistakes that render one of Marx’s main arguments considerably less cogent.  I will attempt to describe the errors.  Thanks to the reader for the tip, the mathematical analysis, and even some of the original language.

The 6th Paragraph:

In addition, the financial benefit would be minimal. Colleges and universities spent, on average, 4.6% of their endowment funds in 2007, according to the National Assn. of College and University Business Officers. With $411.2 billion collectively in those funds, that works out to about $18.9 billion being spent on operating expenses, which includes student aid. Forcing the spending rate up to 5% would generate an additional $1.6 billion…

Our concern is with Marx’s conclusion that “forcing the spending rate up to 5% would generate an additional $1.6 billion.”  Here’s why:

——————————————————————-

Flawed Assumption #1

For simplicity, assume that the total endowment amount of $411.2B is constant across n schools, i.e., each school has an endowment of $411.2B/n .

So if every school spends 4.6%, then the amount to be gained by raising all schools to the 5% minimum would be:

n x (5% - 4.6%) x ($411.2B/n) = 0.4% x $411.2B = $1.6B .

Now suppose that one of the schools spends 4.7%. To keep the average at 4.6% there must be another school that spends 4.5%, but the amount to be gained by setting a 5% minimum would be the same:

[(5% - 4.7%) x $411.2B/n] + [(5% - 4.5%) x $411.2B/n] + [(n-2) x (5% - 4.6%) x ($411.2B/n)] = 0.4% x $411.2B = $1.6B .

Essentially each pair of schools would generate a total of 0.8% = 2 x 0.4% of incremental spending on their endowments.

This pattern continues on up to one school spending 5%. Then another must be spending 4.2%, and the amount to be gained still would be:

[(5% - 5%) x $411.2B/n] + [(5% - 4.2%) x $411.2B/n] + [(n-2) x (5% - 4.6%) x ($411.2B/n)] = 0.4% x $411.2B = $1.6B .

Each pair continues to generate 0.8% = 2 x 0.4% of incremental spending.

But if one of the schools spends 5.1%, then another must be spending 4.1%, and assuming that the first school continues to spend above the 5% minimum there would be no offset to the 0.9% of incremental spending required of the second school. The incremental spending thus would grow to:

[(5% - 4.1%) x $411.2B/n] + [(n-2) x (5% - 4.6%) x ($411.2B/n)] = (0.4% x $411.2B) + (0.1% x $411.2B/n) = $1.6B + (0.1% x $411.2B/n) .

This demonstrates that, to the extent schools are starting out above 5%, incremental spending would exceed $1.6B. Put another way, $1.6B is a lower bound on the incremental spending to be gained from the 5% threshold requirement.

For example, if half of current endowment value is held by schools that spend at a rate of 5.4% and half by those who spend at 3.8%, then the average is indeed 4.6%, but the incremental spend resulting from a 5% requirement would be:

($411.2B x 1/2) x (5% - 3.8%) = $2.5B .

So if there are any schools currently spending over 5% of their endowment - which there almost certainty are - Marx understates the impact of the spending requirement.  Any math major that comes up with a more elegant proof gets a free Barack Obama yard sign.

Flawed Assumption #2

President Marx is using data from NACUBO to suggest that colleges and universities spent, on average, 4.6% of their endowment funds in 2007.  This is correct.  However, that number is an unweighted average.  In other words, NACUBO surveyed every college for its numbers and then took a simple mean of the data.  They did not weight the percentage by endowment size - a critical distinction.

Furthermore, endowment spending percentage and endowment size are not independent.  It’s not hard to conjecture that schools with smaller endowments are likely to spend a greater percentage of their endowments, an assumption that is supported by NACUBO’s table.  And I don’t think that the table fully displays this because its highest bracket is simply limited to endowments greater than $1 Billion.

Let’s consider Princeton and Brown.  I would bet that Princeton (enrollment 7,500 / endowment 17B) spends a smaller percentage of its endowment annually than does Brown (enrollment 7,500 / endowment 3B).  With all of the fixed costs of educating students, how could it be otherwise?

If we accept that better endowed schools tend to spend at a lesser rate, we can reach the conclusion I’m trying to posit.  Under a spending requirement, the better endowed school will have to undergo a larger spending increase by percentage than will the lesser endowed school.  The percentage increase of the better endowed school will result in a significantly larger expenditure than the same percentage increase of the lesser endowed school.  .6% at Princeton is different than .6% at Brown.  And because Princeton is more likely to be further away from 5%, $1.6B is not an adequate figure to describe reality.

By failing to address this factor, Marx again likely understates the impact of a spending requirement.  Free Barack Obama rally sign for an elegant proof.

——————————————————————-

Before declaring that President Marx is irrefutably incorrect, I’ll give him a chance to squeeze out of this predicament.  My analysis is predicated on the assumption that schools spending over 5% will not change their behavior under a spending requirement.  It seems silly to think that they would.  Nevertheless, there is a theoretical possibility that establishing a legal minimum essentially would institutionalize a target that all players would gravitate toward, even those previously higher.  We could imagine a situation in which a  school - already above the goalposts - becomes aware that it is in fact not saving as much as it probably should and chooses to decrease spending to the 5% mark.

If this were the case, a calculation to predict the incremental spending from a 5%  requirement would likely overstate the impact of that spending requirement.  I remain skeptical, however.

In any case, it does not seem likely that the figure of $1.6 Billion accurately describes the sum of additional expenditures likely to be taken by colleges and universities under Senator Grassley’s proposal.  I would recommend that President Marx revisit his calculation.  Alternatively, if he wants to defend his analysis, we’d of course be thrilled to publish it here on AmhPub.