The annual NACUBO report on tuition discounts was covered in Inside Higher Ed back in April, including a figure showing historical rates. (More recently this was covered in the Economist.)
The discount rate is the fraction of posted tuition that is "given back" by the institution in financial aid awards. Usually these are unfunded awards, meaning there was never any money involved. For a primer on this idea, see Paul Tough's account in the New York Times.
In private higher ed, where financial aid leveraging is ubiquitous, high discount rates are seen as a sign of failure (not at the elites, which are not as dependent on tuition revenue). Over my three decades in the business, I've seen a lot of hand-wringing about discount rates. Despite all the angst, the causes of discount rates do not seem to be well understood. High discounts are generally seen as good for students, e.g. "institutions are devoting a lot of their own resources to make education more affordable relative to the tuition price" per a NACUBO official quoted in the IHE article. But that benefit is seen as a drag on institutional finances, e.g. "[...] a challenge for many private colleges, since tuition and fee revenue is a large share of their overall funding [...]", from the same source.
Both of these sentiments miss the point. Tuition discounts go up because they mathematically must go up given the business model of private higher ed. Talking about discount rates from year to year is largely a waste of time. Considering the long term effects, on the other hand, should be a concern.
Discounting Freshmen
The most important thing to know is that it's primarily the first year students who get the discount. Like in any other service industry where there's competition for new customers, it's harder to attract new ones than to retain existing ones. This is why streaming services, newspapers, and so on offer discounts for for an initial period, after which the rates increase. These businesses can't survive on the revenue they would get at the discounted rate, so the increase is necessary. Colleges have a similar situation: because of competition they can only collect an average net tuition revenue per student from freshmen that's lower than what they need per-student to make budget. The difference between the two lines in Figure 1 suggests that private colleges need about a 5% lower discount rate, averaged over all students, than the freshman discount.
Colleges aren't in the kind of business where it's culturally acceptable to overtly offer a discount for the first year to new customers, like a Netflix trial offer. Instead, we raise the price on returning students. If we didn't raise tuition, everyone would pay the low-low introductory rate, and we'd go broke. But raising tuition has a bit of complicated math, because rate hikes compound over time. A sophomore will have one tuition increase, a junior two, and a senior three. These increases are generally not adjusted with increased financial aid, or that would defeat the purpose of increasing net revenue per student up to the level we need.
Raising Tuition
So how much do we need to raise tuition every year to increase the net revenue from returning students so that the total average net revenue per student is high enough to support operations? First, it's useful to notice that if enrollment patterns are stable, the tuition increase needed to maintain net revenue can be a fixed amount, rather than a percentage of current tuition. To specify the enrollment pattern, let's call \( p_1 \) the fraction of undergraduates who are first year, \( p_2 \) the fraction who are second year students, and so on. Average net revenue per student, \( N_a \) has a relationship to the average net revenue per freshman \( N_f \) per the formula
$$ N_a = N_f + (p_2 + 2p_3 + 3p_4)T' $$
where \( T' \) is the constant amount we raise tuition each year. The formula assumes that no matter what the nominal tuition rate is, the amount of net revenue per freshman stays constant due to market forces. The returning classes pay more because the market forces (mostly) no longer apply. The second year students see one tuition increase, the third year students two, and the seniors have seen three increases. This is a simplification that doesn't consider transfers in, students who remain longer than four years, and so on.
Note that the fractions \( p_1 \) , \( p_2 \) , \( p_3 \) , and \( p_4 \) sum to one (100%) and describe a student population that relates to admissions standards and completion rates. A large first year attrition will enlarge \( p_1 \) and put more pressure on tuition increases, since there are relatively fewer returners.
So under strong competition for freshmen, we'd expect to see that tuition increases are approximately linear (as opposed to exponential from percentage compounding).
NACUBO Data
Let's see what we can learn from the NACUBO numbers in Figure 1. It's useful to begin by converting discount rates to "realized rates," or net tuition revenue divided by gross tuition. If tuition is $30,000 and the total average discount rate D is 33%, then the average net revenue per student is $20,000 and the realized rate R is 20,000/30,000 = 67%, or 100% - discount rate. With that idea, we can calculate the ratio of realized rates from the chart. The ratio \(r\) of the realized rates is the same as the ratio of average net revenue, i.e.
$$ r = \frac{R_a}{R_f} = \frac{\frac{N_a}{G}} {\frac{N_f}{G}}= \frac{N_a}{N_f} $$
where the a subscript means all students and f means only freshmen.
In a competitive market for freshmen, we'd expect \(r\) to stay greater than one. In 2015, this ratio was (100 - 43)/(100 - 48) = 1.096. In 2022 it was
(100 - 50.9)/(100 - 56.2) = 1.121. The ratio has been remarkably stable
over the ten year period shown in Figure 1, with an average of 1.11. So
over the last decade, the average undergraduate paid about 11% more than
an average freshmen. We can use that information to find out how much
we need to raise tuition by each year to make budget.
Figure 2. NACUBO and IPEDS realized revenue ratios are nearly constant at about 1.1 in recent years.
For convenience, I'm jumping ahead a little to include IPEDS ratios on the same chart in Figure 2. The IPEDS data is discussed in the next section.
The equation for \( N_a \) in the previous section assumes a constant tuition increase each year, but since we have the ratio of \( N_a / N_f \) here instead of the difference \( N_a - N_f \), it's convenient to fudge a little on the constant tuition increase in order to make the math work. This entails using a percentage increase on net revenue each year instead of a fixed amount.
Assume that for each dollar in average net tuition revenue we get from a new student we actually need r dollars, averaged over all undergraduates. The NACUBO ratios suggests that r = 1.11 is reasonable. Because the increase happens each year, returning students have two increases as third year students and three increases as fourth years. We have
$$r = p_1 + p_2(1+t') + p_3(1+t')^2 + p_4(1+t')^3$$
where \( t' \) is the fraction of freshman net revenue per student that we raise price by, \( t' = T' / N_f \).
A college with a high retention rate might have class proportions of .3, .25, .23, and .22 (ignoring fifth-year and beyond). In that case, solving for r = 1.11 gives t' = 0.076, which means that we need to increase the net tuition revenue for returners by about 7.6% each year. This is a fixed fractional multiplier to \( N_f \) that becomes a tuition increase each year if there is no additional discounting for returners. For context, if discount rates are already in the 50% range, then that 7.6% of extra revenue can be generated by a 7.6/2 = 3.8% tuition hike that gets passed on to returners without additional discounting. That calculation is complicated by the fact that students with the leanest aid packages are usually the first to leave, so the tuition increase may need to be a little higher to accommodate.
This calculation shows how to maintain a constant flow of net revenue, and any increases to the budget would incur larger tuition hikes. If we must raise tuition just to keep the same amount of net revenue coming in, then by definition discount rates must increase.
In this scenario where we raise tuition rates fractionally every year to get r = 1.11, fourth-year students are paying about 26% more than freshmen \( (1.08^3 - 1) \). In practice, this could be lower if the average net revenue paid by freshmen increased each year. However, the historical NACUBO data shows that 11% gap being steady over time, so even if freshmen are paying more, the budget must be growing at the same rate.
IPEDS Data
The
NACUBO data can't be exactly reproduced using IPEDS data, but we can
get close. The finance data includes net tuition revenue, posted tuition amounts, and number of undergraduates, from which we can estimate \( N_a \) as net tuition revenue divided by the number of undergraduates, and hence the discount rate by \( D = 1 - N_a/T \).
I selected
private colleges with at least 1000 undergraduates and with a complete history
of the data I needed, then filtered to those that were heavily
undergraduate, so as to avoid bias from graduate school tuition. This left 223 institutions. I'm not sure how well this overlaps with the NACUBO set.
If the model assumptions are correct, we'd expect that tuition increases look linear over time, and a divergence between net revenue and posted tuition that causes increasing discount rates.
Figure 3. IPEDS history of selected private colleges, showing average tuition (blue) with annual percentage tuition increase labeled and average net revenue per student (red) with discount rates labeled. Figures are median values over 223 four-year undergraduate institutions.
Average tuition rates in Figure 3 are about linear over time, as would
be expected from the model above, and the annual percentage increases in
tuition decline over time as a consequence.
This historical data shows the average tuition in 2007 at around $23,000, and large annual increases (6.5% the first year), while net tuition revenue per student was about $15,000 with a discount rate of 37%. For revenue maintenance, we need linear tuition growth, which is what we see here (blue). Over the eleven years, average tuition increased by $1,225 per year. These increases generated more than steady-state net revenue, as shown in the red line tracking net revenue per student: from 2006 to 2020, it grew from $15k per undergraduate to $20k, a 33% increase. The cumulative inflation rate over that period was about 30%, so in real dollars, net revenue per student was nearly flat. This aligns with what we saw in Figure 2: freshmen pay more over time, but what the college needs to operate increases at about the same rate, so that tuition increases stay relatively constant instead of declining.
Freshman Discount
Under the model assumptions, if we know the annual increase in price and the average net revenue over all undergraduates--both of which IPEDS provides--we can estimate the freshman discount using
$$ N_f = N_a - (p_2 + 2p_3 + 3p_4)T' $$
I estimated the class proportions using available data, but it's not very good, and needs some work. Fortunately, the result isn't that sensitive to the distribution. Once we have the net revenue rates, the discount rates can be calculated.
Figure 4. Comparative discount rates between NACUBO and IPEDS
The discount rates in Figure 4 track well, but there's about a 6% difference in total rates and a slightly large gap for the freshman rates. This is a
big enough discrepancy to demand explanation. A proper analysis needs a
hierarchical model with better empirical estimates of the class proportions.
I'll eventually post the code and data for this so you can try it out
for yourself. In the meantime, email me if you're interested, at
deubanks.office@gmail.com.
Discussion
It's not a good look for higher education to advertise higher and higher prices in order to maintain, in some sectors, a subsistence level of revenue. Headlines talk about the posted tuition rates "skyrocketing" at the same time that small colleges are closing up shop.
The effects of tuition leveraging, where institutional aid is used for differential pricing, is a flexible tool that can be used for good purposes. However, it seems unfair to have large gaps between what freshmen pay and what seniors pay, escalating the financial pressure each year just to stay in college. I doubt that this was an intended result of the leveraging idea, but it must be a common effect.
The leveraging model, combined with competition for first-year students, causes ineluctable rise in tuition and discount rates. In the long run, this business model isn't sustainable because posted
tuition rates will become ridiculously high. Colleges will have to reset
their nominal rates eventually (and take a short-term revenue hit), or find some other way to manage first-year discounts. There are creative solutions to this problem, but they require
re-imagining net revenue streams. Some colleges have tried flat pricing (no discounting), but that's not going to work if you have to compete for students, because it doesn't solve the first-year discount problem, so just jettisoning aid leveraging won't solve the main problem.
Tuition increases should be decided in dollar amounts, not percent increases, with estimated impacts on each returning class, and there should be sensitivity to how constant these increases over time. A per-class retention analysis can inform the decision. Some students may need financial escape valves, which might claw back some of the increase for returners. Of course, if a college can cut spending enough to make do with the average net revenue generated per freshman, it doesn't have to raise tuition at all.
Update: There's a 9/15/2023 article in Inside Higher Ed about cutting tuition that's relevant here: link.
Try It Yourself
I built a simulation so you can try out various scenarios. You can find in on ShinyApps.io. This is a free account, so the monthly usage allotment may dry up. Email me if you want to the code to run locally. When it launches, you should see this
Figure 5. Screenshot of discount simulator
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