# Voucher Math

One of the biggest controversies surrounding the whole voucher program is cost. Depending on who you talk to, the program with either save a ton or cost a ton with little room left in the middle. One of the big questions for me has centered around the impartial analysis and whether the numbers were net or gross. This should have been answered when the updated document got posted by Green Jello, but it only provided insight as to the assumptions being made about the program. I'm going to break down the numbers as I know them to give a clear picture of how we make our assumptions.

What we know is that the voucher amount will be between \$500 and \$3000 with the average being around \$2000 per enrolled student. We know that about 3% of Utah's students current attend private schools. We also know that the national average for private school attendance is around 12%, about four times what we have in our own state. The education fund is about 45% income taxes, 45% property and general fund taxes and about 10% miscellaneous (if memory serves me correctly). State per-pupil expenditures are \$7500. These are the facts that we can all agree on.

We also know that the projections show just over 13,700 17,400 non-switcher private school students in FY 2008 and  just over 18,900 26,000 total private school enrollment in FY2020. Now we're starting to get into assumption territory which can be tricky to navigate. This comes to about 5200 8600 new private school students over the course of 12 years or around 435 720 new private school students per year. Getting this baseline is important for establishing base costs of the program and determining overall student growth (about 3% per year in this case).

From this, we can figure the base costs of the program for each year it's in effect. In Years 1-5, there's no direct cost from current enrollees only a cost from new enrollees, about 1/13 of the student population per year. In Year 6 and beyond, we phase in 1/13 of the private school student population (K-12) as voucher eligible. Because of the "hold harmless" provisions of Years 1-5, all vouchers are considered to be a cost since the money is slated to come from non-education monies in the general fund. Since we don't know the switcher rate, I'll go with the LFA estimate of about 3700 students. This comes to a cost of \$7.6M per year (with \$200K per year in administrative costs) or \$40M during the course of the first five years. The education fund, meanwhile, gets to keep the \$7500 per student in education spending without the overhead associated with said student. Update: The cost of new students is about \$18.5M. That comes to a \$27.75M windfall for schools each of the first five years or of \$92.5M in total. In other words, schools do pretty well on the funding end of things during the first five years should the projections prove true.

So what happens when the private school students start becoming eligible mitigation funds go away? In Year 6, 1/13 6/13 of the private school population of non-switchers become eligible. Based on the projections above, that's about 1300 9800 students which will cost about \$2.6M \$19.5M that year. Figuring that the switcher numbers grow at a similar rate, we have about 4300 4500 switchers costing about \$8.6M \$9M for a total cost (with administration) of \$11.2M \$28.5M. The 4300 4500 switchers will leave behind \$5500 per student in the education fund or \$23.7M \$24.8M. Given that the switchers vouchers are already paid for, we can exclude them we calculating net cost or savings. So far, this looks like a pretty good deal for the state at a \$21.1M \$5.3M windfall and we haven't even factored in new construction at all.

By Year 13, we have over half all of the non-switchers eligible or about 10,700 27,900 students. This has a cost of \$21.4M \$55.7M. Switchers will now number about 5500 5900 for a cost of \$11.2M \$12M. The total voucher cost is now \$32.6M \$67.7M. The money left behind by the switchers is about \$30.3M \$32.5M, so we still have a windfall to the state loss of \$8.9M \$23.2M. It's not the billions of dollars exclaimed by legislators, but it's nothing to sneeze at either.

Now let's take a hard look at Year 18, when all students are eligible. We now should have around 21,800 non-switchers costing \$43.6M. Switchers are up to 6300 students for a cost of \$12.6M. The switchers now leave behind around \$34.7M, a loss to the state. Update: Given how off my projections were, Year 18 doesn't really figure anything anymore.

Over the life of the program, we're looking at a total cost of \$335.9M and a total savings of \$338.4M for a net savings of \$2.5M. This doesn't factor in the building costs saved by not having all of those switchers in private school campuses. Using numbers from the Senate Site, we can expect to save somewhere in the neighborhood of an additional \$118.5M in building costs.

The problem with this scenario, however, is that we presume the switchers never surpass 1% 0.65% of the total student population and that we never exceed more than 4% 3.65% of the total student population in private schools. Given the national average of 12%, this seems like a very low-ball figure. Consider that most of our student growth is coming from out-of-state and that these new residents are, on average, four times as likely to opt for a private school as native Utahns.

Let's work backwards: let's figure out how many switchers we need in Year 18 13 to achieve a break even. We've already established our baseline cost of \$43.6M \$55.7M. We also know that each switcher saves about \$5500. This means we need about 7900 10,100 switchers to hit the break even in Year 18 13 compared to an estimate of 6300 5900. In other words, instead of 1% of the student population being comprised of switchers, we'd need 1.25% 1.7% of them to be switchers, still well within the national average.

As you can see, the estimates are little more than best guesses; they could be higher, they could be lower. But we will never know without the five-year pilot. Think of the pilot as a research project of sorts to gather data. You might not like the process, but I would hope we can all agree that the data itself is very valuable.

BIG UPDATE: Jeremy kindly pointed out a flaw in my figures regarding when private school students start becoming eligible. This is why we should leave match to Excel and not our four-function calculators. I've stuck-though all of the old text and updated as appropriate as well as putting new content in italics. You all get to see my mistakes.

UPDATE 2: I made some additional corrections to the percentage of students who are switchers.

UPDATE 3: Here's a handy Excel spreadsheet you can use to see the calculations on a year-by-year basis.

UPDATE 4: I made a new copy of the spreadsheet with a graph to show costs versus savings. I also made the switchers a ratio of current private school enrollments so you can play with the
numbers. Just adjust the value in cell G2 from 0.65 to the new percentage of switchers. It'll auto-update the graph as well so you can get visuals on the changes and where the break-even point lies. Based on these adjustments, the break-even is around 0.96% switchers with building costs factored in and 1.09% with building costs excluded, a pretty reasonable figure to be met. I'm hoping this is the end of my updates.

While we're on the topic of math, I also want to take a moment to address the criticisms of The Sutherland Institutes's figures on private school tuition. A lot of folks have been howling over the "average" that has been calculated, but it belies a lack of understanding of proper statistical analysis. One of the biggest points of contention is that they shouldn't have excluded the highest end schools when calculating the arithmetic mean (which is what we really mean when we say average). That, however, flies in the face of statistical analysis.

For an arithmetic mean to be worth anything, you have to exclude aberrations in the numbers that would skew it too heavily in one direction. Many hosting companies do this for billing: they lop off the top 5% of traffic spikes when calculating your peak bandwidth. This is standard practice. (See Wikipedia for more.) Granted, it would have been more meaningful to get the mode or median either instead of or in addition to an arithmetic mean, but that's water under the bridge at this point. Remember, kids: math doesn't like to be abused.

### 19 Responses

1. Jeremy says:

Section 7. Section 53A-1a-807 is enacted to read:
53A-1a-807. Mitigation monies.
(1) As provided in this section, a school district shall receive monies to mitigate potential impacts due to the unplanned reduction in revenues resulting from the transfer of scholarship students from the district to a private school.
(2) A school district shall retain in enrollment for purposes of receiving funds under the minimum school program each scholarship student that:
(a) transfers from the school district to a private school; and
(b) resides within the boundaries of the school district.
(3) Each transferred scholarship student meeting the criteria of Subsection (2) shall be retained in enrollment:
(a) for five years following the transfer or until the student would have graduated from high school if within five years of the date or transfer; or (b) until the student transfers back to the school district, if the student transfers back to the school district within five years.
(4) For each year a school district retains in enrollment transferred scholarship students as provided in this section, the board shall deduct from minimum school program funds distributed to the school district an amount equal to the number of transferred scholarship students retained in enrollment times the average scholarship amount for scholarship students statewide.
(5) The monies deducted from a school district’s minimum school program funds as provided in Subsection (4) shall be deposited into the Uniform School Fund.

Sorry for the long quote but I had a question that relates to it and your post. New private school students (kindergarten) will be able to use vouchers from day one. According to the clip above from HB148 it seems that the mitigation money doesn’t apply to these people. Is that figured into your calculations?

Sorry if this is a dumb question but I’m not sure I understand your assertion that private school costs for new enrollees don’t get factored in until after the first 5 years of the program. It was my understanding that at year 13 the program was running at maximum cost.

I didn’t get much sleep last night so it is likely my fault if I’ve misunderstood.

2. Nice analysis. The demand for other “choice” provisions added to families’ educational options (e.g., charter schools, Carson Smith special needs scholarships) suggests to me that demand would be higher than 1%. I’d predict around 3% — which would represent a significant savings. But, as you so clearly state, we simply don’t know what the switch rate will be. In any event, the doomsday financial scenarios that are being tossed around just aren’t accurate — however politically effective they might be.

3. Jesse says:

Looking at this in the morning, I’m realizing more than a few math errors from my end. Expect some corrections shortly.

4. Tim Beagley says:

Nice Job Jesse.

Your analysis shows very clearly how much of an “experiment” this is or would have been. The variables you outline are unknowable without more data. The reason scientist do experiments is that they don’t know for sure what the answer will be. We can guess and estimate all we want but the experiment still needs to be done or we never actually know the answer. In many ways, I would love to see the experiment done (although I still think the “policy” is bad) just to find out what those numbers actually are.

Two other variables that would be involved (and I believe would be heavily weighted in the experimental outcome) are the type (cost wise) of transfer students and the ultimate destination of mitigation monies.

If “cheap” students transfer relative to “expensive” students, or vice versa, the numbers change a lot and the outcome changes as well.

Likewise, if mitigation money simply evaporates in the dry soil of the districts, schools that lose voucher students could really get hit hard. Alternatively, if all the mitigation money goes to the school losing the voucher students, their loss could have minimal effect.

Again, nice job.

5. rmwarnick says:

I have a suggestion. Let the voucher proponents take their little experiment to another state, and let that state bear the costs (or reap the benefits, if any). Then we’ll have a better idea if taking taxpayer money and giving it to private schools is a smart move here in Utah.

6. Jesse says:

Okay, I think I’ve made all the corrections I need to. I also posted an Excel spreadsheet so you can see the math for yourselves.

rmwarnick: If every state took that approach, we’d never have the data because nobody would do it. It’s tough to be a pioneer or to “take one for the team”, but it has to be done at some point. Getting all NIMBY on it won’t do much for us.

7. DaveB says:

Let’s assume for a moment that your numbers are correct, along with the optimistic numbers in terms of number of people using vouchers and actual savings.

Under your figures, it takes 15 years to net a little over \$2 million in savings. Given that these are projections, which become even more unrealistic the further they go out, this to me actually makes the argument against vouchers.

Two million is a rounding error when you are dealing with hundreds of millions over a 15 year period, where lots of things– good or bad– could happen in between. If this is the best reasonable voucher supporters like yourself can prove, I have to say the numbers don’t favor voting for Referendum 1.

8. Jesse says:

The numbers I’m using come from the LFA, so they seem in-line with the numbers we’d all be willing to accept. With the current assumptions, no, it doesn’t look favorable at all. That’s why I think the five-year pilot is so important, so we can replace assumptions with projections based on hard data.

As I noted, even slight changes in the percentage of switchers means the difference between a wide loss or wide gain. Even getting up to half the national rate of private school usage would result in a 13-year windfall of \$1.8B. On the flip side, a marginal increase in private school enrollments means a total gain of *maybe* \$2M. I feel it’s well within a level of acceptable risk though others might not feel the same way. Believe me when I say that I’d drop support for the program if it wasn’t going to make financial sense regardless of my ideology.

I certainly think that during or near the end of the pilot period, we should eliminate the \$500 voucher (because it’s nothing but a lightning rod for criticism and is just plain silly) and look at Green Jello’s proposal to require a certain number of years of attendance in a public school before allowing someone to use a voucher. Both of those would greatly reduce the cost of the program and make it much more equitable.

9. Don says:

Aren’t “building savings” already included in the \$5500 savings per switcher? Afterall, the \$7500 cost per public school student includes building costs, right?

10. Are you assuming no fixed costs in your analyses?

11. Jesse says:

Don: As I understand it, construction is paid for primarily through bonding which is paid for through new taxation. The SL Trib recently ran an article on Jordan’s latest bond for west-side schools that backs this up.

Craig: If I read the LFA document correctly, then yes, I believe I included the \$200K per year in administration costs.

12. Don says:

But isn’t the cost of paying off those bonds included in the UTA figure of \$7500 per pupil spending?

I think Craig was asking about fixed costs to run the schools, not the program. In other words, he’s challenging the \$5500 savings per pupil figure.

13. Don says:

Jesse,
I like the overall structure of your spreadsheet. It’s a simple format that clearly shows the effect of the switch rate on the cost/benefit factor, especially once the program is fully implemented.

You mention the LFA’s analysis and assumptions as the basis for some of your assumptions, so I have questions about some of your figures. You yourself said the student growth rate is about 3% but you used 4% in your spreadsheet. Why?

Where does the initial figure of 3700 switchers come from? The LFA sheet shows 2006 for FY2008.

I also don’t understand this statement from your year 6 analysis above, “Given that the switchers vouchers are already paid for, we can exclude them we calculating net cost or savings.”

On the question of estimating the switch rate, I think it is pie-eyed optimism to think it would ever be larger than 1% of the entire student population. Granted, this is just my opinion, but 1% is three times the LFA’s estimate, which is actually .34%. Furthermore, it is based on the price elasticity relative to the average cost reduction in tuition, so it’s not like it’s just a guess. I would say the LFA’s average tuition estimate of \$8024 for FY2008 is too high, but even if we drop it to \$5000/year, the relative switch rate only jumps to .6% of the entire student population (19% of the private student population).

14. Jesse says:

As I’ve been revisiting the numbers, I’ve been seeing more than a few mistakes in what I plugged into the spreadsheet. I had accidentally added in the 1,700 or so kids below the poverty level already attending public schools, so it looks like the assumptions change yet again. The breakout didn’t seem to make it clear if those students were being shown once or twice. (Seriously, this is a project better suited to a collaborative workspace than a blog. We can catch each other’s errors more quickly.) I’m starting to believe that I need to spend more than a few hours doing crunching to get some more data out of this.

My goal in all of this was to present some kind of starting point for us to crunch our own numbers and understand what the LFA’s numbers mean. I think that succeeded, though my specific implementation of it was a bit on the botched side. Mea culpa.

My intention of the statement is that I did not take the cost of non-switchers and add in the cost of switchers, then deduct from savings. The savings amount is net after paying for the vouchers for switchers.

One thing that amazed me is that about 10% of the students in private schools are below the poverty line. That’s the same percentage of the general population living below the poverty line. I guess we can throw out that “rich kids” argument after all.

Anyway, I’m seeing that, should the program be maintained, we definitely need some tweaks to prevent the costs from spiraling out of control and erasing all financial gains by Year 9 based on the 0.35% switcher assumption. Again, the pilot gives us data and that’s what we really need right now.

15. Bill Fox says:

Don, I think if the cost of retiring a bond were included in the cost per student we would not be so hesitant to approve bonds. Even if it were included, the cost of servicing a bond only pays the interest on such bond and the principal is still left at the end of the term.

To All, as interesting as the math of this issue is, it does not get at what is the heart of the issue. Competition makes everyone better. Schools are no exception and as of this point in time, without vouchers, they have none

16. Jeremy says:

I’m a bit late on this but I wanted to confirm that the \$7,500 UTA cost per pupil figure does include operations, facility construction, and interest costs. They explained their figures in this post on their blog:

http://utahtaxpayer.blogspot.com/2007/08/7500-per-utah-k-12-student-in-fy2008.html

Competition is a worthy goal but this program seems to be more focused on creating a new entitlement for families that never would have considered sending their kids to public schools than on creating viable competition that could improve Utah’s education system.

17. Bill Fox says:

Thank you, I stand corrected

18. Daniel says:

Great post Jesse. I appreciate the work you did to put together the spreadsheet and that you made your assumptions explicit.

1. October 21, 2007

[…] number crunching done in the last Voucher Math post really helped me see how big of a difference the switch rate can be even with just minor changes. […]