The other day I was reading research from way back in 2011: Chetty, Friedman and Rockoff’s fascinating and multifaceted study on the long-term financial impacts of teacher quality on students. The study draws some conclusions on the impact of teachers on lifetime earnings of their students, referencing the concept of present value. Read on:

At age 28, the oldest age at which we currently have a sufficiently large sample size to estimate earnings impacts, a 1 SD increase in teacher quality in a single grade raises annual earnings by 1.3%. If the impact on earnings remains constant at 1.3% over the lifecycle, students would gain approximately $39,000 on average in cumulative lifetime income from a 1 SD improvement in teacher VA in a single grade. Discounting at a 5% rate yields a present value gain of $7,000 at age 12, the mean age at which the interventions we study occur.

What does this mean, exactly? It turns out it’s simple: discounting some future value to present value means we must account for the fact that money now is worth more than money later, because now-money can be invested at some rate to make more later-money. (Khan Academy, as always, has a great intro on the concept.)

Consider the passage above: at the end of her career, a student who experienced the higher-quality teacher will have made $39,000 more, on average. But what’s that money worth today, to an (on average) 12-year-old?

One plausible (but wrong) answer goes like this: If we (like the authors) assume that 12-year-old could get a 5% annual return by investing that cash, the future value after t years will be FV = PV * 1.05 ** t. Using the values from the paper and solving for t, we get that 39,000 = 7,000 * 1.05 ** t => t = 35.2, which tells us Chetty et. al. assume the student will have a career of about 35 years.

The problem with this math is that it assumes the $39,000 difference will arrive as a lump sum at the end of the lucky student’s career. In fact, this difference will accrue gradually, paycheck by paycheck. So the algorithm is slightly different: the authors first compute the average present value (at age 12) of total lifetime earnings to be $522,000. Then, assuming the the 1.3% difference observed at age 28 will hold over one’s whole career, they compute $522,000 * 0.013 = $7,000.

So how do they compute that $522,000? Back to Chetty et. al.: We calculate this number using the mean wage earnings of a random sample of the U.S. population in 2007 to obtain an earnings profile over the lifecycle, and then inflate these values to 2010 dollars. I don’t have access to that dataset, but 2007 wage data aggregates are available from the US Census. If we make a couple low-fi assumptions (average people will clock in at average wages for the whole period in the Census age range, then get bumped to the next range’s wage; average people will work until they die at an average age of 78) we can come up with our own computation of present value of lifetime earnings:

```
avg_lifespan = 78
initial_age = 12
present_value = 0
for i in range(15, avg_lifespan):
if 15 <= i <= 24:
wage = 13580 # wages here and below come from the Census
elif 25 <= i <= 34:
wage = 35071
elif 35 <= i <= 44:
wage = 44028
elif 45 <= i <= 54:
wage = 47215
elif 55 <= i<= 64:
wage = 42854
elif 64 < i:
wage = 24328
present_value += wage / (1.05 ** (i - initial_age))
```

That gives us a value of about $499,000, which comes out to about $525,000 in 2010 dollars. This isn’t quite the computation that’s performed in the paper (they assume 2% wage growth in addition to the 5% discount rate, rather than simply using the average value for each year in a given age range), but it produces a similar value. Not bad!