### Public Pensions Primer: Time-Weighted vs. Dollar-Weighted Returns

by meep

You were told there would be no math?

Oh yes, there will be math.

Before I start diving into numbers, here’s a nice little timeline re: private pensions. It’s far from comprehensive, but I thought some of y’all would be interested.

Thanks to my top referrers this week:

- The Other McCain
- Pension Tsunami
- Wirepoints
- My mystery facebook linker who’s really into Kentucky issues
- And somebody on LinkedIn

The time for frivolity is over.

School is back in session!**MEASURING PENSION FUND PERFORMANCE**

In a prior post, I said that the valuations we see of pension liabilities are just approximations of various sorts. There are all sorts of approximations that go into getting to that end number, yadda yadda. I’m not trying to denigrate that — I’ve been in valuation myself for annuities, life insurance, and reinsurance. It’s all approximation, and some approximations are much worse than others. That’s what all this brou-ha-ha is about.

But rather than try to predict the future, I want to see how well things did in the past.

We saw some of this in my prior post on measuring investment returns. I showed that investment returns should be measured as a geometric average.

Thing is, yes, the funds generally do measure geometric averages, and that is an appropriate way of measuring asset management performance.

But how about measuring whether the return average matches up with valuation assumptions?

I will show below that the straight geometric average of returns is not what we should be looking at, if we want is to check whether our return on assets assumption is proper for our pension funds.

**NOT ALL RETURNS SHOULD BE WEIGHTED EQUALLY**

I want you to look at the below situations.

So the assets go down 10%, an additional $100K is put in to make up for the loss, the next year markets go up 10%, and you end with $1.1 million.

Here it’s opposite: the markets go up 10%, you remove the gains, then the market goes down 10% – you end with $900K.

How should we measure the average returns for these situations?

If you remember that prior post, we had situations where no money was put in or taken out in the interim. We could still ignore the cashflows, and do the geometric average.

This is called the **TIME-WEIGHTED AVERAGE RETURN**.

The time-weighted average return is appropriate to use if you’re simply evaluating the asset manager (who we presume isn’t the one putting in or taking out money). In both these cases, the time-weighted average return is -0.50%.

But if I wanted to see how the pension fund performs? Then what?

This is where the **DOLLAR-WEIGHTED AVERAGE RETURN** comes in.

The concept is that we assume all money accumulates/discounts at a single rate of return, which will be the dollar-weighted average return. Assume that money put into the funds is positive, money taken out is negative, and you’re setting the equation equal to the final accumulation.

**HERE IS THE MATH**

So let’s go to those two situations.

First, let’s look at the case where money is added after the market drops. I’ll use something nicer than my handwriting for the equations:

First off, notice we end up with a quadratic equation. If you remember your high school algebra, that gives you the possibility of two solutions. We get two numerical solutions, but one makes no sense from the point of view of interest rates (negative 200%? How can you do that?).

But notice the one solution we do end up with: 0%. By putting in money after the market drops, we’ve essentially put in the same amount (initial $1 million + later $100K … and end up with $1.1 million at the end). We broke even. Makes sense, right?

What about when we take money out?

Again, we end up with a quadratic equation where one of the solutions makes no sense, so we end up with 0% as our average.

I put in $1 million in total at the beginning, and got out a million in total. We broke even. Makes sense.

Let us compare that against the time-weighted return: -0.5% — so now because we made ourselves whole at time 1 (by either removing money or by adding money to get us back to $1 million), we ended up breaking even because the same dollar amount is being exposed to the up 10%/down 10% movements.

Let us do some thought experiments (I will just tell you the results, and leave the math as an exercise for the reader).

Situation 1: Have $1 million at beginning, do not take money out til end.

Whether it’s +10%, -10% or -10%, +10%, you end up with the dollar-weighted return equalling the time-weighted return.

Situation 2: Remove all the money at time 1.

The dollar-weighted return will be whatever the first period return is.

Situation 3: Remove $100K at time 1 for the first return pattern (down 10% then up 10%)

The dollar-weighted average return is = -1.06%

Situation 4: Add $100K at time 1 for the second return pattern (up 10% then down 10%)

The dollar-weighted average return is = -0.96%

Think about what’s going on in situations 3 & 4: not only are we not breaking even, we’re getting even worse returns than what time-weighted averages will show us. That’s because in both cases, you have larger amounts exposed to the negative returns compared to the positive returns.

**SUMMING UP**

So here’s my lovely artisanal handwriting:

With the given fact pattern of returns, I could get you dollar-weighted returns running anywhere from -10% to +10%, while the time-weighted average stays put.

So let’s think what these means for pension funds: if the days of great returns were when you had a lot of money (in the past), and you’re taking money out now for benefits (and thus don’t have as much weighted for recent good returns)… the dollar-weighted average return may not look good though the time-weighted average says everything is a-okay.

Remember this sketch:

This is what I’m working up to. I want to calculate the dollar-weighted average returns, and compare to the time-weighted returns, for the pensions in the Public Plans Database. It can be that the disparities will be very little, and perhaps that will not be the case. I intend to try it out. It’s going to involve some assumptions on my part (such as timing of cash flows), but I will try a few different things to see if it works.

Stay tuned!

**AN EXTRA FOR FRIDAY**

I didn’t write about this before, but remember this factoid:

That is, the normal average you think of is always greater than the geometric mean (which we should use for measuring returns).

The arithmetic and geometric means are equal **only** when all the returns are equal (i.e. constant).

But what if the returns are very volatile?

Remember how I said volatility kills?

This is what happens:

The greater the volatility, the less the geometric mean is (recall a geometric interpretation of the geometric mean of two numbers: have the two numbers as lengths to add together to the diameter of a circle, and then take the chord from that diameter perpendicular to the circle — that distance is the mean.) The more the geometric mean falls short of the arithmetic mean.

Let me make this concrete. I took the returns for the Public Plans Database from 2001 – 2014, and calculated their arithmetic and geometric means, and took the difference. I plotted this against the volatility of returns for each fund.

Well. I think that’s rather clear.

You may be curious as to who is that little low-volatility fund in the corner. That’s the Texas Municipal pension fund, with an average return of 7.44% over 2001 – 2014. If you look, they’ve been hanging around 85% funded for years. Hmmm. The annualized volatility is about 5%, and the difference between the geometric and arithmetic means is only 0.1%.

If you want to know which fund has the highest volatility, it’s Alameda County Employees. At an annualized volatility of near 15%, the difference between the arithmetic and geometric means is over 1.0%. Its geometric average return is 6.87% from 2001-2014.

Think of the foo-for-raw over dropping a discount rate 50 basis points.

Mmmhmm.

Just for fun, here’s a graph of the volatility against the geometric average returns by plan:

Hmmm, doesn’t look like that extra volatility is translating into extra returns. I just calculated a regular correlation and got a negative 13% correlation coefficient (higher volatility associated with lower returns); I calculated a rank correlation and got a positive 7% correlation. Average it out — essentially no correlation.

Basically, there’s no there there. Higher volatility does not necessarily mean higher returns. It just means higher volatility.

But those are all time-weighted averages. I need to calculate the dollar-weighted, and then we’ll be cooking with gas!

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