### Primer: Average Investment Returns - Which Average?

by meep

I will be doing a series of posts, where I’m trying to get at what the long-term asset performance for public pensions actually has been.

The first step is understanding how we measure average investment returns. In this post, I’m looking at what is meant by average return.

I’m going to start with a very artificial situation: a two-year investment period, where asset performance is -10% the first year and +10% the second year.

Here’s our basic situation:

Look at that craftswomanship.

So what’s the average return?

Should be easy right — average -10% and +10%, and we get 0%.

But wait, what happens when we start following what happens to the assets in this scenario? Let’s pretend we start with $1 million:

Wait a second… we **lost** money? But how can that be with an average return of 0%?

**PLAYING WITH NUMBERS**

Let’s try something else, just to check out our intuition. We’ve already seen that going down 10% and then up 10% is not the same as staying in place.

Let’s reverse the returns – what if we go up 10% and then down 10%:

Good, the order of the returns doesn’t matter. The two moves still don’t cancel each other out, but at least it doesn’t matter that the loss came before the gain, or vice versa.

Let me show how the arithmetic is done:

When an amount changes by s%, then it’s the same as multiplying that amount by (1 + s/100).

The total return for the 2-year period is -1%. Not zero, but close. But what if we wanted an average? How does that work?

**COMPOUNDING INTEREST**

What we do is ask: what’s the rate **r** such that if both years the return was r, we’d get the same result as our overall scenario.

If you do this, it’s called the geometric mean or geometric average.

If we take the geometric average for our situation:

Then our average return was approximately -0.5% over the two years.

The way you can tell a geometric average or mean is being used, the language “compound annual growth rate” or “annualized rate” is often used. There’s ways to know.

But when the language says “average return”?

That’s not so clear. And it can be very deceptive.

**GEOMETRIC MEAN ALWAYS SMALLER THAN ARITHMETIC MEAN**

I just gave away the result in this title, but at first glance, while we can see that doing something that involves taking the square root of two numbers multiplied together [geometric mean] versus adding two numbers together and dividing by two [arithmetic mean] won’t be the same… will one always be bigger than another?

Yes, yes it will. The arithmetic mean will always be smaller.

Now, you don’t have to take my word for it. Check out this geometric proof!

Okay, I’m being a little nasty there, with my handcrafted circle, eye-balled diameter line & radius… not to mention my “right” angle.

The point is that there is a geometric interpretation of the geometric mean (thus the name), and if you want to learn more, go here. You’ll find there are **loads** of geometric interpretations of the geometric mean.

The point is that if somebody is simply averaging in the “normal” way (arithmetic mean), they are going to be biasing your idea of what the returns are.

So let’s look at some real life returns to see what happens.

**COMPARING AVERAGES**

First, let’s just start out with straight market index returns.

I will go to my favorite market return source, and if you look at the bottom, you see he’s already calculated the averages:

Arithmetic Average | S&P 500 | 3-month T.Bill | 10-year T. Bond |
---|---|---|---|

1928-2015 | 11.41% | 3.49% | 5.23% |

1966-2015 | 11.01% | 4.97% | 7.12% |

2006-2015 | 9.03% | 1.16% | 5.16% |

And here’s the geometric averages:

Geometric Average | S&P 500 | 3-month T.Bill | 10-year T. Bond |
---|---|---|---|

1928-2015 | 9.50% | 3.45% | 4.96% |

1966-2015 | 9.61% | 4.92% | 6.71% |

2006-2015 | 7.25% | 1.14% | 4.71% |

and this is the difference:

Arithmetic – Geometric | S&P 500 | 3-month T.Bill | 10-year T. Bond |
---|---|---|---|

1928-2015 | 1.91% | 0.04% | 0.27% |

1966-2015 | 1.40% | 0.05% | 0.41% |

2006-2015 | 1.78% | 0.02% | 0.45% |

That’s quite a difference in returns, especially over decades worth of performance.

All that said, generally funds do not announce average returns using an arithmetic average. I bet there’s something in CFA ethics that would get people rapped on the knuckles for using the arithmetic mean.

That said, reporters tend not to be CFAs. Or particularly numerate. Some who have reported average returns did an arithmetic return.

**PUBLIC PLANS DATABASE**

So I decided to do my own calculation based on information from the Public Funds Database. I filtered it so that I had only those plans with information for 1-year investment returns from fiscal year 2001 to fiscal year 2014.

That left me with 136 pension plans.

Looking at the 14-year (2001 – 2014) results, then you see that the difference between the arithmetic and geometric averages is, in general, between 0.6% and 0.8%:

No big deal, right?

Well, what’s the difference in fund accumulations if we use the two different averages for 14 years?

Oh.

You get a fund that’s 25% higher if you use the arithmetic average.

Maybe you understand why public pensions don’t want to change their asset return assumptions by 0.5%… because there can be a huge difference in valuation, especially if you’re discounting over decades.

I happened to have run into this myself when I was recommended by a senior investment guy to drop my average return assumption down 2 full percentage points when I was doing modeling for a lifecycle mutual fund. I didn’t know about the geometric v. arithmetic mean issue at that point.

**ONE PLAN: CALPERS**

So let’s look at Calpers specifically.

The Public Plans Database (and my independent calculation) give the following results:

5-year annualized return: 10.7%

10-year annualized return: 6.2%

As of 2015.

What did Calpers report?

Hmmm, well, that’s tough to get at. There’s this:

Myth: CalPERS 7.5 percent assumed annual rate of investment return is too high and cannot be achieved.

Facts:CalPERS investments earned 13.2 percent in Fiscal Year 2012-13, 18.4 percent in Fiscal Year 2013-14, 2.4 percent in Fiscal Year 2014-15, and 0.6 percent in Fiscal Year 2015-16.

CalPERS assumed rate of investment return is a long-term average. Any given year is likely to be higher or lower than the assumed rate.

CalPERS investments have earned an average annual return of 8.3 percent since the Total Fund inception date of July 1, 1988.

Well, I don’t know that last one.

Let’s look at press releases. This is the fiscal year 2015 press release:

Over the past three and five years, the Fund has earned returns of 10.9 and 10.7 percent, respectively. Both longer term performance figures exceed the Fund’s assumed investment return of 7.5 percent, and are more appropriate indicators of the overall health of the investment portfolio. Importantly, the three- and five-year returns exceeded policy benchmarks by 59 and 34 basis points, respectively. A basis point is one one-hundredth of a percentage point.

….

It marks the first time since 2007 that the CalPERS portfolio has performed better than the benchmarks for the three- and five-year time periods, and is an important milestone for the System and its Investment Office. CalPERS 20-year investment return stands at 7.76 percent.

So I match their 5-year average return, and I don’t have enough info for the 20-year return.

So the issue?

It’s not arithmetic versus geometric averages.

I think it may be something else.

But that’s for a different post.

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