Wednesday, January 25, 2012

Is Arithmetic Return Bias Basis of Low Vol Anomaly?

I created an index of the highest beta stocks from 1962 to present. Every 6 months I took those 100 highest beta stocks, excluding the lowest 20% in market cap (to get rid of dumb stocks you can't trade). The results are in the chart above, and summary data are as follows:

The top line, "AnnReturn", is the arithmetic return, and here the monthly returns for the high beta stocks are about 0.14% higher than the S&P500, which when multiplied by 12 is a 1.7% difference. But looking at the chart which shows a total return chart, and the geometric annualized return, we see a very different picture, with the high beta stocks underperforming by 3.5% annually.

The basis for this is the difference between geometric and arithmetic returns, which is

Geometric Return =Arithmetic Return - Variance/2

Thus, the differential annualized variance (in this case, 12% vs. 2%), generates the differential annualized return. Interestingly, the return rankings for these data are different depending on the horizon!

Mutual funds and individual investor holding horizons average about 1 year, and I think that's a good assumption for an investment horizon. It seems that 1 year would be the obvious horizon to apply data against, but the problem is there is so little of it. There's like twelve times as much monthly data! A simple fix would be to use log returns, but this doesn't always happen, and I think those who still find the Security Market Line to have a positive slope in general are looking at monthly percent return data, and this is why they see what they do.


Anonymous said...

What if you were able to look ahead. For instance, if you knew at the beginning of the year what the high beta stocks over the upcoming year would be and created a portfolio from those.

Anonymous said...

I am not sure this is the right way of looking at it.

At the end of the month, let us say you sell 1 stock and at the beginning of the next month you can buy either one stock or you can buy no of stocks for exact dollar value at which you sold on the previous month end.
I think the diff between these two strategies will explain the anamoly.

Anonymous said...

Good post. One could wonder if 1-year investment horizon is enough. For example high-beta did very poor in 2008 but very good in 2009. Using annual returns would not show how poor high-beta stocks actually did over this 2-year period. Therefore the compounding really kick in with 3-year or even 7-year evaluation horizons.

It seems that the market becomes more inefficient, if the evaluation horizon increases.

Anonymous said...

In a monthly rebalanced strategy (as a fraction of the portfolio) it's the arithmetic return that counts.

This is not a good post, it's a quite silly one. Way below he usually high standard.

Eric Falkenstein said...

anon: Ok, help me understand this because I don't. You have $200 at time 0, and are looking at two alternative portfolios: one with high variance, one low. Their monthly returns rank order them different than their annual returns. How you you invest $100 in the highly volatile portfolio to generate a greater total return than the lower volatility portfolio?

Anonymous said...

Well Eric, you've written clearer paragraphs, but I'll try :)

Your logic implies a higher vol version (say more leverage) of the same strategy is always a worse return for risk. Not true. That is only viewing the strategy alone (as your whole portfolio). If the strategy is a small part of an overall portfolio, and you rebalance (take away from it when it's up, add to it when it's down) its contribution looks much more like its arithmetic than its geometric average.

Another way to view it is imagine you do put all your money in only this strategy, but you're much more risk averese, so you do it at less octane (say by adding cash). The artihmetic and geometric get much closer. Did we solve the conundrum?

Eric Falkenstein said...

I'm assuming neither strategy is forecastable, so market timing wouldn't work from some sort of value effect here (indeed, there is no autocorrelation in monthly returns).

Assume two strategies available to you where returns are r~ln(mu,sigma), geometric brownian motion, with parameters such that the expected return in one year generates a different rank ordering than the expected return over a month. I'm confident that if your goal is to maximize your end-of-year portfolio value, the amount of weight put on the higher volatility strategy is zero. I don't have a proof, but perhaps it is a corollary of Samuelson's paper on the log of wealth, Why we should not make mean log of wealth big though years to act are long.

Aaron Brown said...

"Ok, help me understand this because I don't. You have $200 at time 0, and are looking at two alternative portfolios: one with high variance, one low. Their monthly returns rank order them different than their annual returns. How you you invest $100 in the highly volatile portfolio to generate a greater total return than the lower volatility portfolio?"

Suppose portfolio 1 returns 1% every month. Portfolio 2 returns 10% half the time and -7.5% the other half.

If you put $100 in each and let it grow, portfolio 1 gives you $112.68 at the end of a year, portfolio 2 gives you $110.97 if you get 6 up months and 6 down months as expected.

If you put $100 in each and take out any profit (or replace any losses) each month, portfolio 1 makes $1 per month, $12 in all. Portfolio 2 makes $60 in 6 up months, loses $45 in 6 down months, for a total profit of $15.

However, eventually leaving the money in portfolio 1 will do better than keeping a constant $100 in strategy 2. After 4 years, letting $100 grow in strategy 1 produces $161.22, while keeping a constant $100 at risk in portfolio 2 produces only $160 (assuming exactly 24 up months and 24 down months).

Eric Falkenstein said...

I still don't really understand this. If you could make more in the highly volatile asset via market timing with a cash account, there would appear to be inevitable arbitrage.

I think a key is how to treat the cash account used to sweep profits and bolster the principal in the drawdowns, in terms of the internal rate of return.

Anonymous said...

Why don't you report "excess return" instead of raw return? A high beta portfolio requires less cash to get the same market exposure, and the leftover cash earns interest. It ought to be credited with that interest, as it is in the calculation of the calculation of excess return.

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Anonymous said...

I am a professional in finance and I am staggered by the number of incredibly smart and educated people who don't know about the relationship between arithmetic return, geometric return, and standard deviation. I call it volatility drag: the higher the volatility for a given arithmetic return, the lower the long run growth rate (which is what the geometric return is). When volatility gets large enough the geometric return starts to decline and for really large volatilities, the geometric return can be negative. This is why leverage can be so dangerous: the high arithmetic return with high volatility is actually a very low geometric return.

This is a mathematic relationship, not an economic or financial one. It applies to the growth rate of rabbits or carbon dioxide in the atmosphere as well.

Here is my simple example: you start with one dollar that you invest. After one one year the investment is worth $2 dollars giving a 100 percent return. In the second year you you have a 50 percent loss, and you end up with $1. The arithmetic return is (100 + -50)/2 = 25%, an impressive number. However, the geometric return is zero: you started with $1 and ended with $1.

There are some other interesting nuances that know of about the mathematics of finance. I have had people ask why this stuff isn't taught. It is a good question.