Abstract

We consider arguments about the impact of volatility on the ability to accumulate wealth. In that context, we explore some properties of arithmetic and continuously compounded returns.

Does increasing volatility reduce wealth accumulation?

Sometimes we hear an argument that, other things equal, assets with higher volatility accumulate less wealth than assets with low volatility.

The argument goes something like this. Suppose you have a risk free asset (asset 1) that always returns exactly 10%. You accumulate wealth with complete predictability. In R code you can readily calculate your wealth over a horizon of 10 years.

1.1 ^ (0:10)
##  [1] 1.000 1.100 1.210 1.331 1.464 1.611 1.772 1.949 2.144 2.358 2.594

Now, let’s consider a process with the same average annual return but more dispesion. A process that produces either a 20% return or a 0% return with equal probability is such a process (we’ll call it asset 2). The mean annual return is still 10%. The expected accumulation of wealth from such a process over 10 periods would be based on 5 periods of 0% returnn and 5 periods of 20% return. Here it is in R code.

cumprod(rep(c(1,1.2),5))
##  [1] 1.000 1.200 1.200 1.440 1.440 1.728 1.728 2.074 2.074 2.488

This result is then interpreted that you have two processes with the same average return producing different terminal wealths. The process with more volatility has a lower terminal and you condlude “Volatility is the enemy of wealth accumulation!”

What’s wrong with this analysis?

The mistake is in the first step. If you have a discrete process that produces a sequence of returns for consecutive periods and you then chain them together, the arithmetic mean of the process is not the expected return of the process. As you learned in finance classes or your CFA training, the correct way to calculate a period return on a process of chained single period returns is to calculate the geometric mean of the returns.

In the first case, the mean of the process and the expected return are the same because all the values are the same. As soon as we add variability, the geometric mean is guaranteed to be lower than the arithmetic mean. This is an example of a mean-preserving spread. The lower spread process is stochastically dominant, i.e. has a higher payoff structure, compared to the wider spread process with the same mean.

If you want to calculate the expected return of the process, you calculate the geometric mean. For asset 1, it is obviously 1.1. In the case of asset 2, the R code to calculate it is:

geomean=prod(rep(c(1,1.2),5))^(1/10)
geomean
## [1] 1.095

The expected return of asset 2 is 9.5%. If you want to know the expected wealth after n periods of owner this asset, the calculation is to 1.095 to the nth power. So, in the example of a 10 year hold of asset 2, the expected wealth after 10 years is:

geomean^10
## [1] 2.488

So, we’ve analytically reproduced the outcome of the 10 period simulation above. Asset 2 has a lower expected return than asset 1. The fact that the arithmetic mean returns of asset 1 and asset 2 are the same is irrelevant, because that’s not the correct way to calculate return in multiperiod context of chained single period returns. And since asset 2 has a lower expected return then, obviously, you will accumulate less wealth owning it.

Log returns to the rescue

Assets exist continuously, can be traded (more or less) continuously and have constantly changing values. By convention (and as required by accounting standards and GIPS), we measure their performance by taking periodic snapshots of the value of the asset. In the most typical method, we then calculate the ratio at time n+1 to time n and that ratio (minus one) is called the return of the asset for the period of time from n to n+1. As time goes by, we then calculate a sequence of returns.

We might then be tempted to perform statistics on these returns. The problem is that we have already shown that the mean of a time series of arithmetic returns is not the expected return of that time series. Let’s do an example.

In the R code below, I create a random arithmetic return sequence with mean of 10% and standard deviation of 15%. Because this is a random generator, the series will not have a mean equal to 10%. I then calculate the expected return and chart both as a growth of a dollar.

set.seed(1234)
series1=rnorm(n=20, mean=0.1, sd=.15)
expected.return.1=-1 + prod(1+series1) ^ (1/20)
data.frame(mean.return=mean(series1),expected.return=expected.return.1)
##   mean.return expected.return
## 1      0.0624         0.05236
plot(0:20,cumprod(c(1,1+series1)),col='red',type='l',ylab='Growth of a $',xlab='')
lines(0:20,(1+expected.return.1)^(0:20),col='blue',type='l')
legend('topleft',legend=c('Asset Value','Growth at Constant Expected Return'),col=c('red','blue'),
       lwd=2)

plot of chunk unnamed-chunk-5

So, in this case we have a series with an arithmetic mean return of 6.2% and an expected return of 5.2%.

Now, let’s say you want to do some statistics and calculate the standard deviation of the returns, correlation with other assets, etc. All these calculations work with the difference betwen observations and the arithmetic mean of the sample to do their work. But you have the undesirable feature that the measure of central tendancy you are working with (the sample mean) is not a good predictor of the return of the asset.

Is there a better way? Indeed, there is. If you work with log (continuously compounded) returns, these issues go away. When working with log returns the mean return and expected return are the same thing. Briefly reviewing the properties of logarithms, we recall that log(a*b) is log(a) + log(b). So, with log returns you calculate compounded returns by adding them and you can simply average the log returns to get their expected return. If we do statistics on log returns, the measure of central tendancy (the arithmetic sample mean) that we use as an anchor for all the calculations is also the expected return of the process.

Here is some R code that shows how to convert the same arithmetic returns we generated above to log returns and then repeats the same chart using the log returns.

series1.log=log(1+series1)
expected.return.log=mean(series1.log)
data.frame(mean.return=mean(series1.log),expected.return=expected.return.log)
##   mean.return expected.return
## 1     0.05103         0.05103
plot(0:20,exp(cumsum(c(0,series1.log))),col='red',type='l',ylab='Growth of a $',xlab='')
lines(0:20,exp(expected.return.log*(0:20)),col='blue',type='l')
legend('topleft',legend=c('Asset Value','Growth at Constant Expected Return'),col=c('red','blue'),
       lwd=2)

plot of chunk unnamed-chunk-6

Stated in log returns, the mean return and the expected return are both 5.1%. The growth of a dollar chart is generated by exponentiating the cumulative sum of these returns. It is identical to the graph generated by calculating the cumulative products on the arithmetic returns.

Conclusions

Arithmetic returns are a fact of life. They are the most common way of reporting returns and are required by accounting standards and GIPS.

Arithmetic returns have the undesirable feature that their sample mean is not the expected return. If you want to calculate the expected return, you have to use the geometric mean. When you are working with arithmetic returns, you need to keep this in mind.

If you have two process with the same arithmetic mean return but different dispersion, the one with less dispersion is stocastically dominant, i.e. has higher payoff. If these two processes are investible assets, then the one with more dispersion has lower expected return with higher volatility and is obviously inferior.

If you happened to have two assets with the same expected return, the expected wealth from owning those assets after n periods will be the same regardless of their volatility. In evaluating assets based on their periodic arithmetic returns, you just need to be careful not to confuse their expected return (geometric mean) with their arithmetic mean. If two assets have the same expected return, the one with higher volatility with have a higher arithmetic mean. This is a property of the measurement method, not a property of the asset itself.

Log returns are a useful antidote to these issues. With log returns, the mean return is the expected return of the asset. As a heuristic for investment decisions, you can perform statistics on arithmetic returns. If you are comparing two assets, the biases work in the same direction and you won’t be too far off. But if you want to make mathematical claims about the properties of assets, you will be better off working with log returns.