In our previous impact of volatility article we made an argument for diversification:
- Large stock market drawdowns occur relatively frequently
- It takes higher positive returns, or a longer period of positive returns, to overcome a large drawdown
- Therefore it’s rational to attempt to reduce volatility within a portfolio
- The most common way of reducing volatility is by diversification: investing in assets that have no (or negative) correlation
- However, correlation between many assets and markets has been increasing over the last decade, and increases more rapidly during violent market falls
- Therefore many investors have been seeking alternative, non-mainstream assets to achieve diversification
In this article we will go one step further and consider ways of measuring the success of a diversification strategy in reducing volatility and explore how risk adjusted returns can help us rationally assess how much risk we have taken on to achieve a certain level of return.
Measuring Risk (Volatility)
In financial mathematics, risk is defined as volatility. An asset or portfolio with unpredictable returns that vary wildly would be considered more risky than one with more stable returns, clustered in a tight range.
There are a number of ways of measuring volatility. One of the simplest is to calculate the range: the difference between the highest and lowest return in a given period.
More useful is the standard deviation: the square root of the average of the squared differences from the mean return.
This might look and sound complex, but it’s not:
- Work out the simple average of the returns (the mean return)
- Then for each return, subtract the mean return and square the result
- Then work out the simple average of the squared differences
Note: we use the square of the differences to ensure that we are working with positive numbers only – that way we can be sure that positives and negatives don’t cancel each other out in the equation.
- Then take the square root of that result
Note: taking the square root puts the result back into the original units – this would normally be percentage returns when measuring the performance of an investment.
Example
Portfolio A has returns of 2,2,4,3,1 and 5% over a 6 month period
The mean return is 2.84% (add up all the returns and divide by the total number of observations)
The differences between the returns and the mean are -0.84, -0.84, 1.16, 0.16,-1.824 and 2.16% (subtract the mean return from the monthly return)
The differences squared are 0.69, 0.69, 1.36, 0.03, 3.36 and 4.69 (multiply the number by itself)
The sum of the differences is 10.83
The variance is 1.805 (the sum of the differences divided by the total number of observations)
The standard deviation is 1.34% (the square root of the variance)
This tells us that if returns are within one standard deviation of the mean, they will fall within a range between 1.5% and 4.18% (2.84% + or – 1.34%)
Statistically, for a normal distribution of returns we would expect the result to be within one standard deviation of the mean around 68% of the time
If the returns are within two standard deviations of the mean, they will fall within a range between 0.16% and 5.52% (2.84% + or – 1.34% x 2)
Statistically, for a normal distribution of returns we would expect the result to be within two standard deviations of the mean around 95% of the time
So by finding the standard deviation we have a simple, easily understood measure of risk. Of course, it is what is known as an ‘ex-post’ measure of risk, which in plain English means it is after the fact – we can only use this to look at historical data, not to make predictions about the future, and of course it is only theoretical. Reality can turn out to be much more unpredictable as the crash in 2008 showed.
Comparing Returns Between Portfolios to Reduce Volatility
Now that we have a quantifiable measure of risk, we can compare how much risk two different portfolios took on. Modern Portfolio Theory (and, frankly common sense) suggests that a rational investor would choose the less risky portfolio for the same level of return.
So in the example above, the portfolio achieved a total return of 18.2% with a standard deviation of 1.34%. This would be preferable to a portfolio that achieved 18.2% with, say, a standard deviation of 5%. In this new portfolio, even though the total return is the same as the original portfolio, there was a much greater range of returns and it was much more volatile. There would be a much greater chance of underperformance.
Risk Adjusted Returns
So how can we compare portfolios with different levels of return and different standard deviations? The most common method is known as the Sharpe Ratio.
Again, this looks more complicated than it is.
The Sharpe Ratio drills down to greater detail by using a concept known as the ‘risk free rate’. Now let’s be totally clear – in reality there is no such thing as a risk free return, but in financial mathematics the risk free rate is used in a similar way to a ‘control’ in a science experiment. It’s the return you could achieve for the lowest possible level of risk. The return that could be achieved by investing in short term government debt is usually used as the risk free rate – such as UK government 3 month gilts or US 3 month treasury bills.
The Sharpe Ratio subtracts the risk free rate from the return the portfolio actually achieved. This means that the equation is only looking at the additional return achieved by taking on the additional risk of investing in something other than the risk free investment. We then divide that return by the standard deviation to give us the Sharpe Ratio.
This gives us a single number we can calculate for any portfolio or any investment and then use to make comparisons. The higher the Sharpe Ratio, the better its risk adjusted return – or to put it another way, the less volatility investors were exposed to in order to achieve that return. A negative Sharpe Ratio would indicate that the risk free asset performed better than the investment being analysed – not a good investment at all!
Conclusions
If investors want to measure if their diversification strategy is successful and how much volatility they’re exposing themselves to, then standard deviation is one useful measure. If they want to assess if the returns they are achieving are worth the amount of risk they are exposing themselves to, then the Sharpe Ratio is a good place to start. Having a grasp of these concepts allows investors to start to construct a portfolio that minimises volatility.
A Post-Script
Note that it isn’t realistic to try and rid a portfolio of all volatility. Some volatility is inevitable if a kind of return is to be achieved. The objective is to earn the required level of return for the lowest level of volatility.