Standard Deviation And Its Limitations As A Measurement of Risk

Risk measurement using standard deviation

One of the defining moments in the career of an investor is when they shed their misunderstanding of “risk” as “the likelihood of losing money” and replace that definition with “unforeseeable movements in price”. We commonly equate the “risk” of an investment with its standard deviation (as volatility), which offers us a useful insight into the return profile of the asset; i.e how much the sequence of investment returns move around over a period of time, and how far away they move from the asset’s most “common” return based on its past performance.

manage risk with standard deviationThis will often enable an investor to narrow their search during the preliminary asset selection process, by eliminating asset classes which are outside of their risk tolerance. Standard deviation alone is however gravely insufficient to inform the portfolio construction process for anything beyond traditional 60/40 Equity/Bond investment portfolios…

Two of the largest criticisms of standard deviation as risk

1) That it is not forward looking; i.e it is based entirely upon historical performance data, of which there is no guarantee of repetition in the future. Of course, most assets, particularly passive investments or trackers, do not tend to deviate from their historic performance profile by a great degree. However, active investments (e.g. actively managed funds) show a much larger scope for variance owing to their ability to change their investment strategy at the discretion of their managers.

2) Standard deviation as a performance metric does not differentiate between upward and downward variance. i.e if an asset shows meteoric growth during a year, it would show a high SD reading, as would an asset which moved down and made large losses that year. Accordingly, without context, standard deviation could lead to some dangerous misunderstandings and the wrongful inclusion of an asset in an investment portfolio.

Key Determinants Of Risk: Position Sizing

The standard deviation of an asset is secondary to position sizing in the context of risk. This is a key concept. In isolation, standard deviation alone does not allow us to make any inference about portfolio risk. If we have two assets, asset A and asset B, and the first has a standard deviation of 15% over a calendar year, and asset B has a standard deviation of 30% over a calendar year, you could be forgiven for making the statement that “asset B is more risky than asset A”. This statement would be correct, but only half the story if both investments are constituents of a portfolio- we can make no inference as to their overall risk contribution to the portfolio unless we understand their weighting in the portfolio. If we work on the premise that both asset A and asset B have a perfect correlation with each other (i.e they move up and down in tandem) then when our allocation to asset A is half of the size of our allocation to asset B, despite their differing levels of volatility, they contribute an equal amount of risk to the portfolio.

Key Determinants Of Risk: Correlation Management

risk calculationAs we have touched upon, assets in any given portfolio will have a degree of correlation and covariance between themselves. As a result of this, it is theoretically possible that the introduction of an asset with a high volatility profile will actually decrease the total amount of portfolio risk if it is negatively correlated to the other assets in the portfolio.

The measure used to indicate the extent to which two random variables change in tandem is known as covariance.
The measure used to represent how strongly two random variables are related known as correlation.

Correlation is dimensionless, i.e. it is a unit-free measure of the relationship between variables (in this case, the returns of the two assets). Unlike co-variance, where the value is obtained by the product of the units of the two variables. Conclusively, a +1 reading denotes a perfect positive correlation (assets move up and down together in lockstep) and a -1 reading denotes a perfect negative correlation (assets move in opposite directions). Failure to consider correlation between portfolio assets could undermine the portfolio construction process, increase non-systematic risk, and reduce returns.

Standard Deviation In A Meaningful Context: An S&P500 Case Study

Nowhere is correlation as meaningful as in a market crash or correction scenario. Increasingly, correlation between asset classes can be seen to move towards +1 in periods of stress, but there are still methods to reduce drawdowns during outlier events. The go-to for professional investors will be the inclusion of alternative investments of absolute return strategies which thematically have low-to-no correlation to traditional asset classes.

Let us consider the impact of having included an allocation to managed futures strategies, or CTA‘s during the global financial crisis. To what extent would it have mitigated losses in an equity portfolio, and what is the overall effect of correlation management on portfolio risk?

For the sake of keeping things simple, we shall use the Societe Generale Trend Index ( CTA funds or Commodity Trading Advisors) for our allocation to absolute return strategies. The Index is designed to track the 10 largest (by AUM) trend following CTAs and be seen as representative of the trend-followers in the managed futures space. Managed futures strategies historically provide a significant diversification effect to traditional portfolios.

For our equity investment, we shall use the SPDR S&P500 index tracker (NYSE:SPY). The ETF tracks the investment performance of the 500 companies composing the S&P500 index- historically used as a barometer for the US stock market.

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Standard deviations of SPY and SG Trend from 2001-2007.Now, the standard deviation for the S&P500’s returns from the period of 2001 – 2007 (preceding the market crash) was 15.15%. The standard deviation for the SG Trend Index was 9.03%. We could speculate that the S&P500 index is capable of producing larger returns. We could further speculate that the S&P500 is 40.46% more volatile than the ST Trend Index during this period. If we ignore the relevance of correlation, we could be fooled into thinking that with 50% of a portfolio in the S&P500, and 50% in the SG Trend Index gives a portfolio with a resulting portfolio standard deviation of (15.15 + 9.03) 24.18%….

Covariance and standard deviation of SPY and SG Trend Note that both the covariance and the correlation between the two assets are significantly negative. Accordingly, we can suppose that when one if losing value, the other will be gaining value and vice versa. With this in mind, a large standard deviation reading for either of the two assets in the pair would indicate a higher degree of positive performance during a time when the other asset is losing value (as both assets in this case produce positive returns more frequently than they produce negative returns). This is the basis of the diversification effect.

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Standard deviations and variance of SPY and SG Trend.Looking at the returns of both indices during 2008 we can see that that the SG Trend index is considerably less volatile, shows less variance in its returns, and is also net positive at year end. Based on its positive performance profile and diversifying properties when coupled with the S&P500 index tracker, we can deduce that a portfolio split equally between the two assets (50/50) would have produced more favorable returns during the market events of 2008.

S&P 500 index and SG trend index drawdown and growth in 2008.

We can see from the table that a 100,000 USD investment into the S&P500 index tracker at the beginning of 2008 would have drawn down to 61,069 USD, producing a total loss of -38.9%, a 100% allocation to the SG Trend index would have grown 20.9% to 120,874 USD and a 50/50 allocation to both assets would have drawn down to 90,972 USD with a total return of -9%.

Some may say, “well why not invest all the money into the SG Trend Index?”, to which the answer would be – diversification. The real conclusion is in the ability of alternative investments (and correlation management among assets in general) to reduce portfolio risk. By spreading their portfolio across two assets instead of one, and investor could have avoided 77% of the drawdown experienced by index investors in the global financial crisis. Now, in reality, it is impossible to simply buy the SG Trend index in the same way that somebody might buy NYSE:SPY. Far from making this exercise a moot point, it should go to show that correlation management is an active, not passive exercise and that the deliberate reduction of portfolio risk should occur before investments are made.

Standard deviation and percentage change of SPY and SGT. Reducing standard deviation alone provides no assurance that risk has been reduced if there is a high degree of correlation between portfolio assets. However, if we are able to reduce standard deviation whilst maintaining a desired balance in portfolio asset correlation, investing in assets which produce positive returns over time, we begin to have the foundation of a truly robust portfolio.

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Sources and Further Reading