Is “Risk” Standard Deviation?
Investors place their money into capital markets with the goal of earning a return for their investment commensurate with the risk they’ve assumed. Return is relatively easy to quantify. For example, if you put $100 dollars into a stock and it’s worth $130 dollars one year later, the return is +30%.
Risk is harder to quantify. Modern Portfolio Theory classically defines risk as the standard deviation of investment returns. Even financial advisors commonly use the terms “risk” and “standard deviation” interchangeably. Standard deviation shows how much variation exists between the actual returns and the average returns; higher standard deviation indicates there is greater variance between the returns and the average. For an example, consider two series of investment returns: the first is the actual series of returns of the S&P 500 (20022011), and the second is a fictional investment that oscillates between bad years (5% return), flat returns (0% return) and extremely good years (37% return).
EXAMPLE 1
These two return series have the same negative element – the same “standard deviation” – but clearly the second, utterly fictional investment is a better deal because the average returns are substantially better. There is no additional volatility to the S&P, but clearly it is a riskier investment. In other words, standard deviation only makes sense as a negative element when placed in context to the expected returns. Given this principle, many investors consider more information in their analyses like the Sharpe Ratio which measures the excess return of an investment per unit of risk
EXAMPLE 2
Even accounting for this, standard deviation is still an arbitrary formula that doesn’t perfectly reflect real world risk of loss. In fact, we can generate a fictional investment with a fictional series of returns with marginally smaller returns and higher standard deviation, but still achieves better realworld results.
EXAMPLE 3
We achieve these better results (despite a lower average return and higher volatility) by splitting the deeply negative years into a greater number of lessnegative years.
As an extreme variation of the same principle, imagine a $1.00 dollar investment that loses nothing over 9 years and then 100% of its value in the tenth year, so its final value drops to $0. Alternatively, imagine a $1.00 dollar investment that loses 10% of its value, annually, over 10 years; that investment would still have a residual value of $0.35.
It is this very problem that pushes some market optimization programs to prefer geometric return (which infers the total return based upon the real world effect, as derived via the “$1 invested” column) instead of the classic arithmetic return (which is just the average).
No matter how a statistician accounts for the mathematic distortion, the underlying problem is standard deviation is an arbitrary measurement of “risk” that means nothing without context. Standard deviation is a valuable standard as “risk” because it can be derived in a universally accepted, replicable way, but that doesn’t mean it should be the primary risk consideration for any particular investor – whether an individual or institution.
When are investment risks “Normal”?
Furthermore, there are risks that are not perfectly captured by the modeling system itself. For instance, modern portfolio theory operates under the classical presumption that the returns of an investment have a single average and that the distribution of annual returns surround that average in a symmetrical, bell shaped curve – a “normal distribution”. A normal distribution has a single peak of expected results, right on top of the average return; in other words, the average return should be the most likely occurrence.
However, real world returns don’t follow the guidelines of a normal distribution. The arithmetic long term average of the S&P 500 is about 12.2%, the median number is 14.3%, but the most common returns are between 20% and 25%. In other words, the mode, median and mean of the S&P results do not match; in a normal distribution, they would.
Actual distribution of S&P Returns 19262012

Hypothetical returns for a “Normal” Distribution

Modeling a nonnormal distribution of returns can yield surprising results. For example, the 2008 financial crisis is sometimes called by economists as a “sixsigma event”. To explain, we’ll need to review some statistics. If you look up at the normal distribution of returns above, you can see that the average return (the μ symbol – the Greek letter mu  in the chart above) is at the center of the bell curve. In other words, in a normal distribution of returns over time, the most common result is the average of all results, the point noted μ on the chart. Next, let’s look at sigma  the σ symbol in the chart above  which means standard deviation. Note the “34.1%” marks on both sides of the bell curve. The 34.1% represents the probability of getting a result between the average and 1 standard deviation of the results. This means, for a normal distribution of returns, you have a 68.2% (i.e. 34.1% + 34.1%, because we’re looking above or below the mean) probability of getting a result that assured of getting a return that is one standard deviation from the mean. So, for example, if you have a normal stock investment with a return of 12% and a standard deviation of 3%, that means you have a 68.2% chance of getting a return between 9% and 15%. Next note the 13.6% marker between 1 σ and 2 σ. It tells us that the probability of having a result that falls within a larger range (2 standard deviations away from the mean) – so, any result between 6% and 18%  is even more probable. Specifically, the probability of getting a result between 6% and 18% is 95.4% (i.e. 13.6% +13.6% +34.1% + 34.1%). Going back to the beginning, a “sixsigma event” is another way of saying a result that incredibly improbable; specifically, 99.99966% of all results should fall within six standard deviations of the average in a normal distribution.
For another example, the S&P 500 has had 1 pretty bad year – 1974  with returns between 25% and 35%. On the other hand, it has had 3 incredibly bad years with returns between 35% and 45%  specifically 2008, 1931 and 1937. This reflects a real world possibility of very poor circumstances gaining momentum and snowballing into catastrophic market environment like the Great Recession or the Great Depression. On a histogram of annual returns, this is a decidedly nonnormal feature of a separate “peaks” towards the far left edge of the data.
Beyond a single investment’s distribution of returns, financial analysts have noticed that the covariance between different investments can get lot stronger in times of stress. For example, in most time periods, there may be only a modest correlation between stock prices for US and Canadian companies. However, if some global shock has knocked the US stock market by more than 20%, it is significantly more probable that the Canadian stock market is falling by 20% as well. The correlations of different investments can vary depending on market conditions, but traditional financial models don’t account for this effect.
One of our recent blog entries – called “Picking up Pennies in front of a Steamroller”  on Westminster Consulting’s website discusses nonnormal return distributions. Again, modern portfolio theory only quantifies risk as historical variability of returns, but that doesn’t always capture true risk borne by any investment. To paraphrase the blog entry, there was a hedge fund investment that had solid repeatable trend of positive results until the past few years. Investors looked at the history of returns and presumed that the investment had little or no downside risk. There is a phrase in finance, “picking up pennies in front of a steamroller”, that describes an investment with deceptively low risk and modest returns that is prone to occasional disastrous results. Rather than a normal distribution of returns, these return series are better known as “Taleb distributions”, named for the statistician describing them. Nassim Taleb also gets the credit for coining ostensibly unlikely results (such as the 2008 financial crisis) as “black swan” events. Historically, black swans were thought not to exist and now it is known that they are actually quite common in Australia and New Zealand; similarly, real world financial markets results frequently, often spectacularly, fail to match their predicted projections.
It is unnecessary to delve deeply into the esoteric attributes of nonnormal return distributions (with multiple peaks, asymmetrical skew, nonnormal kurtosis) or their various solutions (such as “bootstrapping” and other resampling methods) within this short paper. The key point to acknowledge is there are limits to the commonly used financial models and the real world is not always perfectly represented by these financial models. There are alternative modeling techniques that attempt to adjust for the shortfalls of classic modeling techniques, but these revisions continue to be a work in progress for analysts and statisticians. Moreover, using strictly numeric measures of risk may not the best measure of risk; we will continue this conversation in our next newsletter.