Standard Deviation

According to the Capital Asset Pricing Model (CAPM), risk can be used interchangeably with standard deviation with respect to well diversified portfolios. In other words, risk can be viewed as a measure of portfolio volatility. We have seen that Standard Deviation increases exponentially with Potential Return, but what does this translate to in real world application? Again, it is best explained with examples. Let us consider two different portfolios.

Portfolio A has a potential return of 6% and a standard deviation of 6%.
Portfolio B has a potential return of 8% and a standard deviation of 10%.

Let us begin by examining Portfolio A. The long term expected rate of return is 6%. With a standard deviation of 6% it is expected that 68.3% of the time the annual return of the portfolio will be within one standard deviation of 6%, or 6% +/- 6%. This means that roughly 14 years out of every 20 years the annual portfolio return will be between 0% and +12%. Further, it is expected that the portfolio will occasionally have some annual returns outside this range and this can be further expected to lie between one and within two standard deviations from 6% roughly 5 years out of 20, for a range of 6% +/- 12% (two standard deviations), or in other words between  -6% and 18%. Finally, there would be some very extreme cases where the portfolio would expected to have some outlying data points, occurring perhaps once in every 20 years where the portfolio’s annual return would be greater than two and within three standard deviations from 6%. This means that on occasion, the portfolio may have an annual return ranging between -12% and +24%.

(You can click on the figure for a larger view)

While the standard deviation in the first few years may result in performance that is dramatically different than the expected long term return of the portfolio, over time this is expected to balance out and revert to the expected average (as depicted by the heavy black arrows).

Portfolio B, which has a higher expected return (8%) and higher standard deviation (10%), will be more volatile and it is expected that the potential return is higher – although it should be noted that this is not guaranteed. For example, referring to the graph above – while the expected return line would be higher (by 2 %), the range of ending values (as depicted by the ends of the black arrows) would have a greater distance between them, with the lower bound expected to be lower than the higher bound of the 6% portfolio – in other words, it is possible that Portfolio B will have a lower long term return than Portfolio A.

In summary, Portfolio A with a long term expected return of 6% and a standard deviation of 6% may behave as follows:

14 Years out of 20 the annual return will be between  0% and 12%
5 Years out of 20 the annual return will be between -6% and +18%
1 Year out of 20 the annual return will be between -12% and +24%

After 30 years the portfolio may have returned between 5% and 7% on an annualized basis.

If we apply the same calculations to Portfolio B which has a slightly higher long term expected rate of return of 8%, and a much higher standard deviation of 10% we will see the following possibilities:

14 Years out of 20 the annual return will be between -2% and +18%
5 Years out of 20 the annual return will be between -12% and +28%
1 Year out of 20 the annual return will be between -22% and +38%

After 30 years the portfolio may have returned between 6% and 10% on an annualized basis.

In conclusion, a portfolio with a higher expected rate of return may in fact return a lower long term average than a less risky portfolio. In the above examples, it is possible that Portfolio A would have returned 7%, and it is possible that Portfolio B would have returned 6%. The point is that more risk does not guarantee higher returns, it only offers the potential of higher returns.