*Note: Please read the disclaimer. The author is not providing professional investing advice.*

Probably the most common metric used when deciding between various investing options is to examine what rate of return (ROR) each option has delivered in the past. But if option A has historically returned +10.2% while option B has returned +10.6%, does that automatically make B the wisest choice?

**Table 1**

Rate of Return | ||
---|---|---|

5-Yr Average | ||

Option A | +10.2% | |

Option B | +10.6% |

We’re often told that generally more money can be made if we’re willing to take on more risk, but when trying to decide between a selection of mutual funds, or a variety of individual stocks, or various investing algorithms, how exactly do we *quantify* or *measure* the risk of each so as to be able to compare our options intelligently (or automatically, if we’re programming a computer to decide for us)?

There are a variety of factors that contribute to risk and it’s impossible to account for them all, but we may be able to at least get a first-order feel for an option’s risk by performing an additional calculation using its historical data. Then, just as the average historical ROR gives us a (potentially misleading) preview into what returns may be coming in the future, we’ll also gain insight into what type of risk each investment option may carry.

Going back to our original example, what if we not only know each option’s average ROR over, say, the last 5 years, but we also know the breakdown by year (or quarter, etc.). For example, consider the following list of the annual RORs for our two options:

**Table 2**

Rate of Return | ||||||
---|---|---|---|---|---|---|

2001 | 2002 | 2003 | 2004 | 2005 | Average | |

Option A | +10% | +12% | +9% | +11% | +9% | +10.2% |

Option B | +100% | -4% | -23% | -15% | -5% | +10.6% |

We’re now probably starting to get more of an intuitive feel for how we would measure risk. When we look at Option A, we see a ROR that is fairly steady from year to year – with no surprises. Option B, though it has a slightly higher 5-year average, is all over the place – and in fact in most past years, it has *lost* money.

But we still would like a *number* for risk – and to get one we can use a common statistic called the standard deviation. Standard deviation gives us a quantity *proportional to the variability* between our individual samples (in our example, the annual RORs). The set with a smaller standard deviation has *individual* returns that are closer to the *average* return.

The standard deviation is the root mean square distance of individual set values from the set average. For example, to compute the standard deviation for Option A

1. Start with the set of RORs, **A = [10 12 9 11 9]**.

2. Compute the average of that set => **ave(A)=10.2**.

3. Compute the squared distance of each set member from this average => **(A-ave(A))^2 = [.04 3.24 1.44 0.64 1.44]**.

4. Compute the average of that squared set => **1.36**

5. Take the square root of that average => **1.2**.

Or, in Matlab syntax, the standard deviation is:

*sqrt(mean((A-mean(A)).^2))*

Now let’s re-examine our two choices, but this time we’ll include the extra standard deviation statistic.

**Table 3**

Rate of Return | |||||||
---|---|---|---|---|---|---|---|

2001 | 2002 | 2003 | 2004 | 2005 | Average | Std Dev | |

Option A | +10% | +12% | +9% | +11% | +9% | +10.2% | 1.2% |

Option B | +100% | -4% | -23% | -15% | -5% | +10.6% | 45.2% |

By computing the standard deviation we no longer have to rely on “eyeballing” the year-to-year variability, but instead we get an actual number. But what does this number mean and where should we set our threshold for “too risky”? Is a standard deviation of 10% safe but 50% risky?

In order to be able to use the standard deviation to make predictions about future RORs, we’ll first need to make an assumption that the RORs follow a normal (a.k.a. bell-curve or Gaussian) distribution. Is this a reasonable assumption? Well according to the central limit theorem, if you have a parameter that is dependent upon a large number of independent random variables, the parameter’s distribution is approximately the normal distribution *(even if those independent random variables have non-normal distributions!)*. As a fund (or stock, or algorithm) is a function of a variety of factors – whether the economy is in recession or expansion, influence of economic cycles, changes in technology or management, etc. – it does indeed seem reasonable to assume that a normal distribution is what we’re dealing with.

So, given a normal distribution we can immediately use some well-known properties of this distribution’s standard deviation. That is, 68.3% of the values should be within 1 standard deviation of the mean, and 95.4% of the values should be within 2 standard deviations. We can use this to help us get a clearer meaning for what sort of returns our various options may offer in the future. ** BUT** we must keep in mind that as with all projections based on historical data, we may not be able to trust what the past may predict about the future.

*In applying the 68% and 95% “rules” in our example here, we’re estimating the standard deviation from a very small sample size (i.e. our estimate of standard deviation may be way off). Even if our estimate is dead-on, there is no guarantee that the future distribution will match that of the past. A new factor such as a lawsuit, change in Fed policy, etc. can come out of left field and greatly alter “typical” RORs – especially for individual stocks. *

*Could we go back further in time to get more data points for a possibly more accurate standard deviation estimate? Yes, but the further we go back in time the more we risk including data from a time when the economic landscape was vastly different than today. How far back you go, or whether you use some other temporal division rather than yearly (e.g. monthly, quarterly) is up to you.*

So what is implied by the standard deviations above? If all of Option A’s future numbers indeed track the distribution of those of the past, we may predict that most (95%) of the future annual returns should be between +7.8% and +12.6% (computed from ** 10.2% +/- 2*1.2%**). Compare that to Option B’s much larger range of -79.8% to +101%. In this example, it is fairly clear that you are taking on quite a lot of extra risk just to hopefully gain an extra 0.4% ROR by choosing Option B. Therefore, Option A is probably more attractive.

Let’s now say that we come across a third option (C) and add its statistics to our table.

**Table 4**

Rate of Return | |||||||
---|---|---|---|---|---|---|---|

2001 | 2002 | 2003 | 2004 | 2005 | Average | Std Dev | |

Option A | +10% | +12% | +9% | +11% | +9% | +10.2% | 1.2% |

Option B | +100% | -4% | -23% | -15% | -5% | +10.6% | 45.2% |

Option C | +12% | +11% | +14% | +12% | +13% | +12.4% | 1.0% |

We can clearly see that not only does Option C offer an additional +2.2% ROR, but historically it has less variability compared with Option A! Using the same computation as above, we might expect next year’s return to be between +10.4% and +14.4% – quite good compared to A and B. Therefore, if you have no other information besides these annual numbers, it’s hard to think of any good reason not to put your money into option C as it has historically not only had the highest return, but also achieved it with the lowest standard deviation (i.e. risk).

Usually when picking between our investing options though, it’s not as black and white as above. Let’s add a final option (D). Let’s also eliminate options A and B, since C is more attractive as described above (*though if we’re not putting all our eggs in one basket – as we’re doing in this example – we may want to diversify by investing in more than just one option*).

**Table 5**

Rate of Return | |||||||
---|---|---|---|---|---|---|---|

2001 | 2002 | 2003 | 2004 | 2005 | Average | Std Dev | |

Option C | +12% | +11% | +14% | +12% | +13% | +12.4% | 1.0% |

Option D | +15% | +4% | +16% | +15% | +17% | +13.4% | 4.8% |

We may predict that Option C will next year deliver a ROR between +10.4% and +14.4%. Likewise Option D may yield a ROR between +3.8% and +23% (computed from ** 13.4% +/- 2*4.8%**).

Is the extra riskiness of Option D worth it ? That depends on your personality, financial situation, and confidence in how closely Option D’s future may track its past. But at least by examining the standard deviation you get *some* sort of feel for what range of returns you might expect.

In conclusion, picking between investment options – whether the choice is between mutual funds, individual stocks, or investment algorithms or trading systems – is a game of pursuing the highest rate of return given your level of acceptable risk. While historical returns are in no way a guarantee of future results if one is making a decision between options based on how they performed in the past, using the standard deviation can be insightful.

In the absence of other information such as changing economic cycles, changes in fund management, etc., the standard deviation offers a way to estimate future risk by examining past variability between annual (or quarterly, etc.) returns. This can be helpful in estimating probable upper and lower bounds for future returns with the understanding that these are *predictions* but never *guarantees*.

Just wondering: why do you calculate the average of the squared set (step 4 above) by dividing by 5, rather than 4? I think you lose a degree of freedom when you calculate the average of the deviations from the mean return (because now you center the distribution you are examining around zero), and it’s particularly important to do this in small samples. In your August 29, 2012 post, you have calculated sigma using the correct denominator (8, which is 9 minus 1 for the mean you calculated in the differences). What am I missing — can you comment?

tresia – thanks for the spot. that post was written long ago and i (erroneously) didn’t used to differentiate between sample and population standard deviation – i think b/c i usually worked with large sample sets such that dividing by n or n-1 didn’t make much difference. i should have used n-1 for an unbiased estimate. thanks!