Volatility, annualising and why diversification lowers risk

Volatility is the SD of returns, not of prices

The input is a series of returns: the percentage change from one period to the next. You would not take the standard deviation of the prices themselves, because a price that trends steadily upward would show a large SD while being perfectly predictable. Returns remove the trend and leave the period-to-period wobble, which is what the SD measures. If the standard deviation itself is new to you, start there; this post assumes you know what it measures.

Here are twelve monthly returns for a hypothetical fund. They are made-up numbers for illustration, not data for any real security.

The returns add up to 9.2%, so the mean monthly return is 0.767%. The sample standard deviation is 2.71% per month. That is the fund's monthly volatility: a typical month lands a couple of percentage points either side of its average.

Try it: standard deviation calculator

The twelve monthly returns are already filled in below; replace them with your own fund's returns to get its monthly volatility, then multiply by √12 as the next section explains.

Open the full sample standard deviation calculator for the n − 1 formula and every working step.

Annualising: why the square root

Monthly and daily volatilities are not comparable with each other or with the yearly figures fund factsheets quote, so everything gets converted to an annual basis. The rule is to multiply by the square root of the number of periods in a year:

σ(annual) = σ(monthly) × √12 σ(annual) = σ(daily) × √252

The square root comes from how independent random changes add. If each month's return is independent of the last, the variance of a year's total is the sum of twelve monthly variances, 12σ². The standard deviation is the square root of that, √12 · σ. The same logic gives √252 for daily data, 252 being the usual count of trading days in a year.

So the hypothetical fund above has an annualised volatility of 2.71% × 3.464 = 9.39%. Note what the rule does not say. It does not say a year is 12 times as volatile as a month; it is only about 3.5 times. Risk grows with time, but more slowly than time does. The rule also rests on the independence assumption. If returns trend (a good month tends to follow a good month), the true annual figure is higher than √12 predicts; if they mean-revert, it is lower.

Two assets: the portfolio formula

Now hold two assets. Suppose, again as an illustration, asset 1 has an annual volatility of 20% and asset 2 has 8%, and you put 60% of your money in the first and 40% in the second. The portfolio's return is the weighted sum of the two returns, and the variance of a weighted sum is:

σp² = w₁²σ₁² + w₂²σ₂² + 2 w₁ w₂ ρ σ₁ σ₂

where ρ is the correlation between the two assets' returns. The first two terms are each asset's own risk, scaled down by its weight squared. The third is the covariance term, and it is the only place the correlation appears. The weighted average of the two SDs is 0.6 × 20 + 0.4 × 8 = 15.2%. Here is what the formula gives for different correlations:

Work the ρ = 0.5 row by hand to see the pieces: 0.36 × 400 = 144, plus 0.16 × 64 = 10.24, plus 2 × 0.6 × 0.4 × 0.5 × 20 × 8 = 38.4. The total is 192.64, and √192.64 = 13.88%. You can check any row, or try your own weights, with the portfolio standard deviation calculator.

Why the portfolio sits below the average

At ρ = 1 the formula collapses to a perfect square, (w₁σ₁ + w₂σ₂)², so the portfolio SD is exactly the weighted average. The two assets rise and fall in lockstep and there is nothing to offset. Any correlation below 1 shrinks the cross term, and because the other two terms do not depend on ρ, the portfolio variance falls. The gap between 15.2% and the actual figure is the diversification benefit.

The intuition: when asset 1 has a bad year, asset 2 is not guaranteed to have one too. Some of the swings cancel. Even with zero correlation, the portfolio here is at 12.42%, which is 2.78 points less risk for the same weighted-average expected return. Returns average linearly; risk does not. That asymmetry is the reason spreading money across assets is worth anything at all.

The table also shows the limits. Perfectly negative correlation (ρ = −1) would let you pick weights that remove all risk, but real assets almost never behave like that. Correlations are estimated from past data and tend to move around, and they can rise in stressed markets exactly when you want them low. Treat a correlation estimate as uncertain. You can compute one from two return series with the correlation coefficient calculator, but a figure from one calm period is a poor guide to a turbulent one.

What volatility does not tell you

Standard deviation treats a surprise gain and a surprise loss the same way, and most investors do not. It also summarises the whole distribution in one number, which works well for roughly bell-shaped data and poorly for returns with occasional large crashes. A fund with a 10% volatility is not promised to stay within ±10% in a given year; with fat tails, moves of three or four SDs happen more often than a normal distribution would predict. Use volatility to compare assets and to understand diversification, and pair it with other measures, such as the worst historical drawdown, before drawing conclusions about how much you could lose.