Variance of a Security

If you have landed here looking for variance computed from a fund's past monthly returns, you want variance. This page is the forward-looking twin of that one.

Risk, in Modern Portfolio Theory, is the variability of return. So the theory needs a number for how much a single security's return is likely to swing.

It builds that number from a forecast, not from history. You list the states the world might be in. You put a return and a probability against each state. Then you measure how far each return sits from the average, and weight those gaps by how likely each state is.

Squaring the gaps is the trick that makes it work. Without squaring, a return 3% above the mean would cancel a return 3% below it, and every security would look risk-free.

The workbook calls this the ex-ante variance. Its square root is the ex-ante risk, better known as standard deviation. A larger variance means a wider spread, and a wider spread means more risk.

Section 4.5 gives the formula:

Variance (σ²) = Σ [Ri − E(Ri)]² × Pi

where Ri is a possible rate of return, E(Ri) is the expected rate of return, and Pi is the probability of that rate of return. The square root of this measure is the Ex-Ante Risk.

The workbook's worked table (Table 4.2) uses four equally likely outcomes:

Expected rate of return = (8% × 0.25) + (10% × 0.25) + (12% × 0.25) + (14% × 0.25) = 11%. Ex-ante variance = 0.00050. Ex-ante standard deviation = 0.0224, or 2.24%.

Why the security-level number matters. It is the input the portfolio formula needs. Section 4.7 computes portfolio risk as σport = √(Σ Wi²σi² + ΣΣ WiWj Covij), so each security's own variance enters squared by its weight, and every pair contributes a covariance term as well.

Two places the same quantity reappears.

• In the utility function of section 4.3, U = E(r) − 0.5 × A × σ², the σ² is this variance. With a risk aversion index of 3, an expected return of 15% and a variance of 10%, the workbook computes a utility score of 13.5%.
• In CAPM, the denominator of beta is the variance of return on the market portfolio: β = Covi,m ÷ σ²mp.

A risk-free asset is defined by this number being zero. Chapter 5 lists the characteristics: an asset whose returns yield zero standard deviation and zero correlation with all other risky assets. Because both its variance and its covariance with any risky asset are zero, the portfolio risk formula collapses, and the workbook's example reduces to (1 − 70%)² × (10%)² = 0.0009, a portfolio risk of 3%.

A fund manager forecasts next year for Deccan Auto Ltd across three states.

Step 1 — expected return. (0.30 × 24%) + (0.50 × 10%) + (0.20 × −8%) = 7.2 + 5.0 − 1.6 = 10.6%.

Step 2 — squared deviations, weighted.

Step 3. σ = √0.012324 = 0.1110, or 11.10%.

So Deccan Auto has an expected return of 10.6% with an ex-ante risk of 11.10%. The workbook's own illustration, at 11% expected return, carried a standard deviation of only 2.24% — a fifth of this. Same expected return, five times the spread. That difference is exactly what an investor is being asked to be paid for.

Put it into the utility test. Take an investor with a risk aversion index of 3. U = 0.106 − (0.5 × 3 × 0.012324) = 0.106 − 0.0185 = 0.0875, or 8.75%. If the risk-free rate is 6%, the excess utility is 2.75%, so the investment is worth making. Raise the investor's risk aversion index to 7 and U = 0.106 − 0.0431 = 6.29% — barely above the risk-free rate, and this investor should probably pass.

Nothing about Deccan Auto changed. Only the investor did.

Chapter 4 (Introduction to Modern Portfolio Theory), section 4.5, is titled Variance of return for individual security (Ex-Ante Risk) and carries the formula and Table 4.2. The same quantity then powers section 4.3 (the utility score), section 4.7 (portfolio risk) and Chapter 5's beta denominator.

Expect a calculation question giving three or four states with probabilities and returns, asking for the variance or the standard deviation. Expect also the definitional pair: square root of variance is standard deviation, and ex-ante is forecast while ex-post is historical. The chapter's own sample question about which correlation gives the best diversification is a reminder that this security-level number is only half the portfolio story.

• Multiply by the probability, do not average the squares. Equal probabilities make the two look the same. Unequal ones do not, and that is where the marks go.
• Variance is in squared units; standard deviation is its square root. Only the standard deviation is comparable with a return.
• Ex-ante uses forecast states and probabilities. Ex-post uses realised returns. This page is the ex-ante one; variance computed from a scheme's past monthly returns is the ex-post one.
• A portfolio's variance is not the weighted average of its securities' variances. Every pair adds a covariance term, which is why diversification works.
• Both upside and downside count. A return far above the mean adds as much to variance as one far below it. If you want only the downside, the workbook's measure is semi variance.
• In the utility formula σ² is variance, not standard deviation. Substituting the standard deviation there gives the wrong utility score.