Variance

Two schemes can post the same average return and still feel completely different to own. One edges up by roughly the same amount every month. The other lurches up and down. The second is riskier, and the workbook's first tool for putting a number on that is variance.

The idea behind all the workbook's risk measures is that fluctuation in returns is risk. Periodic returns — daily, weekly, fortnightly or monthly — are worked out first, and then their fluctuation is measured against the average. Fluctuation on the upside and on the downside are both treated as risk.

Variance does this in relation to the scheme itself: it measures how far each period's return strays from that scheme's own average. It is relevant for both debt and equity schemes.

Its close relative is standard deviation, which is simply the square root of variance. Standard deviation is easier to read because it is in the same units as the returns, and it is what the workbook goes on to annualise and use in the Sharpe ratio.

The workbook's two schemes:

Working Scheme 1: deviations from the average of 5 are 0, −1, 0, +1. Squared, they sum to 2. Divided by (4 − 1), that is 0.67.

Working Scheme 2: deviations from 3.75 are +1.25, −8.75, +6.25, +1.25. Squared: 1.5625 + 76.5625 + 39.0625 + 1.5625 = 118.75. Divided by 3, that is 39.58.

The workbook computes it in MS Excel with =var(range of periodic returns), which divides by the number of observations minus one — the arithmetic above reproduces its figures exactly.

From variance to standard deviation is one step: √0.67 ≈ 0.82 and √39.58 ≈ 6.29. A footnote adds that at least 30 observations are needed to compute an accurate statistical value of standard deviation — the four-month table is for illustration only.

Variance = Σ (each periodic return − average return)²
           ÷ (number of periods − 1)

Standard deviation = √Variance

In MS Excel:  =var(range)     =stdev(range)

Put Rs 1,00,000 into each of the workbook's two schemes and let the four monthly returns compound:

Scheme 1's average is higher (5 against 3.75) and its variance is tiny (0.67 against 39.58). An investor in Scheme 2 carried roughly 59 times the variance and ended Rs 6,328 poorer.

At the end of month 2 the Scheme 2 investor was sitting on Rs 99,750 — below what she put in — while the Scheme 1 investor had never seen a loss. That dip is what variance is measuring. An investor who panics and redeems at month 2 locks in a loss that Scheme 1 would never have offered.

Now the caution. Variance only describes the past pattern of returns. It says nothing about which scheme will be steadier next year, and with only four observations it is statistically weak — the workbook wants at least 30.

Chapter 10 (Risk, Return and Performance of Funds), section 10.7.1, introduces variance with the two-scheme table, followed immediately by standard deviation in 10.7.2 and the annualisation rules. Chapter 11 uses standard deviation in the Sharpe ratio. Expect a question on what variance measures (fluctuation against the scheme's own average), its relationship to standard deviation, and whether it applies to debt, equity or both.

• Variance compares a scheme with its own average, not with an index. Fluctuation against a market index is beta.
• Relevant for both debt and equity schemes. Beta is relevant only for equity schemes.
• Standard deviation = √variance, not variance squared.
• Upside and downside fluctuation both count as risk.
• A higher average return does not mean lower variance, or vice versa — they measure different things.
• At least 30 observations are needed for an accurate standard deviation; historic risk measures have limited predictive value.