Risk-adjusted return

The last chapter of the subject, and the one that completes the measurement. Chapter 2 showed CAGR deliberately hides the path; this chapter puts some of it back.

A scoping note. The Portfolio theory subject derives these ratios properly and takes apart the ways they are misquoted. Here the concern is narrower and practical: how to use a risk-adjusted figure to interpret a return you are looking at.

The problem, stated plainly

Two portfolios, both 10% a year over a decade. One moved in a narrow band. The other halved twice.

By every measure so far they are identical, and no sensible person would call them equally good. What is missing is not information about the destination but about the journey — and the journey matters for two concrete reasons rather than aesthetic ones.

You might have needed the money mid-journey. A 50% fall is only survivable if you were not forced to sell into it. Chapter 4's money-weighted return is where that shows up.

You might not have stayed. The behaviour gap is largest in the most volatile holdings, because volatility is what triggers the selling. A return you would not have held through is not a return available to you.

The standard adjustment

Divide the return in excess of a risk-free alternative by the volatility taken to get it:

$S = \frac{R_p - R_f}{\sigma_p}$

Three components, each requiring a decision:

$R_p$ — the portfolio return. Use the geometric figure from chapter 2, net of costs.

$R_f$ — the risk-free rate. What you could have earned without taking risk. In India, a short government security or a comparable administered rate; it should sit near the policy corridor, so it moves as the RBI moves. The subtraction matters: a portfolio returning 7% when risk-free money pays 6.5% has earned almost nothing for its risk, and a return figure alone conceals that completely.

$\sigma_p$ — volatility. The standard deviation of returns, derived in Quantitative methods. Annualised, and computed from the same frequency for anything being compared.

The ratio answers: how much excess return per unit of variability? A larger number means the return was obtained more efficiently.

Note that nothing in the published disclosure regime gives you this. SEBI requires CAGR against a Total Return Index over standard periods — a return-only basis. The risk figure is yours to add, and its absence from factsheet headlines is why return-chasing is so easy.

What volatility misses

Worth being explicit, because the ratio is routinely over-trusted.

It treats upside and downside alike. Standard deviation punishes a portfolio for rising unexpectedly. Nobody experiences an unexpected gain as risk. The Sortino ratio, covered in Portfolio theory, uses only downside deviation for this reason.

It assumes a distribution that does not hold. Volatility describes a bell-shaped spread well, and financial returns have fatter tails — extreme events occur far more often than the normal distribution implies. The lognormal case chapter in Quantitative methods sets out why. So the single most dangerous risk, a rare catastrophic loss, is the one volatility measures worst.

It can be lowered by illiquidity rather than by safety. An asset that is rarely repriced shows low measured volatility because the price is stale, not because the risk is small. Unlisted holdings and some property measures flatter themselves this way.

It says nothing about the worst case. Two series can share a volatility and have very different maximum drawdowns.

And it is estimated from a sample, so it carries the sampling error any estimate does — which is why small differences in a ratio should not be treated as rankings.

The simpler measures worth reading alongside

For practical use these are often more informative than the ratio, and much harder to misread:

Maximum drawdown. The largest peak-to-trough fall. It answers the question people actually care about — how bad did it get? — in one number.

The worst rolling twelve-month return. The worst single year anybody holding this experienced.

Time to recover. How long the worst drawdown took to make back. This is the number that predicts whether someone holds on.

None of these requires a distributional assumption, which is their advantage.

Working the problem

A: 14% a year, 22% volatility. B: 11% a year, 9% volatility. Risk-free about 6.5%.

Ratios:

$S_A = \frac{14 - 6.5}{22} = 0.34 \qquad S_B = \frac{11 - 6.5}{9} = 0.50$

B produced about 0.50 of excess return per unit of volatility against A's 0.34 — roughly half again as much return per unit of risk. On this measure B performed better, despite A's higher return.

Notice how much of A's return was not compensation for risk. A's excess over risk-free is 7.5 points for 22 points of volatility; B's is 4.5 points for 9. A took nearly two and a half times the variability to earn two-thirds more excess return.

The qualifications I would attach. A difference between 0.34 and 0.50 is meaningful but both are estimates from a sample; I would want the maximum drawdown and worst rolling year for each before treating it as settled. And if A's volatility is fat-tailed while B's is not, the ratio understates the gap.

An investor for whom A was right. Someone with a long horizon, no possibility of being forced to sell, and the demonstrated temperament to hold through a large fall. Concretely: a thirty-year-old investing for retirement, with a separate emergency fund covering a year of expenses, no debt, secure income, and a record of having held through a previous crash without selling.

For that person the volatility is close to irrelevant, because the two mechanisms that make volatility costly are both switched off — they will not be forced to sell, and they will not choose to. The higher return is then simply a higher return, and over thirty years three extra points a year is transformative.

The general conclusion, which is the subject's conclusion. The risk-adjusted figure tells you which portfolio earned its return more efficiently. It does not tell you which to hold, because that depends on facts about you — your horizon, your obligations, and your behaviour under stress. Measurement narrows the question to a judgement; it does not replace the judgement.

The point

Two portfolios with the same return differ if one was far more volatile, because volatility can force a sale and can cause a voluntary one — the two mechanisms that make a paper return unavailable to you. Dividing excess return over a risk-free rate by volatility gives a comparable efficiency figure, and the subtraction matters because a 7% return when safe money pays 6.5% has earned almost nothing for its risk. But volatility punishes upside, assumes thinner tails than markets have, can be faked by illiquidity, and says nothing about the worst case — so read maximum drawdown, worst rolling year and time to recover alongside it.