Sharpe, Sortino and the ratios that get misquoted

A scoping note. The Measuring your return subject covers how to use a risk-adjusted figure to interpret a return you are looking at. This chapter does the other half: where the ratio comes from, and the specific ways it is quoted wrongly.

It is the slope of the line

$S = \frac{\mu_P - r_f}{\sigma_P}$

Chapter 6 derived exactly this expression as the slope of the capital market line — the line from the riskless rate through portfolio $P$.

So the ratio is not a scoring system bolted onto portfolio theory. It is the theory's objective function. Finding the tangency portfolio is maximising the Sharpe ratio; they are one operation seen from two directions.

And that gives the ratio its meaning. A higher Sharpe ratio means a steeper line, and a steeper line is better at every level of risk, because any point on the lower line can be beaten by scaling the higher one. The ratio answers: if I can lever or de-lever this freely with cash, how good is it?

Sharpe states the decision rule that follows:

the investor should choose the desired level of risk (k), then obtain that level of risk by using the fund (F) with the greatest excess return Sharpe Ratio. Correlation does not play a role since the remaining holdings are riskless.

Note the final clause. It is a condition, not an aside, and misquote 4 below is what happens when it is ignored.

Misquote 1: dropping the risk-free rate

Return divided by volatility is not the Sharpe ratio, and Sharpe is emphatic:

it is essential that the Sharpe Ratio be computed using the mean and standard deviation of a differential return... Otherwise it loses its raison d'être.

His own example shows why, and it is the chapter's worked problem.

	Expected return	Standard deviation	Return ÷ volatility	Sharpe ratio
Fund X	5%	10%	0.50	0.20
Fund Y	8%	20%	0.40	0.25

The two measures rank the funds oppositely, and only one of them is right.

Why the subtraction is not cosmetic. The riskless rate is available without taking any risk, so it is not something the fund earned. A fund returning 7% when cash pays 6.5% has produced half a point of compensation for its risk, and a ratio that does not subtract treats it as though it produced seven.

Misquote 2: comparing ratios computed over different periods

A Sharpe ratio is meaningless without a period attached. Sharpe derives the scaling:

$S_T = S_1 \times \sqrt{T}$

So a monthly ratio of 0.10 becomes an annual ratio of $0.10 \times \sqrt{12} = 0.346$.

A factor of 3.46 between two correct numbers for the same fund. Quote one against the other and the comparison is nonsense.

The convention is to annualise, and Sharpe recommends measuring over short intervals and then scaling: "To maximize information content, it is usually desirable to measure risks and returns using fairly short (e.g. monthly) periods. For purposes of standardization it is then desirable to annualize the results."

But the scaling assumes no serial correlation, which he flags immediately — multiperiod returns compound, and "even if the underlying process does not involve serial correlation, a specific ex post sample may." A strategy whose returns are smooth by construction — illiquid holdings, infrequent marks — has its measured volatility understated and its annualised ratio inflated, which is the Measuring your return subject's illiquidity point arriving through the scaling instead of the level.

Misquote 3: reading a historical ratio as a prediction

Sharpe's own warning is blunter than most of its users:

Certainly, the use of unadjusted historic (ex post) Sharpe Ratios as surrogates for unbiased predictions of ex ante ratios is subject to serious question.

The ratio is a decision tool defined on expectations. The number on a factsheet is a sample statistic from a past window, and chapter 5's estimation-error argument applies in full — the mean in the numerator is the least reliably estimated quantity in finance.

A useful benchmark for how big these numbers actually are, from the same paper: a broad developed stock market with a 6% mean annual excess return and 15% standard deviation has a Sharpe ratio of 0.40.

Hold that 0.40 next to anything advertised. Ratios well above it, sustained, are rare enough that the first question should be what is smoothing the denominator.

Misquote 4: using it when the holding is not your whole risky portfolio

This is the subtlest and the most common, and Sharpe devotes his summary to it:

Whatever the application, it is essential to remember that the Sharpe Ratio does not take correlations into account. When a choice may affect important correlations with other assets in an investor's portfolio, such information should be used to supplement comparisons based on Sharpe Ratios.

The decision rule quoted at the top of this chapter holds "since the remaining holdings are riskless." If your other holdings are not riskless — if you already own equities and are choosing what to add — then the correct criterion is the effect on the whole portfolio's risk, which depends on correlation, and the ratio does not contain it.

The practical form of the error. Ranking five funds by Sharpe ratio and buying the top one is right if that fund will be your entire risky portfolio. It is wrong if you will hold all five, because the best addition to a portfolio is often not the best stand-alone holding — chapter 3's whole argument.

Sharpe allows the common case with an explicit condition: choosing among funds by predicted ratio makes sense "as long as the correlations of the funds with other relevant asset classes are reasonably similar."

Sortino, and the other variants

The Sortino ratio replaces the denominator with downside deviation — the standard deviation computed only over returns below a chosen threshold:

$\text{Sortino} = \frac{\mu_P - T}{\sigma_{\text{downside}}}$

It fixes a real defect. Variance counts an unexpectedly good year as risk, and nobody experiences a gain as risk.

And it introduces two of its own. The threshold is a free choice that changes the answer and is rarely disclosed, and discarding the upside observations roughly halves the data used to estimate the denominator, so the estimate is noisier than the Sharpe ratio's from the same sample. A measure that is conceptually better and statistically worse is a genuine trade, not an upgrade.

The information ratio uses excess return over a benchmark rather than over cash, and measures it against tracking error. It is the right tool for judging an active manager against the index they are paid to beat, and the wrong one for deciding how much of something to own. Sharpe notes the terminology here is genuinely confused in the literature, with the same name used for different constructs.

The Treynor ratio divides excess return by beta instead of volatility. That makes it the correct measure precisely when misquote 4 bites — when the holding is one part of a diversified portfolio, so its contribution is measured by beta rather than by total risk.

Which to use is decided by one question: is this holding my whole risky portfolio, or an addition to it? Sharpe for the first, Treynor for the second, information ratio for judging a manager, Sortino when the distribution is badly asymmetric and you have enough data to afford the noise.

The Indian numbers

Five years to 30 September 2026, with the policy repo rate of 5.50% standing in for the riskless rate:

	Return	Std dev	Sharpe ratio
Nifty 50	6.38%	13.82%	0.064
Nifty200 Momentum 30	9.28%	19.44%	0.194
Nifty200 Quality 30	6.40%	13.54%	0.066

Every one of these is far below Sharpe's 0.40 benchmark, because the window contained a poor five years for Indian equities — 6.38% against a 5.50% policy rate.

The ranking is robust to the risk-free assumption, which is worth checking rather than assuming:

Risk-free assumed	Nifty 50	Momentum 30	Quality 30
5.0%	0.100	0.220	0.103
5.5%	0.064	0.194	0.066
6.0%	0.027	0.169	0.030
6.5%	−0.009	0.143	−0.007

Momentum ranks first at every assumption, and the other two are indistinguishable from each other throughout. The ordering is a conclusion; the levels are not, since they swing by a factor of three across a plausible range of one input.

The honest reading. Momentum earned more excess return per unit of volatility over this window. Chapter 5's standard-error argument says a five-year sample cannot establish that it will continue to, and chapter 10 is about how seriously to take findings of exactly this shape.

Working the problem

X: 5% and 10%. Y: 8% and 20%. Riskless 3%.

Step 1 — what the two measures say. Return over volatility gives X 0.50 and Y 0.40, favouring X. Sharpe ratios give X $(5-3)/10 = 0.20$ and Y $(8-3)/20 = 0.25$, favouring Y.

Step 2 — build the comparison at equal risk. The investor wants 10% volatility.

With X: hold it outright. Volatility 10%, expected return 5.0%.

With Y: hold half in Y and half in cash. Volatility is $0.5 \times 20\% = 10\%$, and the expected return is

$0.5 \times 8\% + 0.5 \times 3\% = 5.5\%$

Step 3 — the verdict. At identical 10% volatility, the Y-and-cash strategy returns 5.5% against X's 5.0%. Sharpe's conclusion: the ratio "provides the correct answer (a strategy using Y is preferred to one using X), while the 'return information ratio' provides the wrong one."

Step 4 — which step the naive measure got wrong. It assumed the funds had to be held as they are. Once cash is available, any fund can be scaled to any risk level, and what matters is the rate at which risk converts into excess return — the slope — not the fund's own coordinates.

Y's higher volatility is not a cost here; it is simply a scale. Dividing it by two costs nothing, because de-levering with cash is free. What cannot be fixed by scaling is a poor ratio of excess return to risk, and that is the only thing the Sharpe ratio measures.

Step 5 — when the naive measure's answer would have been right. If the investor could not hold cash, could not scale the position, and had to take one fund whole, then X at 10% volatility and Y at 20% are genuinely different propositions and the comparison is a different problem. The Sharpe ratio's superiority depends on the scalability assumption — the same assumption that gives chapter 6 its straight line, and the same one that fails above the tangency point when borrowing is not available at the riskless rate.

The point

The Sharpe ratio is the slope of the capital market line, so maximising it is portfolio choice rather than a scorecard bolted onto it. It is misquoted in four ways: dropping the risk-free subtraction, which reverses Sharpe's own X-versus-Y ranking; comparing periods, when $S_T = S_1\sqrt{T}$ makes a monthly and an annual figure differ by 3.46 times; reading a historical ratio as a prediction, which its author calls subject to serious question; and applying it to a holding that is one part of a larger risky portfolio, where it omits the correlation that decides the answer. Sortino fixes the upside-as-risk defect and pays for it in estimation noise; Treynor is the right measure when the holding is an addition rather than the whole. A developed stock market's ratio is about 0.40, and the Nifty 50's over the five years to September 2026 was 0.064.