Benchmark-relative performance and paired risk

What you will learn

Derive benchmark-relative reward-to-variability ratios from paired state differences, construct a benchmark-driven ranking reversal and a fixed-marginal coupling witness, and diagnose scaling and zero-denominator limits.

Prerequisites: Linked returns and external cash flows · Portfolio covariance and minimum variance

Declare what performance is relative to

An account return answers how much one invested unit changes over a specified interval. A performance comparison asks what that unit earned relative to an alternative. Those questions need different random variables. Let R be a strategy return and C the return on a declared benchmark in the same state and over the same period. Define the differential return D=R−C. Its mean measures expected added return against that alternative. Its standard deviation measures uncertainty in that difference. Dividing the mean by that standard deviation gives a dimensionless reward-to-variability ratio when the denominator is positive. Against a riskless cash return, this is commonly called a Sharpe ratio. Against a risky target, mean active return divided by tracking-error volatility is commonly called an information ratio. Naming conventions do not change the need to subtract paired returns before computing dispersion. A ratio without a benchmark definition is an incomplete statement.

Assume one period, four equally probable population states, simple returns measured in percentage points, and identical state labels for all strategies and benchmarks. There are no investor contributions, withdrawals, fees or intraperiod trades. Strategies A=(0,2,4,6) and B=(1,2,3,4) are stipulated return laws, not estimated observations or recommendations. Cash C=(1,1,1,1) is riskless within this model. A second benchmark H=(0,1,3,4) is risky. These alternatives serve different evaluation mandates. Changing the benchmark changes the question being asked; it cannot establish that a manager acquired skill.

Derive differential risk before taking a ratio

Write muR=E[R] and muC=E[C]. Centering the difference gives D−E[D]=(R−muR)−(C−muC). Squaring, averaging and expanding produces Var(D)=Var(R)+Var(C)−2Cov(R,C). The covariance uses simultaneous state pairs. The denominator is sqrt(Var(D)), rather than Var(D), and its units are percentage points. Thus S=E[D]/sqrt(Var(D)) has no units. If C is constant, Var(C)=0 and Cov(R,C)=0, so differential volatility equals strategy volatility. This special simplification does not survive an arbitrary risky benchmark.

The numerator obeys E[D]=E[R]−E[C] without independence. The denominator cannot generally be computed from two separate marginal distributions. Reassigning the same benchmark returns to different strategy states preserves both individual variances but changes their covariance. This is why sorting two historical return series separately before subtracting is invalid: it invents a new joint experiment. We use a population divisor of four because the four states exhaust the assumed law. An empirical sample standard deviation with divisor n−1 belongs to a different estimation convention, which would need to be stated explicitly.

Work a benchmark-driven ranking reversal

For A, the mean is 3 and deviations are (−3,−1,1,3), so variance is (9+1+1+9)/4=5. For B, the mean is 5/2 and deviations are (−3/2,−1/2,1/2,3/2), so variance is (9/4+1/4+1/4+9/4)/4=5/4. Relative to cash at one, A has differential mean 2 and ratio 2/sqrt(5), about 0.894427. B has differential mean 3/2 and ratio (3/2)/(sqrt(5)/2)=3/sqrt(5), about 1.341641. B therefore has the larger cash-relative ratio despite A having the larger raw mean.

Against H, A−H=(0,1,1,2), with mean 1 and variance (1+0+0+1)/4=1/2. Its ratio is sqrt(2), about 1.414214. B−H=(1,1,0,0), with mean 1/2 and variance 1/4, giving ratio 1. The ranking reverses. Independently, H has mean 2, variance 5/2, Cov(A,H)=7/2 and Cov(B,H)=7/4. Expansion gives A differential variance 5+5/2−7=1/2 and B differential variance 5/4+5/2−7/2=1/4. These two calculations check the pairing and the factor of two. Using raw strategy volatility would miss both denominator changes.

Construct a coupling witness with fixed marginal laws

Now exchange the benchmark returns attached to states one and four, leaving every strategy payoff untouched. The benchmark becomes Hprime=(4,1,3,0). It still has mean 2 and variance 5/2. A−Hprime=(−4,1,1,6), with mean 1, deviations (−5,0,0,5) and variance 25/2. B−Hprime=(−3,1,0,4), with mean 1/2, deviations (−7/2,1/2,−1/2,7/2) and variance 25/4. The resulting ratios are sqrt(2)/5, about 0.282843, and 1/5. A still leads in this particular rearrangement, but the magnitudes collapse even though every marginal law is unchanged. A coupling edit need not reverse a ranking to refute a marginal-only calculation.

In the workshop, choose cash or the risky benchmark, then swap adjacent benchmark-state assignments. Adjacent swaps can construct every permutation of the four risky outcomes. Compute both differential means and both differential variances before choosing the larger ratio. The audit reports ratios only after these intermediate quantities and the comparison are correct. Save a cash certificate, an original risky certificate and a rearranged certificate to compare the claims side by side. Edits clear answers; saved certificates retain their original state pairing. The construction is visit-local practice and does not certify investment competence.

Respect scaling, zero risk and horizon limits

If a differential payoff D is multiplied by positive k, its mean becomes kE[D] and its standard deviation becomes k sqrt(Var(D)), so the ratio is unchanged. This is scaling of the entire differential payoff, not multiplication of just R while leaving an unrelated benchmark unchanged. At k=0 both numerator and denominator vanish and the ratio is undefined. Negative scaling reverses the mean while multiplying standard deviation by the absolute value of k, so the ratio changes sign. A constant positive differential payoff also has zero differential volatility; do not silently turn division by zero into a finite ranking.

All ratios here describe one-period population laws. Historical estimates can be noisy, and the highest observed ratio among many searched strategies can reflect selection. Multiplying by the square root of periods per year requires assumptions about serial covariance and how returns are aggregated; none are supplied here. The ratio is not a compounded growth rate or a probability of winning. In particular, a value above one need not mean a return above 100%; it compares a mean and a standard deviation in the same units. A larger ratio alone also does not choose an allocation alongside a risky existing portfolio, where additional correlations matter. Report the benchmark, joint state model, period, population convention and positive-denominator condition with the number. Fees, leverage feasibility, tail exposure and the credibility of estimated inputs require separate analysis.

Hands-on: Construct a paired benchmark certificate

Reassign benchmark returns across simultaneous states while preserving its marginal law. Certify both differential means, variances and their ratio comparison; retain contrasting constructions.

Common mistakes

• Using raw mean as added return or raw volatility with a risky benchmark.
• Subtracting standard deviations or omitting twice the covariance.
• Sorting return columns independently before subtracting.
• Confusing a variance denominator with a standard deviation denominator.
• Reporting a finite ratio at zero differential risk or annualizing without serial assumptions.

Keep these ideas

• Subtract a declared benchmark within each simultaneous state.
• Average differential returns and square centered differential deviations.
• Check differential variance independently with both marginal variances and covariance.
• Report benchmark and horizon; treat zero risk, signed scaling and historical uncertainty explicitly.

Further reading

• William F. Sharpe: The Sharpe Ratio (1994) ↗

  Read Definitions (ex ante and ex post), Time Dependence, scale independence discussion, and Summary on the author’s Stanford page. Supports differential-return definition, horizon and correlation caveats. Our four-state strategies, pairing manipulations, ranking reversal, arithmetic and questions are original. Population law rather than empirical estimate; no video or whole-paper review claimed. Checked 2026-10-25.