Sharpe, Information Ratio, or Treynor — which should I use?

Sharpe measures total return per unit of total risk and is right when you can hold the strategy as a stand-alone investment. Information Ratio measures excess return over a benchmark per unit of tracking error and is right for active managers benchmarked to an index. Treynor measures excess return per unit of systematic (market) risk and is right when the position is one piece of a well-diversified portfolio.

What is the formula for each?

Sharpe = (r_p − r_f) / σ_p, where r_p is portfolio return, r_f is risk-free rate, σ_p is total volatility. Information Ratio = (r_p − r_b) / σ_(p−b), where r_b is benchmark return and σ_(p−b) is tracking error (std of the active return series). Treynor = (r_p − r_f) / β_p, where β_p is the portfolio beta to the market.

When does Sharpe lie about an active manager?

When the manager is benchmarked to an index but holds positions correlated with it. A long-only equity manager who runs at 95% beta to S&P 500 will have a Sharpe similar to the index itself even if they add zero alpha — the index volatility dominates the manager volatility. Information Ratio strips out the benchmark exposure and tells you whether the active bets are adding value.

When does Sharpe lie about a position in a portfolio?

When the position has high idiosyncratic volatility but low market correlation. A single biotech stock might have Sharpe = 0.3 because of high total vol, but a diversified portfolio holding it as 2% of NAV doesn't see most of that idiosyncratic risk — it averages out. Treynor uses only beta, which captures the part of the position's risk that doesn't diversify away.

What is a "good" value for each ratio?

Sharpe: 1.0 is good, 2.0 is excellent, 3.0+ is suspicious and likely overfit. Information Ratio: 0.5 is good for an active equity manager, 0.75 is strong, 1.0+ is exceptional and rare. Treynor: harder to anchor without market context; in equilibrium Treynor of every asset should equal the market Treynor.

Can a strategy have great Sharpe but bad Information Ratio?

Yes, easily. A buy-and-hold S&P 500 fund has Sharpe near 0.5 (matching the market) but Information Ratio near zero — by construction it doesn't deviate from its benchmark, so there's no active return to scale. Anyone charging active-management fees for that profile is selling index exposure at active prices.

What is "tracking error" exactly?

Tracking error is the standard deviation of the active return series — the daily / monthly differences between portfolio return and benchmark return. An index fund has tracking error under 0.5% per year. A core active fund: 2-4%. An aggressive active fund: 5-10%. A long/short hedge fund vs cash: 10-20%. Higher tracking error means more concentrated active bets, not necessarily worse — IR normalizes for this.

How do I compute beta for the Treynor ratio?

Run a simple linear regression: portfolio_excess_return = α + β · market_excess_return + ε. Beta is the slope. With daily data over a year (~252 obs) the standard error is meaningful — point estimates wobble by 0.1-0.2 just from sampling noise. Use rolling windows or shrinkage estimators for production.

What about Sortino, Calmar, and the other ratios?

Sortino is Sharpe but divides by downside deviation instead of total volatility. Calmar is annualized return divided by max drawdown. These complement Sharpe rather than replace it. See the separate Sharpe vs Sortino vs Calmar comparison for that detail.

Which QuantOracle calculators do these?

Sharpe Ratio Calculator does standalone Sharpe with confidence intervals. The full Probabilistic Sharpe Ratio calculator extends to "is the Sharpe estimate statistically significant given skew/kurtosis/sample size." Information Ratio and Treynor are computable via the API but no dedicated calculator yet — request via GitHub.

Sharpe vs Information Ratio vs Treynor: Three Risk-Adjusted Return Metrics

The 30-second decision rule

"Is this strategy worth holding by itself?" → Sharpe. Total return per unit of total volatility.
"Is this active manager beating their benchmark efficiently?" → Information Ratio. Benchmark-relative return per unit of tracking error.
"Is this position adding value in a diversified portfolio?" → Treynor. Excess return per unit of beta (systematic risk only).

Side-by-side

Metric	Sharpe	Information Ratio	Treynor
Numerator	r_p − r_f	r_p − r_b	r_p − r_f
Denominator	σ_p (total vol)	σ(r_p − r_b) (tracking error)	β_p (beta)
Risk being penalized	All volatility	Active-bet volatility	Systematic only
Right when…	Held stand-alone	Compared to benchmark	Part of diversified portfolio
"Good" value	1.0+ (2.0 excellent)	0.5+ (1.0 exceptional)	Match the market's
Penalty for benchmark-hugging	None — could match index	Severe — near zero	None directly
Penalty for idiosyncratic risk	Full	Full	None (assumed diversified away)

Sharpe ratio — the universal default

The Sharpe ratio (William Sharpe, 1966) is the most widely-used risk-adjusted return metric. The formula:

Sharpe = (r_p - r_f) / σ_p

r_p = portfolio return (annualized)
r_f = risk-free rate (typically 3-month T-bill or SOFR)
σ_p = standard deviation of portfolio returns (annualized)

The numerator measures excess return — how much you earned above just sitting in cash. The denominator measures total volatility — every wiggle of your equity curve, whether it came from your own positioning or just being long the market.

Sharpe is the right metric when you're evaluating a strategy as a stand-alone investment. The classic use cases: a hedge fund LP comparing two managers offered as replacements for cash, a retail investor picking between two ETFs, a quant comparing backtested strategies in isolation.

Where Sharpe lies: it cannot tell whether a manager is beating their benchmark by skill or simply holding the benchmark with leverage. A long-only equity manager at 95% beta to S&P 500 will have almost exactly the index's Sharpe, because the index volatility dominates the manager volatility. The Sharpe ratio doesn't care where the volatility comes from.

Information Ratio — the active-management metric

Information Ratio (IR) is what allocators actually use to evaluate active managers:

IR = (r_p - r_b) / σ(r_p - r_b)

r_p − r_b = active return (excess over benchmark, annualized)
σ(r_p − r_b) = tracking error — the std deviation of the active return series (annualized)

The crucial move: instead of comparing to cash, IR compares the portfolio to its declared benchmark. Instead of penalizing total volatility, IR penalizes only the volatility of the active bets. A manager who runs 100% S&P 500 with no deviations has an IR of zero — there is no active return to scale, no matter how high their absolute Sharpe.

This is exactly the question an LP cares about: "If I'm paying for active management, is the manager actually doing active management efficiently?" A manager with IR = 0.5 is genuinely adding alpha to the benchmark. A manager with the same Sharpe as the benchmark but IR = 0.05 is selling index exposure at active prices.

Where IR lies: it assumes the benchmark is the right reference. A small-cap manager benchmarked to large-cap will look brilliant on IR for cap reasons, not skill. A long/short fund benchmarked to cash will look brilliant because tracking error is effectively just the fund's standalone volatility. Always check what benchmark IR is computed against, and whether that benchmark genuinely represents the investable alternative.

Treynor ratio — the diversified-portfolio metric

The Treynor ratio (Jack Treynor, 1965) scales excess return by beta instead of total volatility:

Treynor = (r_p - r_f) / β_p

Where β_p is the portfolio's beta to the market — the slope from regressing portfolio excess returns on market excess returns. Beta measures only systematic risk; the idiosyncratic part is assumed to be diversified away in the larger portfolio.

The intuition: if you're adding a position to an already-diversified portfolio, its standalone volatility doesn't matter — only the part that survives diversification (the beta exposure) actually contributes risk to your combined book. A single biotech stock might look terrible on Sharpe because of high idiosyncratic volatility, but be a perfectly reasonable 2% portfolio position if its beta is moderate.

Treynor is the CAPM-native metric. In equilibrium, every asset's Treynor ratio should equal the market's Treynor ratio; deviations represent alpha. It's the standard ratio in risk-budgeting frameworks at institutional asset managers and insurance company general accounts where everything is held in a much bigger pool.

Where Treynor lies: when the position isn't actually being held in a diversified portfolio. A retail investor putting 50% of their net worth into one tech stock cannot count on idiosyncratic risk diversifying away — for them, Sharpe is the right metric. Also: beta estimates are noisy (point estimate ±0.1-0.2 from sampling noise alone), so small differences in Treynor ratios should not drive allocation decisions.

Worked example: the same strategy under three lenses

Consider a long-only US equity manager with these annualized statistics:

Portfolio return: 12%
Benchmark (S&P 500) return: 10%
Risk-free rate: 4%
Portfolio volatility: 16%
Tracking error: 4%
Beta to S&P 500: 1.05

Sharpe = (12% − 4%) / 16% = 0.50 — looks comparable to the S&P 500 itself (also around 0.4-0.5 historically).
Information Ratio = (12% − 10%) / 4% = 0.50 — this is genuine, sustained alpha for an equity manager. LPs would consider this very good.
Treynor = (12% − 4%) / 1.05 = 7.6% per unit of beta. The market's Treynor is (10% − 4%) / 1.0 = 6.0%, so this manager adds about 1.6% per unit of beta over the market — that's alpha.

Same strategy. Three numbers. Sharpe says "okay, similar to the index." IR says "genuinely good active manager." Treynor says "adding alpha per unit of systematic risk." All three are technically correct; they just answer different questions about the same returns.

Which one does an allocator actually use?

In practice, allocators compute all three and check that they tell a consistent story. A manager with high Sharpe but low IR is selling index exposure; the conversation gets harder. A manager with low Sharpe but high IR is a high-tracking- error strategy that still adds genuine value relative to benchmark — the IR justifies the fees.

The honest practitioner reports all three. Mutual fund prospectuses are required to report Sharpe but rarely do IR voluntarily because most active managers don't beat their benchmarks on a tracking-error-adjusted basis. Hedge funds quote both because they want to show absolute performance (Sharpe) and skill-relative-to- benchmark (IR). Pension funds compute Treynor for their factor-based portfolios.

Sharpe Ratio Calculator — standalone Sharpe with Lo (2002) confidence intervals
Probabilistic Sharpe Ratio Calculator — is the observed Sharpe statistically significant given skew + kurtosis?
Sharpe vs Sortino vs Calmar — three risk-adjusted metrics that emphasize different aspects of risk
Value at Risk Calculator — parametric and historical VaR for the same return series
Drawdown Calculator — the metric investors actually feel

References

Sharpe, W. F. (1966). "Mutual fund performance." Journal of Business 39(1), 119-138. — original Sharpe ratio paper.
Treynor, J. L. (1965). "How to rate management of investment funds." Harvard Business Review 43(1), 63-75.
Goodwin, T. H. (1998). "The Information Ratio." Financial Analysts Journal 54(4), 34-43. — IR as the right metric for active management.
Grinold, R. C. & Kahn, R. N. (1999). "Active Portfolio Management." McGraw-Hill. — the bible of information-ratio-based active management.
Lo, A. W. (2002). "The statistics of Sharpe ratios." Financial Analysts Journal 58(4), 36-52. — sampling distribution of Sharpe estimates.