Methodology

One methodology, applied to every investment

OpenFactsheet uses the same published rules for every investment, so two reports can be compared line by line. This page explains those rules, first in plain words and then in full technical detail.

Browse all 97 measures Try it on an example

In plain words

The same rules for every investment. Nobody, including us, can pick the period or the measures that flatter a result.
The data is checked first. Gaps, repeated values and suspiciously smooth returns are flagged before anything is measured.
Every result says how much to trust it, with a confidence label and the reason behind it.
Nothing is hidden. Every measure is always calculated, and one that can't be calculated is still listed, with the reason.

Technically: every number follows OpenFactsheet Methodology 0.1.0. The details are below.

The rules

1.
Every metric, every time.
All 97 metrics are computed for every report. There are no switches to hide a metric, change a window or pick a confidence level.
2.
Data is checked before it is measured.
Missing months, duplicate dates, stale or smoothed returns and extreme values are detected and disclosed. Nothing is silently filled or removed.
3.
Every result says how much to trust it.
Each metric carries a confidence label and the warnings that apply to it, based on the length and quality of the data.
4.
Every result can be traced.
Each metric has a formula identifier and a methodology version. When a convention changes, the version changes, and old reports stay reproducible.

Confidence labels

A metric can be computed and still not mean much. Confidence describes the reliability of an estimate, not the quality of the investment. Every downgrade comes with a warning that says why.

High confidence: The history meets this metric's minimum-data policy and no diagnostic reduces it.
Moderate confidence: Usable with care. The history is shorter than the policy's full threshold, or a diagnostic caps it: moderate smoothing, many zero-return months, missing periods, or a benchmark covering under 90% of the fund's months.
Low confidence: Treat with caution. The history is too short for this kind of estimate, a partial year is being annualized, or high smoothing makes volatility-based figures unreliable.
Not assessable: The metric cannot be computed for this data, for example a ratio whose denominator is zero. The reason is shown.

Key conventions

Methodology version 0.1.0. The reasoning behind each choice is recorded in the library's decision records.

Risk-free rate: 0%. Sharpe-type ratios are therefore total-return ratios, which reads higher than an excess-return version when cash pays interest. Every affected metric says so.

Risk-free rate: 0% per period in methodology 0.1.0 for all excess-return metrics (Sharpe, Sortino, Calmar, Martin, alpha, Treynor). Ratios are therefore total-return ratios; this is disclosed on every affected metric.
Annual return: geometric (CAGR): (prod(1 + r))^(periods_per_year / n) - 1; the arithmetic mean x periods_per_year is reported separately
Annualization: square-root-of-time for volatility; linear for arithmetic means. Monthly data uses 12 periods per year.
Volatility: sample standard deviation (n - 1 denominator)
Downside risk: target (MAR) = 0% per period; downside deviation = sqrt(sum(min(r - MAR, 0)^2) / n) using all n observations
Drawdowns: wealth index starts at 1 before the first return, so a first-period loss is a drawdown; drawdowns reported as signed returns (negative = loss)
Tail risk: historical 95% VaR = 5% empirical quantile (linear interpolation, R type 7); 95% ES = mean of returns at or below that quantile; both reported as signed returns (negative = loss)
Missing data: missing returns are never replaced by zero or forward-filled; leading and trailing gaps are trimmed, interior gaps are disclosed and the remaining observations are analyzed
Outliers: robust z-score |r - median| / (1.4826 * MAD) above 5 is flagged (never removed)
Benchmark alignment: intersection of periods; never filled

What we measure

97 measures in 11 families. Select a family to jump to its measures in the catalog.

Returns11How much the investment grew, measured several ways.
Volatility and downside7How much returns swung around, and how much of that swing was downward.
Drawdowns15How far it fell from a previous high, and how long recovery took.
Risk-adjusted performance12Return earned per unit of risk taken. The ratios most often quoted in pitches.
Tail risk7What the bad months look like: losses that are rare but large.
Shape of returns3Whether returns follow a bell curve or hide lopsided surprises.
Consistency10How often it made money, by month, by year and over rolling twelve months.
Serial dependence6Whether one month predicts the next, a sign that returns may be smoothed.
Versus benchmark18Behaviour relative to a benchmark you supply: beta, alpha, capture ratios.
Factor exposure4Exposure to common market factors, when factor data is supplied.
Statistical confidence4How much the track record can really tell you, given its length.

Metric catalog

What each of the 97 metrics measures, in plain English and in formula form, with the minimum data it needs and the literature it comes from. 6 of them are supplementary: they add context, such as parametric alternatives to a historical estimate, rather than a headline figure.

97 metrics

Returns

How much the investment grew, measured several ways.

Cumulative returnHHigher is better
How much 1 unit invested at the start would have gained or lost by the end, with all returns reinvested.
Path-independent summary of realized compounded performance over the full track record; not comparable across different history lengths.
Formula
prod_{t=1..n}(1 + r_t) - 1
Convention
Geometric compounding of simple returns; no annualization.
Minimum data
Observed fact about the supplied history; any non-empty history suffices.
References
Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
cumulative_return@v1
Annualized return (CAGR)HHigher is better
The average yearly growth rate, taking compounding and losses into account.
Geometric mean return annualized with the canonical periods-per-year factor; the standard GIPS-style annualized return. Always below the annualized arithmetic mean when returns vary.
Formula
(prod(1 + r_t))^(P / n) - 1, where P = periods per year and n = observations
Convention
Observation count (not calendar span) with the canonical periods-per-year factor (decision 0004).
Minimum data
Mean-type estimates: low confidence below 3 years, moderate below 5 years.
References
Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.; CFA Institute (2020). Global Investment Performance Standards (GIPS) for Firms, provisions on annualized returns.
Identifier
cagr@v1
Mean return per periodHHigher is better
The typical return in one period (e.g. one month), ignoring compounding.
Sample mean; the numerator of Sharpe-type ratios. An estimate of the expected return with standard error roughly sigma / sqrt(n).
Formula
(1/n) * sum(r_t)
Convention
Plain sample mean of per-period returns.
Minimum data
Mean-type estimates: low confidence below 3 years, moderate below 5 years.
References
Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
arithmetic_mean@v1
Geometric mean return per periodHHigher is better
The average compounded return per period (e.g. per month).
Per-period geometric mean; CAGR is its annualized form. Lies below the arithmetic mean by roughly half the variance (volatility drag). Needs no frequency.
Formula
(prod(1 + r_t))^(1/n) - 1
Convention
Geometric compounding of simple returns; -100% after a total loss.
Minimum data
Mean-type estimates: low confidence below 3 years, moderate below 5 years.
References
Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
geometric_mean_return@v1
Annualized mean return (arithmetic)HHigher is better
The average return per year if yearly results were simply added up; usually higher than the compounded annual return.
Linear annualization of the sample mean; the return component of the canonical annualized Sharpe ratio. Differs from CAGR by roughly half the variance.
Formula
P * mean(r_t)
Convention
mean x periods per year (decision 0004).
Minimum data
Mean-type estimates: low confidence below 3 years, moderate below 5 years.
References
Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
annualized_arithmetic_mean@v1
Median return per periodHHigher is better
Half of the periods did better than this and half did worse.
Robust location estimate, insensitive to a few extreme periods. A median well above the mean indicates occasional large losses (negative skew).
Formula
median(r_t)
Convention
Sample median (average of the two middle values when n is even).
Minimum data
Mean-type estimates: low confidence below 3 years, moderate below 5 years.
References
Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
median_return@v1
Best periodHHigher is better
The best single period (e.g. month) the investment had.
Observed upper extreme; context for the dispersion and tail statistics.
Formula
max(r_t)
Convention
Per observation period of the supplied data (not calendar aggregated).
Minimum data
Observed fact about the supplied history; any non-empty history suffices.
References
Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
best_period@v1
Worst periodHHigher is better
The worst single period (e.g. month) the investment had.
Observed lower extreme; a model-free loss statistic that is always shown.
Formula
min(r_t)
Convention
Per observation period; signed (negative = loss).
Minimum data
Observed fact about the supplied history; any non-empty history suffices.
References
Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
worst_period@v1
Gain-to-pain ratioHHigher is better
How much net return was earned for each unit of losses suffered along the way.
Schwager's gain-to-pain ratio on per-period returns; a simple, distribution-free complement to volatility-based ratios.
Formula
sum(r_t) / |sum(r_t for r_t < 0)|
Convention
Computed on the supplied observation frequency; undefined (not applicable) when no period is negative.
Minimum data
Risk-adjusted ratios: at least 2 observations; low confidence below 3 years, moderate below 5 years.
References
Schwager, J. D. (2012). Market Sense and Nonsense. Wiley (Gain-to-Pain ratio).
Identifier
gain_to_pain_ratio@v1
Payoff ratio (average win / average loss)HHigher is better
How big the typical gain is compared with the typical loss. Read it together with the share of positive periods: a low payoff with a high hit rate can still be profitable, and vice versa.
Win/loss (payoff) ratio on the supplied frequency; zero-return periods are excluded from both averages. The expectation per period equals hit rate x average win - loss rate x average loss.
Formula
mean(r_t | r_t > 0) / |mean(r_t | r_t < 0)|
Convention
Arithmetic averages of strictly positive and strictly negative periods.
Minimum data
Risk-adjusted ratios: at least 2 observations; low confidence below 3 years, moderate below 5 years.
References
Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
payoff_ratio@v1
Profit factor (gross gains / gross losses)HHigher is better
How many units of gains the history produced for every unit of losses; above 1 means total gains exceeded total losses.
Gross gains over gross losses on per-period returns (a trading-system statistic). It equals one plus the gain-to-pain ratio, so it adds no information beyond it; reported because it is widely quoted. Not compounded and frequency dependent.
Formula
sum(r_t | r_t > 0) / |sum(r_t | r_t < 0)|
Convention
Computed on the supplied observation frequency; zero returns count in neither sum (decision 0029).
Minimum data
Risk-adjusted ratios: at least 2 observations; low confidence below 3 years, moderate below 5 years.
References
Pardo, R. (2008). The Evaluation and Optimization of Trading Strategies (2nd ed.). Wiley (profit factor: gross profit / gross loss).; Schwager, J. D. (2012). Market Sense and Nonsense. Wiley (Gain-to-Pain ratio).
Identifier
profit_factor@v1

Volatility and downside

How much returns swung around, and how much of that swing was downward.

Volatility per periodLLower is better
How much returns typically swing from one period to the next; higher means a bumpier ride.
Sample standard deviation of per-period returns; symmetric, treats upside and downside deviations alike.
Formula
sqrt( sum((r_t - mean)^2) / (n - 1) )
Convention
Sample standard deviation (n - 1), decision 0003.
Minimum data
Requires at least 2 observations; low confidence below 3 years, moderate below 5.
References
Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
volatility@v1
Annualized volatilityLLower is better
The typical size of yearly swings implied by the period-to-period variability.
Square-root-of-time scaling assumes serially uncorrelated returns; positive autocorrelation makes it understate true annual volatility (Lo, 2002).
Formula
std(r_t, ddof=1) * sqrt(P)
Convention
Sample standard deviation x sqrt(periods per year).
Minimum data
Requires at least 2 observations; low confidence below 3 years, moderate below 5.
References
Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.; Lo, A. W. (2002). The Statistics of Sharpe Ratios. Financial Analysts Journal, 58(4), 36-52.
Identifier
annualized_volatility@v1
Mean absolute deviationLLower is better
The typical size of a period's deviation from the average, without squaring (so single extreme periods weigh less than in volatility).
Mean absolute deviation around the mean (the L1 dispersion used in Konno-Yamazaki portfolio models); for normal returns MAD = sqrt(2/pi) sigma. Per period.
Formula
(1/n) * sum |r_t - mean|
Convention
Deviation from the sample mean, denominator n, per period.
Minimum data
Requires at least 2 observations; low confidence below 3 years, moderate below 5.
References
Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
mean_absolute_deviation@v1
Interquartile rangeLLower is better
How far apart typical good and typical bad periods are, ignoring the extremes.
Robust dispersion measure, insensitive to the tails (for normal returns IQR = 1.349 sigma). Contrast with volatility to see how much the tails drive risk. Per period.
Formula
Q_0.75(r) - Q_0.25(r), linear interpolation
Convention
Linear quantiles, per period (decision 0025).
Minimum data
Requires at least 2 observations; low confidence below 3 years, moderate below 5.
References
Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
interquartile_range@v1
Downside deviation (annualized)LLower is better
How large the losing periods typically are; gains do not count as risk here.
Lower partial moment of order 2 (root), full-sample denominator as in Sortino & Price (1994); the Sortino ratio denominator.
Formula
sqrt( (1/n) * sum(min(r_t - MAR, 0)^2) ) * sqrt(P), MAR = 0
Convention
MAR = 0; denominator n (all observations), not the number of losing periods (decision 0006).
Minimum data
Requires at least 2 observations; low confidence below 3 years, moderate below 5.
References
Sortino, F. A., & Price, L. N. (1994). Performance Measurement in a Downside Risk Framework. Journal of Investing, 3(3), 59-64.; Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
downside_deviation@v1
Semi-deviation (annualized)LLower is better
How large the below-average periods typically are.
Downside deviation with the sample mean as target (Markowitz semi-variance, square root), full-sample denominator. Unlike downside deviation it measures dispersion below the mean, not losses.
Formula
sqrt( (1/n) * sum over r_t < mean of (r_t - mean)^2 ) * sqrt(P)
Convention
Full-sample denominator n, target = mean, annualized by sqrt(P) (decision 0016).
Minimum data
Requires at least 2 observations; low confidence below 3 years, moderate below 5.
References
Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
semi_deviation@v1
Downside potential (LPM 1)LLower is better
The average loss per period, counting gains as zero.
First lower partial moment at target 0; the denominator of the Omega ratio (Omega = 1 + mean / LPM1 at threshold 0). Per period, not annualized.
Formula
(1/n) * sum max(tau - r_t, 0), tau = 0
Convention
Target 0, full-sample denominator n, per period (decision 0016).
Minimum data
Requires at least 2 observations; low confidence below 3 years, moderate below 5.
References
Sortino, F. A., & Price, L. N. (1994). Performance Measurement in a Downside Risk Framework. Journal of Investing, 3(3), 59-64.; Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
downside_potential@v1

Drawdowns

How far it fell from a previous high, and how long recovery took.

Maximum drawdownHHigher is better
The worst loss an investor would have suffered buying at a high point and selling at the following low point.
Path-dependent realized loss; always shown. Expected maximum drawdown grows with history length and volatility (Magdon-Ismail & Atiya, 2004), so compare only across similar horizons.
Formula
min_t( W_t / max_{s<=t} W_s - 1 ), W_0 = 1, W_t = prod_{i<=t}(1 + r_i)
Convention
Signed (negative = loss); the inception value counts as a peak so a first-period loss is a drawdown (decision 0007).
Minimum data
Path statistics (drawdowns, streaks) are computable from one observation but are unrepresentative below 3 years (low) and 5 years (moderate); longer histories tend to reveal deeper drawdowns.
References
Magdon-Ismail, M., & Atiya, A. (2004). Maximum Drawdown. Risk, 17(10), 99-102.; Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
max_drawdown@v1
Average drawdown (per episode)HHigher is better
How deep the typical decline from a high point has been.
Episode-based average (PerformanceAnalytics AverageDrawdown convention); less dominated by a single event than maximum drawdown. Compare with the Pain Index, which averages over time instead of over episodes.
Formula
(1/K) * sum_k depth_k over K peak-to-recovery episodes
Convention
Mean of episode depths, signed (negative = loss); 0 when no drawdown occurred.
Minimum data
Path statistics (drawdowns, streaks) are computable from one observation but are unrepresentative below 3 years (low) and 5 years (moderate); longer histories tend to reveal deeper drawdowns.
References
Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
average_drawdown@v1
Current drawdownHHigher is better
How far below its previous high the investment stands at the end of the data.
State variable at the reporting date; zero when the series ends at a high-water mark.
Formula
W_n / max_{s<=n} W_s - 1
Convention
Signed (negative = loss).
Minimum data
Observed fact about the supplied history; any non-empty history suffices.
References
Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
current_drawdown@v1
Longest drawdown durationLLower is better
The longest stretch an investor would have waited to get back to a previous high.
Peak-to-recovery duration in observation periods; an ongoing episode is a lower bound and is flagged as such.
Formula
max_k (recovery_k - peak_k) in periods; ongoing episodes use n - peak_k
Convention
Measured in observation periods from the peak to the recovery period inclusive; details report the equivalent in years when the frequency is regular.
Minimum data
Path statistics (drawdowns, streaks) are computable from one observation but are unrepresentative below 3 years (low) and 5 years (moderate); longer histories tend to reveal deeper drawdowns.
References
Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
longest_drawdown_duration@v1
Number of drawdown episodesNNeutral
How often the investment fell below a previous high.
Context for average drawdown and durations; depends strongly on data frequency and history length.
Formula
K = number of maximal runs with W_t below its running peak
Convention
An ongoing episode at the end of the data is counted.
Minimum data
Path statistics (drawdowns, streaks) are computable from one observation but are unrepresentative below 3 years (low) and 5 years (moderate); longer histories tend to reveal deeper drawdowns.
References
Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
drawdown_count@v1
Time under waterLLower is better
The share of the time the investment was below a previous high.
Complements depth measures with persistence; high values with shallow drawdowns indicate grinding rather than crash risk.
Formula
(1/n) * #{t : D_t < 0}
Convention
Counted over observations 1..n (the inception point is excluded).
Minimum data
Path statistics (drawdowns, streaks) are computable from one observation but are unrepresentative below 3 years (low) and 5 years (moderate); longer histories tend to reveal deeper drawdowns.
References
Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
time_underwater_pct@v1
Ulcer IndexLLower is better
A single number for how deep and how long-lasting the declines were; 0 means the investment never fell below a previous high.
Martin & McCann's Ulcer Index on the canonical drawdown path; the denominator of the Martin ratio (Ulcer Performance Index).
Formula
sqrt( (1/n) * sum_t D_t^2 )
Convention
Non-negative magnitude in decimal units, computed from the canonical drawdown path over observations 1..n.
Minimum data
Path statistics (drawdowns, streaks) are computable from one observation but are unrepresentative below 3 years (low) and 5 years (moderate); longer histories tend to reveal deeper drawdowns.
References
Martin, P. G., & McCann, B. B. (1989). The Investor's Guide to Fidelity Funds. Wiley (Ulcer Index and Ulcer Performance Index).; Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
ulcer_index@v1
Pain IndexLLower is better
How far below its previous high the investment was, on average, over the whole history.
Zephyr Pain Index (see Bacon); time-weighted, unlike the episode-based average drawdown.
Formula
(1/n) * sum_t |D_t|
Convention
Non-negative magnitude in decimal units over observations 1..n.
Minimum data
Path statistics (drawdowns, streaks) are computable from one observation but are unrepresentative below 3 years (low) and 5 years (moderate); longer histories tend to reveal deeper drawdowns.
References
Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
pain_index@v1
Median drawdown (per episode)HHigher is better
The depth of a typical decline from a previous high; unlike the average it is not pulled down by one extreme episode.
Robust counterpart of the episode-based average drawdown; many shallow episodes make it close to zero.
Formula
median_k depth_k over K episodes (0 if K = 0)
Convention
Median of episode depths, signed (decision 0025).
Minimum data
Path statistics (drawdowns, streaks) are computable from one observation but are unrepresentative below 3 years (low) and 5 years (moderate); longer histories tend to reveal deeper drawdowns.
References
Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
median_drawdown@v1
Average drawdown durationLLower is better
How long, on average, the investment took to get back to a previous high.
Episode-weighted average duration in observation periods; an open episode contributes a lower bound (flagged).
Formula
(1/K) * sum_k duration_k, duration as in longest_drawdown_duration (0 if K = 0)
Convention
Peak to recovery inclusive; open episodes measured to the last observation (decision 0025).
Minimum data
Path statistics (drawdowns, streaks) are computable from one observation but are unrepresentative below 3 years (low) and 5 years (moderate); longer histories tend to reveal deeper drawdowns.
References
Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
average_drawdown_duration@v1
Longest decline (peak to trough)LLower is better
The longest stretch during which losses kept deepening before the turnaround.
Maximum peak-to-trough length in observation periods; complements the total duration, which also includes the recovery.
Formula
max_k (trough_k - peak_k) in periods (0 if K = 0)
Convention
Periods from the peak to the trough (decision 0025).
Minimum data
Path statistics (drawdowns, streaks) are computable from one observation but are unrepresentative below 3 years (low) and 5 years (moderate); longer histories tend to reveal deeper drawdowns.
References
Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
max_peak_to_trough_duration@v1
Average recovery time (trough to recovery)LLower is better
Once the bottom was reached, how long it typically took to climb back to the previous high.
Average recovery length of completed episodes only; open episodes are excluded because their recovery time is unknown (disclosed in details).
Formula
mean_k (recovery_k - trough_k) over recovered episodes
Convention
Recovered episodes only (decision 0025).
Minimum data
Path statistics (drawdowns, streaks) are computable from one observation but are unrepresentative below 3 years (low) and 5 years (moderate); longer histories tend to reveal deeper drawdowns.
References
Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
average_recovery_duration@v1
Drawdown-at-Risk (95%)HHigher is better
In about 1 period out of 20 the investment was at least this far below its previous high.
Drawdown-at-Risk (Chekhlov, Uryasev & Zabarankin 2005) on the compounded drawdown path, with the same linear quantile as historical VaR.
Formula
Q_0.05(D_1..D_n), linear interpolation (signed)
Convention
Compounded drawdowns, linear quantile, signed (decision 0021).
Minimum data
Requires at least 2 observations; low confidence when fewer than 5 observations are expected in the 5% tail (n < 100), moderate when fewer than 10 (n < 200).
References
Chekhlov, A., Uryasev, S., & Zabarankin, M. (2005). Drawdown Measure in Portfolio Optimization. International Journal of Theoretical and Applied Finance, 8(1), 13-58 (Drawdown-at-Risk and Conditional Drawdown-at-Risk).
Identifier
drawdown_at_risk_95@v1
Conditional Drawdown-at-Risk (95%)HHigher is better
How deep the drawdowns were, on average, during the worst 5% of periods.
CDaR (Chekhlov, Uryasev & Zabarankin 2005): Expected Shortfall of the drawdown path, a coherent path-dependent risk measure between DaR and the maximum drawdown. Estimated exactly like historical ES (decision 0010).
Formula
mean(D_t | D_t <= Q_0.05(D))
Convention
Compounded drawdowns; mean of observations at or below the linear 5% quantile; signed (decision 0021).
Minimum data
Requires at least 2 observations; low confidence when fewer than 5 observations are expected in the 5% tail (n < 100), moderate when fewer than 10 (n < 200).
References
Chekhlov, A., Uryasev, S., & Zabarankin, M. (2005). Drawdown Measure in Portfolio Optimization. International Journal of Theoretical and Applied Finance, 8(1), 13-58 (Drawdown-at-Risk and Conditional Drawdown-at-Risk).
Identifier
cdar_95@v1
Entropic Drawdown-at-Risk (95%)HHigher is better
A conservative measure of how deep drawdowns get, sitting between the Conditional Drawdown-at-Risk and the maximum drawdown.
EDaR (Cajas 2021): Entropic VaR (Ahmadi-Javid 2012) of the drawdown magnitudes; the tightest Chernoff-type bound on DaR, coherent and more tail-sensitive than CDaR. Computed by one-dimensional convex minimization.
Formula
-EVaR_0.05(-D), EVaR_a(X) = inf_{z>0} z * ln( mean(exp(X/z)) / a )
Convention
Compounded drawdowns, 95%, signed (decision 0021).
Minimum data
Requires at least 2 observations; low confidence when fewer than 5 observations are expected in the 5% tail (n < 100), moderate when fewer than 10 (n < 200).
References
Cajas, D. (2021). Entropic Portfolio Optimization: A Disciplined Convex Programming Framework. SSRN working paper 3792520 (Entropic Drawdown-at-Risk).; Ahmadi-Javid, A. (2012). Entropic Value-at-Risk: A New Coherent Risk Measure. Journal of Optimization Theory and Applications, 155(3), 1105-1123.
Identifier
edar_95@v1

Risk-adjusted performance

Return earned per unit of risk taken. The ratios most often quoted in pitches.

Sharpe ratioHHigher is better
The Sharpe ratio compares excess return with total return variability. Higher values indicate more return per unit of measured volatility, but the statistic can be misleading when returns are smoothed, strongly skewed, serially correlated, or have significant tail risk.
Ex-post Sharpe (1994) with arithmetic mean and square-root-of-time annualization. Serial dependence biases the annualized value (Lo, 2002); smoothed returns inflate it (Getmansky, Lo & Makarov, 2004). Standard error is roughly sqrt((1 + SR^2/2) / years) under i.i.d. normality.
Formula
(mean(r_t) - rf) / std(r_t, ddof=1) * sqrt(P), rf = 0
Convention
Arithmetic mean / sample std x sqrt(P); rf = 0 (decisions 0005, 0008).
Minimum data
Risk-adjusted ratios: at least 2 observations; low confidence below 3 years, moderate below 5 years.
References
Sharpe, W. F. (1966). Mutual Fund Performance. Journal of Business, 39(1), 119-138.; Sharpe, W. F. (1994). The Sharpe Ratio. Journal of Portfolio Management, 21(1), 49-58.; Lo, A. W. (2002). The Statistics of Sharpe Ratios. Financial Analysts Journal, 58(4), 36-52.; Getmansky, M., Lo, A. W., & Makarov, I. (2004). An Econometric Model of Serial Correlation and Illiquidity in Hedge Fund Returns. Journal of Financial Economics, 74(3), 529-609.
Identifier
sharpe_ratio@v1
Sortino ratioHHigher is better
Like the Sharpe ratio, but only losing periods count as risk.
Sortino & Price (1994) with MAR = 0 and full-sample downside deviation. Downside deviation from few losses is noisy; the ratio is undefined without losses.
Formula
P * mean(r_t - MAR) / ( sqrt((1/n) sum min(r_t - MAR, 0)^2) * sqrt(P) ), MAR = 0
Convention
MAR = 0 per period; full-sample denominator; annualized by sqrt(P) (decision 0006).
Minimum data
Risk-adjusted ratios: at least 2 observations; low confidence below 3 years, moderate below 5 years.
References
Sortino, F. A., & Price, L. N. (1994). Performance Measurement in a Downside Risk Framework. Journal of Investing, 3(3), 59-64.; Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
sortino_ratio@v1
Calmar ratioHHigher is better
How much yearly return was earned relative to the worst peak-to-trough loss.
Young (1991) defined Calmar on a trailing 36-month window; OpenFactsheet canonically uses the full history so that no window is chosen. Maximum drawdown grows with history length, so values fall as track records lengthen.
Formula
CAGR / |max drawdown|
Convention
Full history, geometric CAGR, signed max drawdown in absolute value (decision 0009).
Minimum data
Risk-adjusted ratios: at least 2 observations; low confidence below 3 years, moderate below 5 years.
References
Young, T. W. (1991). Calmar Ratio: A Smoother Tool. Futures, 20(1), 40.; Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
calmar_ratio@v1
Martin ratio (Ulcer Performance Index)HHigher is better
Return relative to how deep and prolonged the declines were.
Ulcer Performance Index (Martin & McCann, 1989); penalizes both depth and duration of drawdowns rather than volatility.
Formula
(CAGR - rf) / UlcerIndex, rf = 0
Convention
Geometric CAGR over the canonical Ulcer Index.
Minimum data
Risk-adjusted ratios: at least 2 observations; low confidence below 3 years, moderate below 5 years.
References
Martin, P. G., & McCann, B. B. (1989). The Investor's Guide to Fidelity Funds. Wiley (Ulcer Index and Ulcer Performance Index).; Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
martin_ratio@v1
Omega ratioHHigher is better
Total gains divided by total losses; above 1 means gains outweighed losses.
Keating & Shadwick (2002) Omega at threshold 0, using the full empirical distribution (all moments). Equals 1 + mean / lower partial moment of order 1.
Formula
sum(max(r_t - L, 0)) / sum(max(L - r_t, 0)), L = 0
Convention
Threshold 0 per period; simple empirical estimator; not annualized.
Minimum data
Risk-adjusted ratios: at least 2 observations; low confidence below 3 years, moderate below 5 years.
References
Keating, C., & Shadwick, W. F. (2002). A Universal Performance Measure. Journal of Performance Measurement, 6(3), 59-84.; Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
omega_ratio@v1
Kappa 3 ratioHHigher is better
Return per unit of downside risk, where large losses count much more than small ones.
Kaplan & Knowles (2004) Kappa of order 3 (Sortino is Kappa 2; Omega - 1 is Kappa 1). The cubic lower partial moment emphasizes tail losses. Per-period, not annualized.
Formula
(mean(r_t) - tau) / ( (1/n) sum max(tau - r_t, 0)^3 )^(1/3), tau = 0
Convention
Target 0, full-sample denominator n, per period (decision 0016).
Minimum data
Risk-adjusted ratios: at least 2 observations; low confidence below 3 years, moderate below 5 years.
References
Kaplan, P. D., & Knowles, J. A. (2004). Kappa: A Generalized Downside Risk-Adjusted Performance Measure. Journal of Performance Measurement, 8(3), 42-54.; Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
kappa_3_ratio@v1
Upside potential ratioHHigher is better
How much upside the investment delivered for each unit of downside it exposed investors to.
Sortino, van der Meer & Plantinga (1999): first upper partial moment over the square root of the second lower partial moment, both at target 0 with full-sample denominators. Per-period, not annualized.
Formula
( (1/n) sum max(r_t - tau, 0) ) / sqrt( (1/n) sum min(r_t - tau, 0)^2 ), tau = 0
Convention
Target 0, full-sample denominators (decision 0016).
Minimum data
Risk-adjusted ratios: at least 2 observations; low confidence below 3 years, moderate below 5 years.
References
Sortino, F. A., van der Meer, R., & Plantinga, A. (1999). The Dutch Triangle. Journal of Portfolio Management, 26(1), 50-57 (upside potential ratio).; Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
upside_potential_ratio@v1
Burke ratioHHigher is better
Return relative to the drawdowns suffered, where deep drawdowns weigh more than shallow ones.
Burke (1994): penalizes the square of every drawdown episode rather than only the worst one (Calmar) or their time-average (Martin, Pain).
Formula
(CAGR - rf) / sqrt( sum_k depth_k^2 ), rf = 0
Convention
Episode depths from the canonical drawdown engine; geometric CAGR (decision 0017).
Minimum data
Risk-adjusted ratios: at least 2 observations; low confidence below 3 years, moderate below 5 years.
References
Burke, G. (1994). A Sharper Sharpe Ratio. Futures, 23(3), 56.; Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
burke_ratio@v1
Sterling ratioHHigher is better
Like the Calmar ratio, but the fixed 10% added to the worst loss keeps the ratio from exploding for investments with tiny drawdowns.
Sterling-type ratio in the form popularized by Kestner (1996) and Bacon, using the full-history maximum drawdown. The 10% adjustment is a historical convention, not an estimate; definitions in the literature vary (e.g. averages of annual maximum drawdowns).
Formula
CAGR / ( |max drawdown| + 0.10 )
Convention
CAGR / (|MDD| + 10%), full history (decision 0017).
Minimum data
Risk-adjusted ratios: at least 2 observations; low confidence below 3 years, moderate below 5 years.
References
Kestner, L. N. (1996). Getting a Handle on True Performance. Futures, 25(1), 44-46 (Sterling ratio).; Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
sterling_ratio@v1
Pain ratioHHigher is better
Return relative to how far below its previous highs the investment typically was.
Zephyr Pain ratio (see Bacon): like the Martin ratio but with the mean rather than the root-mean-square drawdown, so deep drawdowns weigh less.
Formula
(CAGR - rf) / PainIndex, rf = 0
Convention
Geometric CAGR over the canonical Pain Index (decision 0017).
Minimum data
Risk-adjusted ratios: at least 2 observations; low confidence below 3 years, moderate below 5 years.
References
Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
pain_ratio@v1
Recovery factorHHigher is better
How many times over the investment earned back its worst decline.
Net compounded profit over the maximum drawdown (a practitioner measure used in trading-system evaluation). Not annualized, so it grows with the length of the track record; compare only across equal histories.
Formula
cumulative return / |max drawdown|
Convention
Compounded cumulative return over |MDD| (decision 0025).
Minimum data
Risk-adjusted ratios: at least 2 observations; low confidence below 3 years, moderate below 5 years.
References
Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
recovery_factor@v1
Sharpe ratio (serial-correlation adjusted, Lo 2002)HHigher is better
The Sharpe ratio after allowing for gains or losses that persist from one period to the next. When returns are smoothed (common for hedge funds and private assets) this is usually lower than the conventional Sharpe ratio.
Lo (2002) annualization of the per-period Sharpe ratio: eta(q) replaces sqrt(q); with no autocorrelation the two coincide. Positive autocorrelation (for example from smoothing) lowers it. The adjustment rests on q - 1 estimated autocorrelations and is itself noisy for short histories; details report eta, sqrt(q) and the variance ratio.
Formula
(mean(r_t) / std(r_t, ddof=1)) * q / sqrt(q + 2 sum_{k=1}^{q-1} (q - k) rho_k), q = round(P)
Convention
Horizon one year (q = round(P)); per-period Sharpe ratio as in decision 0008; autocorrelations of the dependence metrics (decision 0029).
Minimum data
One-year-horizon serial-correlation statistics need at least two years of data (undefined below); their standard error is about sqrt(4 / (3 y)) for y years, so confidence is low below 5 years and moderate below 10 years.
References
Lo, A. W. (2002). The Statistics of Sharpe Ratios. Financial Analysts Journal, 58(4), 36-52.; Getmansky, M., Lo, A. W., & Makarov, I. (2004). An Econometric Model of Serial Correlation and Illiquidity in Hedge Fund Returns. Journal of Financial Economics, 74(3), 529-609.; Sharpe, W. F. (1994). The Sharpe Ratio. Journal of Portfolio Management, 21(1), 49-58.
Identifier
sharpe_ratio_autocorrelation_adjusted@v1

Tail risk

What the bad months look like: losses that are rare but large.

Value at Risk (historical, 95%)HHigher is better
In about 1 period out of 20, the return was this bad or worse.
Historical-simulation VaR at 95% for one observation period; says nothing about losses beyond the threshold (see Expected Shortfall).
Formula
Q_0.05(r), empirical quantile with linear interpolation (Hyndman-Fan type 7)
Convention
95% confidence, per observation period, signed return (negative = loss), linear interpolation (decision 0010).
Minimum data
Requires at least 2 observations; low confidence when fewer than 5 observations are expected in the 5% tail (n < 100), moderate when fewer than 10 (n < 200).
References
Jorion, P. (2006). Value at Risk (3rd ed.). McGraw-Hill.; Hyndman, R. J., & Fan, Y. (1996). Sample Quantiles in Statistical Packages. The American Statistician, 50(4), 361-365 (definition 7 = linear interpolation).
Identifier
var_historical_95@v1
Expected Shortfall / CVaR (historical, 95%)HHigher is better
When things went badly (the worst 5% of periods), this was the average return.
Historical Expected Shortfall (CVaR); a coherent risk measure (Acerbi & Tasche, 2002) that accounts for the severity of tail losses.
Formula
mean(r_t | r_t <= Q_0.05(r))
Convention
Mean of returns at or below the linear-interpolated 5% quantile; signed.
Minimum data
Requires at least 2 observations; low confidence when fewer than 5 observations are expected in the 5% tail (n < 100), moderate when fewer than 10 (n < 200).
References
Acerbi, C., & Tasche, D. (2002). On the Coherence of Expected Shortfall. Journal of Banking & Finance, 26(7), 1487-1503.; Jorion, P. (2006). Value at Risk (3rd ed.). McGraw-Hill.; Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
expected_shortfall_historical_95@v1
Tail ratio (95th / 5th percentile)HHigher is better
Above 1: the best periods were larger than the worst periods were bad.
Quantile-based asymmetry measure, robust to single extremes but noisy in short samples.
Formula
|Q_0.95(r)| / |Q_0.05(r)|
Convention
Absolute values of the 95th and 5th percentiles (decision 0010).
Minimum data
Requires at least 2 observations; low confidence when fewer than 5 observations are expected in the 5% tail (n < 100), moderate when fewer than 10 (n < 200).
References
Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
tail_ratio@v1
Value at Risk (Gaussian, 95%)HHigher is better
The 1-in-20 loss threshold if returns followed a bell curve. Compare it with the historical VaR: a large gap means the bell curve does not fit.
Parametric (delta-normal) VaR with population moments. Supplementary and model-dependent: it ignores skewness and fat tails, which the historical and Cornish-Fisher VaR capture.
Formula
mean + z_0.05 * sqrt(m2), m2 = (1/n) sum (r_t - mean)^2, z_0.05 = -1.645
Convention
Population standard deviation sqrt(m2), signed return, per period (decision 0018).
Minimum data
Requires at least 2 observations; low confidence below 3 years, moderate below 5.
References
Jorion, P. (2006). Value at Risk (3rd ed.). McGraw-Hill.; Boudt, K., Peterson, B. G., & Croux, C. (2008). Estimation and Decomposition of Downside Risk for Portfolios with Non-Normal Returns. Journal of Risk, 11(2), 79-103.
Identifier
gaussian_var_95@v1
Modified VaR (Cornish-Fisher, 95%)HHigher is better
A 1-in-20 loss threshold that accounts for lopsided and fat-tailed returns.
Cornish-Fisher modified VaR (Zangari 1996; Favre & Galeano 2002) with population moments as in Boudt, Peterson & Croux (2008). Reduces to the Gaussian VaR when S = K = 0. Supplementary and model-dependent.
Formula
mean + h * sqrt(m2), h = z + (z^2-1) S/6 + (z^3-3z) K/24 - (2z^3-5z) S^2/36
Convention
Population moments, signed return, per period (decision 0018).
Minimum data
Requires at least 4 observations; kurtosis estimates are highly unstable, so low confidence below 120 and moderate below 240 observations.
References
Zangari, P. (1996). A VaR Methodology for Portfolios That Include Options. RiskMetrics Monitor, First Quarter, 4-12 (Cornish-Fisher VaR).; Favre, L., & Galeano, J.-A. (2002). Mean-Modified Value-at-Risk Optimization with Hedge Funds. Journal of Alternative Investments, 5(2), 21-25.; Boudt, K., Peterson, B. G., & Croux, C. (2008). Estimation and Decomposition of Downside Risk for Portfolios with Non-Normal Returns. Journal of Risk, 11(2), 79-103.
Identifier
modified_var_95@v1
Modified Expected Shortfall (Cornish-Fisher, 95%)HHigher is better
The average loss in the worst 5% of periods implied by a skew- and fat-tail-adjusted distribution. Compare with the historical Expected Shortfall.
Modified ES of Boudt, Peterson & Croux (2008), the Cornish-Fisher analogue of the historical ES, with population moments and the full second-order expansion. Supplementary and model-dependent; for extreme skewness or kurtosis the expansion can make it less severe than the modified VaR, which is flagged.
Formula
mean - sqrt(m2)/a * phi(h) * [1 + h^3 S/6 + (h^6 - 9h^4 + 9h^2 + 3) S^2/72 + (h^4 - 2h^2 - 1) K/24], h = Cornish-Fisher quantile, a = 0.05
Convention
Population moments, full expansion, signed return, per period (decision 0022).
Minimum data
Requires at least 4 observations; kurtosis estimates are highly unstable, so low confidence below 120 and moderate below 240 observations.
References
Boudt, K., Peterson, B. G., & Croux, C. (2008). Estimation and Decomposition of Downside Risk for Portfolios with Non-Normal Returns. Journal of Risk, 11(2), 79-103.
Identifier
modified_es_95@v1
Entropic Value at Risk (95%)HHigher is better
A conservative loss threshold for bad periods, at least as severe as the Expected Shortfall and never worse than the worst period.
Entropic VaR (Ahmadi-Javid 2012) of the empirical loss distribution: the tightest Chernoff bound on VaR, coherent, and more sensitive to extreme losses than ES. Computed by one-dimensional convex minimization; signed return.
Formula
-EVaR_0.05(-r), EVaR_a(X) = inf_{z>0} z * ln( mean(exp(X/z)) / a )
Convention
Empirical distribution of per-period returns, 95%, signed (decision 0021).
Minimum data
Requires at least 2 observations; low confidence when fewer than 5 observations are expected in the 5% tail (n < 100), moderate when fewer than 10 (n < 200).
References
Ahmadi-Javid, A. (2012). Entropic Value-at-Risk: A New Coherent Risk Measure. Journal of Optimization Theory and Applications, 155(3), 1105-1123.
Identifier
evar_95@v1

Shape of returns

Whether returns follow a bell curve or hide lopsided surprises.

SkewnessCDepends on context
Negative values mean occasional large losses and many small gains; positive values mean occasional large gains.
Bias-corrected sample skewness (Joanes & Gill type G1). Negative skew combined with a high Sharpe ratio is a classic signature of hidden tail risk.
Formula
G1 = sqrt(n(n-1))/(n-2) * m3 / m2^(3/2), m_k = (1/n) sum (r_t - mean)^k
Convention
Adjusted Fisher-Pearson G1, as in pandas and Excel SKEW (decision 0011).
Minimum data
Requires at least 3 observations; standard error is roughly sqrt(6/n), so low confidence below 60 and moderate below 120 observations.
References
Joanes, D. N., & Gill, C. A. (1998). Comparing Measures of Sample Skewness and Kurtosis. Journal of the Royal Statistical Society D, 47(1), 183-189.
Identifier
skewness@v1
Excess kurtosisCDepends on context
Positive values mean extreme periods (good or bad) happen more often than a bell curve would suggest.
Bias-corrected sample excess kurtosis (G2). Fat tails make volatility-based risk measures understate tail risk.
Formula
G2 = (n-1)/((n-2)(n-3)) * ((n+1) * m4/m2^2 - 3(n-1))
Convention
Excess (normal = 0) G2, as in pandas and Excel KURT (decision 0011).
Minimum data
Requires at least 4 observations; kurtosis estimates are highly unstable, so low confidence below 120 and moderate below 240 observations.
References
Joanes, D. N., & Gill, C. A. (1998). Comparing Measures of Sample Skewness and Kurtosis. Journal of the Royal Statistical Society D, 47(1), 183-189.
Identifier
excess_kurtosis@v1
Jarque-Bera normality p-valueCDepends on context
A small value (below 0.05) indicates the returns are not bell-shaped, so volatility-based statistics tell only part of the story.
Jarque & Bera (1987) test with population moments and its asymptotic chi-squared(2) p-value. A diagnostic, not a performance measure: in small samples the test has low power and poor size; in very large samples trivial departures are significant.
Formula
JB = n/6 * (g1^2 + g2^2 / 4), g1 = m3/m2^1.5, g2 = m4/m2^2 - 3; p = 1 - F_chi2(2)
Convention
Population (biased) skewness and excess kurtosis, as in the original test (decision 0020).
Minimum data
Requires at least 3 observations; standard error is roughly sqrt(6/n), so low confidence below 60 and moderate below 120 observations.
References
Jarque, C. M., & Bera, A. K. (1987). A Test for Normality of Observations and Regression Residuals. International Statistical Review, 55(2), 163-172.
Identifier
jarque_bera_pvalue@v1

Consistency

How often it made money, by month, by year and over rolling twelve months.

Positive periodsHHigher is better
How often the investment made money in a period.
Win rate on the supplied frequency; says nothing about the size of wins and losses (see gain-to-pain, Omega). A very high hit rate with small gains can mask rare large losses.
Formula
#{t : r_t > 0} / n
Convention
Denominator includes all observations (zeros count as not positive).
Minimum data
Mean-type estimates: low confidence below 3 years, moderate below 5 years.
References
Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
positive_period_pct@v1
Negative periodsLLower is better
How often the investment lost money in a period.
Loss frequency on the supplied frequency; together with positive periods it also reveals the share of exactly-zero periods.
Formula
#{t : r_t < 0} / n
Convention
Denominator includes all observations.
Minimum data
Mean-type estimates: low confidence below 3 years, moderate below 5 years.
References
Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
negative_period_pct@v1
Longest run of positive periodsNNeutral
The longest stretch of back-to-back gaining periods.
Descriptive path statistic; unusually long streaks relative to the hit rate can indicate smoothing.
Formula
max length of consecutive runs with r_t > 0
Convention
Strictly positive returns only.
Minimum data
Path statistics (drawdowns, streaks) are computable from one observation but are unrepresentative below 3 years (low) and 5 years (moderate); longer histories tend to reveal deeper drawdowns.
References
Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
longest_winning_streak@v1
Longest run of negative periodsLLower is better
The longest stretch of back-to-back losing periods.
Descriptive path statistic complementing drawdown duration.
Formula
max length of consecutive runs with r_t < 0
Convention
Strictly negative returns only.
Minimum data
Path statistics (drawdowns, streaks) are computable from one observation but are unrepresentative below 3 years (low) and 5 years (moderate); longer histories tend to reveal deeper drawdowns.
References
Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
longest_losing_streak@v1
Best calendar yearHHigher is better
The best full January-to-December year.
Maximum complete-calendar-year return; partial first and last years are excluded and listed in the details, together with every complete year's return.
Formula
max over complete years y of prod_{t in y} (1 + r_t) - 1
Convention
Complete years only; compounded monthly/quarterly/daily returns (decision 0023).
Minimum data
Path statistics (drawdowns, streaks) are computable from one observation but are unrepresentative below 3 years (low) and 5 years (moderate); longer histories tend to reveal deeper drawdowns.
References
CFA Institute (2020). Global Investment Performance Standards (GIPS) for Firms, provisions on annualized returns.; Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
best_calendar_year@v1
Worst calendar yearHHigher is better
The worst full January-to-December year.
Minimum complete-calendar-year return; always shown, including when it is unfavorable.
Formula
min over complete years y of prod_{t in y} (1 + r_t) - 1
Convention
Complete years only (decision 0023).
Minimum data
Path statistics (drawdowns, streaks) are computable from one observation but are unrepresentative below 3 years (low) and 5 years (moderate); longer histories tend to reveal deeper drawdowns.
References
CFA Institute (2020). Global Investment Performance Standards (GIPS) for Firms, provisions on annualized returns.; Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
worst_calendar_year@v1
Positive calendar yearsHHigher is better
How many full years ended with a gain.
Calendar-year hit rate; the details list every complete year's return and every excluded partial year.
Formula
#{y : R_y > 0} / #Y over complete calendar years
Convention
Complete years only; a zero year counts as not positive (decision 0023).
Minimum data
Path statistics (drawdowns, streaks) are computable from one observation but are unrepresentative below 3 years (low) and 5 years (moderate); longer histories tend to reveal deeper drawdowns.
References
CFA Institute (2020). Global Investment Performance Standards (GIPS) for Firms, provisions on annualized returns.; Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
positive_calendar_years_pct@v1
Best rolling one-year returnHHigher is better
The best twelve months (or one year of periods) an investor could have experienced, whatever the start date.
Maximum of all overlapping one-year compounded returns; every start date is considered, so no favorable start date can be chosen.
Formula
max_t prod_{i=t-w+1..t} (1 + r_i) - 1, w = round(P)
Convention
Window round(P) observations, step 1 (decision 0024).
Minimum data
Path statistics (drawdowns, streaks) are computable from one observation but are unrepresentative below 3 years (low) and 5 years (moderate); longer histories tend to reveal deeper drawdowns.
References
Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
best_rolling_year@v1
Worst rolling one-year returnHHigher is better
The worst twelve months (or one year of periods) an investor could have experienced, whatever the start date.
Minimum of all overlapping one-year compounded returns; a start-date-free worst case that is often much worse than the worst calendar year.
Formula
min_t prod_{i=t-w+1..t} (1 + r_i) - 1, w = round(P)
Convention
Window round(P) observations, step 1 (decision 0024).
Minimum data
Path statistics (drawdowns, streaks) are computable from one observation but are unrepresentative below 3 years (low) and 5 years (moderate); longer histories tend to reveal deeper drawdowns.
References
Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
worst_rolling_year@v1
Positive rolling one-year windowsHHigher is better
If you had invested at any point and held for one year, how often you would have ended with a gain.
Rolling one-year hit rate over all start dates. Windows overlap, so it is a descriptive statistic, not a probability estimate.
Formula
#{t : R_t^(w) > 0} / (n - w + 1), w = round(P)
Convention
Window round(P) observations, step 1; zero counts as not positive (decision 0024).
Minimum data
Path statistics (drawdowns, streaks) are computable from one observation but are unrepresentative below 3 years (low) and 5 years (moderate); longer histories tend to reveal deeper drawdowns.
References
Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
positive_rolling_year_pct@v1

Serial dependence

Whether one month predicts the next, a sign that returns may be smoothed.

Autocorrelation (lag 1)CDepends on context
Whether good or bad periods tend to be followed by similar periods. Values near zero are typical for liquid, market-priced investments.
Sample autocorrelation (full-sample variance denominator). Significant positive low-lag autocorrelation is a hallmark of return smoothing or illiquidity (Getmansky, Lo & Makarov, 2004) and biases square-root-of-time annualization. Approximate 95% significance bound: +/- 1.96 / sqrt(n).
Formula
rho_1 = sum_(t>1) (r_t - m)(r_(t-1) - m) / sum_t (r_t - m)^2
Convention
Standard biased-denominator estimator; not bias-corrected.
Minimum data
Requires at least 5 observations; standard error is roughly 1/sqrt(n), so low confidence below 36 and moderate below 60 observations.
References
Getmansky, M., Lo, A. W., & Makarov, I. (2004). An Econometric Model of Serial Correlation and Illiquidity in Hedge Fund Returns. Journal of Financial Economics, 74(3), 529-609.
Identifier
autocorrelation_lag1@v1
Autocorrelation (lag 2)CDepends on context
Whether good or bad periods tend to be followed by similar periods. Values near zero are typical for liquid, market-priced investments.
Sample autocorrelation (full-sample variance denominator). Significant positive low-lag autocorrelation is a hallmark of return smoothing or illiquidity (Getmansky, Lo & Makarov, 2004) and biases square-root-of-time annualization. Approximate 95% significance bound: +/- 1.96 / sqrt(n).
Formula
rho_2 = sum_(t>2) (r_t - m)(r_(t-2) - m) / sum_t (r_t - m)^2
Convention
Standard biased-denominator estimator; not bias-corrected.
Minimum data
Requires at least 5 observations; standard error is roughly 1/sqrt(n), so low confidence below 36 and moderate below 60 observations.
References
Getmansky, M., Lo, A. W., & Makarov, I. (2004). An Econometric Model of Serial Correlation and Illiquidity in Hedge Fund Returns. Journal of Financial Economics, 74(3), 529-609.
Identifier
autocorrelation_lag2@v1
Autocorrelation (lag 3)CDepends on context
Whether good or bad periods tend to be followed by similar periods. Values near zero are typical for liquid, market-priced investments.
Sample autocorrelation (full-sample variance denominator). Significant positive low-lag autocorrelation is a hallmark of return smoothing or illiquidity (Getmansky, Lo & Makarov, 2004) and biases square-root-of-time annualization. Approximate 95% significance bound: +/- 1.96 / sqrt(n).
Formula
rho_3 = sum_(t>3) (r_t - m)(r_(t-3) - m) / sum_t (r_t - m)^2
Convention
Standard biased-denominator estimator; not bias-corrected.
Minimum data
Requires at least 5 observations; standard error is roughly 1/sqrt(n), so low confidence below 36 and moderate below 60 observations.
References
Getmansky, M., Lo, A. W., & Makarov, I. (2004). An Econometric Model of Serial Correlation and Illiquidity in Hedge Fund Returns. Journal of Financial Economics, 74(3), 529-609.
Identifier
autocorrelation_lag3@v1
Ljung-Box p-value (3 lags)CDepends on context
A small value (below 0.05) suggests returns are not independent from one period to the next.
Ljung-Box portmanteau test with 3 lags (matching the reported autocorrelations). A diagnostic, not a performance measure; with very long samples trivial dependence becomes 'significant'.
Formula
Q = n(n+2) sum_{k=1..3} rho_k^2 / (n - k); p-value from chi-squared(3)
Convention
3 lags, no degrees-of-freedom adjustment; supplementary status.
Minimum data
Requires at least 10 observations for a 3-lag test; low below 36, moderate below 60.
References
Ljung, G. M., & Box, G. E. P. (1978). On a Measure of Lack of Fit in Time Series Models. Biometrika, 65(2), 297-303.
Identifier
ljung_box_pvalue@v1
Zero-return periodsCDepends on context
Many exactly-zero periods can mean prices were not updated (stale pricing).
Stale-price indicator; exact zeros are rare for marked-to-market portfolios and suppress measured volatility.
Formula
#{t : r_t = 0} / n
Convention
Exact equality with 0.0 after normalization.
Minimum data
Observed fact about the supplied history; any non-empty history suffices.
References
Getmansky, M., Lo, A. W., & Makarov, I. (2004). An Econometric Model of Serial Correlation and Illiquidity in Hedge Fund Returns. Journal of Financial Economics, 74(3), 529-609.
Identifier
zero_return_pct@v1
Variance ratio (one-year horizon)CDepends on context
Whether risk over a whole year is larger or smaller than monthly (or daily) volatility suggests. Above 1: gains and losses tend to persist, so yearly swings are bigger than the annualized volatility shows; below 1: they tend to reverse.
Lo-MacKinlay variance ratio at the annualization horizon, written in terms of the sample autocorrelations, so it equals (q + 2 sum (q - k) rho_k) / q, the quantity that corrects square-root-of-time annualization (Lo 2002). Details report the horizon, the asymptotic i.i.d. standard error, z-statistic and p-value. A diagnostic of the annualization assumption, not a performance measure.
Formula
VR(q) = 1 + 2 sum_{k=1}^{q-1} (1 - k/q) rho_k, q = round(P)
Convention
Autocorrelation form of the overlapping variance ratio on simple returns, horizon one year, i.i.d. standard error; supplementary (decision 0029).
Minimum data
One-year-horizon serial-correlation statistics need at least two years of data (undefined below); their standard error is about sqrt(4 / (3 y)) for y years, so confidence is low below 5 years and moderate below 10 years.
References
Lo, A. W., & MacKinlay, A. C. (1988). Stock Market Prices Do Not Follow Random Walks: Evidence from a Simple Specification Test. Review of Financial Studies, 1(1), 41-66.; Lo, A. W. (2002). The Statistics of Sharpe Ratios. Financial Analysts Journal, 58(4), 36-52.; Getmansky, M., Lo, A. W., & Makarov, I. (2004). An Econometric Model of Serial Correlation and Illiquidity in Hedge Fund Returns. Journal of Financial Economics, 74(3), 529-609.
Identifier
variance_ratio_1y@v1

Versus benchmark

Behaviour relative to a benchmark you supply: beta, alpha, capture ratios.

BetaCDepends on context, needs a benchmark
A beta of 1 means the fund tended to move one-for-one with the benchmark; 0.5 means about half as much.
OLS beta against the supplied benchmark with rf = 0; details include the standard error and t-statistic (conventional i.i.d. errors).
Formula
slope b in OLS: r_fund,t = a + b * r_bench,t + e_t
Convention
Total returns (rf = 0), OLS with intercept (decision 0012).
Minimum data
Requires at least 3 aligned observations; low confidence below 3 years of overlap, moderate below 5 years.
References
Jensen, M. C. (1968). The Performance of Mutual Funds in the Period 1945-1964. Journal of Finance, 23(2), 389-416.; Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
beta@v1
Alpha (annualized, CAPM regression)HHigher is better, needs a benchmark
The part of the fund's yearly return that the benchmark does not explain; it depends heavily on which benchmark is used.
Jensen-style single-index alpha against the supplied benchmark with rf = 0, annualized arithmetically. Details report the per-period intercept, its standard error, t-statistic and p-value. Statistical significance, not the point estimate, indicates whether alpha is distinguishable from zero.
Formula
P * a, with a the intercept of OLS: r_fund,t = a + b * r_bench,t + e_t
Convention
Per-period OLS intercept x periods per year (decision 0012).
Minimum data
Requires at least 3 aligned observations; low confidence below 3 years of overlap, moderate below 5 years.
References
Jensen, M. C. (1968). The Performance of Mutual Funds in the Period 1945-1964. Journal of Finance, 23(2), 389-416.; Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
alpha@v1
R-squaredNNeutral, needs a benchmark
How closely the fund's ups and downs followed the benchmark (1 = perfectly).
Low R-squared means beta and alpha say little about the fund; the benchmark may be inappropriate.
Formula
1 - SSR/SST of the OLS regression (equals correlation^2 for one regressor)
Convention
Regression with intercept.
Minimum data
Requires at least 3 aligned observations; low confidence below 3 years of overlap, moderate below 5 years.
References
Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
r_squared@v1
Correlation with benchmarkNNeutral, needs a benchmark
+1 means the fund moved exactly with the benchmark, 0 no linear relation, -1 exactly opposite.
Pearson correlation; sensitive to outliers and to regime changes.
Formula
Pearson correlation of aligned returns
Convention
Pearson product-moment correlation.
Minimum data
Requires at least 3 aligned observations; low confidence below 3 years of overlap, moderate below 5 years.
References
Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
correlation@v1
Tracking error (annualized)CDepends on context, needs a benchmark
How much the fund's results differed from the benchmark from period to period.
Ex-post tracking error (active risk); the Information Ratio denominator.
Formula
std(r_fund,t - r_bench,t, ddof=1) * sqrt(P)
Convention
Sample standard deviation of arithmetic active returns x sqrt(P).
Minimum data
Requires at least 2 aligned observations; low confidence below 3 years of overlap, moderate below 5 years.
References
Grinold, R. C., & Kahn, R. N. (2000). Active Portfolio Management (2nd ed.). McGraw-Hill.; Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
tracking_error@v1
Active return (annualized, arithmetic)HHigher is better, needs a benchmark
How much the fund returned above (or below) the benchmark per year, on average.
Arithmetic active return, consistent with the Information Ratio numerator; differs from the difference of geometric annualized returns (the PerformanceAnalytics ActivePremium).
Formula
P * mean(r_fund,t - r_bench,t)
Convention
Arithmetic mean of active returns x periods per year (decision 0012).
Minimum data
Requires at least 2 aligned observations; low confidence below 3 years of overlap, moderate below 5 years.
References
Grinold, R. C., & Kahn, R. N. (2000). Active Portfolio Management (2nd ed.). McGraw-Hill.
Identifier
active_return@v1
Information ratioHHigher is better, needs a benchmark
How consistently the fund beat (or lagged) its benchmark relative to how much it deviated from it.
Arithmetic ex-post Information Ratio (Grinold & Kahn), equal to the Sharpe ratio of active returns. PerformanceAnalytics divides the difference of geometric annualized returns instead.
Formula
P * mean(a_t) / (std(a_t, ddof=1) * sqrt(P)), a_t = r_fund,t - r_bench,t
Convention
Arithmetic active mean over tracking error (decision 0012).
Minimum data
Requires at least 2 aligned observations; low confidence below 3 years of overlap, moderate below 5 years.
References
Grinold, R. C., & Kahn, R. N. (2000). Active Portfolio Management (2nd ed.). McGraw-Hill.; Goodwin, T. H. (1998). The Information Ratio. Financial Analysts Journal, 54(4), 34-43.
Identifier
information_ratio@v1
Treynor ratioHHigher is better, needs a benchmark
Return earned per unit of market (benchmark) risk taken.
Treynor (1965) with arithmetic annualized mean on the aligned sample; only meaningful for well-diversified portfolios with a stable, clearly positive beta.
Formula
P * mean(r_fund,t - rf) / beta, rf = 0
Convention
Arithmetic annualized mean / OLS beta on aligned periods (decision 0012).
Minimum data
Requires at least 3 aligned observations; low confidence below 3 years of overlap, moderate below 5 years.
References
Treynor, J. L. (1965). How to Rate Management of Investment Funds. Harvard Business Review, 43(1), 63-75.; Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
treynor_ratio@v1
Upside capture ratioHHigher is better, needs a benchmark
Above 1 (100%): the fund gained more than the benchmark when the benchmark rose.
Morningstar-style geometric upside capture; read together with downside capture.
Formula
annualized geometric fund return over {t : r_bench,t > 0} / same for benchmark
Convention
Geometric, annualized over the up periods (decision 0012).
Minimum data
Requires at least 2 aligned observations; low confidence below 3 years of overlap, moderate below 5 years.
References
Morningstar, Inc. Upside/Downside Capture Ratio methodology (geometric compounding of returns in up/down benchmark periods).; Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
upside_capture@v1
Downside capture ratioLLower is better, needs a benchmark
Below 1 (100%): the fund lost less than the benchmark when the benchmark fell. Negative: the fund gained on average when the benchmark fell.
Morningstar-style geometric downside capture; read together with upside capture.
Formula
annualized geometric fund return over {t : r_bench,t < 0} / same for benchmark
Convention
Geometric, annualized over the down periods (decision 0012).
Minimum data
Requires at least 2 aligned observations; low confidence below 3 years of overlap, moderate below 5 years.
References
Morningstar, Inc. Upside/Downside Capture Ratio methodology (geometric compounding of returns in up/down benchmark periods).; Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
downside_capture@v1
Capture ratio (upside / downside capture)HHigher is better, needs a benchmark
Above 1: the fund participated more in rising markets than in falling ones. A fund with 80% upside and 60% downside capture has a capture ratio of about 1.33.
Ratio of the Morningstar-style geometric capture ratios of this report. It hides the level of participation (a fund with 10% / 5% capture has the same ratio as one with 120% / 60%); always read it with both components.
Formula
upside_capture / downside_capture
Convention
Geometric annualized capture ratios; downside capture must be positive (decision 0027).
Minimum data
Requires at least 2 aligned observations; low confidence below 3 years of overlap, moderate below 5 years.
References
Morningstar, Inc. Upside/Downside Capture Ratio methodology (geometric compounding of returns in up/down benchmark periods).; Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
capture_ratio@v1
Batting average (share of periods beating the benchmark)HHigher is better, needs a benchmark
How often the fund beat its benchmark: 0.6 means in 6 periods out of 10. It says nothing about by how much it won or lost.
Frequency of positive active returns (ties count as not beating). Details give the up- and down-market batting averages (PerformanceAnalytics UpDownRatios 'Percent'). A high batting average with a negative active return indicates small frequent wins and large occasional losses.
Formula
(1/m) * #{t : r_fund,t > r_bench,t}
Convention
Strict outperformance; all aligned periods in the denominator (decision 0027).
Minimum data
Requires at least 2 aligned observations; low confidence below 3 years of overlap, moderate below 5 years.
References
Morningstar, Inc. Batting average: share of periods in which a fund outperformed its benchmark (Morningstar Direct data definitions).; Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
batting_average@v1
Residual (specific) volatility (annualized)CDepends on context, needs a benchmark
How much the fund moved for reasons unrelated to its benchmark. Near zero means the fund behaves like the benchmark (scaled by beta).
Specific (idiosyncratic, non-systematic) risk: the regression standard error of the single-index model, annualized by sqrt(P). Details give the systematic risk |beta| x benchmark volatility. Unlike tracking error it is measured after scaling by beta.
Formula
sqrt( sum(e_t^2) / (m - 2) ) * sqrt(P), e_t the OLS residuals
Convention
Regression standard error (m - 2 degrees of freedom) x sqrt(P) (decision 0027).
Minimum data
Requires at least 3 aligned observations; low confidence below 3 years of overlap, moderate below 5 years.
References
Treynor, J. L., & Black, F. (1973). How to Use Security Analysis to Improve Portfolio Selection. Journal of Business, 46(1), 66-86 (appraisal ratio).; Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
residual_volatility@v1
Appraisal ratioHHigher is better, needs a benchmark
How much benchmark-independent return the fund earned for each unit of benchmark-independent risk it took.
Treynor & Black (1973) appraisal ratio: annualized OLS alpha over annualized residual volatility (the Information ratio of the beta-adjusted active return). It is approximately the alpha t-statistic divided by the square root of the overlap in years (exactly so when the benchmark's mean return is zero); details give that t-statistic and its p-value.
Formula
(P * a) / (sigma_e * sqrt(P)) = a / sigma_e * sqrt(P)
Convention
Arithmetic annualized OLS alpha / residual volatility, rf = 0 (decision 0027).
Minimum data
Requires at least 3 aligned observations; low confidence below 3 years of overlap, moderate below 5 years.
References
Treynor, J. L., & Black, F. (1973). How to Use Security Analysis to Improve Portfolio Selection. Journal of Business, 46(1), 66-86 (appraisal ratio).; Grinold, R. C., & Kahn, R. N. (2000). Active Portfolio Management (2nd ed.). McGraw-Hill.; Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
appraisal_ratio@v1
M-squared (Modigliani risk-adjusted return)HHigher is better, needs a benchmark
Puts the fund and the benchmark on the same risk footing: the return the fund would have delivered at the benchmark's volatility. Compare it with the benchmark's own return (details).
Modigliani & Modigliani (1997) M-squared with rf = 0 on the aligned overlap: the aligned-sample Sharpe ratio times the benchmark's annualized volatility. It ranks funds exactly as the Sharpe ratio does, but in return units. Details give the benchmark's annualized mean and the difference (M-squared excess).
Formula
P * (mean(r_fund,t) / std(r_fund,t)) * std(r_bench,t), aligned periods, rf = 0
Convention
Arithmetic annualization, sample standard deviations on aligned periods, rf = 0 (decision 0027).
Minimum data
Requires at least 2 aligned observations; low confidence below 3 years of overlap, moderate below 5 years.
References
Modigliani, F., & Modigliani, L. (1997). Risk-Adjusted Performance. Journal of Portfolio Management, 23(2), 45-54 (M-squared).; Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
m_squared@v1
Bull-market betaCDepends on context, needs a benchmark
How strongly the fund followed the benchmark when the benchmark went up.
Single-index beta estimated on the benchmark-up subsample (Fabozzi & Francis 1977), rf = 0; compare with bear-market beta for asymmetric participation. Details give the standard error and number of periods.
Formula
slope of OLS r_fund,t = a + b * r_bench,t + e_t over {t : r_bench,t > 0}
Convention
OLS with intercept on periods with benchmark return > 0 (decision 0027).
Minimum data
Requires at least 3 aligned observations; low confidence below 3 years of overlap, moderate below 5 years.
References
Fabozzi, F. J., & Francis, J. C. (1977). Stability Tests for Alphas and Betas over Bull and Bear Market Conditions. Journal of Finance, 32(4), 1093-1099.; Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
bull_beta@v1
Bear-market betaCDepends on context, needs a benchmark
How strongly the fund followed the benchmark when the benchmark went down; lower means better protection in falling markets.
Single-index beta estimated on the benchmark-down subsample (Fabozzi & Francis 1977), rf = 0. Details give the standard error and number of periods.
Formula
slope of OLS r_fund,t = a + b * r_bench,t + e_t over {t : r_bench,t < 0}
Convention
OLS with intercept on periods with benchmark return < 0 (decision 0027).
Minimum data
Requires at least 3 aligned observations; low confidence below 3 years of overlap, moderate below 5 years.
References
Fabozzi, F. J., & Francis, J. C. (1977). Stability Tests for Alphas and Betas over Bull and Bear Market Conditions. Journal of Finance, 32(4), 1093-1099.; Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.
Identifier
bear_beta@v1
Timing ratio (bull beta / bear beta)HHigher is better, needs a benchmark
Above 1: the fund was more sensitive to rising than to falling markets.
Ratio of subsample betas (PerformanceAnalytics TimingRatio). Both betas are noisy, so the ratio is very noisy; it is descriptive, not evidence of timing skill. Formal timing tests (Treynor-Mazuy, Henriksson-Merton) are not part of methodology 0.1.0.
Formula
bull_beta / bear_beta, bear_beta > 0
Convention
Ratio of OLS subsample betas; bear beta must be positive (decision 0027).
Minimum data
Requires at least 3 aligned observations; low confidence below 3 years of overlap, moderate below 5 years.
References
Bacon, C. R. (2021). Practical Risk-Adjusted Performance Measurement (2nd ed.). Wiley.; Fabozzi, F. J., & Francis, J. C. (1977). Stability Tests for Alphas and Betas over Bull and Bear Market Conditions. Journal of Finance, 32(4), 1093-1099.
Identifier
timing_ratio@v1

Factor exposure

Exposure to common market factors, when factor data is supplied.

Factor-model alpha (annualized)HHigher is better, needs factor data
The part of the fund's yearly return that the supplied factors (for example the market, size and value) do not explain. It depends entirely on which factors are used.
Multi-factor (Jensen-style) alpha, annualized arithmetically. Details report the per-period intercept with its Newey-West standard error, t-statistic, p-value and 95% confidence interval (annualized). Judge alpha by its uncertainty, not its point estimate; with smoothed returns it is biased upward.
Formula
P * a, with a the intercept of OLS: y_t = a + sum_j b_j f_j,t + e_t
Convention
Per-period OLS intercept x periods per year (decision 0028).
Minimum data
Requires at least 3 aligned observations and at least k + 2 for k factors (k + 1 coefficients and one residual degree of freedom); low confidence below 3 years of overlap, moderate below 5 years.
References
Jensen, M. C. (1968). The Performance of Mutual Funds in the Period 1945-1964. Journal of Finance, 23(2), 389-416.; Fama, E. F., & French, K. R. (1993). Common Risk Factors in the Returns on Stocks and Bonds. Journal of Financial Economics, 33(1), 3-56.; Carhart, M. M. (1997). On Persistence in Mutual Fund Performance. Journal of Finance, 52(1), 57-82.; Fama, E. F., & French, K. R. (2015). A Five-Factor Asset Pricing Model. Journal of Financial Economics, 116(1), 1-22.; Newey, W. K., & West, K. D. (1987). A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix. Econometrica, 55(3), 703-708.; Newey, W. K., & West, K. D. (1994). Automatic Lag Selection in Covariance Matrix Estimation. Review of Economic Studies, 61(4), 631-653 (lag rule floor(4 (T/100)^(2/9))).; Asness, C., Krail, R., & Liew, J. (2001). Do Hedge Funds Hedge? Journal of Portfolio Management, 28(1), 6-19 (stale prices bias betas down and alphas up).
Identifier
factor_alpha@v1
Factor-model alpha t-statistic (Newey-West)CDepends on context, needs factor data
A rough gauge of whether the unexplained return could be luck: values beyond about +2 or -2 are unlikely to arise by chance alone if the model is right.
Student-t statistic of the intercept with m - k - 1 degrees of freedom and heteroskedasticity- and autocorrelation-consistent standard errors; the p-value and interval are in the alpha details. Multiple testing across funds or models inflates apparent significance.
Formula
a / SE_HAC(a)
Convention
Newey-West lag floor(4 (m/100)^(2/9)), Bartlett kernel, small-sample factor m / (m - k - 1), no prewhitening (decision 0028).
Minimum data
Requires at least 3 aligned observations and at least k + 2 for k factors (k + 1 coefficients and one residual degree of freedom); low confidence below 3 years of overlap, moderate below 5 years.
References
Newey, W. K., & West, K. D. (1987). A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix. Econometrica, 55(3), 703-708.; Newey, W. K., & West, K. D. (1994). Automatic Lag Selection in Covariance Matrix Estimation. Review of Economic Studies, 61(4), 631-653 (lag rule floor(4 (T/100)^(2/9))).; Fama, E. F., & French, K. R. (1993). Common Risk Factors in the Returns on Stocks and Bonds. Journal of Financial Economics, 33(1), 3-56.; Carhart, M. M. (1997). On Persistence in Mutual Fund Performance. Journal of Finance, 52(1), 57-82.; Fama, E. F., & French, K. R. (2015). A Five-Factor Asset Pricing Model. Journal of Financial Economics, 116(1), 1-22.
Identifier
factor_alpha_t_statistic@v1
Factor-model R-squaredNNeutral, needs factor data
How much of the fund's ups and downs the factors account for (1 = all of it).
A low R-squared means the factor loadings and alpha describe little of the fund's risk; the factor set may be inappropriate for the strategy. Adjusted R-squared is in the details.
Formula
1 - SSR / SST of the factor regression
Convention
Centred R-squared of the OLS fit with an intercept (decision 0028).
Minimum data
Requires at least 3 aligned observations and at least k + 2 for k factors (k + 1 coefficients and one residual degree of freedom); low confidence below 3 years of overlap, moderate below 5 years.
References
Fama, E. F., & French, K. R. (1993). Common Risk Factors in the Returns on Stocks and Bonds. Journal of Financial Economics, 33(1), 3-56.; Carhart, M. M. (1997). On Persistence in Mutual Fund Performance. Journal of Finance, 52(1), 57-82.; Fama, E. F., & French, K. R. (2015). A Five-Factor Asset Pricing Model. Journal of Financial Economics, 116(1), 1-22.; Fung, W., & Hsieh, D. A. (2004). Hedge Fund Benchmarks: A Risk-Based Approach. Financial Analysts Journal, 60(5), 65-80.
Identifier
factor_r_squared@v1
Factor-model residual volatility (annualized)NNeutral, needs factor data
How much the fund moves for reasons the factors do not capture; the size of the manager's independent bets, whether they pay off or not.
Regression standard error annualized by the square root of time; the denominator of a factor-model appraisal ratio. Understated when returns are smoothed.
Formula
sqrt(SSR / (m - k - 1)) * sqrt(P)
Convention
Residual degrees of freedom m - k - 1, square-root-of-time annualization (decision 0028).
Minimum data
Requires at least 3 aligned observations and at least k + 2 for k factors (k + 1 coefficients and one residual degree of freedom); low confidence below 3 years of overlap, moderate below 5 years.
References
Fama, E. F., & French, K. R. (1993). Common Risk Factors in the Returns on Stocks and Bonds. Journal of Financial Economics, 33(1), 3-56.; Carhart, M. M. (1997). On Persistence in Mutual Fund Performance. Journal of Finance, 52(1), 57-82.; Fama, E. F., & French, K. R. (2015). A Five-Factor Asset Pricing Model. Journal of Financial Economics, 116(1), 1-22.
Identifier
factor_residual_volatility@v1

Statistical confidence

How much the track record can really tell you, given its length.

Probabilistic Sharpe Ratio (vs 0)HHigher is better
How confident one can be, statistically, that the investment's risk-adjusted return is genuinely positive rather than luck. Values near 1 mean strong evidence; 0.5 means no evidence either way.
Bailey & Lopez de Prado (2012) PSR with benchmark SR* = 0 on the per-period (non-annualized) Sharpe ratio, using moment skewness and non-excess kurtosis. Unlike PerformanceAnalytics' default, kurtosis is not ignored: fat tails reduce confidence.
Formula
Phi( (SR - SR*) * sqrt(n - 1) / sqrt(1 - g3*SR + (g4 - 1)/4 * SR^2) ), SR* = 0
Convention
Reference Sharpe ratio 0, per-period Sharpe, population moments (decision 0019).
Minimum data
Asymptotic Sharpe-ratio inference needs at least 4 observations for the moments it uses; the normal approximation is poor below 36 observations (low) and 60 (moderate).
References
Bailey, D. H., & Lopez de Prado, M. (2012). The Sharpe Ratio Efficient Frontier. Journal of Risk, 15(2), 3-44 (Probabilistic Sharpe Ratio, Minimum Track Record Length).; Mertens, E. (2002). Comments on Variance of the IID Estimator in Lo (2002). Working paper, University of Basel.
Identifier
probabilistic_sharpe_ratio@v1
Sharpe ratio standard error (annualized)NNeutral
How precisely the Sharpe ratio is known. A Sharpe ratio of 1.0 with a standard error of 0.5 could plausibly be anywhere from about 0 to 2.
Non-normal i.i.d. asymptotic standard error (Mertens 2002; Opdyke 2007), scaled by sqrt(P) like the annualized Sharpe ratio. The details report the two-sided 95% interval (z = 1.96); PerformanceAnalytics' interval uses the one-sided quantile and is therefore a 90% interval.
Formula
sqrt(P) * sqrt( (1 - g3*SR + (g4 - 1)/4 * SR^2) / (n - 1) ); CI = SR_ann +/- z_0.975 * SE
Convention
Two-sided 95% interval, annualized by sqrt(P), population moments (decision 0019).
Minimum data
Asymptotic Sharpe-ratio inference needs at least 4 observations for the moments it uses; the normal approximation is poor below 36 observations (low) and 60 (moderate).
References
Mertens, E. (2002). Comments on Variance of the IID Estimator in Lo (2002). Working paper, University of Basel.; Opdyke, J. D. (2007). Comparing Sharpe Ratios: So Where Are the p-Values? Journal of Asset Management, 8(5), 308-336.; Lo, A. W. (2002). The Statistics of Sharpe Ratios. Financial Analysts Journal, 58(4), 36-52.
Identifier
sharpe_ratio_standard_error@v1
Minimum track record lengthNNeutral
How long a track record this investment would need before its risk-adjusted return could be told apart from luck. If it exceeds the actual history, the record is too short to be conclusive.
Bailey & Lopez de Prado (2012) MinTRL at 95% one-sided confidence with SR* = 0, in observation periods (details give years and whether the actual history suffices). Depends on the observed Sharpe ratio, skewness and kurtosis.
Formula
1 + (1 - g3*SR + (g4 - 1)/4 * SR^2) * ( z_0.95 / (SR - SR*) )^2, SR* = 0
Convention
95% one-sided, SR* = 0, per-period Sharpe, population moments (decision 0019).
Minimum data
Asymptotic Sharpe-ratio inference needs at least 4 observations for the moments it uses; the normal approximation is poor below 36 observations (low) and 60 (moderate).
References
Bailey, D. H., & Lopez de Prado, M. (2012). The Sharpe Ratio Efficient Frontier. Journal of Risk, 15(2), 3-44 (Probabilistic Sharpe Ratio, Minimum Track Record Length).
Identifier
minimum_track_record_length@v1
Mean return standard error (per period)NNeutral
How precisely the average return is known. If the confidence interval includes zero, the history cannot distinguish the average return from zero.
Student-t interval and two-sided t-test for the mean with n - 1 degrees of freedom (exact under normal i.i.d. returns, asymptotically valid otherwise). Serial correlation widens the true interval.
Formula
s / sqrt(n); CI = mean +/- t_(0.975, n-1) * s / sqrt(n)
Convention
Student-t, two-sided 95%, per period (decision 0020).
Minimum data
Requires at least 2 observations; low confidence below 3 years, moderate below 5.
References
Student [Gosset, W. S.] (1908). The Probable Error of a Mean. Biometrika, 6(1), 1-25.
Identifier
mean_return_standard_error@v1