Quantitative Metrics

MesoMetrics is Deltaray's performance-analysis library used by MesoSim. It calculates the quantitative statistics in backtest results, portfolio analysis, and tearsheets from NAV, benchmark prices, and exposure history.

Use this reference to understand what each metric measures and how it is calculated. The percentage-return formulas below assume a strictly positive NAV series. Currency-PnL and drawdown metrics use their stated definitions and denominators.

Inputs and notation

Symbol	Meaning
V₀, VT	Initial and terminal NAV
Vᵢ	NAV at the end of interval i; Vᵢ₋₁ is NAV at its start
C	Funded initial capital
n	Number of usable return or daily-PnL observations
M	Number of NAV observations in the full drawdown path
P	Annualization factor: 252 for index/equity options; 365 for legacy crypto options
D, Y	Elapsed calendar days and years, with Y = D / 365.2425
F, A	Annual effective risk-free rate and minimum acceptable return (MAR)
fᵢ, mᵢ	Risk-free rate and MAR converted to interval i
bᵢ	Benchmark return over the same interval as rᵢ
Φ	Standard normal cumulative distribution function

Returns are decimals: 0.10 means 10%. Return and probability results are displayed as percentages. Ratios are dimensionless unless another unit is stated.

$$r_i=\frac{V_i}{V_{i-1}}-1p_i=V_i-V_{i-1}$$

$$f_i={(1+F)}^{{\Delta Y}_i}-1m_i={(1+A)}^{{\Delta Y}_i}-1a_i=r_i-f_i$$

The interval year fraction ΔYᵢ uses actual elapsed time divided by 365.2425 days. Risk-free rate and MAR are distinct inputs; their library defaults are both zero.

$$\overline{x}=\frac{\sum _{i=1}^n{x}_i}{n}s(x)=\sqrt{\frac{\sum _{i=1}^n{(x_i-\overline{x})}^2}{n-1}}$$

Daily estimators omit intervals with missing scheduled observations. Partial opening or closing intervals carry partial-coverage information. Benchmark relationships use exactly matched interval starts and ends. Monthly and yearly summaries use complete calendar periods unless noted below.

Returns and risk

Cumulative Return

Total percentage change between the initial and terminal NAV. Requires two endpoints.

$$R=\frac{V_T}{V_0}-1$$

CAGR

Compound annual growth rate using elapsed calendar time. Requires at least 30 calendar days and positive NAV throughout the path.

$$\mathrm{CAGR}={(\frac{V_T}{V_0})}^{\frac{1}{Y}}-1$$

Annualized Volatility

Sample standard deviation of usable daily returns, annualized. Requires at least two returns.

$$\mathrm{Volatility}=s(r)\sqrt{P}$$

Sharpe

Annualized mean excess return divided by its sample standard deviation. Requires at least two observations and nonzero excess-return variation.

$$\mathrm{Sharpe}=\sqrt{P}\frac{\overline{a}}{s(a)}$$

Sortino

Uses risk-free-adjusted return in the numerator and downside deviations below MAR in the denominator. All n usable observations enter the downside average, including zero downside for returns at or above MAR.

$$\mathrm{Sortino}=\frac{\sqrt{P}\overline{a}}{\sqrt{\frac{\sum _{i=1}^n{\min (r_i-m_i,0)}^2}{n}}}$$

Omega

Ratio of gains above MAR to shortfalls below MAR. Requires at least one usable return.

$$\mathrm{Omega}=\frac{\sum _{i=1}^n\max (r_i-m_i,0)}{\sum _{i=1}^n\max (m_i-r_i,0)}$$

Probabilistic Sharpe Ratio

Probability estimate for exceeding a specified nonannualized Sharpe benchmark S₀, which defaults to zero. Sₒᵦₛ = mean(a) / s(a); γ₃ is adjusted sample skewness of excess returns, and γ₄ is their adjusted Pearson kurtosis (excess kurtosis + 3). Requires at least four returns and a positive variance term.

$$\mathrm{PSR}=\Phi (\frac{(S_{\mathrm{obs}}-S_0)\sqrt{n-1}}{\sqrt{1-{\gamma }_3{S}_{\mathrm{obs}}+\frac{{\gamma }_4-1}{4}{S_{\mathrm{obs}}}^2}})$$

This PSR calculation uses an IID approximation and is not annualized. It differs from the stationary-bootstrap positive-mean-return test in StatSim. For the methodology, see Bailey and López de Prado’s The Sharpe Ratio Efficient Frontier.

Return distribution and tail risk

For the following moment formulas, μₖ is the central moment with divisor n. The displayed Skew and Excess Kurtosis use raw returns; PSR applies the same moment estimators to excess returns.

$${\mu }_k=\frac{\sum _{i=1}^n{(r_i-\overline{r})}^k}{n}$$

Skew

Bias-corrected Fisher–Pearson sample skewness. Requires at least three returns with nonzero variance.

$$\mathrm{Skew}=\frac{\sqrt{n(n-1)}}{n-2}\frac{{\mu }_3}{{{\mu }_2}^{\frac{3}{2}}}$$

Excess Kurtosis

Bias-corrected Fisher excess kurtosis. A normal distribution has excess kurtosis zero. Requires at least four returns with nonzero variance.

$$\mathrm{Kurtosis}=\frac{(n²-1)\frac{{\mu }_4}{{{\mu }_2}^2}-3(n-1)²}{(n-2)(n-3)}$$

Empirical quantiles Q use Type 7 interpolation. For sorted values x₍₁₎ through x₍ₙ₎ and quantile level q:

$$h=1+(n-1)qj=\lfloor h\rfloor w=h-j$$

$$Q_q(x)=(1-w)x_{\left(j\right)}+w{x}_{(j+1)}$$

At the endpoints, Q₀ and Q₁ are the sample minimum and maximum; when h is an integer, use x₍ₕ₎.

Historical Daily VaR 95%

The 95th percentile of signed daily losses, −r. A negative value is possible when the observed outcomes are all gains. Requires one return; fewer than 20 observations are marked as a small sample.

$${\mathrm{VaR}}_{95}=Q_{0.95}(-r)$$

Historical Daily Expected Shortfall 95%

Average of the worst 5% of signed-loss mass. Sort losses ℓ = −r from largest to smallest; u = 0.05n, k = floor(u), and w = u − k. The boundary observation receives fractional weight w; when w = 0, omit that term. Requires one return; fewer than 20 are marked as a small sample.

$${\mathrm{ES}}_{95}=\frac{\sum _{i=1}^k{\ell }_{\left(i\right)}+w{\ell }_{(k+1)}}{u}$$

Tail Ratio

Absolute upper-tail return divided by absolute lower-tail return. Requires at least 20 returns.

$$\mathrm{Tail}=\frac{|Q_{0.95}(r)|}{|Q_{0.05}(r)|}$$

Outlier Win Ratio

99th-percentile return divided by the mean strictly positive return. Requires at least 100 returns and a positive observation.

$$\mathrm{OutlierWin}=\frac{Q_{0.99}(r)}{\mathrm{mean}(r_i\text{where}{r}_i>0)}$$

Outlier Loss Ratio

Absolute 1st-percentile return divided by the absolute mean strictly negative return. Requires at least 100 returns and a negative observation.

$$\mathrm{OutlierLoss}=\frac{|Q_{0.01}(r)|}{|\mathrm{mean}(r_i\text{where}{r}_i<0)|}$$

Drawdowns and recovery

Let Hⱼ be the running high-water mark, dⱼ the peak-relative drawdown, and dⱼᶜ the drawdown relative to funded initial capital. Drawdowns are zero or negative.

$$H_j={\max }_{u\le j}{V}_u{d}_j=\frac{V_j}{H_j}-1$$

$$d_{jᶜ}=\frac{V_j-H_j}{C}$$

Max Drawdown from Peak

Deepest drawdown as a fraction of the running peak.

$$\mathrm{MaxDD}={\min }_j{d}_j$$

Max Drawdown on Initial Capital

Deepest peak-to-trough currency decline divided by funded initial capital. Initial capital is this metric’s fixed denominator by definition.

$$\mathrm{MaxDDonInitial}={\min }_j{d}_{jᶜ}$$

A drawdown episode begins at a peak and ends when NAV recovers that peak. Let eⱼ be episode j’s deepest peak-relative drawdown and τⱼ its elapsed calendar days. An unrecovered episode ends at the report’s terminal timestamp for duration measurement. Let E be the number of episodes.

Average Drawdown

Arithmetic mean of episode trough depths. This averages episodes, rather than every underwater observation; requires at least one episode.

$$\mathrm{AverageDD}=\frac{\sum _{j=1}^E{e}_j}{E}$$

Longest Drawdown Days

Longest peak-to-recovery duration, including an open episode through the report end. Returns zero when no drawdown episode exists.

$$\mathrm{LongestDDDays}={\max }_j{\tau }_j$$

Average Drawdown Days

Mean calendar duration of the episodes. Requires at least one episode.

$$\mathrm{AverageDDDays}=\frac{\sum _{j=1}^E{\tau }_j}{E}$$

Recovery Factor

Net currency PnL divided by the absolute maximum currency drawdown.

$$\mathrm{Recovery}=\frac{V_T-V_0}{|{\min }_j(V_j-H_j)|}$$

Calmar

CAGR divided by the absolute maximum peak-relative drawdown. Inherits the CAGR history requirement.

$$\mathrm{Calmar}=\frac{\mathrm{CAGR}}{|\mathrm{MaxDD}|}$$

Ulcer Index

Root mean square of peak-relative drawdowns over the full NAV path, including zero-drawdown observations. Uses divisor M.

$$\mathrm{UI}=\sqrt{\frac{\sum _{j=1}^M{d_j}^2}{M}}$$

Ulcer Performance Index

Annual excess CAGR divided by the Ulcer Index.

$$\mathrm{UPI}=\frac{\mathrm{CAGR}-F}{\mathrm{UI}}$$

Daily PnL and exposure

These metrics use daily NAV changes, including changes in open-position value. Win Days, Payoff Ratio, and Profit Factor here are not closed-trade statistics. Let W contain days with pᵢ > 0 and L contain days with pᵢ < 0; vertical bars around a set denote its count.

Win Days

Fraction of nonzero daily-PnL observations that are positive. Zero-PnL days are excluded from the denominator.

$$\mathrm{WinDays}=\frac{|W|}{|W|+|L|}$$

Max Consecutive Winning Days

Longest run of strictly positive daily PnL. A zero or negative value ends a winning streak.

$$\mathrm{WinningStreak}=\max (\text{length of consecutive pᵢ > 0 runs})$$

Max Consecutive Losing Days

Longest run of strictly negative daily PnL. A zero or positive value ends a losing streak.

$$\mathrm{LosingStreak}=\max (\text{length of consecutive pᵢ < 0 runs})$$

Payoff Ratio

Average winning-day PnL divided by the absolute average losing-day PnL.

$$\mathrm{Payoff}=\frac{\mathrm{mean}(p_i,i\in W)}{|\mathrm{mean}(p_i,i\in L)|}$$

Profit Factor

Gross positive daily PnL divided by absolute gross negative daily PnL. For Payoff Ratio and Profit Factor, gains-only samples are unbounded; losses-only samples give zero; all-zero samples are unavailable.

$$\mathrm{PF}=\frac{\sum _{i\in W}^{}{p}_i}{|\sum _{i\in L}^{}{p}_i|}$$

Common Sense Ratio

Currency-PnL Profit Factor multiplied by normalized-return Tail Ratio. This is a mixed-domain heuristic because its components use different input domains. It is available only when both component metrics are available.

$$\mathrm{CommonSense}=\mathrm{PF}\times \mathrm{Tail}$$

Kelly Sizing Multiple

Nonnegative capital multiple maximizing empirical log growth, subject to 1 + f pᵢ/C > 0 for every observed day. C is the explicit sizing base. For an interior optimum, the derivative below is zero.

$$\mathrm{Kelly}={\mathrm{argmax}}_{f\ge 0}\frac{\sum _{i=1}^n\ln (1+f\frac{p_i}{C})}{n}$$

$$\frac{\sum _{i=1}^n\frac{\frac{p_i}{C}}{1+f\frac{p_i}{C}}}{n}=0$$

A nonpositive mean normalized PnL gives 0x; an all-nonnegative sample with a gain has an unbounded optimum. This is a historical sizing statistic, not a prescription for live leverage.

CPC Index

Product of Profit Factor, the Win Days fraction, and Payoff Ratio.

$$\mathrm{CPC}=\mathrm{PF}\times \mathrm{WinDays}\times \mathrm{Payoff}$$

Time in Market

Iⱼ is 1 when invested during exposure segment j and 0 otherwise. Integrates actual invested time, including nights, weekends, and holidays. Requires the initial exposure state and complete state-change history.

$$\mathrm{TimeInMarket}=\frac{\sum _{j=1}^J{I}_j{\Delta t}_j}{t_{\mathrm{end}}-t_{\mathrm{start}}}$$

Calendar and horizon performance

For a calendar period or selected window B, compound the included simple returns. Each move belongs to the calendar period containing its interval end. Missing periods are not inserted as zero returns.

$$R_B=\prod _{i\in B}^{}(1+r_i)-1$$

Realized Geometric Mean Daily

Usable daily returns compounded into an average period return. K is the number of included periods; requires at least one.

$$\mathrm{GeometricMean}=\exp (\frac{\sum _{j=1}^K\ln (1+r_j)}{K})-1$$

Realized Geometric Mean Monthly

Complete calendar-month returns compounded into an average period return. K is the number of included periods; requires at least one.

$$\mathrm{GeometricMean}=\exp (\frac{\sum _{j=1}^K\ln (1+{Rᴹ}_j)}{K})-1$$

Realized Geometric Mean Yearly

Complete calendar-year returns compounded into an average period return. K is the number of included periods; requires at least one.

$$\mathrm{GeometricMean}=\exp (\frac{\sum _{j=1}^K\ln (1+{Rʸ}_j)}{K})-1$$

MTD

Return from the available beginning of the current calendar month through the report end.

$$\text{MTD}=R_{\text{current month}}$$

YTD

Return from the available beginning of the current calendar year through the report end.

$$\text{YTD}=R_{\text{current year}}$$

3M

Return over the trailing three calendar months.

$$\text{3M}=R_{\text{trailing 3 months}}$$

6M

Return over the trailing six calendar months.

$$\text{6M}=R_{\text{trailing 6 months}}$$

1Y

Return over the trailing calendar year.

$$\text{1Y}=R_{\text{trailing 1 year}}$$

MTD and YTD retain partial-period coverage information. Trailing 3M, 6M, and 1Y require complete scheduled coverage. When a calendar boundary is not a trading observation, the window uses the interval spanning that boundary.

3Y (annualized)

Geometrically annualized return over a complete trailing 3-calendar-year window.

$$\text{3Y annualized}={(1+R_{\text{trailing 3 years}})}^{\frac{1}{3}}-1$$

5Y (annualized)

Geometrically annualized return over a complete trailing 5-calendar-year window.

$$\text{5Y annualized}={(1+R_{\text{trailing 5 years}})}^{\frac{1}{5}}-1$$

10Y (annualized)

Geometrically annualized return over a complete trailing 10-calendar-year window.

$$\text{10Y annualized}={(1+R_{\text{trailing 10 years}})}^{\frac{1}{10}}-1$$

Best Day

Largest return among usable daily returns.

$$\text{Best Day}={\max }_j{r}_j$$

Worst Day

Smallest return among usable daily returns.

$$\text{Worst Day}={\min }_j{r}_j$$

Best Month

Largest return among complete monthly returns.

$$\text{Best Month}={\max }_j{Rᴹ}_j$$

Worst Month

Smallest return among complete monthly returns.

$$\text{Worst Month}={\min }_j{Rᴹ}_j$$

Best Year

Largest return among complete yearly returns.

$$\text{Best Year}={\max }_j{Rʸ}_j$$

Worst Year

Smallest return among complete yearly returns.

$$\text{Worst Year}={\min }_j{Rʸ}_j$$

Average Up Month

Mean of strictly positive complete monthly returns. Requires at least one such month.

$$\text{Average Up Month}=\mathrm{mean}({Rᴹ}_j\text{where}{Rᴹ}_j>0)$$

Average Down Month

Mean of strictly negative complete monthly returns. Requires at least one such month.

$$\text{Average Down Month}=\mathrm{mean}({Rᴹ}_j\text{where}{Rᴹ}_j<0)$$

Win Month

Positive complete months divided by nonzero complete months. Flat periods are excluded.

$$\text{Win Month}=\frac{\mathrm{count}({Rᴹ}_j>0)}{\mathrm{count}({Rᴹ}_j\ne 0)}$$

Win Quarter

Positive complete quarters divided by nonzero complete quarters. Flat periods are excluded.

$$\text{Win Quarter}=\frac{\mathrm{count}({Rᑫ}_j>0)}{\mathrm{count}({Rᑫ}_j\ne 0)}$$

Win Year

Positive complete years divided by nonzero complete years. Flat periods are excluded.

$$\text{Win Year}=\frac{\mathrm{count}({Rʸ}_j>0)}{\mathrm{count}({Rʸ}_j\ne 0)}$$

Gain/Pain (1M)

Sum of all complete monthly returns divided by the absolute sum of negative complete monthly returns. The numerator includes both gains and losses.

$$\mathrm{GainPain}=\frac{\sum _{j=1}^K{Rᴹ}_j}{|\sum _{j=1}^K\min ({Rᴹ}_j,0)|}$$

Benchmark comparisons

Use exactly paired strategy and benchmark return intervals. Let yᵢ = rᵢ and xᵢ = bᵢ. MesoMetrics fits ordinary least squares with an intercept to raw returns, not risk-free-adjusted returns. The following sums run over the n matched pairs.

$$S_{\mathrm{xx}}=\sum _{i=1}^n{(x_i-\overline{x})}^2{S}_{\mathrm{xy}}=\sum _{i=1}^n(x_i-\overline{x})(y_i-\overline{y})$$

$$S_{\mathrm{yy}}=\sum _{i=1}^n{(y_i-\overline{y})}^2$$

Beta

Sensitivity to benchmark returns. Requires at least two matched pairs and a nonconstant benchmark.

$$\beta =\frac{S_{\mathrm{xy}}}{S_{\mathrm{xx}}}$$

Alpha

Annualized regression intercept. The auxiliary per-period intercept is c. Alpha is displayed as a raw annual fraction: 0.03 means 3% annual alpha; beta remains a dimensionless coefficient.

$$c=\overline{y}-\beta \overline{x}\alpha =\mathrm{Pc}$$

Let residuals εᵢ = yᵢ − c − βxᵢ, and SSE = ∑ εᵢ².

R²

Fraction of strategy-return variation explained by the fitted benchmark regression. Requires nonconstant strategy returns.

$$R²=1-\frac{\mathrm{SSE}}{S_{\mathrm{yy}}}$$

Correlation

Pearson correlation of the exact-paired raw returns. Both series must have nonzero variation.

$$\rho =\frac{S_{\mathrm{xy}}}{\sqrt{S_{\mathrm{xx}}{S}_{\mathrm{yy}}}}$$

For regression standard errors, use residual variance σ̂² = SSE/(n − 2), requiring at least three matched pairs.

Alpha Standard Error

Standard error of annualized alpha. The per-period intercept standard error is SE(c) = SE(α)/P.

$$\mathrm{SE}\left(\alpha \right)=P\sqrt{\frac{\mathrm{SSE}}{n-2}(\frac{1}{n}+\frac{{\overline{x}}^2}{S_{\mathrm{xx}}})}$$

Beta Standard Error

Standard error of the regression slope.

$$\mathrm{SE}\left(\beta \right)=\sqrt{\frac{\frac{\mathrm{SSE}}{n-2}}{S_{\mathrm{xx}}}}$$

Information Ratio

Annualized mean active return divided by tracking error, using the same exact-paired intervals. Requires at least two pairs and nonzero variation in active returns.

$$\mathrm{IR}=\sqrt{P}\frac{\overline{r-b}}{s(r-b)}$$

Treynor Ratio

Annualized mean excess strategy return per unit of beta on the matched sample. Requires nonzero beta. Unit: annual return per beta.

$$\mathrm{Treynor}=\frac{P\overline{a}}{\beta }$$

Rolling and structured results

The full report also contains time series and distribution summaries. These reuse the metric definitions above rather than introducing a different return model.

Rolling 6M and 12M metrics

At each endpoint t, calculate each metric from the observations in the preceding six or twelve calendar months:

$${\mathrm{Metric}}_w\left(t\right)=\mathrm{Metric}(\text{observations in window ending at t})w\in (6M,12M)$$

The outputs are annualized volatility, Sharpe, Sortino, alpha, and beta. The benchmark also has its own rolling volatility, Sharpe, and Sortino. Windows require complete scheduled coverage; alpha and beta use exact-paired intervals within each window.

Calendar series and yearly benchmark comparisons

Monthly and yearly returns use the calendar compounding formula above. Each value retains whether its period is complete. For a complete matched year:

$$\text{Active return}=R_{\mathrm{strategy}}-R_{\mathrm{benchmark}}$$

$$\text{Relative wealth}=\frac{1+R_{\mathrm{strategy}}}{1+R_{\mathrm{benchmark}}}$$

Relative wealth is a multiple, so equal performance gives 1. It does not subtract 1. Drawdown episode records include peak and trough timestamps, trough depth, recovery status, and elapsed duration as defined in Drawdowns and recovery.

Return quantiles and box-plot summaries

Daily, ISO-weekly, monthly, quarterly, and yearly aggregated returns have descriptive quantile summaries:

$$Q_1=Q_{0.25}(R)\mathrm{Median}=Q_{0.50}(R)Q_3=Q_{0.75}(R)$$

$$\mathrm{IQR}=Q_3-Q_1$$

$$\text{Lower fence}=Q_1-1.5\mathrm{IQR}\text{Upper fence}=Q_3+1.5\mathrm{IQR}$$

These are Tukey display fences. They describe the distribution and do not remove observations from the performance calculations. Distribution summaries include partial calendar periods when present.

Reading the reported values

• N/A means the metric’s data or mathematical requirements are not met. A missing value is not zero.
• Some ratios can be unbounded when the denominator is zero and the numerator is nonzero. Undefined expressions such as 0/0 are unavailable. Sharpe, Information Ratio, and other metrics with explicit variation requirements remain unavailable for constant input series.
• Coverage matters: compare metrics over the same dates and check whether their inputs contain incomplete periods.
• Trade count, adjustment count, profit-target/stop-loss hits, settlements, and average days in trade come from MesoSim’s trade execution records. They are separate from the MesoMetrics NAV-based formulas on this page.

To inspect the values, open MesoSim backtests, portfolio results, or the Tearsheet tab. For alternative paths from observed returns, use StatSim.

For background on risk-adjusted performance, see William Sharpe’s The Sharpe Ratio. Deltaray’s Portfolio Construction Methods provides a separate research example of comparing strategy and portfolio performance.