Add Adjusted Sharpe Ratio and Adjusted CAGR to portfolio statistics

[AlexCatarino]

Expected Behavior

PortfolioStatistics exposes two additional risk-adjusted return statistics in the backtest/live report:

• Adjusted Sharpe Ratio (ASR) — the Sharpe Ratio penalised for non-normality of the return distribution, per Pezier & White (2006):

  ASR = SR * (1 + (S / 6) * SR - ((K - 3) / 24) * SR^2)

  where S is the skewness and K the kurtosis of the return series. Negative skew and fat tails reduce the ratio, which is the point of the metric for mean-reversion and short-volatility strategies whose plain Sharpe flatters them.

• Adjusted CAGR — a risk/quality-adjusted compounding annual return, alongside the existing raw CompoundingAnnualReturn. The exact definition is worth pinning down before implementation: candidates are CAGR scaled by a drawdown or volatility penalty, or CAGR restated on the skew/kurtosis-adjusted return distribution.

Actual Behavior

Neither exists. Common/Statistics/PortfolioStatistics.cs currently exposes SharpeRatio, ProbabilisticSharpeRatio, SortinoRatio, CompoundingAnnualReturn, InformationRatio, TreynorRatio, ValueAtRisk95/99, and friends — nothing skew/kurtosis-adjusted, and no adjusted variant of CAGR.

Potential Solution

The inputs for ASR are already computed in the codebase: Statistics.ProbabilisticSharpeRatio (Common/Statistics/Statistics.cs) calls listPerformance.Skewness() and listPerformance.Kurtosis() from MathNet, so ASR is a small addition on top of the existing ObservedSharpeRatio / annualised Sharpe plumbing rather than new machinery.

Points to settle:

• Whether ASR is built from the annualised SharpeRatio or the non-annualised ObservedSharpeRatio.
• Whether the kurtosis term is excess or raw. The existing PSR code uses (kurtosis - 1) against MathNet's Kurtosis(), which is worth confirming against the convention chosen for ASR so the two statistics agree on the same input.
• A concrete definition for "Adjusted CAGR", since the term is not standardised.

Reproducing the Problem

N/A — feature request.

System Information

N/A

Related

• New Summary Statistics #8197 (New Summary Statistics) asks for raw Skew and Kurtosis among other statistics. Adjacent, but it does not cover a skew/kurtosis-adjusted Sharpe or an adjusted CAGR.
• Feature Request: Smart Sharpe Ratio Backtest Statistic #4682 (Smart Sharpe Ratio) was closed as not planned. That is the autocorrelation-penalised variant, a different metric from the Pezier & White adjustment requested here.

Requested by a long-time user who has been maintaining their own implementation (Intercom conversation 215476003778646).

Checklist

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• I have confirmed that this issue exists on the current master branch
• I have confirmed that this is not a duplicate issue by searching issues

[abhijeetvichare76]

Hi @AlexCatarino, I'd like to take this. Could I be assigned? Here's the plan and a few questions before I start.

Kurtosis convention. MathNet's Kurtosis() returns excess kurtosis. On 200,000 standard-normal samples it gives 0.0005, not 3. Pezier & White write the penalty as (K - 3), where K is raw kurtosis, so ASR can use MathNet's value directly: ASR = SR * (1 + (S / 6) * SR - (Kexcess / 24) * SR^2).

This affects the existing PSR. Bailey & López de Prado's variance term is (γ4 - 1) / 4 * SR^2, where γ4 is raw kurtosis. Statistics.ProbabilisticSharpeRatio puts MathNet's excess kurtosis into that slot. That makes the term 3/4 * SR^2 too small, so the standard error of the Sharpe estimate is understated and PSR comes out too high. The fix is (kurtosis + 2) / 4. It would change the expected PSR in every regression algorithm, so I'd send it as a separate PR. Do you want me to open an issue for it?

Which SR. I plan to adjust the non-annualised ObservedSharpeRatio on excess returns (the same inputs PSR uses since #9583), because the skew and kurtosis are measured on those daily returns. Then I'd annualise it the same way SharpeRatio is. Does that work for you?

Where it shows up. Adding it to the StatisticsBuilder summary means updating the expected statistics in every regression algorithm (#8865 touched 613 files). Should I do that, or only add it to PortfolioStatistics?

Adjusted CAGR. The term has no standard definition. I'd leave it out of the first PR unless you have one in mind.