Orthogonality Is an Acceptance Test
A portfolio can look good on the usual scorecard and still answer the wrong question. One line says return was high. Another says risk-adjusted performance was acceptable. A third says drawdown stayed inside a tolerable range. Then the market turns, the benchmark starts recovering, and the thing I actually care about is different: how efficiently did the portfolio catch up? That is where a new metric can fool its own author. If I build a recovery measure and it moves almost exactly like an existing ratio, I have created a longer name for the same signal. The right acceptance test is geometric: a useful metric should cast a different shadow. This is the rule I used while validating Hyperlogarithmic Benchmark Catch-Up Ratio (HBCR): orthogonality to existing measures is a first-class test, not a chart for the appendix. 1. A new metric has to earn its axis HBCR was built to measure benchmark-relative recovery dynamics. The research page states the motivation plainly: traditional benchmark-relative metrics often fail to capture the true dynamics of investment performance, especially during market recoveries [ A New Metric for Private Equity Risk Adjusted Returns , Calibration of Risk and Correlation in Private Equity ]. That framing matters because the obvious validation path is tempting and weak. You compare the new number with familiar performance measures, find a comforting relationship, and declare victory. But a high correlation with a well-known score can be a warning. If HBCR strongly tracked Sharpe Ratio, it would probably be an expensive synonym for risk-adjusted return. The acceptance test I wanted was sharper. HBCR should have some relationship with performance, because recovery has economic content. It should also avoid collapsing into the same direction as Sharpe Ratio, Beta, Volatility, Alpha, Total Return, or Max Drawdown. Written as a predicate, the test has two sides. Let $\mathcal{T}$ be the set of metrics already on the scorecard, $\rho_{n,m}$ the corr