Case 016Strategy evaluation and backtestsCore

A book has a Sharpe ratio of 1.0. A candidate strategy has a Sharpe of 0.8 and correlation 0.3 with the book. What is the Sharpe of the best combination, and does the weaker strategy earn a place?

The situation

Morvikant Partners runs a multi-strategy book with a Sharpe ratio of 1.0. A researcher proposes a new strategy with a Sharpe ratio of 0.8, measured over three years, and a correlation of 0.3 with the existing book's returns.

The CIO's first reaction is that a 0.8 strategy will only dilute a 1.0 book. Treat both return streams as scalable, so the question is purely how to split risk between them.

Your task

What is the highest Sharpe ratio the combination can reach, at what split, and does the new strategy deserve capital?

Quick check

Can adding a Sharpe 0.8 strategy raise a Sharpe 1.0 book's Sharpe ratio?

Worked solution

Try it on paper, then open one step at a time.

30-second answerThe answer to give first

Yes: the best mix reaches a Sharpe of about 1.13, with about 40% of the risk in the new strategy. A strategy adds value when its Sharpe exceeds the correlation times the book's, 0.8 against 0.3. The gain is modest and the 0.8 is estimated over three years, so start at about 20% of risk, which already lifts the Sharpe to 1.09, and grow it as live data confirms.

Step 1Why does the CIO's intuition fail?

Because returns add but risks only partly add. When two streams are imperfectly correlated, some of their bad days cancel, so the combined risk is less than the sum. The weaker strategy lowers average return per unit of capital, but it lowers risk by more, provided its correlation with the book is low enough. Two shops in a town, one selling umbrellas and one selling ice cream, each have a mediocre year on their own; owned together, their takings are steadier than either, and steadiness is what the Sharpe ratio rewards.

The relationship
add if S2>ρ S1Smax⁡=S12+S22−2ρS1S21−ρ2=1+0.64−0.480.91≈1.129\text{add if } S_2 > \rho\,S_1 \qquad S_{\max} = \sqrt{\frac{S_1^2 + S_2^2 - 2\rho S_1 S_2}{1-\rho^2}} = \sqrt{\frac{1 + 0.64 - 0.48}{0.91}} \approx 1.129
S_1Sharpe ratio of the existing book, 1.0
S_2Sharpe ratio of the candidate strategy, 0.8
\rhocorrelation between the two return streams, 0.3
What it says in wordsA strategy earns a place when its Sharpe beats the correlation times the book's Sharpe, and the best combination of two streams has the Sharpe shown.
Step 2What is the best split, and how much does it gain?

Scale both streams to the same volatility and vary the share of risk in the new one. The combined Sharpe rises from 1.0 to a peak of 1.13 with about 40% of the risk in the new strategy, then falls back towards 0.8. The curve is flat near the top: at 20% of the risk the Sharpe is already 1.09, which captures about 70% of the possible gain with half the allocation.

A weaker strategy can still lift the book's Sharpe ratio0.850.900.951.001.051.101.15book alone, Sharpe 1.0peak 1.13 at 40% of risk20%: 1.09correlation 0.3correlation 0.8: no gain0%20%40%60%80%Share of total risk given to the new strategy (both scaled to equal volatility)
With correlation 0.3, the combined Sharpe ratio rises from 1.0 to a peak of 1.13 when 40% of the risk goes to the 0.8 strategy and is already 1.09 at 20%; with correlation 0.8 the same strategy adds nothing, because 0.8 no longer beats 0.8 times 1.0.
Correlation with the bookBest share of risk in the new strategyBest combined Sharpe
0.044%1.28
0.340%1.13
0.533%1.06
0.80%1.00
0.90%1.00
The same 0.8 strategy is worth up to 1.28 of combined Sharpe at zero correlation and 1.13 at 0.3, and nothing at 0.8 or above, where the best long-only weight is zero.
Step 3How much should you trust the 0.8?

Less than the arithmetic suggests. A Sharpe ratio measured over three years has a standard error of roughly 0.66, so the true figure could easily be 0.3 or 1.3. The hurdle is only 0.3, so the decision to add some risk survives most of that uncertainty, but the size of the optimal weight does not. On a normal approximation there is about a 23% chance the true Sharpe is below the hurdle. That argues for the flat part of the curve: about 20% of risk, then grow it if the live Sharpe and the live correlation hold.

Name what the formula leaves out. Correlations measured in calm years tend to rise in a sell-off, which is when diversification is wanted most. The new strategy may have less capacity than the book, and it carries its own costs and operational risk. The answer is a yes to a measured allocation, not a yes to the peak.

Where candidates lose it

The common loss is agreeing with the CIO: a lower Sharpe must dilute a higher one. That treats Sharpe ratios as if they averaged, and ignores the risk that cancels when correlation is below one.

The second is quoting the optimal 40% split as the answer. The optimum depends on two estimated inputs, and a candidate who sizes to the peak without discussing the error in a three-year Sharpe has missed what the interviewer is really testing.

What the interviewer asks next

  • At what correlation does the new strategy add exactly nothing?
  • The new strategy's correlation with the book rises to 0.7 in sell-offs. How does that change your view?
  • How would you combine three strategies with a correlation matrix instead of two?

Company names and figures are illustrative.