# "Risk Matters Hypothesis" - does it really?

Risk.net has recently run a story about the "risk matters hypothesis" which refers to Sharpe’s Arithmetic and the Risk Matters Hypothesis by Haghani, Ragulin and White (2023).

If I understand its message correctly, the claim is that an average active managed portfolio will achieve worse risk-adjusted returns (measured with Sharpe ratio, I presume) than the passive portfolio. Indexers everywhere rejoice! But, since the algebra used in the article looks less rigorous than desired, I tried to validate this result with some simple concrete numbers, and arrived at, let's say, divergent conclusion.

Let the risk free rate be 0. There is a market portfolio that yields an expected return of $$\mu_m$$=5% at the risk of $$\sigma_m$$=10%. One can also invest in an active managed portfolio, which returns in expectation $$\mu_a$$=10% but with commensurately increased risk $$\sigma_a$$=20%. The two should be independent so $$\rho_{ma}$$=0, otherwise without loss of generality we could just count the correlated part of the active portfolio as part of the passive market portfolio and treat only the remainder as "active" - this can be relaxed anyway, the calculations below work also for $$\rho_{ma}$$≠0.

There is a fund 1 that has invested $$w_{1m}$$=1 000 000 into the market and $$w_{1a}$$=100 000 into the active instrument. It earns expected return $$\mu_m w_{1m} + \mu_a w_{1a}$$ = 60 000 and bears risk $$\sqrt{\mu^2_m w^2_{1m} + \mu^2_a w^2_{1a}}$$=101 980, so it achieves Sharpe ratio $$SR_1$$=58,8%.

Since in aggregate only the market can be invested into, there must also be a fund 2 that has gone short with $$w_{2a}$$=-$$w_{1a}$$=-100 000 into the actively managed portfolio, and also lets say it put $$w_{2m}$$=2 000 000 into the market. This fund earns expected return of $$\mu_m w_{2m} + \mu_a w_{2a}$$ = 90 000 in exchange for risk $$\sqrt{\mu^2_m w^2_{2m} + \mu^2_a w^2_{2a}}$$=200 998, with $$SR_2$$=44,8%.

This means that an "average" Sharpe ratio across existing funds is (58,8%+44,8%)/2=51,8%. Compare this to a passive investor who just puts everything into the market portfolio, getting $$SR$$=$$\mu_m$$/$$\sigma_m$$=50%. It would seem the "average" active manager can outperform on the risk-adjusted return basis (before costs ofc, and for this set of parameters).

Or did I misunderstand something very simple and fundamental about the article?

## 2 Answers

I didn't read the referenced paper, but I did try to replicate your numbers. I agree with your calculations for the individual Sharpe ratios for the two portfolios.

However, I do not agree with your approach to "average Sharpe" calculation. You are using a simple average but the second portfolio is nearly 2x the first.

If you take a weighted average, you will get a Sharpe ratio of 49.931%, which is below the 50% Sharpe of your market portfolio.

• Thank you, you're right that the fund size should be taken into account when calculating "average Sharpe". This doesn't seem to ameliorate the problem, though. Running the same numbers but with $w_{2m}$ = 500 000 instead, we arrive at the fund size weighted "average Sharpe" of 50,6%, still superior to a passive-only investment. I'm starting to believe the Sharpe ratio is not a right measure to do cross-section aggregates with. Commented Apr 26 at 20:49

I agree with AlRacoon that the construction of the average matters a lot here. And thinking about what matters I would like to focus, as they do in the paper, on the aggregate outcome. You can either hold the market portfolio directly, or you can hold it through active and passive managers that collectively holds it at all times by privately holding different parts of it at different times (imagine that you buy all shares of all funds and that they together hold the market portfolio). The trading among and between them does not alter your returns. Not as long as they don't alter the market porfolio that is.

How could the active funds alter the market portfolio? Maybe if they spot an undervalued company with good investment opportunities. They buy the company's share from passive hodlers thus bidding up the price giving a signal to the company it should issue more shares and invest in their business. If the active funds have made the right decision, the market portfolio and the real investments in the economy change in the right directions.

Given a static market portfolio though, the active funds can add nothing in aggregate, just some of them outperforming some others. Or as it is stated in the paper: "Logic dictates that investors cannot in aggregate be rewarded for the extra risk they incur in owning concentrated stock portfolios."