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I am a math student and I am trying to understand the budget constraint in Sharpe Ratio optimization for portfolio design. Recall the budget constraint requires that the sum of the portfolio weights is 1.

Now suppose all of the expected returns are negative and identical. Then the budget constraint will result in a negative Sharpe ratio. However without this constraint the Sharpe ratio will be positive.

It seems to me that the less restrictive budget constraint that the sum of portfolio weights not equal to 0 should be used instead.

Is there a reason why the sum of portfolio weights has to be 1 ( or even positive).

Thanks

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3 Answers 3

Portfolio management is about solving problems in the real world.

In the real world, it is highly unlikely that EVERY asset has a negative expected return. If all the assets in your universe have negative returns, expand your universe to include a short-term fixed income security that is bound to produce a return greater than (or at a minimum equal to) zero. Alternately, assume it is a recession and every asset has a negative return over a short horizon. Even then, it is highly unlikely that the negative return is identical for all. This would suggest that some long-only managers could lose less money by focusing on the securities that are likely to fall less in a recession. Long-short managers could go short some of the other securities with even weaker expected returns.

There's no mathematical reason you can't have a budget constrained between 0 and 1, even if you don't consider a risk-free or short-term fixed income instrument. Then, the optimal portfolio would invest in nothing. In the real world, budget constraints are driven more by business reasons than any mathematical reason. Investors don't want to pay equity (or hedge fund) manager fees for investment management companies to not invest their money in anything. Further, many managers are not judged on an absolute basis, but a relative one. What matters is what their benchmark is doing or what their peers are doing. So even if the benchmark declines, they can still make large institutional managers happy if they decline by less than the benchmark.

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I see your argument with the math. "1" is an arbitrary choice of positive numbers, and you could choose anything. In the end, you're going to scale the whole thing to fit your capital anyway.

If you are using a numerical optimizer, it will be happier with something noticeably away from 0 and away from infinity, so I recommend choosing a specific positive number, rather than just a >0 constraint.

The real world doesn't present situations where all assets have negative returns and you have a positive budget constraint. John's idea of adding cash/t-bills should fix the problem from a mechanics/math point of view.

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In many cases, clients want to be fully invested and don't want their assets lying around in cash. Hence the budget constraint $\sum_i w_i = 1$ is fairly common in practice.

By the way, there are also cases where the constraint $\sum_i w_i = 0$ is applied: the result is a dollar neutral portfolio with long and short positions, but no net investment (short positions with $w_i < 0$ are allowed in this case). Finally, it is also possible to discard the budget constraint altogether and constrain the volatility or the tracking error instead. In that case, the portfolio has a risk budget constraint.

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