Why Sharpe Maximization Can Use a Normalized Return Constraint

Summary

The note asks why the maximum-Sharpe portfolio problem can be reformulated as minimizing portfolio variance subject to a fixed excess-return target. Its central idea is scale invariance: multiplying portfolio weights by a positive constant leaves the Sharpe ratio unchanged, so a convenient normalization can select one representative from all portfolios on the same ray. Setting expected excess return to one then turns the ratio objective into a variance minimization problem under that constraint.

The discussion sketches this transformation but does not establish all conditions needed for equivalence. In particular, the normalization requires a portfolio with positive expected excess return, and the treatment of additional constraints such as budget, long-only weights, or borrowing limits can affect whether the rescaling is valid. The post raises the question but provides no worked proof or empirical evidence, so the derivation should be checked against the precise portfolio constraints being used.

Key ideas

• The Sharpe ratio is unchanged when portfolio weights are scaled by a positive constant.
• Fixing expected excess return to one can convert Sharpe maximization into variance minimization.
• The normalization only applies when the portfolio has positive expected excess return.
• Other constraints may not remain unchanged under rescaling.

Tags

# sharpe ratio, convert into convex function, not understand that constraint,


# sharpe ratio, convert into convex function, not understand that constraint,












I am reading about tranforming sharpe ratio into convex problem

After some following, its converted into `min xTxy s.t. (u-rf e)x = 1`

```
X=x+/k+

X+ = x/(u-rf e)x
K+ = 1/ (u-rf e)T x
```

In the book, it mention adding the normalizing constraint `(u-rf)x = 1` does not affect the equation

I understand that f(x) is same as f(kx)

However, i get some questions here.

Why `(u-rf)x = 1` can confirm its maximum sharpe ratio ?

So why this constraint

## Answer by andy (score 0)

https://quant.stackexchange.com/a/79206

After reading about A Signal Processing Perspective on Financial Engineering.

I am trying to answer my question. But I am not sure that if I am correct or not. and show my question again.

I would like to ask why we add this `(µ − rf1) = 1` constraint.

```
min xT Σx
subject to xT
(µ − rf1) = 1,
xT 1 > 0.
```

First, we do homogenizing `x = kx, as f(x) = f(kx)`, so the objective function becomes `wT Σw`

```
x = x+ / k+
x+ = x / (µ − rf1)
k+ = 1 / (µ − rf1)
```

this paramater given in max, so in min k become (µ − rf1)

after `x = x+ / k+`, it will be same as x

as `x = k+x+`

as we scale constraint `wT 1 = 1` can be relaxed to `wT 1 > 0` so we can assume one arbitrarily set as `wT(µ − rf1) = 1`

```
k = (µ − rf1) `in min that` wT(µ − rf1) = 1
```

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