Why Minimum-Variance Portfolios Use Return Covariances

Summary

The discussion asks whether a minimum-variance portfolio of currency derivatives can be optimized using the variance and correlation of each instrument’s value instead of its returns. The answer explains why covariance of asset values is unsuitable: changing an allocation’s scale would change the measured variance of that holding, even though the underlying asset’s risk has not changed. Portfolio weights should combine measures of asset risk that are independent of the amount invested.

For derivatives, the response recommends modeling the underlying instruments, pricing the derivatives, and then measuring portfolio risk. It does not provide a worked example, specify a particular return convention, or explain how to estimate derivative sensitivities or covariance inputs. The key distinction is between the value of a position, which depends on allocation size, and an asset’s risk measure, which should support meaningful comparison and portfolio construction.

Key ideas

• Covariance of asset values changes with the scale of the investment and is not an appropriate standalone risk input.
• Return-based risk measures allow portfolio weights to combine asset risks consistently.
• Derivative portfolio risk requires modeling the underlyings and pricing the derivatives before measuring risk.
• The discussion does not specify an estimation method or a detailed derivative risk model.

Tags

# Portfolio Theory: Must VarCovar Matrix be based on return var/covar?


# Portfolio Theory: Must VarCovar Matrix be based on return var/covar?












I am trying to estimate the minimum variance portfolio where the assets are currency derivatives. In the specific case it does not make sense to base correlations or variance on asset returns. I am interested in getting a low variance in the actual value of the portfolio and not the returns. Can I just calculate the variance and correlation on the back of the individual asset value or does it have to be returns for the MVP theory to make sense?

Thanks in advance!

## Answer by user20429 (score 1)

https://quant.stackexchange.com/a/25568

It doesn't make sense to use the (co)variance(s) of asset values; if you did, by cutting an investment's share of the allocation by half, you would also cut its variance by a factor of 4. In a meaningful portfolio design, the volatility (variance) of an asset, by itself, is the same no matter how much or how little of your portfolio you put in it.

Why doesn't it make sense to use correlations and variance of RETURNS for currency derivatives?

## Answer by user2183336 (score 0)

https://quant.stackexchange.com/a/25575

You need to model the underlyings, price the derivatives, and then measure risk.

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