Scaling Portfolio Variance When Weights Sum to One
Summary
The note asks how to interpret the quadratic term in a long-only mean–variance optimization problem when portfolio weights are normalized to sum to one. With a return covariance matrix, the expression formed by multiplying weights on both sides of the matrix is the variance of portfolio returns. The question then distinguishes those normalized weights from dollar holdings in an investment of a specified size.
The key scaling principle is that multiplying every holding by the investment amount scales the variance of dollar profit or loss by the square of that amount, while expected dollar return scales linearly. Thus a portfolio fully allocated to one asset with a stated return variance has dollar-profit variance equal to the asset’s variance times the square of the investment. The note raises this as a mathematical concern, but does not resolve whether dollar variance is the most useful risk measure; interpretation also depends on whether the covariance matrix describes returns or dollar changes and on the units used.
Key ideas
- The quadratic form of portfolio weights and the return covariance matrix gives portfolio return variance.
- Scaling normalized weights by the investment amount scales dollar profit variance by the square of that amount.
- Expected dollar return scales linearly with investment size, unlike variance.
- The covariance matrix’s units determine whether the result represents return variance or dollar-profit variance.
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Full text
# Mean-Variance portfolio: How do I compute the variance when the portfolio is normalized
# Mean-Variance portfolio: How do I compute the variance when the portfolio is normalized
Let's consider the very basic of a Mean-Variance Portfolio:
$$ \text{max}_{x} (1-\lambda)\sum_i^n\mu_ix_i-\lambda\sum_i^n\sum_j^n x_i Q_{ij}x_j $$ $$\text{ s.t. }\sum_i^nx_i=1 \text{ , } x_i \geq 0 \text{ (No shorting) }$$ where $\lambda, \mu, Q$ is the risk averse parameter, expected return vector and variance-covariance matrix for all the assets.
I Interpret $\sum_i^n\sum_j^n x_i Q_{ij}x_j$ as the variance of my protfolio, am I correct?
Let's assume I invest 10\$ and my first constraint is normalizing the 10\$ to 1 so I have to multiply my results by 10. Let's assume I use software to find the optimal portfolio with the weights $x_1...x_n$ so the value of asset $i$ is $x_i*100\%$ of my total investment.
My expected return is then: $10\$\sum_i^n\mu_ix_i$
QUESTION: What is the variance (risk) of the portfolio then? Becuase of the non-linearity I cant simply multiply by 10.
My first thought is of course to multiply my weights $x_i$ with 10 and simply compute the variance by $\sum_i^n\sum_j^n x_i Q_{ij}x_j$:
EDIT
But consider this case: I have a low $\lambda$ (I am "risk loving") so my solution to the problem is to invest all my money in asset $c$ where the expected return is $\mu_c=23\%=0.23$ and variance of the return is $Q_{c,c}=0.9$. This suggests that my variance is $10^2*0.9=90$ while expected to have a profit of 2.3\$. Am I correct? My concern is then that my variance or risk is so much higher initial investment so it makes me question if my math is correct.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.