Comparing Markowitz Optimization with Sample Quadratic Utility
Summary
The document compares a conditional Markowitz optimization, using estimated expected returns and a covariance matrix to choose next-period weights, with an approach that maximizes average realized portfolio returns penalized by squared deviations from the sample mean. It asks whether these represent the same quadratic utility objective and whether they estimate weights differently. The second formulation evaluates portfolio returns across the historical sample directly, while the first expresses the decision through expected return and portfolio variance estimates.
With consistent return observations, a matching variance definition, and aligned scaling for the risk-aversion parameter, the sample objective can correspond to a mean-variance objective: maximizing average return while penalizing sample variance. Differences can arise from variance normalization, the treatment of the mean, constraints on weights, or using data and parameters differently. The document presents the question and equations but supplies no answer, derivation, or empirical comparison, so those equivalences and implementation choices remain unresolved within the source.
Key ideas
- Markowitz optimization uses expected returns and a covariance matrix to select portfolio weights.
- The alternative objective penalizes dispersion of realized portfolio returns around their sample mean.
- The objectives can align when the return sample, variance convention, and risk-aversion scaling match.
- Differences in normalization, constraints, or estimation inputs can produce different weights.
- The document poses the comparison but does not provide a derivation or conclusion.
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Full text
# Two approaches to optimizing quadratic utility
# Two approaches to optimizing quadratic utility
My understanding of the traditional Markowitz portfolio optimization process is as follows:
Let’s say I have data from year 1 to year 10. At the end of year 10 (having information about year 10), I want to optimize the portfolio weights for the year 11. I derive my optimal portfolio weights conditional on some utility function. Generally, one can describe it as follows:
\begin{equation} \max\limits_{\{w_{i,10}\}^{N}_{i=1}} E_{10}\left[u\left(\sum\limits_{i=1}^{N}w_{i,10}r_{i,11}\right)\right], \end{equation}
where $r$ denotes returns, $w$ denotes weights, $i \in N$ denotes the stocks, and $u$ denotes a utility function. In this case, MV is a quadratic utility so that:
\begin{equation} \max\limits_{\{w_{i,10}\}^{N}_{i=1}} E_{10}\left[\sum\limits_{i=1}^{N}w_{i,10}r_{i,11}-\frac{\lambda}{2}V_{11}\right], \end{equation}
where $V_{11}$ denotes the expected portfolio variance (for which I need the covariance of the stocks) and $\lambda$ denotes the risk aversion parameter.
In the simplest case, $r_{i,11}$ are just the sample averages from year 1 to year 10 and $V_{11}$ is the sample covariance matrix. One then simply derives the weights by maximizing the function. From my understanding there is a closed form solution.
Then, I have recently seen the following approach in a paper titled “Parametric Portfolio Policies: Exploiting Characteristics in the Cross Section of Equity Returns“ by Brandt et al (2005). The authors optimize portfolio weights as follows. For the sake of simplicity, $w$ denotes the vector of weights:
\begin{equation} \max\limits_w \frac{1}{10}\sum_{t=1}^{10}\left(r_{p,t}(w) - \frac{\lambda}{2}\left(r_{p,t}(w) - \frac{1}{10}\sum_{t=1}^{10}r_{p,t}(w)\right) ^2\right). \end{equation}
where the subscript $p$ denotes the portfolio (return). Again, the sample spans across 10 years of data. They maximize this with respect to $w$.
I have difficulty connecting the dots here. Are the two approaches equivalent? Do they differ in the way they estimate the weights? Are the two approaches different despite them describing the same utility function?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.