# 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:

$$$$\max\limits_{\{w_{i,10}\}^{N}_{i=1}} E_{10}\left[u\left(\sum\limits_{i=1}^{N}w_{i,10}r_{i,11}\right)\right],$$$$

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:

$$$$\max\limits_{\{w_{i,10}\}^{N}_{i=1}} E_{10}\left[\sum\limits_{i=1}^{N}w_{i,10}r_{i,11}-\frac{\lambda}{2}V_{11}\right],$$$$

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:

$$$$\max\limits_w \frac{1}{10}\sum_{t=1}^{10}\left(r_{p,t}(w) - \frac{\lambda}{2}\left(r_{p,t}(w) - \frac{1}{10}\sum_{t=1}^{10}r_{p,t}(w)\right) ^2\right).$$$$

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?

• What’s the source of the optimization approach with gradient descent? Sep 23, 2022 at 15:05
• papers.ssrn.com/sol3/papers.cfm?abstract_id=661343 this paper. They replace weights with a linear function. However, the baseline Markowitz scenario they are describing is as abovementioned. Sep 24, 2022 at 21:02
• IMO it is the same idea, but they differ in how they calculate the portfolio variance. Method 1 explicitly estimates the covariance matrix V, while method 2 is simpler because it just computes the variance of the chosen portfolio (and the candidate portfolio during each iteration) so the V matrix is not used. (I suppose not storing the V matrix saves some storage, but I am not sure if that is an important consideration). Sep 26, 2022 at 7:27
• @nbbo2 okay I get that. However - what is a reason to not do it as in method 2? Especially now that you can choose from a plethora of easily implementable optimizers, such as gradient descent? Sep 27, 2022 at 11:09