Markowitz Optimization and the Choice of Return Series

# Confusion about Assumptions in Markowitz Optimization


# Confusion about Assumptions in Markowitz Optimization












### Setup and Definition of Terms

Supposed that we have a universe of possible securities $\mathcal{S}$. We wish to construct an "optimal" portfolio, which will be represented by proportional weights $\{w_i\}_{i\in\mathcal{S}}$. The word "proportional" simply means that the weights sum to one:

$$\sum_i w_i = 1$$

Suppose further that we have some past price data on each stock from time $t=0$ to time $t=t_f$, and the data are sampled at discrete points $t_n$. We thus have, for each security, and at each timepoint $n$, a price value

$$P_{i,n} = P_i(t_n).$$

Let us define the "floating" returns of security $i$ as the fractional difference in price between timesteps:

$$R^{(f)}_{i,n} = \frac{P_{i,n}-P_{i,n-1}}{P_{i,n-1}}.$$

Let us define "cumulative" returns as the fractional difference in price since the beginning of our data:

$$R^{(c)}_{i,n} = \frac{P_{i,n}-P_{i,0}}{P_{i,0}}.$$

Let $V_0$ denote the initial investment put into the portfolio, and $V_n$ denote the value of the portfolio at time $t_n$. Note that

$$V_n = V_0 \sum_{i\in\mathcal{S}}w_i\frac{P_{i,n}}{P_{i,0}} = V_0 + V_0 \sum_{i\in\mathcal{S}}w_iR^{(c)}_n.$$

We will reserve the subscript $p$ to refer to our total portfolio. Using this convention, the floating returns of the portfolio are denoted by

$$ R^{(f)}_{p,n} = \frac{V_n-V_{n-1}}{V_{n-1}}$$

and the cumulative returns are denoted by

$$R^{(c)}_{p,n} = \frac{V_n-V_0}{V_0}.$$

### The Essential Observation

Observe the following important fact: the floating returns of the portfolio are not linear in the weights. That is to say,

$$R^{(f)}_{p,n} \neq \sum_{i}w_i R^{(f)}_{i,n}.$$

This can be seen by expanding the expressions for portfolio and individual security-level floating returns. In contrast, the cumulative returns ARE linear in the weights:

$$R^{(c)}_{p,n} = \sum_{i} w_i R^{(c)}_{i,n}.$$

This can be easily seen from our definition of portfolio cumulative returns.

### Statement of the Optimization Problem

In order to use linear algebra notation, we'll let $w$ refer to the column vector of weights, $\overline{R}$ refer to the column vector of time-averaged returns (I'm being intentionally ambiguous here) and $\Sigma$ the covariance matrix of the same returns.

To my understanding, the major idea behind Markowitz optimization is to minimize the objective function

$$f(w) = w^T\Sigma w - \gamma \,w^T\overline{R}$$

subject to the constraints $\sum w = 1$ and $w_i >0$. Here, $\gamma$ is some real parameter that accounts for risk tolerance.

My question is the following: do we use cumulative returns $R^{(c)}_n$ or floating returns $R^{(f)}_n$ in computing $\overline{R}$ and $\Sigma$?

It seems the most likely choice would be floating returns. Since the distribution of the cumulative returns is highly autocorrelated, we are not making independent draws from a sample, so the interpretation of the sample mean and variance $\overline{R}$ and $\Sigma$ is not clear to me. However, if we use floating returns, we run into the problem that our objective function does not represent the expected returns (or variance) of the portfolio! Instead it represents some statistical property of the portfolio, the mean expected returns and variance. This seems strange to me, as I assume we would want to optimize the actual returns of our portfolio.

Can anyone clear this up for me? I've been banging my head against this for a while now and haven't found anything that has clarified things for me.

As a side question, the terms "floating returns" and "cumulative returns" are things that I use to keep things straight for myself; are there standard terms for these objects?