Modern Portfolio Theory and Mean-Variance Optimization › Portfolio Returns and Risk

Before we can optimize portfolios, we need precise definitions of portfolio return and risk. These definitions form the foundation for Modern Portfolio Theory. We need accurate measures of portfolio return and risk to compare investment strategies and identify optimal allocations.

Suppose we allocate wealth across $n$ assets. The fundamental question is: how should we combine individual asset characteristics to describe the portfolio as a whole? Let $w_i$ denote the fraction of wealth invested in asset $i$, where the weights must sum to one:

$$\sum_{i=1}^{n} w_i = 1$$

where:

• $w_i$: fraction of wealth invested in asset $i$
• $n$: total number of assets in the portfolio

This constraint ensures we account for all invested capital. A weight of 0.3 means 30% of the portfolio is allocated to that asset. We allow short selling, meaning weights can be negative (representing borrowed positions). Shorting an asset involves selling borrowed shares with the obligation to return them later, creating a negative position that profits when the price falls.

The expected return of a portfolio follows naturally from the linearity of expectation, a concept we explored in our probability foundations. If each asset $i$ has expected return $\mu_i = E[R_i]$, the portfolio's expected return is simply the weighted average:

$$\mu_p = E[R_p] = \sum_{i=1}^{n} w_i \mu_i = \mathbf{w}^\top \boldsymbol{\mu}$$

where:

• $\mu_p$: expected return of the portfolio
• $E[R_p]$: expected value of portfolio return
• $w_i$: weight of asset $i$
• $\mu_i$: expected return of asset $i$
• $\mathbf{w}$: vector of portfolio weights, $(w_1, \ldots, w_n)^\top$
• $\boldsymbol{\mu}$: vector of expected returns, $(\mu_1, \ldots, \mu_n)^\top$

Portfolio expected return is linear in the weights. Doubling an asset's weight doubles that asset's contribution $w_i\mu_i$; the change in total portfolio return also depends on which holdings fund the increase. Vector notation provides a compact way to express this relationship for many assets.

The portfolio variance requires more care and reveals how diversification works mathematically. Unlike expected return, variance does not simply average across assets. The portfolio return is:

$$R_p = \sum_{i=1}^{n} w_i R_i$$

where:

• $R_p$: return of the portfolio
• $w_i$: weight of asset $i$
• $R_i$: return of asset $i$
• $n$: number of assets

To find the variance of this weighted sum, we must account for how each pair of assets moves together. Taking the variance and using bilinearity of covariance:

$$\begin{aligned} \sigma_p^2 &= \text{Var}(R_p) \\ &= \text{Var}\left(\sum_{i=1}^{n} w_i R_i\right) \\ &= \sum_{i=1}^{n} \sum_{j=1}^{n} w_i w_j \text{Cov}(R_i, R_j) && \text{(bilinearity of covariance)} \end{aligned}$$

where:

• $\sigma_p^2$: variance of the portfolio
• $w_i, w_j$: weights allocated to assets $i$ and $j$
• $\text{Cov}(R_i, R_j)$: covariance between returns of assets $i$ and $j$

The double summation captures all pairwise interactions between assets. When $i = j$, the covariance term becomes the variance of asset $i$, contributing $w_i^2 \sigma_i^2$ to portfolio variance. When $i \neq j$, the covariance measures how assets $i$ and $j$ move together, and this is where diversification benefits emerge.

Let $\Sigma$ denote the $n \times n$ covariance matrix, where $\Sigma_{ij} = \text{Cov}(R_i, R_j)$. Then the portfolio variance is:

$$\sigma_p^2 = \mathbf{w}^\top \Sigma \mathbf{w}$$

where:

• $\sigma_p^2$: variance of the portfolio
• $\mathbf{w}$: vector of portfolio weights
• $\Sigma$: covariance matrix of asset returns

This compact matrix notation, which we developed in Part I's linear algebra chapter, makes optimization tractable. The quadratic form $\mathbf{w}^\top \Sigma \mathbf{w}$ appears throughout financial mathematics because it elegantly encapsulates how portfolio risk depends on individual asset variances (the diagonal entries $\Sigma_{ii}=\sigma_i^2$) and pairwise covariances (the off-diagonal entries $\Sigma_{ij}=\text{Cov}(R_i,R_j)$ for $i\ne j$).