December 5, 2022. A quick, first-principles derivation of optimal portfolios with risk.

Introduction

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Suppose there are $n$ bets I can make, with the return per dollar for bet $k$ represented by a random variable $B_k$ with expected value $\mu_k = \mathbb{E}[B_k]$. If I invest $\omega_k$ on bet $k$, and have a fixed total $\Omega$, then I can view the value of my portfolio as a random variable

\[P(\omega_k) = \sum_{k = 1}^n \omega_k B_k.\]

If I try to optimize the expected return $\mu_P = \mathbb{E}[P]$, I get a boring linear function

\[\mu_P(\omega_k) = \sum_{k = 1}^{n} \omega_k \mu_k.\]

To maximize this, introduce a Lagrange multiplier $\gamma$ to enforce the fixed total:

\[\mu_P(\omega_k, \gamma) = \sum_{k = 1}^{n} \omega_k \mu_k + \gamma \left(\Omega - \sum_{k=1}^{n-1}\omega_k\right).\]

This is linear and has no local maxima, so the maximal return must lie at the edge of the feasible region. In fact, it’s clear that we just invest all our money in the bet with maximum return:

\[P^* = \Omega B_{k^*}, \quad k^* = \text{argmax}_k\,\mu_k.\]

But putting all your eggs in one basket seems like a bad idea. My intuition is that to optimize my portfolio, my investment should include a spread of high-risk, high-return and low-risk, low-return bets. What have we missed?

Derisky business

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Maximizing expected return ignores the risk altogether! The simplest way to measure the risk of our portfolio is the total variance,

\[\sigma^2_P = \mathbb{E}[(P - \mu_P)^2].\]

If the bets are independent random variables, then the variance is additive, with

\[\sigma^2_P(\omega_k) = \sum_{k = 1}^{n} \omega_k^2 \sigma^2_k,\]

where $\sigma^2_k$ is the variance of $B_k$. If they are not independent, then we add covariance terms:

\[\sigma^2_P(\omega_k) = \sum_{k = 1}^{n} \omega_k^2 \sigma^2_k + \sum_{j \neq k} \omega_j\omega_k\text{cov}(B_j, B_k), \quad \text{cov}(B_j, B_k) = \mathbb{E}[(B_j - \mu_j)(B_k - \mu_k)].\]

Instead of just maximizing expected return, we should balance return and risk. A simple way to do this is to maximize the convex combination of $\mu_k$ and $-\sigma^2_P$, which we’ll call the $\lambda$-derisked return:

\[R_\lambda(\omega_k) = (1 - \lambda) \mu_P(\omega_k) - \lambda \sigma^2_P(\omega_k).\]

The expected return is $0$-derisked, while $1$-derisked return minimizes the variance of the portfolio and ignores the return completely.

Distributing eggs

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For simplicity, let’s assume our bets are independent. Otherwise, we simply diagonalize the covariance matrix and go to a basis of orthogonal bets. Then adding a Lagrange multiplier as above, we get

\[\begin{align*} R_\lambda(\omega_k, \gamma) & = \sum_{k=1}^n \left[(1-\lambda) \omega_k\mu_k - \lambda \omega_k^2\sigma_k^2\right] + \gamma \left(\Omega - \sum_{k=1}^n\omega_k\right). \end{align*}\]

The partial derivatives are

\[\partial_{\omega_k} R_\lambda = (1-\lambda)\mu_k + \gamma - 2\lambda \omega_k \sigma_k^2,\]

so we have an extremum at

\[\omega_k = \frac{(1-\lambda)\mu_k + \gamma}{2\lambda \sigma_k^2}.\]

To determine the value of $\gamma$, note from our constraint that

\[\begin{align*} \Omega & = \sum_{k=1}^n\omega_k \\ & = \sum_{k=1}^n\frac{(1-\lambda)\mu_k + \gamma}{2\lambda \sigma_k^2} \\ \Longrightarrow \quad \gamma & = 2\lambda\left(\sum_{k=1}^n\frac{1}{\sigma_k^2}\right)^{-1}\left[\Omega - \sum_{k=1}^n\frac{(1-\lambda)\mu_k}{2\lambda \sigma_k^2}\right]. \end{align*}\]

Since $\gamma \simeq \lambda$, for small $\lambda$ (a return-oriented investor) we can ignore that $\gamma$ term. Then the investments

\[\omega_k \approx \frac{(1 - \lambda)\mu_k}{2\lambda\sigma^2_k} \propto \frac{\mu_k}{\sigma^2_k},\]

so we weight investments proportional to expected return, but inversely to variance. Sounds sensible! On the other hand, when $\lambda \to 1$ (a risk-averse investor), the Lagrange multiplier $\gamma \gg (1 - \lambda)\mu_k$, so that the investment is proportional to the inverse variance only:

\[\omega_k\approx \frac{\gamma}{2\lambda\sigma_k^2} \propto \frac{1}{\sigma_k^2}.\]

Intermediate values of $\lambda$ interpolate between these two regimes, with a degeneracy at $\lambda = 0$ where only expected return matters. Thus, we have a whole one-parameter family of risk-sensitive ways to value a portfolio!