Appendix: Why the Mean-Variance Objective?

Why is the mean–variance (MV) objective function \[ \max_w \mathbf{w}^T \mathbf{\mu} - \frac{\gamma}{2} \mathbf{w}^T \mathbf{\Sigma} \mathbf{w} \] a valid approximation of investor preferences?

Here is one justification of the Mean-Variance optimization problem using the Taylor Series Approximation. Refer to Levy and Markowitz (1979) for more on this topic.

Levy, H., and H. M. Markowitz. 1979. “Approximating Expected Utility by a Function of Mean and Variance.” The American Economic Review 69 (3): 308–17. http://www.jstor.org/stable/1807366.

The Goal

We start with an investor who wants to maximize the expected utility of their final wealth, \(E[U(W)]\). We want to show that it can be approxmiated by the optimization problem: \(\max_w \mathbf{w}^T \mathbf{\mu} - \frac{\gamma}{2} \mathbf{w}^T \mathbf{\Sigma} \mathbf{w}\).

Step 1: The Setup

Let

  • \(W\) be the random variable representing final wealth
  • \(\mu_W = E[W]\) be the expected final wealth
  • \(\sigma^2_W = E[(W - \mu_W)^2]\) be the variance of final wealth
  • \(U(W)\) be a utility function where \(U'>0\) (more wealth is good) and \(U''<0\) (risk aversion)

We define the Certainty Equivalent (CE), the guaranteed cash amount that provides the same utility as the risky wealth \(W\). \[U(CE) = E[U(W)]\]

Since utility increases with wealth, maximizing Expected Utility is equivalent to maximizing the Certainty Equivalent (CE).

Step 2: Approximating the “Risky” Side (\(E[U(W)]\))

We approximate the utility of the random wealth \(W\) using a second-order Taylor Series expansion around the expected wealth \(\mu_W\).

\[U(W) \approx U(\mu_W) + U'(\mu_W)(W - \mu_W) + \frac{1}{2}U''(\mu_W)(W - \mu_W)^2\]

Now, we apply the expectation operator \(E[\cdot]\) to the entire equation:

\[E[U(W)] \approx E[U(\mu_W)] + E[U'(\mu_W)(W - \mu_W)] + E\left[\frac{1}{2}U''(\mu_W)(W - \mu_W)^2\right]\]

Since \(\mu_W\), \(U(\mu_W)\), and its derivatives are constants, they move outside the expectation:

\[E[U(W)] \approx U(\mu_W) + U'(\mu_W)\underbrace{E[W - \mu_W]}_{0} + \frac{1}{2}U''(\mu_W)\underbrace{E[(W - \mu_W)^2]}_{\sigma^2_W}\]

Note: The middle term is zero because the expected deviation from the mean is always zero.

Result A: \[E[U(W)] \approx U(\mu_W) + \frac{1}{2}U''(\mu_W)\sigma^2_W\]

Step 3: Approximating the “Certainty” Side (\(U(CE)\))

We also approximate the utility of the Certainty Equivalent \(CE\). We expand this around \(\mu_W\) as well. (This is valid assuming the risk premium is small, so \(CE\) is close to \(\mu_W\)).

\[U(CE) \approx U(\mu_W) + U'(\mu_W)(CE - \mu_W)\]

(We stop at the first order here because \(CE\) is a deterministic constant, not a random variable, and we assume the difference \((CE-\mu_W)\) is small enough that the squared term is negligible.)

Result B: \[U(CE) \approx U(\mu_W) + U'(\mu_W)(CE - \mu_W)\]

Step 4: Equating and Solving for CE

Recall our definition: \(U(CE) = E[U(W)]\). Therefore, Result A = Result B.

\[U(\mu_W) + U'(\mu_W)(CE - \mu_W) \approx U(\mu_W) + \frac{1}{2}U''(\mu_W)\sigma^2_W\]

Subtract \(U(\mu_W)\) from both sides:

\[U'(\mu_W)(CE - \mu_W) \approx \frac{1}{2}U''(\mu_W)\sigma^2_W\]

Divide by \(U'(\mu_W)\) to isolate \(CE\):

\[CE - \mu_W \approx \frac{1}{2} \frac{U''(\mu_W)}{U'(\mu_W)} \sigma^2_W\]

\[CE \approx \mu_W + \frac{1}{2} \frac{U''(\mu_W)}{U'(\mu_W)} \sigma^2_W\]

Step 5: Introducing the Risk Aversion Parameter

We substitute the Arrow-Pratt coefficient of absolute risk aversion, defined as \(A = -\frac{U''(\mu_W)}{U'(\mu_W)}\).

\[CE \approx \mu_W - \frac{1}{2} A \sigma^2_W\]

Step 6: Converting Wealth to Portfolio Returns

The equation above is in terms of wealth. We must convert it to portfolio returns.

Let \(W_0\) be initial wealth and \(R_p\) be the random portfolio return with mean \(\mu_p\) and variance \(\sigma^2_p\). Final wealth is: \(W = W_0(1 + R_p)\). We can compute the mean and variance of final wealth \(W\):

  1. Mean: \(\mu_W = W_0(1 + \mu_p) = W_0 + W_0 \mu_p\)
  2. Variance: \(\sigma^2_W = W_0^2 \sigma^2_p\)

Substitute these into the CE equation:

\[CE \approx (W_0 + W_0 \mu_p) - \frac{1}{2} A (W_0^2 \sigma^2_p)\]

Step 7: The Final Optimization

The investor wants to maximize \(CE\). In an optimization problem, we can remove additive constants (\(W_0\)) and divide by positive constants (\(W_0\)) without changing the location of the maximum.

  1. Maximize CE: \[\max \left[ W_0 + W_0 \mu_p - \frac{1}{2} A W_0^2 \sigma^2_p \right]\]

  2. Remove additive constant \(W_0\): \[\max \left[ W_0 \mu_p - \frac{1}{2} A W_0^2 \sigma^2_p \right]\]

  3. Divide by scaling factor \(W_0\): \[\max \left[ \mu_p - \frac{1}{2} (A \cdot W_0) \sigma^2_p \right]\]

  4. Define Relative Risk Aversion (\(\gamma\)): We define \(\gamma = A \cdot W_0\). This is the coefficient of Relative Risk Aversion.

  5. Substitute Vector Notation: \(\mu_p = \mathbf{w}^T \mathbf{\mu}\) and \(\sigma^2_p = \mathbf{w}^T \mathbf{\Sigma} \mathbf{w}\)

We arrive at the final optimization problem:

\[\max_{\mathbf{w}} \quad \mathbf{w}^T \mathbf{\mu} - \frac{\gamma}{2} \mathbf{w}^T \mathbf{\Sigma} \mathbf{w}\]