# What does risk tolerance represent for utility-maximizing optimization with linear constraints?

Referencing Wei Jiao (2003) p. 8, formula (1.12), for $$Ax = b$$ set of linear constraints in a portfolio, the solution for the optimum weights to maximize the utility is: $$w^* = \Sigma^{-1}A^T \left( A \Sigma^{-1} A^T \right)^{-1}b + \lambda \Sigma^{-1} \left( \mu-A^T \left( A \Sigma^{-1} A^T \right) A \Sigma^{-1} \mu\right)$$

Based on a similar question, $$\lambda$$ could be removed by dividing by the sum of the numerator, ie dividing the second component of $$w^*$$ by the sum of the component. Is this the correct way of eliminating the risk-aversion variable?

• Are you familiar with the method of Lagrange multipliers? – Matthew Gunn Nov 1 '18 at 16:46

You cannot eliminate the dependence of a solution on the risk aversion parameter (which this author confusingly calls $$\lambda$$).

### Perhaps a source of confusion?

• Typically $$\lambda$$ is used to denote a Lagrange multiplier in Lagrangian optimization, but the author is using $$\lambda$$ as a risk tolerance parameter. (In your other linked question, $$\lambda$$ denotes a Lagrange multiplier.)

The author uses a $$m \times 1$$ vector $$\boldsymbol{\gamma}$$ as the Lagrange multipliers and the scalar $$\lambda$$ as the risk tolerance parameter for utility specification $$u(\mathbf{w}) = \mu_p - \frac{1}{2 \lambda} \sigma^2_p$$ where $$\mu_p$$ is the expected portfolio return given portfolio weights $$\mathbf{w}$$ and $$\sigma^2_p$$ is the variance of the portfolio return.

For a closed forms solution to the optimization problem, the author's goal is to find an expression for the solution that does not use the multipliers. In this case, that means eliminating $$\boldsymbol{\gamma}$$ (which he does). In your other link, $$\lambda$$ is used in the typical way as a Lagrange multiplier so a closed form solution eliminates $$\lambda$$.

The optimization problem by the way is: $$$$\begin{array}{*2{>{\displaystyle}r}} \mbox{maximize (over \mathbf{w})} & \boldsymbol{\mu}'\mathbf{w} - \frac{1}{2 \lambda} \mathbf{w}'\Sigma \mathbf{w} \\ \mbox{subject to} & A \mathbf{w} = \mathbf{b} \end{array}$$$$

The solution $$\mathbf{w}^*$$ will be a function of expected returns $$\boldsymbol{\mu}$$, covariance matrix $$\Sigma$$, and risk tolerance $$\lambda$$.

This is a simple, pretty standard problem, and you can undoubtedly find other people solving it all over the Internet.

### Motivation for objective $$\boldsymbol{\mu}'\mathbf{w} - \frac{1}{2 \lambda} \mathbf{w}'\Sigma \mathbf{w}$$

Let's assume the agents preferences over various lotteries can be represented by:

Let $$X$$ be some lottery that's normally distributed with mean $$\mu$$ and variance $$\sigma^2$$. $$X \sim \mathcal{N}(\mu, \sigma^2)$$

You can show that this lottery $$X$$ has a certainty equivalent value to our agent given by: $$c(X) = \mu - \frac{1}{2}a\sigma^2$$ (Start with $$\mathbb{E}[-e^{-aX}] = \frac{1}{\sqrt{2\pi}\sigma}\int_{- \infty}^{\infty}-e^{-ax-\frac{1}{2}\left(\frac{x - \mu}{\sigma}\right)^2}$$ and use normal pdf sums to 1 to find certainty equivalent.)

Here, $$a$$ is Arrow-Pratt coefficient of absolute risk aversion. You can also define risk tolerance $$\tau = \frac{1}{a}$$. Writing the certainty equivalent with risk tolerance:

$$c(X) = \mu - \frac{1}{2\tau}\sigma^2$$

Maximizing expected utility with CARA risk aversion over a normally distributed lottery is equivalent to maximizing the certainty equivalent given above. It's a nice, convenient specification that makes the math easy to work with.

You can of course point out all kinds of deficiencies which would motivate richer specifications:

• Portfolio returns covary with other variables agents care about.
• Returns aren't normally distributed
• CARA has problems: would you insure the risk of a 1,000 loss the same way if your wealth was 2,000 as if your wealth was 2,000,000,000? Probably not.
• Thank you for your elaboration, it's helped clarify some things. Using a similar framework as described, would it be a similar process to eliminate the risk tolerance variable so that w* is a function of mu, covar_matrix and the Lagrangian multipliers? – Cameron Cox Nov 2 '18 at 15:48