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{equation}
\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}
\end{equation}
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.
