Interpreting Mean-Variance Portfolio Weights and Lagrange Multipliers

# Mean Variance Analysis: what does the solution of the following exercise tells me?


# Mean Variance Analysis: what does the solution of the following exercise tells me?












I'm new in here and I hope this is the right board to ask this question. I'm at second year of university and in the Informatics II course the lecturer made us solve the following mean variance analysis for two assets in R:

- we have create the following matrix $A$: $$A=\begin{bmatrix} 2\sigma_{11}&2\sigma_{12} &-\overline{r}_1 &-1 \\ 2\sigma_{21}& 2\sigma_{22} & -\overline{r}_2 & -1\\ \overline{r}_1&\overline{r}_2 &0 &0 \\ 1& 1 & 0 & 0 \end{bmatrix}=\begin{bmatrix} 0.0002310968 & 0.0002230081 & -0.0003322979 &-1 \\ 0.0002230081& 0.0003104519 & 0.0004241650 &-1 \\ -0.0003322979& -0.0004241650 & 0 & 0\\ 1& 1& 0& 0 \end{bmatrix}$$ where: $2\sigma_{ij}$ denotes the covariance the rates of return between assets $i$ and $j$, and $\overline{r}_i$ denotes the expected rate of return for assets $i$;

- we have created the following vector $b$: $$b=\begin{bmatrix} 0\\ 0\\ \overline{r}\\ 1 \end{bmatrix}=\begin{bmatrix} 0\\ 0\\ 0.35\\ 1 \end{bmatrix}$$ where: $\overline{r}$ is the expected portfolio return;

- we solved $A^{-1}b$ to find vector $x$ defined as: $$x=\begin{bmatrix} w_1\\ w_2\\ \lambda\\ \mu \end{bmatrix}$$

- we found that: $$x=\begin{bmatrix} 3814.468561\\ -3813.468561\\ -3965.712523\\ 1.348875 \end{bmatrix}$$

What is the financial meaning of these numbers ($w_1$, $w_2$, $\lambda$, $\mu$)? Thanks for the answer.

## Answer by roym00 (score 1, accepted)

https://quant.stackexchange.com/a/15258

From how you outlined your solution, you are computing the mean variance portfolio with minimum risk and with target return $\overline{r}$.

I'd say that you are solving an optimization using Lagrange multiplier method given the values of matrix A.

$\lambda$ and $\mu$ are the Lagrande multipliers: these parameters measure the sensitivity of the Lagrange function with respect to the constraints of the mean variance problem, which are

- the return of the mean variance portfolio must be equal to $\overline{r}$: $$w_1 \overline{r}_1 + w_2 \overline{r}_2 = \overline{r}$$

- the sum of the weights ($w_1$ and $w_2$) must be equal to $1$: $$w_1 + w_2 = 1$$

From the constraints, you can understand that $w_1$ and $w_2$ are the weights of the assets in the mean variance portfolio: you should buy $3814.468561$ units of asset 1 and you should short sell $3813.468561$ units of asset 2.

$\lambda = -3965.712523$ and $\mu = 1.348875$ shows that the Lagrange function has a strong negative sensitivity to constraint 1 and a positive low sensitivity to constraint 2. This means that constraint 1 was the real driver to the solution of the optimization (very high absolute value); the negative sign means that the Lagrangian function decreases when the $w_1 \overline{r}_1 + w_2 \overline{r}_2 - \overline{r}$ increases (this is just constraint 1 rewritten), i.e. a portfolio return higher than the target one, \overline{r}, is rewarded during the optimization.

Summarizing, $w_1$ and $w_2$ have a financial meaning (how many units of each asset you should buy), while $\lambda$ and $\mu$ give hints about the optimization process (how the constraints interact with the objective function).

## Answer by Drew (score 0)

https://quant.stackexchange.com/a/15259

The mean-variance portfolio is given by $\sigma_t^{-1} \theta_t$ where $\theta_t$ is the market price of risk $\frac{\mu_t - r_t}{\sigma_t}$. Here your taking $\Sigma^{-1} (\mu_t - r_t)$ and using block matrices to enforce the constraints as outlined by Arrigo.