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Problem Statement for the definition of the Efficient Frontier

Let $\mu$ be the vector of expected returns and $\Omega$ the returns covariance matrix of $n$ assets. The Markowitz optimization problem is to find the minimum variance portfolio that achieves an expected return $\mu_p$.

$$w^* = {\arg\,\min} \frac{1}{2} w^t \Omega w$$

subject to the sum of weights constraints $u^t w = 1$, and returns constraints $\mu^t w = \mu_p $ where $u$ is the unit vector composed of ones: $u^t=(1, \ldots 1)$.

Note that for simplicity, we show a version of the problem that does not have non-negative weight constraint.

Solution by Lagrange Multipliers

We write the Lagrangian: $$L = \frac{1}{2} w^t \Omega w - \lambda_1 (w^t u - 1) - \lambda_2 (w^t \mu - \mu_p) $$

The solution is obtaining by writing that its derivatives at the optimum are 0:

$$\frac{\partial L}{\partial w_i} = \sum_j \Omega_{ij} w^*_j - \lambda_1 u_i - \lambda_2 \mu_i = 0$$

$$\frac{\partial L}{\partial \lambda_1} = u^t w^* -1 = 0$$

$$\frac{\partial L}{\partial \lambda_2} = \mu^t w^* - \mu_p = 0$$

Cancelling the $n$ gradient w.r. to $w_i$ gives us:

$$w^* = \Omega^{-1} ( \lambda_1 u + \lambda_2 \mu)$$

The cancellation of the gradient w.r. to $\lambda_1$ and $\lambda_2$ give:

$$1 = u^t w^* = \lambda_1 u^t \Omega^{-1} u + \lambda_2 u^t \Omega^{-1} \mu$$

$$\mu_p = \mu^t w^* = \lambda_1 \mu^t \Omega^{-1} u + \lambda_2 \mu^t \Omega^{-1} \mu$$

Introducing the simplifying notations $a,b,c$:

$$a = u^t \Omega^{-1} u$$

$$b = \mu^t \Omega^{-1} u$$

$$c = \mu^t \Omega^{-1} \mu$$

with the determinant $\Delta = ac-b^2$, we have :

$$ \lambda_1 = \frac{c - b \mu_p}{\Delta}$$

$$ \lambda_2 = \frac{a \mu_p - b}{\Delta}$$

Shape of the Efficient Frontier

The portfolio variance $\sigma_p$ verifies:

$$\sigma_p^2 = w^t \Omega w = w^t \Omega (\lambda_1 \Omega^{-1} u + \lambda_2 \Omega^{-1} \mu$$

$$\sigma_p^2 = \lambda_1 w^t u + \lambda_2 w^t \mu = \lambda_1 + \lambda_2 \mu$$

$$\sigma_p^2 = \frac{a \mu_p^2 -2 b \mu_p +c}{\Delta}$$

Therefore, the efficient frontier is a parabola in return/variance space.

Two funds theorem

The minimum variance is obtained for

$$\lambda_1 = 1/a, \lambda_2 =0, \mu=b/a, w_g = \frac{1}{a} \Omega^{-1} u = \frac{\Omega^{-1} u}{u^t \Omega^{-1} u}$$

The other portfolio that cancels the constraint $\lambda_1$ is given by:

$$\lambda_1 = 0, \lambda_2 =1/b, \mu=c/b, w_d = \frac{1}{b} \Omega^{-1} \mu = \frac{\Omega^{-1} \mu}{u^t \Omega^{-1} \mu}$$

Any optimal portfolio with weights $w$ is a linear combination of these 2 portfolio since:

$$ w = (\lambda_1 a) w_g + (\lambda_2 b) w_d $$

The multipliers are picked by the optimizer to achieve the required return $\mu_p$ at minimal variance.

The minimum variance portfolio $w_g$ depends only on the covariance input and not on expected return. The second portfolio $w_d$ depends on expected returns and is required to achieve some expected return.

Tangency Portfolio, unconstrained problem and Kelly ratio

Amongst all the optimal portfolios, one portfolio has a higher Sharpe ratio than the others. This is the tangency portfolio obtained by drawing a tangent to the efficient frontier that goes through from the risk-free rate return point.

There is an intuitive demonstration showing that $w_d$ is the efficient frontier portfolio provided $\mu$ is the vector of return above the risk-free asset.

We define the simple return $R_i = \mu_i \Delta t, \sigma_i \Delta W_t$.

The return over a period of a portfolio is the weighted average return $w^t R$, with $E(w^t R) = w^t \mu \Delta t$ and $Var(w^t R) = \frac{1}{2} w^t \Omega w \Delta t$. The quantity of interest when compounding this return is the geometric return, we need to maximize the expected log return: $$ E(\ln (1+w^t R)) = w^t \mu \Delta t - \frac{1}{2} w^t \Omega w \Delta t + o(\Delta t) $$ Writing that the Lagrangian $L_u$ corresponding to this quantity being maximized is even simpler than the previous problem as there is no constraint on $w$. $$L_u := w^t \mu - \frac{1}{2} w^t \Omega w$$ This leads to the solution: $$ w_u = {\arg\, \min} L = \Omega^{-1} \mu $$ Note that this solution $w_u$ is just the solution $w_d$ (where weights summed to 1) rescaled by the optimal leverage $K$ that ensures that the tangency portfolio compounds at the highest possible rate.

This optimal leverage $K$ is actually the multi-asset Kelly ratio.

Note that this is only an intuitive demonstration, as this relies on $\Delta t$ being small. An actual demonstration based on the geometric properties of the efficient frontier and would be more involved as $w_d$ is more complicated term than $w_u$ can be found in section 5.2 of this optimization in finance advanced lecture course (can also be found here).

Therefore, the 2 fund theorem shows that any optimal portfolio (positioned on the efficient frontier) is a linear combination of the minimum variance portfolio and the optimal return portfolio, and the Kelly ratio is the leverage of the unconstrained optimal weights.

Next Steps

All the above results are classic. Next, we are going to look at a spectral analysis of the problem.