Deriving the Tangency Portfolio from the Efficient Frontier

# Calculation of Market portfolio from efficient frontier


# Calculation of Market portfolio from efficient frontier












I have a specific Portfolio frontier. Can someone provides me with details about how can I calculate the market portfolio from the efficient frontier? I know that I have to draw the tangent line from the risk free asset, but how? is there any specific formula to calculate the risk free asset? Any help will be appreciated.

## Answer by Kermittfrog (score 4, accepted)

https://quant.stackexchange.com/a/60738

As @stans already said in the comments to your question, the existence of the market portfolio hinges on the existence of a risk free rate $r_f$, where risk free, in this context, means that its value can be perfectly contracted for the relevant return horizon, e.g. you will with probability one get that rate for 1 month or 1 year. In theory, we must also be able to lend out and/or borrow at that same risk free rate.

For sake of argument, let us assume that you have queried the LIBOR rates or any other interbank rates panel for the relevant risk free rates.*

What does the tangency condition imply?

Draw a line from the $0,r_f$ point in your diagram such that it is tangent to your efficient frontier. Without knowning the market point ab initio, let us just call that point $M$, and let us denote its expected return and its volatility as $\mu_m$ and $\sigma_M$.

Given this (yet unknown) point, the formula for the capital market line $L$ is:

$$ \mu_L=r_f+\frac{\mu_M-r_f}{\sigma_M}\sigma $$

i.e. if $\sigma = \sigma_M$, the line is at the market point and has an expected return of $\mu_L=\mu_M$. Furthermore, given any investment weight vector $\mathbb{w}$, the assets' expected return vector $\mathbb{\mu}$ and the assets' covariance matrix $\mathbb{\Sigma}$, our portfolio's expected return is:

$$ \mu_p(\mathbb{w})=r_f + \left(\mathbb{\mu}-\mathbb{1}r_f\right)^T\mathbb{w} \qquad $$ NB: With a risk free rate in the mix, we could add it to our portfolio (and in the efficient frontier its weight is simply fixed at zero,though).

... and our portfolio's volatility is: $$ \sigma_p(\mathbb{w})=\left(\mathbb{w}^T\mathbb{\Sigma}\mathbb{w}\right)^{\frac{1}{2}} $$

At the tangency point (market point) the slope of the capital market line $L$ and the slope of the efficient frontier (at portfolio $p$) are equal, i.e.

$$ \left.\frac{\partial \mu_L}{\partial \sigma}\right|_M=\left.\frac{\partial \mu_p}{\partial \sigma_p}\right|_{M} $$ Let's write this out (suppressing the $M$):

$$ \frac{\mu_M-r_f}{\sigma_M}=\frac{\partial \mu_p}{\partial \mathbb{w}}\bigg/\frac{\partial \sigma_p}{\partial \mathbb{w}} \Leftrightarrow \frac{\mu_M-r_f}{\sigma_M}\frac{\partial \sigma_p}{\partial \mathbb{w}}=\frac{\partial \mu_p}{\partial \mathbb{w}} $$

From matrix calculus, we know that $\frac{\partial}{\partial x}a^Tx=a$ and $\frac{\partial}{\partial x}x^TBx=Bx+B^Tx$, and in our case, due to symmetry of $\mathbb{\Sigma}$, $\frac{\partial}{\partial w}w^T\Sigma w =2\Sigma w$. We can thus rearrange the tangency condition and find:

$$ \frac{\mu_M-r_f}{\sigma_M}\frac{1}{\sigma(w)}\mathbb{\Sigma}w=\mathbb{\mu}-\mathbb{1}r_f $$

At $M$, the portfolio volatility and the market volatility coincide, i.e. $\sigma(w)\equiv \sigma_M$. We can hence solve for $w$ as:

$$ w=\frac{\sigma_M^2}{\mu_M-r_f}\mathbb{\Sigma}^{-1}\left(\mathbb{\mu}-\mathbb{1}r_f\right) $$

And as we are looking for a portfolio whose asset weights sum to 100%, we introduce the condition $\mathbb{1}^Tw=1$, yielding finally:

$$ \begin{align} w_M&=\frac{w}{\mathbb{1}^Tw}\\ &=\frac{\mathbb{\Sigma}^{-1}\left(\mathbb{\mu}-\mathbb{1}r_f\right)}{\mathbb{1}^T\mathbb{\Sigma}^{-1}\left(\mathbb{\mu}-\mathbb{1}r_f\right)} \end{align} $$

This is the formula for the market portfolio, derived using the tangency condition. Note that you can also arrive at this result using a Lagrangian ansatz.

HTH?

* NB: In practice, you will also see treasury bill rates as risk free rates as these are the most-risk-free rates available.

Addendum for a problem with positivity constraints

If your problem is bounded by non-negativity constraints, $w_i\geq 0$, one approach could be to formulate a quadratic program with a target return $m^*$:

$$ \min \frac{1}{2} w^T\Sigma w \qquad s.t. \quad w_i \geq 0,\quad w^T(\mu-r_f)=m^* $$

You then vary $m^*$ until $\sum w_i=1$. This results in your tangency portfolio under non-negativity constraints.

Ultimatively, you could use your preferred non-linear optimizer and simply instruct it to maximize the Sharpe ratio s.t. non-negativity and full investment constraints....