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I am working on an efficient frontier problem for my financial theory class. The idea is to select weights that minimize a portfolio's variance subject to 2 constraints: the first constraint is that the weighted average of mean of the assets equals a target mean. The second is that the sum of the weights are one. More formally, we can write the problem as:

$$\text{min}\left(w_{1}^{2}\sigma_{1}^{2}+w_{2}^{2}\sigma_{2}^{2}+2w_{1}w_{2}\right)$$

subject to $$ w_{1}+w_{2}=1 $$ and $$ w_{1}\mu_{1}+w_{2}\mu_{2}=d $$ where $\sigma^{2}$represents the variane of either asset, and the $\mu$ denote the mean of either asset. Here, $d$ is our target mean. It seems to me that this system is singular- the two constraints themselves guarantee a solution for $w_{1}$ and $w_{2}$ that obviates the need for a minimzation problem. In other words, these are two equations and two unknowns and the minimization is redundant. What am I missing here?

ChinG
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  • Question for you: I learnt about the efficient frontier in this video: http://www.investopedia.com/video/play/explaining-efficient-frontier/. Yet my finding seems to disprove it. In case the returns of the investments are all independent, all investments with a positive return rate are worth investing. It's only a matter of how much money to invest, based on the ratio $\mu_i/\sigma_i^2$. Any explanations? – Zhuoran He Oct 13 '17 at 18:15

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Let $x_1$ and $x_2$ be the return rates (i.e. return per invested dollar) of the two investiments. Let $w_1$ and $w_2$ be the money invested into the two investments. Let $1$ be all the money you have. Suppose the return rates $x_1$ and $x_2$ do not depend on the money $w_1$ and $w_2$ invested into them, i.e., no diminishing margin, then $y=w_1x_1+w_2x_2$ is your total return. Assuming the return rates $x_1$ and $x_2$ are not correlated, so the minimum-variance portfolio problem can be formulated as

\begin{align} \mbox{Minimize }\,&\mathrm{var}(y)=w_1^2\sigma_1^2+w_2^2\sigma_2^2\\ \mbox{subject to }\,&w_1\geq 0, w_2\geq 0, w_1+w_2\leq 1,\\ &\langle y\rangle=w_1\mu_1+w_2\mu_2\geq d. \end{align}

Here $\langle x_i\rangle=\mu_i$, $\mathrm{var}(x_i)=\sigma_i^2$, $i=1,2$. You don't have to invest all you money to meet the goal $d$. In case $x_1$ and $x_2$ are correlated, add a covariance term $2w_1w_2\,\mathrm{cov}(x_1,x_2)$ to $\mathrm{var}(y)$.

Using this model, I did some further studies and found that if $\,w_1+w_2\leq 1\,$ is not binding (i.e. the goal $d\,$ is not too ambitious), The optimal portfolio can be seen from Cauchy-Schwarz inequality

$$(w_1^2\sigma_1^2+\cdots+w_n^2\sigma_n^2)\left(\frac{\mu_1^2}{\sigma_1^2}+\cdots+\frac{\mu_n^2}{\sigma_n^2}\right)\geq(w_1\mu_1+\cdots+w_n\mu_n)^2$$

with equal sign obtained when $w_i\propto\mu_i/\sigma_i^2$. This generalizes to $n$ investments with independent risks. If the required money $\,w_1+\cdots+w_n>1$ to reach the goal $d\,$ is beyond budget, then one would have to deviate from this minimum risk portfolio and aim for more risky ones.

Zhuoran He
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