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I am trying to solve the following problem from a Quant exam (abridged):

You have 1000 USD. You can only invest in two (independent) stocks, A and B, with the annualized expected returns and volatilities given. Interest rates are zero. Find the fully invested portfolio that minimizes the portfolio volatility.

Then theres the hint, apparently to make calculations easier: "Ignore also the differences between normal and log-normal distributions."

My question is about the hint, I am not sure if I interpret it correctly: I would use the fact that for an infinitesimal time step $\mathrm d t$ we have for a stock price $$\frac{\mathrm d S}{S} \sim N(\mu\, \mathrm d t, \sigma^2\,\mathrm d t),$$ which follows from the model of a stock price as a geom. Brownian motion. As a consequence, for a sufficiently small time step $\delta t$ this would imply that approximately $$S(\delta t) \sim N((1+\mu) S(0)\delta t, \sigma^2S(0)^2\delta t).$$ The way I would interpret the hint is that I should work with this approximation, instead of using that $S(\delta t)$ is actually (but yet almost inperceivably) log-normally distributed.

Is my interpretation correct? I am interested if I understand the hint correctly from a stochastic point of view.

Then, with $\mu_A = 0.1, \mu_B = 0.15, \sigma_A = 0.1$ and $\sigma_B = 0.2$, I would obtain the following: If I invest $\lambda_A$ USD into stock A, and $1000-\lambda_A$ into stock B, then the portfolio value $\Pi$ after $\delta t$ would approximately be distributed as $$\Pi(\delta t) \sim N\big([1.1\lambda_A +1.15(1000-\lambda_A)]\delta t, [0.01\lambda_A^2+0.04(1000-\lambda_A)^2]\delta t).$$ From this, I would directly obtain:

  • The expected profit of the portfolio is maximized if the portfolio consists entirely of stock B (i.e. $\lambda_A = 0$).
  • The portfolio volatility would be minimized if I invested 800 USD in stock A and 200 USD in stock B (i.e. $\lambda_A = 800$).

I am interested in properly understanding the underlying mathematics, and not so much in applying a formula I don't understand.

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    $\begingroup$ It sounds like they're just asking for standard, Markowitz portfolio optimization? $ \operatorname{minimize}(\text{over } \mathbf{w}) \; \mathbf{w}' \Sigma \mathbf{w}$ subject to $\sum_i w_i = 1$ and $\mathbf{w}' \operatorname{E}[\mathbf{R}] = \mu$. Since you just have 2 variables and a risk free rate of 0, the solution is going to be especially straightforward. $\endgroup$ – Matthew Gunn Nov 17 '17 at 22:56
  • $\begingroup$ That's interesting, thanks! Still, the exam is aimed at people who only have little finance understanding, and I believe they are looking for a basic derivation/solution. If my understanding of the hint is correct, I would actually be able to solve the problem by using the distribution of the portfolio after $\delta t$. $\endgroup$ – DominikS Nov 17 '17 at 23:10
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Super basic Markowitz min variance problem

Portfolio return is $r_p = w_a r_a + w_b r_b$ hence portfolio variance (under assumption of independent assets is $w^2_a \sigma^2_b + w^2_b \sigma^2_b$

\begin{equation} \begin{array}{*2{>{\displaystyle}r}} \mbox{minimize (over $w_a, w_b$)} & w^2_a \sigma^2_b + w^2_b \sigma^2_b \\ \mbox{subject to} & w_a + w_b = 1 \end{array} \end{equation} This is a convex optimization problem where Slater's condition is satisfied. The first order conditions are necessary and sufficient conditions for an optimum. Do a bit of algebra on the first order conditions and you get: $$ w_a = \frac{\sigma^2_b}{\sigma^2_a + \sigma^2_b} \quad \quad w_b = \frac{\sigma^2_a}{\sigma^2_a + \sigma^2_b} $$

Using your values of $\sigma_a = .1$ and $\sigma_b = .2$, then $w_a = .2$ and $w_b = .8$, as you calculated.

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  • $\begingroup$ If you are not sure where the first order conditions come from: they come from the Lagrange Multiplier method for optimization with one equality constraint. $\endgroup$ – Alex C Nov 20 '17 at 23:38
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    $\begingroup$ @AlexC Yeah. The Lagrangian is $\mathcal{L}(w_a, w_b, \lambda) = w_a^2\sigma_a^2 + w_b^2\sigma_b^2 - \lambda(w_a + w_b - 1)$ then $\frac{\partial \mathcal{L}}{\partial w_a} = 2w_a\sigma_a^2 - \lambda$ which equals 0 under the KKT conditions etc.... Then it's just some algebra. As an alternative, you could of course simply substitute $w_b = 1 - w_a$ to make it a high school level optimization problem in a single variable. $\endgroup$ – Matthew Gunn Nov 20 '17 at 23:44
  • $\begingroup$ Thanks for your answer, but this is not exactly what I meant: In what sense should I 'ignore the log-normal distribution'? I guess I am trying to understand this problem from a stochastic point of view. $\endgroup$ – DominikS Dec 15 '17 at 15:24
  • $\begingroup$ @DominikS I honestly don't understand what the purpose of the hint is. Maybe with more context I could guess. To solve a minimum variance problem, all you need is the covariance matrix. Let $R$ be a random vector denoting security returns. Portfolio return with weights $\mathbf{w}$ is given by $R_p = \mathbf{w}' \mathbf{R}$. Hence portfolio variance is $\operatorname{Var}(R_p) = \mathbf{w}' \Sigma \mathbf{w}$ where $\Sigma = \operatorname{Cov}(R) $ is the covariance matrix. What else is there to know besides $\Sigma$? $\endgroup$ – Matthew Gunn Dec 15 '17 at 15:46
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    $\begingroup$ @DominikS That might make sense: basically a minimum variance problem over an infinitesimal time horizon? $\endgroup$ – Matthew Gunn Dec 15 '17 at 17:06

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