Minimum-Variance Weights for Two Independent Stocks

# Ignore the difference between normal and log-normal distributions


# Ignore the difference between normal and log-normal distributions












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.

## Answer by Matthew Gunn (score 1)

https://quant.stackexchange.com/a/36986

#### 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.