When a Two-Asset Minimum-Variance Portfolio Is Long-Only
Summary
The document derives the conditions under which the unconstrained minimum-variance portfolio of two assets has no short positions. Starting from the formula for the weight on the first asset, it requires that weight to be nonnegative and no greater than one. Applying those bounds yields two symmetric inequalities: the correlation must not exceed either asset’s volatility divided by the other’s. Together, the tighter condition is that correlation is no greater than the ratio of the smaller volatility to the larger volatility.
The explanation is an algebraic result, not an empirical study. It assumes the stated two-asset weight formula applies and that both volatilities are nonzero; the source also frames the result for a nondegenerate variance minimum. It does not discuss expected returns, transaction costs, additional assets, or portfolio constraints beyond avoiding short positions. The final answer in the source is brief, so the conditions are more informative than its separate, undeveloped comment.
Key ideas
- For two assets, a long-only minimum-variance allocation requires each portfolio weight to be nonnegative.
- The weight bounds reduce to two symmetric limits on correlation relative to the assets’ volatility ratio.
- The tighter limit is the smaller volatility divided by the larger volatility.
- The derivation assumes nonzero volatilities and the stated unconstrained two-asset variance formula.
Tags
Full text
# Under which conditions the minimum variance portfolio involves no short selling?
# Under which conditions the minimum variance portfolio involves no short selling?
If $\rho_{12} < 1$ or $\sigma_1 \not= \sigma_2$ then $\sigma_v^2$ representing the variance of the portfolio with weights $(w_1, w_2) = (s, 1-s)$ as a function of $s$ attains its minimum value at: $$ s_0 = \frac{\sigma_2^2 - \sigma_1\sigma_2\rho_{12}}{\sigma_1^2+\sigma_2^2-2\sigma_1\sigma_2\rho_{12}} $$ Under which conditions on $\sigma_1$, $\sigma_2$, and $\rho_{12}$ does the minimum variance portfolio involve no short selling?
$\rho$ is correlation coefficient and $\sigma$ is standard deviation. Squared is variance. I'm not sure what this question means.
## Answer by Alex C (score 2)
https://quant.stackexchange.com/a/32700
There are two conditions: $W_1=s_0$ has to be non-negative, which means $\sigma_2^2 - \sigma_1\sigma_2\rho \ge 0$, which simplifies to $\sigma_2 \ge \sigma_1 \rho$. (I assumed $\sigma_2 \ne 0$).
The second condition is that $W_2=1-s_0$ also has to be non-negative, i.e. $s_0 \le 1$. So $\sigma_2^2 - \sigma_1\sigma_2\rho \le \sigma_1^2+\sigma_2^2-2\sigma_1\sigma_2\rho$. Which reduces to $\sigma_1 \ge \sigma_2 \rho$. (Again assuming $\sigma_1 \ne 0$).
The two conditions are nicely symmetric, and can be combined into the following statement
$ \rho\le \frac{\sigma_2}{\sigma_1}, \rho\le \frac{\sigma_1}{\sigma_2} $
(Only one of these two conditions, the one with the lower ratio, will be binding. We could say $\rho \le \frac{\sigma_{small}}{\sigma_{big}}$ once we know which vol is bigger and which is smaller).
## Answer by user26749 (score -3)
https://quant.stackexchange.com/a/32698
when the numerator is positive, then you get rho.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.