Portfolio risk decreased by increasing share of riskiest asset?

Question

In Parker's The Economics of Entrepreneurship he explains how certain theoretical models predict seemingly bizzare things (e.g. people becoming more risk-averse resulting in them taking riskier jobs) by analogy to investment risk. He states that sometimes a portfolio's overall risk can be decreased by increasing the fraction of the portfolio which is invested in its riskiest asset.

He gives no citation for this, and playing around with some sample numbers I can't find an example of when this would be true. Could anyone point me to an example of when this would be true?

Answer 1

This can for example be seen in modern portfolio theory (Harry Markowitz, William Sharpe)

As an example consider a two asset portfolio with a full investment constraint ($w_1+w_2=1$) so we can write the proportion in asset 1 as $w_1=w$ and in asset 2 as $1-w$

The expected portfolio return $E[R_p]=wE[R_1]+(1-w)E[R_2]$

And variance $\sigma_p^2 = w^2\sigma_1^2+(1-w)^2\sigma_2^2+2w(1-w)\rho_{12}\sigma_1 \sigma_2$

Where $\sigma_i$ is asset $i$'s standard deviation and $\rho_{12}$ is the correlation between the two.

If the investor tries to optimize a mean variance tradeoff, where the variance is interpreted as risk. For example where there could be some risk preference parameter included if desired other than the $\frac{1}{2}$ which is included for convinience:

$\min_w L(w) = \min_w (\frac{1}{2}\sigma_p^2 - E[R_p])$ Taking first order conditions by $dL(w)/dw=0$ gives

$\frac{1}{2}\bigl(2w\sigma_1^2+2(w-1)\sigma_2^2+2(1-2w)\rho_{12}\sigma_1 \sigma_2\bigr) - \bigl(E[R_1]-E[R_2]\bigr)=0$

$w(\sigma_1^2+\sigma_2^2-2\rho_{12}\sigma_1 \sigma_2 )-\sigma_2^2+\rho_{12}\sigma_1 \sigma_2 - \bigl(E[R_1]-E[R_2]\bigr)=0$

$w = \frac{E[R_1]-E[R_2]+\sigma_2^2-\rho_{12}\sigma_1 \sigma_2 }{\sigma_1^2+\sigma_2^2-2\rho_{12}\sigma_1 \sigma_2}$

The minimum variance portfolio is found by $\min_w(\frac{1}{2}\sigma_p^2)$ and has first asset weight $w^*$:

$w^* = \frac{\sigma_2^2-\rho_{12}\sigma_1 \sigma_2 }{\sigma_1^2+\sigma_2^2-2\rho_{12}\sigma_1 \sigma_2}$

Now calculating the minimum variance portfolio weight for a couple of cases:

s1=0.1;s2=0.1;rho=-1;print s1, s2, rho, (s2**2-rho*s1*s2)/(s1**2+s2**2-2*rho*s1*s2)

So you can see the minimum variance portfolio weights.

 s1   s2  rho   w*
0.1  0.1  -1    0.5    // equal risk, perfect negative corr -> equal weight
0.2  0.1  -1    0.333  // asset 1 riskier -> lower weight
0.2  0.1   0    0.2    // asset 1 riskier and no hedge -> even lower weight
0.2  0.1   0.2  0.142  // asset 1 riskier and positive corr -> STILL POSITIVE W ! (no negative correlation required for a hedge!)
0.2  0.1   0.5  0.0    // asset 1 riskier and higher pos corr -> everything in asset 2
0.2  0.1   0.8 -0.33   // asset 1 riskier and high corr -> short asset 1
0.2  0.1   1   -1.0    // asset 1 riskier and perfect corr -> fully short ass 1

for example the case $\sigma_1=0.2,\sigma_2=0.1,\rho_{12}=0.2,w^*=0.142$ gives portfolio variance:

w**2*s1**2+(1-w)**2*s2**2+2*w*(1-w)*rho*s1*s2

$\sigma_p^2 = 0.91\%$ which is less than both individual asset variances $\sigma_1^2 = 4\%$ and $\sigma_2^2 = 1\%$ .