So, per the title, why would a combination of two risky assets that have the same exact expected return and standard deviation while being perfectly negatively correlated not return 0%? Why do you just combine the weighted expected return of the two assets to get the expected return at 0 standard deviation?

It seems logical that if asset A goes up by expected return and asset B goes down by the expected return, the portfolio return would be 0.

But in textbooks the expected return of two perfectly negatively correlated assets is just the sum of their weighted expected returns. Why would you not subtract returns since if one goes up, the other goes down (detracting from returns)?

  • $\begingroup$ The key point is that return is not a linear function. $\endgroup$ – Gordon Jul 4 '16 at 2:21
  • $\begingroup$ What does correlation has to do with expectation. Maybe you meant covariance? $\endgroup$ – Quantuple Jul 4 '16 at 10:40
  • $\begingroup$ Correlation. Modern portfolio theory states that two perfectly negatively correlated assets will not return zero percent, rather it will return the combined weighted expected return of both assets. $\endgroup$ – J. Doez Jul 4 '16 at 21:03
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    $\begingroup$ Well yes because expectation is linear and we have $E(A+B)=E (A)+E (B) $ regardless of how random variables $A $ and $B $ are distributed (hence regardless of their correlation). This is one of the most basic results of statistics $\endgroup$ – Quantuple Jul 4 '16 at 23:32

If I understand your question correctly; the (expected) return always depends on the weights that the respective factor has in the portfolio, regardless of the risk.

You are trying to find the optimal portfolio (given risk with highest return, or given return with lowest risk) so in the first step you try to diversify out the risk (and get the weigths for the sought portofilo) once you get your weights, you calculate the return (that does not depend on the risk or covariances at all, just the weights and respective returns).

In extreme case: 50-50 allocation between two perfectly negatively correlated assets of the same return that would result in 0 (expected) return.

But say, typical example - selling ice cream and umbrellas. If it is a rainy season, umbrella stocks goes up 70\$, ice cream loses 30\$ and vice versa if it is sunny. If you invest 50-50, then regardless of the weather you get $0.5*(-30\$)+0.5*(70\$)=40$$ with certainty.

Always the payoff depends on the weights (60-40) and if its a 'bad' year would give you only 10\$ etc.

  • $\begingroup$ But modern portfolio theory simply adds the two expected returns together, despite them being negatively correlated. So instead of subtracting 30$ they add the two together. It just doesn't make sense. $\endgroup$ – J. Doez Jul 4 '16 at 20:54
  • $\begingroup$ Yh it does add them as a liner (convex) combination depending on the weights. Or a product of two vectors $w^{T}R$, where $w$ is the vector of weights and $R$ is the vector of (expected) returns. $w^{T}R=w_{1}R_{1}+w_{2}R_{2}=E_{portfolio}$ and it wont depend on the correlation between them. That only matters for the optimisation task how to find the weights $w$. $\endgroup$ – Jan Sila Jul 4 '16 at 21:14
  • $\begingroup$ I appreciate the answers! But I'm still a little bit confused. Why are the expected returns of the portfolio not affected by correlation? It seems logical that you would take that into account when forecasting returns of a portfolio of the two assets. $\endgroup$ – J. Doez Jul 4 '16 at 22:31
  • $\begingroup$ Yes, you are taking them into account - in the step when you are determining exactly what the portfolio should look like = finding the optimal weights $w^{*}$! Once you got the portfolio (the weights $w^{*}=(0.1, 0.5, 0.4)$ then you just think-> Out of my money, I invested 10\% in a stock with 0\% profit, 50\% in a stock with 10\% and 40\% in something that got me 20\%. Hence I get - $Return=0.1\cdot 0+0.5\cdot 0.1+0.4\cdot 0.2=0.13$ $\endgroup$ – Jan Sila Jul 4 '16 at 22:37
  • $\begingroup$ I think this might be better way: Step 1 - look at the market correlation structure - that will say what is the portfolio that is the least risky, if you want from assets yielding 0.1, 0.3 and 0.5 say 0.25. So you use the correlation matrix to find it and you get $w^{*}$. Step 2 - now you know what is the optimal portfolio to get you 0.25 with lowest possible risk. It is say (0.4,0.5,0.1) or maybe even something like (-0.2, 0.5,0.7) if short sales are allowed. And then you look what return to expect with this portfolio - weights by multiplying the weight with its respective return. $\endgroup$ – Jan Sila Jul 4 '16 at 22:41

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