Intro to Econ: Ninth Lecture – Risk Premia under Non-Independent Risks

Recall the problem we had in the previous two posts (here and here). You are considering undertaking a worthwhile but risky project and need some startup money in order to do it. Investors give your project an 80% chance of succeeding and a 20% chance of failing. The problem can be summarized in the following table, where  x is the repayment amount that you pay back to the investor in case of the project being successful. If it is unsuccessful you pay nothing, because you have nothing. You “default” on your loan in that case. This is the risk the investor takes on when she or he gives you this loan.

 \begin{tabular}{c|ccccc} Scenario & Income & Probability & you get & investor gets \\ \hline good & 200.000 & 80\% & 200.000-x & x \\ bad & -50.000 & 20\% & 0 & -50.000 \\ \end{tabular}

 

In the previous post we considered the case that this risk inherent in your project is stochastically independent of the risks in other potential investment opportunities. In this case we figured out that the interest rate you might get for your project might be as low as 12.5% (but certainly not below that). This is so low that, due to the risk in the investment, investors expect actually a zero return on their investment. The actual interest rate would then probably be a bit higher, determined by supply and demand.

All this depends, however, on the fact the risk is stochastically independent of other risks. Expressed differently, one could say that the financial market generates no risk premium on any stochastically independent risk in an investment opportunity. This is because investors can hedge independent risks away by diversifying their investment portfolio. They can invest small amounts in many such independent risks and then, by force of the law of large numbers, actually have no risk in their diversified portfolio.

In this post, which I am now finally getting to, I want to consider how this analysis changes when the risk inherent in this investment opportunity is not stochastically independent of other risks, but is correlated with them.

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