# 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.

I find it easiest to talk about this considering a concrete example. For the same project as before suppose now that the risk can be split into two components. First there is the “market” risk (at least in the investors’ view). Suppose there is a 50% chance of a global recession and a 50% chance of a global boom period. Then our project can be summarized as follows.

$\begin{tabular}{cc} \begin{tabular}{c|ccccc} Scenario & Income & Probability & you get & investor gets \\ \hline good & 200.000 & 75\% & 200.000-x & x \\ bad & -50.000 & 25\% & 0 & -50.000 \\ \end{tabular} & \begin{tabular}{c|ccccc} Scenario & Income & Probability & you get & investor gets \\ \hline good & 200.000 & 85\% & 200.000-x & x \\ bad & -50.000 & 15\% & 0 & -50.000 \\ \end{tabular} \\ \\ bad world & good world \end{tabular}$

Now suppose that conditional on being in the bad or good world, all risks are again stochastically independent. But note that overall the risks are correlated because there is a 50% chance that all risks are more severe and a 50% chance that all risks are less severe.

If this is how investors see your (and others’) investment opportunity, how would they feel about a 12.5% interest rate for your loan? Well, at 12.5% investors will get an expected (or average) payout of $0,75 * 12.500 - 0,25 * 50.000 = -3125$ in the bad world and $0,85 * 12.500 - 0,15 * 50.000 =+3125$ in the good world. But this means that, if the world happens to go into a recession, the investor loses money on average, and diversification across investment opportunities does not cancel out this risk.

If the investor in our example is extremely risk averse, i.e., she wants to guarantee a non-negative return on her investment in both worlds, she will ask for an interest rate of at least 16.66%, because $0,75 * 16.666,66 - 0,25 * 50.000 = 0.$

This means that only non-independent risk will typically have risk-premia in the financial markets.