We have not covered risk yet, because we did not need to. In a risk-neutral world, all that matters is the expected rate of return, not how uncertain you are about what you will receive. Of course, we can assess risk even in a risk-neutral world, even if risk were to earn no extra compensation (a risk premium).
So, which investment is most risky: full ownership, loan ownership, or levered ownership? As the visual shows, the loan is least risky, followed by the full ownership, followed by the levered ownership. Your intuition should tell you that, by taking the mortgage, the medium-risky project “building” has been split into a more risky project “levered building” and a less risky project “mortgage.” The combined “full building ownership” project therefore has an average risk.
It should not come as a surprise to learn that all investment projects expect to earn a 10%rate of return. After all, 10% is the time-premium for investing money. Recall from the previous posts that the expected rate of return (the cost of capital) consists only of a time-premium and a risk premium. (The default premium is a component only of promised interest rates, not of expected interest rates). By assuming that investors are risk-neutral, we have assumed that the risk premium is zero. Investors are willing to take any investment that offers an expected rate of return of 10%, regardless of risk.
Although our example has been a little sterile, because we assumed away risk preferences, it is nevertheless very useful. Almost all projects in the real world are financed with loans extended by one party and levered ownership held by another party. Understanding debt and equity is as important to corporations as it is to building owners. After all, stocks in corporations are basically levered ownership claims that provide money only after the corporation has paid back its loans. The building example has given you the skills to compute state-contingent, promised, and expected payoffs, and state-contingent, promised, and expected rates of returns—the necessary tools to work with debt, equity, or any other state-contingent claim. And really, all that will happen later when we introduce risk aversion is that we will add a couple of extra basis points of required compensation—more to equity (the riskiest claim) than to the project (the medium-risk claim) than to debt (the safest claim).
We now know how to compute the NPV of state-contingent payoffs—our building paid off differently in the two states of nature. Thus, our building was a state-contingent claim—its payoff depended on the outcome. But it is just one of many. Another state-contingent claim might promise to pay $1 if the sun shines and $25 if a tornado strikes. Using payoff tables, we can work out the value of any state-contingent claims—and in particular the value of the two most important state-contingent claims, debt and equity.