The chapter covered the following major points:

• The "expected value" is the probability-weighted sum of all possible outcomes. It is the "average" or "mean" but applied to the future instead of to a historical data series. It is a measure of "reward."

• The possibility of future default causes promised (quoted) interest rates to be higher than expected interest rates. Default risk is also often called credit risk.

• Most of the difference between promised and expected interest rates is due to default. Extra compensation for bearing more risk—the risk premium—and other premia are typically smaller than the default premium for bonds.

• Credit ratings can help judge the probability of and potential losses in default. Moody's and S&P are the two most prominent vendors of ratings for corporate bonds.

• The key tool for thinking about uncertainty is the payoff table. Each row represents one possible state outcome, which contains the probability that the state will come about, the total project value that can be distributed, and the allocation of this total project value to different state-contingent claims. The state-contingent claims "carve up" the possible project payoffs.

• Most real-world projects are financed with the two most common state-contingent claims— debt and equity. The conceptual basis of debt and equity is firmly grounded in payoff tables. Debt financing is the safer investment. Equity financing is the riskier investment.

• If debt promises to pay more than the project can deliver in the worst state of nature, then the debt is risky and requires a promised interest rate in excess of its expected interest rate.

• NPV is robust to modest errors in the expected interest rate (the discount rate) for near-term cash flows. However, NPV is not necessarily robust with respect to modest errors in either expected cash flows or discount rates for distant cash flows.

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