Measuring risk is crucial to understanding the financial system.

a. To study random future events, start by listing all the possibilities and assign a probability to each. Be sure the probabilities add to one.

b. The expected value is the probability-weighted sum of all possible future outcomes.

c. A risk-free asset is an investment whose future value, or payoff, is known with certainty.

d. Risk increases when the spread (or range) of possible outcomes widens but the expected value stays the same.

e. One measure of risk is the standard deviation of the possible payoffs.

f. A second measure of risk is value at risk, the worst possible loss over a specific time horizon, at a given probability.