The economy's equilibrium level of real interest rates depends on the willingness of households to save, as reflected in the supply curve of funds, and on the expected profitability of business investment in plant, equipment, and inventories, as reflected in the demand curve for funds. It depends also on government fiscal and monetary policy.

The nominal rate of interest is the equilibrium real rate plus the expected rate of inflation. In general, we can directly observe only nominal interest rates; from them, we must infer expected real rates, using inflation forecasts.

The equilibrium expected rate of return on any security is the sum of the equilibrium real rate of interest, the expected rate of inflation, and a security-specific risk premium.

Investors face a trade-off between risk and expected return. Historical data confirm our intuition that assets with low degrees of risk provide lower returns on average than do those of higher risk.

Assets with guaranteed nominal interest rates are risky in real terms because the future inflation rate is uncertain.

Historical rates of return over the 20th century in developed capital markets suggest the U.S. history of stock returns is not an outlier compared to other countries.

Investments in risky portfolios do not become safer in the long run. On the contrary, the longer a risky investment is held, the greater the risk. The basis of the argument that stocks are safe in the long run is the fact that the probability of a shortfall becomes smaller. However, probability of shortfall is a poor measure of the safety of an investment. It ignores the magnitude of possible losses.

Historical returns on stocks exhibit more frequent large negative deviations from the mean than would be predicted from a normal distribution. The lower partial standard deviation (LPSD), the skew, and kurtosis of the actual distribution quantify the deviation from normality. The LPSD, instead of the standard deviation, is sometimes used by practitioners as a measure of risk.

Widely used measures of tail risk are value at risk (VaR) and expected shortfall or, equivalently, conditional tail expectations. VaR measures the loss that will be exceeded with a specified probability such as 5%. The VaR does not add new information when returns are normally distributed. When negative deviations from the average are larger and more frequent than the normal distribution, the 5% VaR will be more than 1.65 standard deviations below the average return. Expected shortfall (ES) measures the expected rate of return conditional on the portfolio falling below a certain value. Thus, 1% ES is the expected return of all possible outcomes in the bottom 1% of the distribution.