Required return, risk premia and excess returns

In short

• The required rate of return is the minimum return that makes an investor give up consumption today; issuers of shares and bonds must offer at least that much.
• The risk-free rate has no default or reinvestment risk; short-term government bills are the usual proxy. Real risk-free rate: $r_{rf} = (1+r_f)/(1+\text{infl}) - 1$, approximated by $r_f - \text{infl}$.
• The subtraction shortcut is fine for quick estimates, small rates or continuously compounded rates, but its error compounds over long horizons.
• Build-up: $r_i \approx r_{rf} + \text{infl} + \sum_j f_{ij}\,rp_j$. Equity premia: market, size, value. Bond premia: default, liquidity, maturity. Premia change over time.
• Excess return: $(1+r_i)/(1+r_f) - 1$, approximately $r_i - r_f$. The benchmark can also be a real rate or an index.