Explain required returns, the risk-free rate and risk premiums
The idea
A required rate of return is the minimum return an investor demands for holding an asset, and it can be built up from its parts. The real risk-free rate rewards deferring consumption. Adding expected inflation gives the nominal risk-free rate, exactly (1 + nominal) = (1 + real) x (1 + inflation), or approximately the sum of the two. On top sit risk premia: a default risk premium for the chance of non-payment, a liquidity premium for the cost of selling quickly, and a maturity premium for the greater price sensitivity of longer-dated claims. Comparing two instruments that differ in only one feature isolates the premium for that feature.
A practice question
A corporate bond investor uses a real risk-free rate of 1.5%, expected inflation of 2.5%, a default risk premium of 1.2%, a liquidity premium of 0.4% and a maturity premium of 0.6%. Using the additive build-up, the required return is closest to:
- 6.2% — correct
- 4.0%Why not: Stopped at the nominal risk-free rate (1.5% + 2.5%) and left out all three risk premia.
- 3.7%Why not: Added the risk premia to the real rate but omitted the 2.5% inflation premium.
Required return = 1.5% + 2.5% + 1.2% + 0.4% + 0.6% = 6.2%. The first two terms give the nominal risk-free rate; the last three compensate for the bond's specific risks.
A harder one
The real risk-free rate is 2% and expected inflation is 3%. Using the exact (multiplicative) relationship, the nominal risk-free rate is closest to:
- 5.00%Why not: Added the two rates, which is the approximation and drops the 2% x 3% cross term the exact form asks for.
- 0.98%Why not: Divided the inflation factor by the real-rate factor instead of multiplying the two factors.
- 5.06% — correct
(1 + nominal) = (1 + real) x (1 + inflation) = 1.02 x 1.03 = 1.0506, so the nominal rate is 5.06%. The additive 5.00% ignores the 0.06% cross term.
The hardest tier
A bond portfolio earned a nominal return of 6.0% over a year in which inflation was 4.5%. The exact real return is closest to:
- 1.50%Why not: Subtracted inflation from the nominal return, the additive shortcut, rather than dividing the growth factors.
- 1.44% — correct
- 10.77%Why not: Multiplied the nominal and inflation factors, which inflates the return instead of deflating it.
Real return = 1.060 / 1.045 - 1 = 1.44%. The approximate 1.50% overstates it because it ignores that inflation also erodes the return itself.
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