Explain how a commodity's current futures price and expected future spot price are affected by systematic risk and by nonsystematic risk.
The idea
A commodity's required expected return, written μ here, is set by its systematic risk, which is its beta against the market portfolio. This μ is not the alpha of a regression. Through CAPM, μ = r + β × (expected market return − r). The forward price relates to the expected future spot price E(S) as F = E(S) × e^(−(μ − r)T). Today's spot price is the expected spot price discounted at the full required return, so S = E(S) × e^(−μT). With positive beta, μ exceeds r so F is below expected spot. With negative beta, F is above expected spot. With zero beta, F equals expected spot. Nonsystematic risk, such as the weather on one farm, can be diversified away and earns no premium, so it does not move the forward relative to expected spot, however volatile the price is. The usual mistake is to link a higher total volatility to a larger risk premium.
Formulas
Commodity's required return under CAPM
μ = r + β × (E(RM) − r)
- r
- the risk-free rate
- β
- the commodity's beta against the market portfolio
- E(RM)
- the expected market return
Only systematic risk enters. A negative beta gives μ below r and a forward above expected spot; idiosyncratic volatility changes nothing.
A practice question
A local pest outbreak affects the crop of a single farm in a diversified market. Its effect on the relation between the crop's forward price and expected spot price is:
- a large increase in the risk premium, because the volatility of the farm's crop increases.Why not: Treats idiosyncratic volatility as priced; only systematic risk earns a premium.
- none for the risk premium, because the risk is nonsystematic and can be diversified. - correct
- a decrease in the risk-free rate, because the crop is riskier.Why not: The risk-free rate is a market-wide rate and does not depend on one farm's crop risk.
- a negative beta for the crop, so the forward rises above the expected spot price.Why not: Idiosyncratic shocks do not change beta, which is based on co-movement with the market.
Only systematic risk, measured by beta, drives the risk premium; a farm-specific shock is diversifiable and leaves the forward-expected spot relation unchanged.
A harder one
A commodity has beta -0.3 against the market, the risk-free rate is 3% and the expected market risk premium is 6%, all continuously compounded. The commodity's expected spot price in two years is USD 100. The two-year forward price is closest to:
- USD 96.46Why not: Reverses the sign: with a negative beta the forward lies above expected spot.
- USD 100.00Why not: Assumes the forward equals expected spot; this holds only at zero beta.
- USD 103.67 - correct
- USD 97.63Why not: Discounts expected spot at the commodity's own return; this gives today's spot price.
mu = 0.03 + (-0.3 x 0.06) = 0.012. F = E x exp(-(mu - r) x T) = 100 x exp(+0.036) = 103.67, above expected spot.
The hardest tier
Commodities A and B each have beta 0.8, and both have an expected spot price of USD 100 in one year. Commodity A's price volatility is 20% per year and B's is 45%, the extra volatility being idiosyncratic. The risk-free rate is 3% and the market risk premium is 5%, continuously compounded. Which statement about the forward prices is correct?
- B's forward is lower than A's, because higher total volatility requires a larger risk premium.Why not: Prices idiosyncratic volatility; only systematic risk earns a premium.
- B's forward is higher than A's, because hedgers pay more for protection against volatility.Why not: Makes the premium depend on volatility rather than beta.
- Both forwards are about USD 96.08, because the gap to expected spot depends on beta, not on total volatility. - correct
- Both forwards equal USD 100, since volatility cancels and the forward equals expected spot.Why not: The forward equals expected spot only when beta is zero; here beta is positive.
mu - r = beta x premium = 0.8 x 0.05 = 0.04. F = 100 x exp(-0.04) = 96.08 for both commodities.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
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