FINC3017 · Investments And Portfolio Management
Consumption-Based Asset Pricing, State Prices & the SDF
Consumption-Based Asset Pricing, State Prices & the SDF (Week 6) is the signature analytic content of FINC3017. Working in a world of discrete states with known probabilities, you price an Arrow-Debreu security that pays $1 in one state, then price any asset as the sum of its payoffs times their state prices. The stochastic discount factor M = ψ/π collapses all of this into one rule, p = E[M·X] or 1 = E[M(1 + R_i)]; risk-neutral probabilities re-weight the states; and the consumption Euler equation with CRRA utility links the SDF to marginal utility, so M is high precisely in bad states.
What this chapter covers
- 01State space: S states, probabilities π_s summing to 1
- 02Arrow-Debreu security and the state price ψ_s (0 to 1)
- 03Pricing any asset: p = Σψ_s·y_s; risk-free bond B = Σψ_s and 1 + rf = 1/B
- 04Stochastic discount factor M_s = ψ_s/π_s, so p = E[M·y] and 1 = E[M(1 + R_i)]
- 05Risk-neutral probabilities π̃_s = ψ_s/B and pricing under Q
- 06Inferring AD prices from option prices via the butterfly: AD(K) = c(K−1) − 2c(K) + c(K+1)
- 07SDF intuition: M high in bad times; risk premium = −Cov(R_i, M)/E[M]
- 08Euler equation with CRRA utility and the two-period consumption choice; Merton optimal risky share α* = (μ − rf)/(γσ²)
State prices, risk-neutral probabilities and the SDF in a two-state economy
- 2 marks(a) Set up two equations: ψ_B + ψ_b = 0.95 (the bond) and 4ψ_B + ψ_b = 2.20 (Stock A).
- 2 marks(a cont.) Subtract the first from the second: 3ψ_B = 1.25, so ψ_B = 0.41667 and ψ_b = 0.95 − 0.41667 = 0.53333.
- 1 mark(b) The risk-free bond price is B = Σψ_s = 0.95, so 1 + rf = 1/0.95 = 1.05263, giving rf ≈ 5.26%.
- 1 mark(c) Risk-neutral probabilities π̃_s = ψ_s/B: π̃_B = 0.41667/0.95 ≈ 0.439 and π̃_b ≈ 0.561.
- 2 marks(c cont.) SDF M_s = ψ_s/π_s: M_B = 0.41667/0.5 = 0.833 and M_b = 0.53333/0.5 = 1.067.
Key terms
- Arrow-Debreu security
- A primitive claim that pays $1 in exactly one future state and nothing otherwise. Its price is the state price ψ_s; once you know every state price you can value any asset as the sum of its state-by-state payoffs times those prices.
- State price (ψ)
- The present value today of $1 delivered in a particular future state, with 0 to 1. State prices encode both time value and risk; their sum is the price of a sure $1, the risk-free bond, so 1 + rf = 1/Σψ_s.
- Stochastic discount factor (SDF / M)
- The pricing kernel M_s = ψ_s/π_s that prices everything through p = E[M·X], equivalently 1 = E[M(1 + R_i)]. It is high in low-consumption (bad) states, so assets paying off there are valuable and command lower required returns.
- Risk-neutral probabilities
- Re-weighted probabilities π̃_s = ψ_s/B under which every asset's price is just its expected payoff discounted at the risk-free rate. They tilt weight toward bad states relative to the true (physical) probabilities, embedding the market's risk aversion.
- Euler equation
- The consumer's optimality condition U'(C_t) = E[β(1 + R_{t+1})U'(C_{t+1})], which links asset returns to marginal utility of consumption. With CRRA utility U'(C) = C^(−γ) it produces the consumption-based SDF and the two-period consumption-saving choice.
Consumption-Based Asset Pricing, State Prices & the SDF FAQ
What is the intuition behind the stochastic discount factor being high in bad states?
In a bad state your consumption is low, so an extra dollar there is worth a lot — marginal utility is high — and the SDF M is exactly that scaled marginal utility. An asset that pays off in bad states is therefore valuable insurance, gets a high price, and earns a low required return; the SDF formalises this 'pay when it hurts most' logic.
How do risk-neutral probabilities differ from real (physical) probabilities?
Physical probabilities π describe how likely states actually are; risk-neutral probabilities π̃ = ψ/B are adjusted so that prices equal expected payoffs discounted at the risk-free rate. Because investors dislike losses, the risk-neutral measure puts MORE weight on bad states than reality does — the gap reflects risk aversion, not a forecast.
How can you read state prices out of option prices?
A butterfly spread — long one call at strike K−1, short two at K, long one at K+1 — pays off close to $1 only when the underlying lands near K, approximating an Arrow-Debreu security for that state. So AD(K) ≈ c(K−1) − 2c(K) + c(K+1), letting you infer the market's state prices (and hence risk-neutral density) directly from traded option quotes.
Exam move
This is the most distinctive material in the unit, so master the mechanical chain: solve state prices from a system of payoff equations, then derive rf = 1/Σψ − 1, the risk-neutral probabilities ψ/B, and the SDF ψ/π. Pair every step with its one-line intuition (M high in bad states, risk-neutral over-weights bad states) and memorise the butterfly formula for state prices from options.
