Consumption Based Model Flashcards
Motivation
Fundamental decision of any investor
how much to save and how much to consume
how to allocate savings across (risky) assets
Most basic pricing equation is the first-order condition for that decision, equalizes marginal utility loss of consuming less today to buy more of the asset
marginal utility gain of consuming more of asset’s payoff in future
Conclusion: we should use an investor’s marginal consumption utility to discount payoffs → this is the consumption-based model of asset pricing
Utility functions and expected utility
A utility function maps outcomes for an individual to a numerical index
utility index measures the felicity or satisfaction of an individual
numerical value itself has no meaning,
U(x) and aU(x) + b with a > 0 describe the same (risk) preferences
If outcomes are uncertain (given by random variable x), we weight utilities for different outcome realizations by their probabilities, leading to expected utility:
common utility functions
Properties of Utility functions
Utility functions often satisfy two important properties Monotonicity (non-satiation): U′(x) > 0
Concavity (risk aversion): U′′(x) < 0
A risk averse individual always prefers the expected outcome to a risky one: U (E[x]) ≥ E[U (x)]
How much risk aversion is empirically plausible?
research on risk preferences suggests that (absolute) risk aversion declines with wealth
consistent with CRRA utility but not CARA and quadratic
(of course, there are other possibilities than these three)
for CRRA parameter γ, evidence typically suggests values in the range 1–10, while
Utility over Dynamic Consumption Streams
E.g. for a two-period investor active in periods t and t + 1
U(ct,ct+1) = u(ct) + βEt[u(ct+1)]
ct: consumption at date t (known at t)
ct+1: consumption at date t + 1 (random from perspective of date t)
u: period utility function (increasing & concave)
β: subjective discount factor (0 ≤ β ≤ 1, captures impatience)
Et[·]: conditional expectation conditional on time-t information
Environment faced by the investor
We want to determine the time-t value of a time-t + 1 payoff xt+1 (random variable)
payoff is the cash flow an investor receives from investing in one unit of the asset
do not confuse with a profit (subtracts cost) or return (divides by initial investment)
Assumptions about the asset market
an asset with payoff xt+1 can be freely traded at date t at market price pt
the investor can hold any fraction of the asset (including negative = short sales)
seeks to maximize utility
u(ct ) + βEt [u(ct+1)]
has some other resources et, et+1 available, e.g.
income (from labor, privately held firms, transfers, etc.)
wealth invested in other assets
than the one we seek to price
if investor buys ξ units of the asset, consumption in the two periods is
ct =et −ptξ ct+1 =et+1 +xt+1ξ
Maximisation problem
The basic pricing equation and derivation
The Stochastic Discount Factor
Why is it called a stochastic discount factor? And what are the ideas behind it
What Do We Gain from Defining m?
Defining m and writing (∗∗) instead of (∗) is just notation
However, it gives us a useful separation
x contains all asset-specific payoff information
(independent of pricing model)
m contains all model-specific pricing content (independent of asset)
When we change the model (e.g. different utility function),
this changes m but not p = E[mx]
Any conclusions we derive from p = E[mx] hold for all asset pricing models
Asset Pricing Equation for Returns
Risk Adjustment Implied by p = E[mx]
Interpretation of Risk Adjustment equation
u′(c) is strictly decreasing in c
pt is lowered (raised) if payoff covaries positively (negatively) with future consumption ct+1
why is u′(c) strictly decreasing in c
risk-averse investors dislike consumption uncertainty
payoff that is positively correlated with consumption: increases consumption volatility
pays off well when consumption is already high (investor feels wealthy)
pays off badly when consumption is low (investor feels poorly)
→ makes consumption more risky - should have low price
payoff that is negatively correlated with consumption: reduces consumption volatility
pays off well when consumption is low, badly when consumption is high → provides insurance, should have high price
Covariance matters, not variance
Risk adjustment term tells us: covariance cov(m,x) matters, not variance var(x)
Why? Investor cares about variation in consumption, not in individual asset payoffs
Contribution of small additional unit of the asset to consumption volatility is measured by the covariance
Idiosyncratic Risk
Because only the covariance matters, there can be unpriced risk:
Risk Adjustment Equation for Returns
Gross returns are just a special form of payoff with price 1 Restating the covariance risk adjustment formula for returns:
This provides us with a convenient formula for the (expected return) risk premium Risk premium is positive for assets that comove negatively with the SDF
Remark 1: Price-Payoff Formulation Is very General
Remark 2: Nominal versus Real Units
Prices and payoffs can be real (denominated in goods) or nominal (in dollars, euros, etc.)
p = E[mx] holds for either case, if we use the correct definition of m
Remark 3: Assumptions We Do not Need for p = E[mx]
We have not assumed complete markets or that there is a representative investor
We have made no assumptions about return or payoff distributions
We have assumed time-separable expected utility preferences, but this is not crucial → p = E[mx] can be derived from more general risk preferences
(form of m changes, interpretation not)
We have made no assumptions on nonmarketable human capital or sources of outside income
Remark 4: Assumptions We Do Need for p = E[mx]
Our pricing equation applies to marginal (small) investments into the asset
it does not apply (without adjustments) to large discrete decisions
e.g. whether to invest a large stake into a private firm or not
We have assumed that the investor can short-sell assets without restrictions
We have assumed that there are no bid/ask spreads or other trading frictions
Multiple periods
Have used a two-period investor for simplicity
If we consider instead a T-horizon investor with preferences
Closing the Model/General Equilibrium
The basic consumption-based pricing equation relates pt to endogenous variables
Not a full solution of the investor’s problem because ct, ct+1 depend on the choice ξ
Interpretation depends on how we close the model to form equilibrium
But for analyzing p = E[mx], this is often irrelevant
p = E[mx] has to hold regardless of how we close the model
unless our specific economic question requires more structure, we do not need to specify it in fact, specifying extra structure, possibly wrongly, may lead to misspecified model
Some Possibilities
The Consumption-based Model Answers All Valuation Questions
At least in principle, the consumption-based model can be used to value any claim
Holding-period returns on any security
Excess returns (differences) formed from holding-period returns
Other things it can price
Security prices for stocks
price formulas
default-free nominal bonds
European call options
How to apply the model in practice
To apply the consumption-based model, we need to make choices
data counterpart of consumption ct, ct+1 in model
e.g. aggregate consumption (“representative agent assumption”)
choice of (marginal) utility
function and parameters
e.g. CRRA utility, u′(c) = c−γ
two parameters β, γ can be estimated (e.g. to minimize historical pricing errors)
These choices yield an explicit function f mapping data to the SDF m = f (data), e.g.
Testing the Model
How Does this Work in Reality? – Summary
Model not hopeless: some correlation between predictions and actual average returns
But pricing errors (actual expected return − predicted expected return) are large
same order of magnitude as variation in excess returns across portfolios
Takeaway
If the model is not useful for pricing in practice, should we discard it?
No, both p = E[mx] and marginal utility interpretation of m are very valuable
conceptually
But: we need better observable indicators for marginal utility than aggregate consumption plugged into power utility function
This motivates alternative asset pricing models
alternative ways of modeling m as a function of data
the marginal utility interpretation valuable to evaluate economic plausibility
plausibility important to generate confidence in out-of-sample predictions of the model
Summary
Derivation Details for Model Test
