Consumption Based Model

Question 1

Motivation

Answer 1

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

Question 2

Utility functions and expected utility

Answer 2

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:

Question 3

common utility functions

Answer 3

Question 4

Properties of Utility functions

Answer 4

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)]

Question 5

How much risk aversion is empirically plausible?

Answer 5

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

Question 6

Utility over Dynamic Consumption Streams

Answer 6

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

Question 7

Environment faced by the investor

Answer 7

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)

Question 8

Assumptions about the asset market

Answer 8

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

Question 9

Maximisation problem

Answer 9

Question 10

The basic pricing equation and derivation

Answer 10

Question 11

The Stochastic Discount Factor

Answer 11

Question 12

Why is it called a stochastic discount factor? And what are the ideas behind it

Answer 12

Question 13

What Do We Gain from Defining m?

Answer 13

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

Question 14

Asset Pricing Equation for Returns

Answer 14

Question 15

Risk Adjustment Implied by p = E[mx]

Answer 15

Question 16

Interpretation of Risk Adjustment equation

Answer 16

Question 17

u′(c) is strictly decreasing in c

Answer 17

pt is lowered (raised) if payoff covaries positively (negatively) with future consumption ct+1

Question 18

why is u′(c) strictly decreasing in c

Answer 18

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

Question 19

Covariance matters, not variance

Answer 19

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

Question 20

Idiosyncratic Risk

Answer 20

Because only the covariance matters, there can be unpriced risk:

Question 21

Risk Adjustment Equation for Returns

Answer 21

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

Question 22

Remark 1: Price-Payoff Formulation Is very General

Answer 22

Question 23

Remark 2: Nominal versus Real Units

Answer 23

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

Question 24

Remark 3: Assumptions We Do not Need for p = E[mx]

Answer 24

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

Question 25

Remark 4: Assumptions We Do Need for p = E[mx]

Answer 25

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

Question 26

Multiple periods

Answer 26

Have used a two-period investor for simplicity
If we consider instead a T-horizon investor with preferences

Question 27

Closing the Model/General Equilibrium

Answer 27

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

Question 28

Some Possibilities

Answer 28

Question 29

The Consumption-based Model Answers All Valuation Questions

Answer 29

At least in principle, the consumption-based model can be used to value any claim

Question 30

Holding-period returns on any security

Answer 30

Question 31

Excess returns (differences) formed from holding-period returns

Answer 31

Question 32

Other things it can price

Answer 32

Security prices for stocks

price formulas

default-free nominal bonds

European call options

Question 33

How to apply the model in practice

Answer 33

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.

Question 34

Testing the Model

Answer 34

Question 35

How Does this Work in Reality? – Summary

Answer 35

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

Question 36

Takeaway

Answer 36

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

Question 37

Summary

Answer 37

Question 38

Derivation Details for Model Test

Answer 38