Consumption-CAPM

#termstructure #economics #capm #finance

Oh, Hyunzi. (email: wisdom302@naver.com)
Korea University, Graduate School of Economics.

Main References

Kim, Seung Hyun. (2024). "Asset Pricing and Term Structure Models". WORK IN PROGRESS.

Fundamentals of Asset Pricing

Basic Terminologies

Assume the following notations:

: a sequence of payoffs for an asset
- Here, is understood to be a payoff the investor would receive at time if she were to invest the asset at time and then sell it at .
: a price to the asset at time
- The goal of asset pricing is to find that the price is a function of its payoff , i.e.

Definition (rate of return).

The rate of return of the asset at time is defined as where the asset's return at is defined as .

Using ^a14380 Definition 1 (rate of return), we have note that both the denominator () and the numerator () is random variables.

the asset is risk-free: the rate of return of the asset is known at time , denoted as .
the asset is risky: the rate of return of the asset () is a random variable.
- amount of risk of the asset at is represented by .
- profitability of the asset at is represented by expected rate of return, .
- an asset is said to be high-risk and high-return if and are both large, and it is low-risk and low-return if the both are small.

Definition (risk premium).

The risk premium, or expected excess return of an asset at time is defined as which is known at time . The risk premium represents the compensation that an investor receives in exchange for taking on risk. Generally, the riskier the asset, the higher its risk premium.

Capital Asset Pricing Model

Assumption (assumptions for CAPM).

Many homogeneous investors
Two-period model: the economy lasts for two period of and , where the investors invest at time and receive the payoffs at time .
Many risky assets and one risk-free asset: mean and covariance of the vector of risky asset rate of return are given by where is a positive definite matrix.
Mean-variance utility: given a portfolio with expected rate of return and variance , the representative investor receives utility equal to where denotes the risk-aversion coefficient.

Note that a portfolio with weights has

rate of return:
expected rate of return:
risk premium:
variance (risk):

Thus the problem is to find the vector of weights from the following maximization problem: Then F.O.C. note that is a maximizing solution since the S.O.C. is where the inequality holds as is a positive definite.

Thus we have
and the -th row is which represents the risk premium of the -th asset.

The optimal portfolio, i.e. the market portfolio has

rate of return:
expected rate of return:
risk premium:
risk (variance): From the driven the risk premium of the market portfolio, we have thus the risk-aversion coefficient can be re-written as Furthermore, for any , we have It follows that the risk premium of the -th asset is where and are called beta term and the market price of risk, respectively.

Definition (beta-term).

From the formula of risk premium of asset , the first term is denoted as beta-term which denotes

the idiosyncratic part of risk premium of the asset .
the coefficient from the regression of on .
determining how much the factor loads onto the risk premium of asset .
how much represents the market risk(systematic risk).
the of asset added(offset) by the market risk.

Definition (market price of risk).

From the formula of risk premium of asset , the second term is denoted as market price of risk which denotes

the common part of the risk premium of each asset.
as denotes the risk-aversion coefficient, it means how much the market-risk be represented in the for all .

Consumption-CAPM

Model and Solution

Assumption (assumptions for C-CAPM).

Many homogeneous investors.
Multi-period model: the economy starts at times , and is populated by infinitely living investors.
Many risky assets and one risk-free asset:
1. there exists one consumption good, whose price normalized as .
2. there exists risky assets with rate of return at time as and one risk-free asset with rate of return .
3. at time , the representative investor is assumed to be endowed with unit of asset .
4. at time , the representative investor holds units of asset .
5. at time , the price and dividend of asset is denoted as and , respectively.
General utility function: the representative investor has instantaneous utility function such that and , implying that the investors are risk-averse and receive positive marginal utilities from consumption.
1. discount factor is given as .
2. wage income is given as for each period.

The investor's problem is

The Lagrangian is F.O.C.

Thus the Euler equation is which implies that the marginal benefit from the additional consumption at () equals to the discounted expected marginal cost received at .

In asset pricing literature, the Euler equation is interpreted as a pricing formula,

Definition (stochastic discount factor).

From Euler equation, the price of asset () is given as the expectation of its payoff discounted by the stochastic discount factor (SDF) which denotes

the discounted marginal utility of future consumption () relative to the marginal utility of current consumption ().
which gets smaller if the investor expects to consume more tomorrow than today.
which discounts the asset 's payoff much more

Beta Representation

From the pricing formula, by dividing both sides by , we have Rearranging the equation, we have and the expected rate of return of the risk-free asset is thus Therefore, the risk premium of the asset is where similar to Capital Asset Pricing Model Capital Asset Pricing Model, denotes the ^c57404 Definition 4 (beta-term) and denotes the ^38ea47 Definition 5 (market price of risk).

Sharpe Ratio

Definition (sharp ratio).

The Sharp ratio (SR) of an asset is an indicator of the profitability of an asset relative to its risk. The SR of asset from time to is defined as which is a risk premium (profit) of as asset divided by its risk (volatility, the standard deviation of the rate of return of asset ).
The higher the SR, the grater the the profitability of the asset compared to other assets with the same amount of the risk.

Using the price formula, we can express in term of the correlation between the asset's rate of return and the SDF.
Then, the SR is As by the definition of correlation, we have where the highest and lowest possible SR is the case when the rate of return is perfectly correlated with SDF.

Also, note that by the ^fd2d45 Definition 8 (sharp ratio), we have which tells us that the expected rate of return of an asset can be understand as the deviation from the risk-free rate of return () by its standard deviation multiplied by the SR. Here, the slope of the line, called as capital allocation line (CAL), is exactly the SR.

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Special Cases of C-CAPM

Case of Log-Normal Returns

Suppose now that the log of the SDF and the return jointly follow a normal distribution conditional on information up to time :

Note that when the random variable follows the normal distribution, the moment generating function is Then the pricing formula is Taking logs on both sides, thus the expected rate of return is

Similarly, for the risk-free asset by taking logs on both sides, Therefore, the risk premium of the asset is Note that the additional variance term is denoted as Jensen's Inequality term, and it is often ignored when talking of about the expected excess returns (risk premium).

We can also re-define the ^fd2d45 Definition 8 (sharp ratio) as

Case of CRRA Utility

From ^187b53 Assumption 6 (assumptions for C-CAPM), now assume for the special utility function of CRRA utility: where is the coefficient of relative risk aversion.

Then, the ^96b178 Definition 7 (stochastic discount factor) is

Defining a consumption growth as , we have where the approximation used the Taylor expansion.

Then, the risk premium of asset is and note that often represents the systematic risk presents in the economy, and the beta term (covariance term) shows how much the systematic risk is embedded in the asset itself, and differs from the asset to asset.