# Mathematics and Economics

## CCAPM I: The Basic Pricing Equation

In Asset Pricing Reading Group, Bubbles and Crashes, Economics, Finance on January 20, 2011 at 1:41 am

1. Introduction

Over a series of two posts, we will discuss the consumption capital asset pricing model (CCAPM) and some of its results concerning expected asset price returns. In particular, we derive a range within which asset returns lie. The model we will be discussing describes the simplified behavior of a trader who chooses to consume a portion of his income and invest the rest in market assets over two time periods of any kind (i.e. days, months, etc.)

In this post, we describe a simplified setting of the model and derive a formula for prices arising from the marginal cost and marginal benefit of the trader’s investment decisions. This equation will then show us how prices are affected by preferences. Furthermore, the resultant equation turns out to be robust to the initial simple setting of the model. Our exposition of the model borrows from John Cochrane’s book, Asset Pricing.

2. Investor Consumption Model

We begin our discussion of the Consumption Capital Asset Pricing Model (CCAPM) by first modeling our trader with a utility function defined over current and future values of consumption:

$\displaystyle U(c_t,c_{t+1}) = u(c_t) + \beta E_t[u(c_{t+1})]. \ \ \ \ \ (1)$

To simplify our discussion, we assume that there exists only one consumption good, denoted by ${c}$, and that all other asset values are quoted in terms of it. The utility function represents the “fundamental pleasure” that the investor derives from consuming the good in question. The period utility function ${u(\cdot)}$ is assumed to be increasing, concave, and continuously differentiable (i.e. smooth). That the function is increasing describes, somewhat expectedly, the desire for greater consumption rather than lesser consumption, while concavity is a result of the decling marginal value of greater consumption. In other words, our investors will be happy to consume more rather than less, but enjoy each additional unit at a decreasing rate.

The second term in (1) captures the trader’s valuation of uncertain future consumption. Here, we assume that the trader discounts his expected utility of future consumption at the rate ${\beta\in(0,1)}$. This captures the tendency of individuals to prefer present consumption to future consumption; the expected utility term, ${E_t[u(c_{t+1})]}$, represents the pleasure that the trader derives from his uncertain future consumption.

In addition, our investor is subject to two constaints. First, his or her income, consisting of an endowment, ${e}$, is to be spent on either consumption or investment in their choice of ${n}$ available assets:

$\displaystyle \begin{array}{rcl} e_t = c_t + \sum_{i=1}^n p^i_t s_t^i \end{array}$

or

$\displaystyle \begin{array}{rcl} c_t = e_t - \sum_{i=1}^n p^i_t s_t^i. \end{array}$

where ${c_t}$ is the amount of the consumption good at time ${t}$; ${e_t}$ is the endownment in terms of the consumption good at time ${t}$; ${s_t^i}$ is the number of purchased shares of asset ${i\in\{1,...,n\}}$ at time ${t}$; and ${p^i_t}$ is the price of asset ${i}$ at time ${t}$.

Second, in the subsequent period, ${t+1}$, he or she will consume the income recieved from these investments and their endowment.

$\displaystyle \begin{array}{rcl} c_{t+1} &=& e_{t+1} + \sum_{i=1}^n x^i_{t+1}s_t^i \end{array}$

where ${x^i_{t+1}}$ is the payoff from their investment in asset ${i}$ at time ${t+1}$. Here, we are allowing for assets to pay dividends, so that ${x^i_{t+1}=p^i_{t+1}+d^i_{t+1}}$ where ${d^i_{t+1}}$ denotes the dividend of asset ${i}$ payed out at time ${t+1}$.

To summarize, our objective, or optimization problem is:

$\displaystyle \begin{array}{rcl} \max_{(s^i_t)_{i=1}^n}u(c_t) &+& E_t\left[\beta u(c_{t+1})\right]\; s.t.\\ c_t &=& e_t - \sum_{i=1}^n p^i_ts_t^i\\ c_{t+1} &=& e_{t+1} + \sum_{i=1}^n x^i_{t+1}s_t^i \end{array} (2)$

where

• ${c_t}$ is the trader’s consumption at date ${t}$
• ${\beta}$ is the trader’s subjective discount factor
• ${e_t}$ is the trader’s endowment at time ${t}$
• ${p_{t}^i}$ is the price of asset ${i}$ in terms of the consumption good at time ${t}$.
• ${ x^i_t}$ is the payoff of asset ${i}$ where ${ x^i_{t+1}}$ is a R.V. and ${ x^i_{t+1} = p_{t+1} + d_{t+1}}$.
• ${s_t^i}$ is the amount of asset ${i}$ that the trader holds (his shares).

3. First Order Conditions and CCAMP Pricing Equation

Since our utility function is strictly increasing, concave, and continuously differentiable, this optimization problem satisfies the Kuhn-Tucker conditions and we are ensured a unique solution to our objective problem which satisfies the first order condition (FOC):

$\displaystyle p_t^i u'(c_t) = E_t\left[\beta u'(c_{t+1}) x^i_{t+1}\right] \; \ \ \ \ \ (3)$

or

$\displaystyle p_t^i = E_t\left[\beta\frac{u'(c_{t+1})}{u'(c_t)} x^i_{t+1}\right] \ \ \ \ \ (4)$

To elucidate further, we have the following interpretation of (3): ${p_t^i u'(c_t)}$ is the loss in the utility if the investor buys another unit of the asset and ${E_t\left[\beta u'(c_{t+1}) x^i_{t+1}\right]}$ is the gain in discounted, expected utility he obtains from the extra payoff at ${t+1}$, ${ x^i_{t+1}}$. The tradercontinues to buy or sell the asset until marginal loss equals marginal gain.

Rearranging (3), we get (4). Pricing equation (4) is of striking importance: it supplies a relationship between prices and the preferences of individual traders.

Now, we define the Stochastic Discount Factor (SDF), or Marginal Rate of Substitution (MRS), ${m_{t+1}}$ as

$\displaystyle m_{t+1} : = \beta \frac{u'(c_{t+1})}{u'(c_t)} \ \ \ \ \ (5)$

and rewrite our pricing equation (4) as

$\displaystyle p_t^i = E_t (m_{t+1}x^i_{t+1}) \ \ \ \ \ (6)$

The SDF has an intuitive economic interpretation: ${m_{t+1}}$ is the rate at which our investor is willing to substitute consumption at time ${t+1}$ with consumtion at time ${t}$.

4. Next Post: Asset Returns and Risk

In our next post, we will discuss asset returns, and reformulate the results in this post in terms of them. We will discuss how risk affects returns, and discuss risk corrections.