Consumption Based Asset Pricing

Lorenzo Naranjo

Fall 2026

Introduction

Time-Line of Cash Flows

• Most of modern asset pricing can be cast in terms of the stochastic discount factor.
• The idea is to find the price p_{t} at time t of a payoff x_{t+1} paid at t+1.
• The payoff will in general be random, and therefore unknown at time t.
• For example, if you purchase a stock at time t your payoff at time t+1 will be the price p_{t+1} at which you can sell the stock plus possibly a dividend d_{t+1}, or x_{t+1} = p_{t+1} + d_{t+1}.

Intertemporal Utility

Intertemporal Utility Theory

• Let’s now think about an investor considering buying slightly more of a stock that pays x_{t+1} at time t+1.
• The investor has already arranged her investments in such a way that she can consume optimally at time t and t+1.
• The optimality of consumption is defined with respect a .
• A simple way to model the tradeoff between consuming today versus tomorrow is to write U(c_{t}, c_{t+1}) = u(c_{t}) + \beta \operatorname{E}_{t}[u(c_{t+1})], where \operatorname{E}_{t}[\cdot] denotes the expectation conditional on the information available at time t, and u(\cdot) is an increasing and concave function of consumption.

Perturbation Analysis

• If the investor purchases \xi shares of the asset, the consumption at time t drops to c_{t} - \xi p_{t} whereas the consumption at time t+1 changes to c_{t+1} + \xi x_{t+1}.
• Given the original levels of consumption C_{t} and C_{t+1}, the new utility can be seen as a function of \xi U(\xi) = u(c_{t} - \xi p_{t}) + \beta \operatorname{E}_{t}[u(c_{t+1} + \xi x_{t+1})].
• The first-order condition (FOC) for the optimal \xi is: U'(\xi) = -p_{t} u'(c_{t} - \xi p_{t}) + \beta \operatorname{E}_{t}[x_{t+1} u'(c_{t+1} + \xi x_{t+1})] = 0.

The Euler Equation

• However, if the original levels of consumption c_{t} and c_{t+1} are already optimal, then we know that the optimal \xi = 0, therefore -p_{t} u'(c_{t}) + \beta \operatorname{E}_{t}[x_{t+1} u'(c_{t+1})] = 0, or P_{t} = \operatorname{E}_{t}\left[\beta \frac{u'(C_{t+1})}{u'(C_{t})} X_{t+1}\right]. \tag{1}

The Stochastic Discount Factor

• Equation (1) is the fundamental asset pricing formula. Intuitively, the price of an asset is high if the asset pays well when marginal utility is high, that is, when consumption is low.
• An asset that pays well when consumption is high and not much when consumption is low is not attractive for a risk-averse investor, carrying a low price.
• It is common to write (1) as p_{t} = \operatorname{E}_{t}(m_{t+1} x_{t+1}), where m_{t+1} = \beta \frac{u'(c_{t+1})}{u'(c_{t})} \tag{2} is called the stochastic discount factor.

The Fundamental Pricing Equation

• To simplify notation, we will typically write the pricing equation as p = \operatorname{E}(m x), \tag{3} where is understood that p denotes the price today of a payoff x paid next period.
• If x is a random variable defined on a finite probability space (\mathcal{S}, q), the pricing equation can be written as p = \sum_{s \in \mathcal{S}} q(s) m(s) x(s), \tag{4} where s \in \mathcal{S} denotes a state of the world, q(s) is the probability of state s occurring, and x(s) is the payoff if state s happens.

The SDF with Power Utility

• Consider an investor with a power utility function u(c) = \frac{c^{1 - \gamma}}{1 - \gamma}.
• The stochastic discount factor is then given by m_{t+1} = \beta \left(\frac{c_{t+1}}{c_{t}}\right)^{-\gamma}.

Potential Payoffs

• The investor’s current consumption is c_{t} = 6.5, and is considering investing in two assets X and Y.
• The table below presents the probabilities of different scenarios, along with the future consumption and payoffs of the assets.

	Probability	Consumption	Payoff X	Payoff Y
Boom	0.30	9.00	9.80	6.00
Normal	0.50	6.70	8.30	5.00
Recession	0.20	5.40	6.50	7.10

Computing the Stochastic Discount Factor

• If \gamma = 4 and \beta = 0.95, we can compute the stochastic discount factor (SDF) for each scenario as

	Probability	SDF
Boom	0.30	0.26
Normal	0.50	0.84
Recession	0.20	1.99

• We find that p(x) = 6.85 and p(y) = 5.4, respectively.
• The expected return of each asset is equal to the expected payoff divided by its price minus one.
• Thus, \operatorname{E}(r^{x}) = 22.57\% and \operatorname{E}(r^{y}) = 5.91\%.

The Real SDF

• So far we have been silent about the currency used to price assets and quantify payoffs.
• It is clear, however, that our derivation of the fundamental pricing equation (1) used units of real consumption to quantify prices and payoffs.
• The stochastic discount factor defined in (2) is a real discount factor.

The Nominal SDF

• Let p_{t} = p_{t}^{*} / \Pi_{t} and x_{t+1} = x_{t+1}^{*} / \Pi_{t+1} where \Pi_{t} denotes the price level and we use asterisks to denote nominal quantities. \frac{p_{t}^{*}}{\Pi_{t}} = \operatorname{E}_{t} \left(m_{t+1} \frac{x_{t+1}^{*}}{\Pi_{t+1}} \right).
• The previous expression implies that p_{t}^{*} = \operatorname{E}_{t} \left(m_{t+1}^{*} x_{t+1}^{*} \right), where m_{t+1}^{*} = m_{t+1} \frac{{\Pi}_{t}}{\Pi_{t+1}}.
• We obtain the same equation as before but now expressed in nominal terms.
• In the analysis, m_{t+1}^{*} plays the role of a nominal discount factor.

Prices and Returns

Gross vs. Net Returns

• We will use R_{t+1} = \frac{x_{t+1}}{p_{t}} to denote the gross rate of return of investing in the stock.
• For example, if an investment of $100 generates $105, the gross return is R = 1.05, whereas the net return is r = 5\%.

The Fundamental Pricing Equation for Returns

• For a given security i, the fundamental pricing equation can also be written as \operatorname{E}(m R^{i}) = 1. \tag{5}
• A risk-free asset pays next period x = 1 no matter what, so that its return R^{f} is constant.
• This implies that \operatorname{E}(m R^{f}) = 1, or R^{f} = \frac{1}{\operatorname{E}(m)}.
• The stochastic discount factor has all the information to recover the behavior of interest rates in the economy.

Zero-Cost Portfolios

• A zero-cost portfolio involves buying asset i and shorting asset j, generating a zero-cost return R^{e} = R^{i} - R^{j}.
• Equation (5) implies that \operatorname{E}(m R^{e}) = 0 for any zero-cost portfolio.
• The price of a zero-cost portfolio is of course zero, since it involves no cash outflow to create it.
• The payoffs, however, need not be equal to zero as they are determined by R^{i} - R^{j}.

Compensation for Time and Risk

Accounting for Time and Risk

• The pricing of risky cash flows should incorporate two dimensions.
  • On the one hand, cash flows paid in the future should be discounted to account for the time value of money.
  • On the other hand, riskier payoffs should be generate lower prices.
• To analyze these issues, we can again start from (3) \begin{aligned} p & = \operatorname{E}(m x) = \operatorname{E}(m) \operatorname{E}(x) + \operatorname{Cov}(m, x) \\ & = \frac{\operatorname{E}(x)}{R^{f}} + \operatorname{Cov}(m, x). \end{aligned} \tag{6}
• The price of the asset is high when R^{f} is low and/or the covariance with the stochastic discount factor is high.

Beta Pricing

• We can use returns instead of prices to write (3) for asset i as 1 = \operatorname{E}(m R^{i}) = \operatorname{E}(m) \operatorname{E}(R^{i}) + \operatorname{Cov}(m, R^{i}).
• Dividing the previous expression by \operatorname{E}(m), we find that \operatorname{E}(R^{i}) - R^{f} = - \frac{\operatorname{Cov}(R^{i}, m)}{\operatorname{E}(m)} = \beta_{i,m} \lambda_{m}, \tag{7} where \beta_{i,m} = \dfrac{\operatorname{Cov}(R^{i}, m)}{\operatorname{V}(m)} and \lambda_{m} = - \dfrac{\operatorname{V}(m)}{\operatorname{E}(m)} < 0.

The Marginal Utility of Consumption

• Equation (2) implies that \beta_{i, m} = \dfrac{\operatorname{Cov}(R^{i}, u'(c_{t+1}))}{\operatorname{V}(u'(c_{t+1}))} and \lambda_{m} = - \dfrac{\operatorname{V}(u'(c_{t+1}))}{\operatorname{E}(u'(c_{t+1}))}.
• In the beta pricing model defined by (7) only the marginal utility of future consumption matters to price assets.
• This is a simplified version of the consumption CAPM of Breeden (1979) and Lucas (1978).

Multi-Period Asset Pricing

A Multi-Period Utility Function

• Consider now a stream of consumption \left\{c_{t+j}\right\}_{j=0}^{\infty}.
• We can extend the utility function defined earlier the following way: V_{t} = \operatorname{E}_{t} \sum_{j=0}^{\infty} \beta^{j} u(c_{t+j}). \tag{8}
• Note that we can also write (8) as \begin{aligned} V_{t} & = u(c_{t}) + \beta \operatorname{E}_{t} \sum_{j=0}^{\infty} \beta^{j} u(c_{t+1+j}) \\ & = u(c_{t}) + \beta \operatorname{E}_{t} \operatorname{E}_{t+1} \sum_{j=0}^{\infty} \beta^{j} u(c_{t+1+j}) \\ & = u(c_{t}) + \beta \operatorname{E}_{t} V_{t+1}. \end{aligned} \tag{9}

Pricing a Stream of Cash Flows

• Consider now a stream of dividends \left\{D_{t+j}\right\}_{j=1}^{\infty}.
• The same perturbation analysis can be used to show that p_{t} = \operatorname{E}_{t} \sum_{j=1}^{\infty} \beta^{j} \frac{u'(c_{t+j})}{u'(c_{t})} D_{t+j}. \tag{10}
• Therefore, the corresponding stochastic discount factor to price a dividend paid at time t+j is: m_{t+j} = \beta^{j} \frac{u'(c_{t+j})}{u'(c_{t})}.

Applications

• Real discount bond expiring with time-to-maturity time T and face value 1 unit of consumption: B_{t} = \operatorname{E}_{t} (m_{t+T}).
• Nominal discount bond expiring with time-to-maturity time T and face value $1: B_{t}^{*} = \operatorname{E}_{t} \left(m_{t+T} \frac{\Pi_{t}}{\Pi_{t+T}}\right) = \operatorname{E}_{t} (m_{t+T}^{*}).
• Call option with strike K and maturity T: c_{t} = \operatorname{E}_{t} (m_{t+T}^{*} (S_{t+T} - K)^{+}).

References

References

Breeden, Douglas T. 1979. “An Intertemporal Asset Pricing Model with Stochastic Consumption and Investment Opportunities.” Journal of Financial Economics 7 (3): 265–96.

Lucas, Robert E. 1978. “Asset Prices in an Exchange Economy.” Econometrica 46 (6): 1429–45.