Consumption-Based Pricing
\[ \def\BB#1{{\mathbb{#1}}} \def\BF#1{{\mathbf{#1}}} \]
yedlu, Winter 2024
Riskiness of Assets
We’ve all heard of the term risk-free/risky assets, but how can we formally define the riskiness of assets?
Consider the following simple state setting:
- \(S = \{s_1, s_2\}\) be the state space;
- \(p_1, p_2 = 1 - p_1\) be the probability of each state happening;
- \(x_1, x_2\) be the state-contingent payoff of the asset.
We will say an asset to be risk-free if the said asset pays the same payoff across every possible states in the state space:
\[\begin{aligned} x_1 = x_2 \end{aligned}\]
Similarly, an asset is risky if the payoff is different across states:
\[\begin{aligned} x_1 \neq x_2 \end{aligned}\]
Pricing Through Consumption
Settings
To begin our analysis in the simpliest way possible, we assume that individual living in this economy only enjoys and values consumption over today (time \(t\)) and tomorrow (time \(t+1\)).
Here are the notations useful in the model:
| Variable | Description |
|---|---|
| \(c_t\) | consumption level at time \(t\) |
| \(u(\cdot)\) | utility function over consumption level with \(u' > 0\) (strictly increasing) and \(u'' < 0\) (concave) |
| \(\beta \in (0, 1)\) | discount factor to future utility (investor impatience) |
| \(p_t\) | asset price (or more formally willingness to pay) at time \(t\) |
| \(x_{t+1}\) | asset payoff at time \(t+1\) |
| \(\xi\) | arbitratily-small unit of the asset an investor decides to buy |
| \(e_t\) | endowment level at time \(t\) |
| \(\BB{E}_t\) | expected utility conditional on all information available at time \(t\) |
The Optimization Problem
Under this two-period setting, every individual is rational and wishes to maximize intertemporal utility through choosing consumption levels across the time horizon:
\[\begin{aligned} \max_{c_t, c_{t+1}} \quad u(c_t) + \beta \BB{E}_t [u(c_{t+1})] \end{aligned}\]
With the asset in the economy, we can further break down the consumption level. The optimization problem:
\[\begin{aligned} \max_{c_t, c_{t+1}} & \quad u(c_t(\xi)) + \beta \BB{E}_t [u(c_{t+1}(\xi))] \\ \text{s.t.} \quad c_t(\xi) &= e_t - p_t \xi \\ c_{t+1}(\xi) &= e_{t+1} + x_{t+1} \xi \end{aligned}\]
now have two budget constraints at both time periods.
Solving the Optimization
Simplify the objective function:
\[\begin{aligned}
\max_{\xi} & \quad u(e_t - p_t \xi) + \beta \BB{E}_t [u(e_{t+1} + x_{t+1} \xi)] \\
\end{aligned}\]
Its First-Order Condition (FOC) w.r.t. \(\xi\) yields:
\[\begin{aligned}
-p_t u'(e_t - p_t \xi) + \beta \BB{E}_t [x_{t+1} u'(e_{t+1} + x_{t+1} \xi)] = 0
\end{aligned}\]
Clean this up and we would have:
\[\begin{aligned}
\beta \BB{E}_t [x_{t+1} u'(c_{t+1})] = p_t u'(c_t)
\end{aligned}\]
Intuition of the FOC
The left-hand side: \[\begin{aligned} \beta \BB{E}_t [x_{t+1} u'(c_{t+1})] \end{aligned}\] describes the future marginal utility gain of buying the asset, discounted to today’s term by \(\beta\).
The right-hand side: \[\begin{aligned} p_t u'(c_t) \end{aligned}\] describes today’s marginal disutility of buying the asset, as we are reducing consumption today.
The optimization amount of asset bought is then met when the marginal loss equals marginal gain.
Intertemporal Asset Pricing Equation
Solving for \(p_t\) of the FOC yields: \[\begin{aligned} p_t = \BB{E}_t [\beta \frac{u'(c_{t+1})}{u'(c_t)} x_{t+1}] \end{aligned}\] where future consumption level \(c_{t+1}\) and asset payoff \(x_{t+1}\) are all random variables (thus need to stay in the expectation).
We can see that the asset’s price (willingness to pay) is dependent to:
- investor-specific utility \(u(\cdot)\);
- future states \(\BB{E}_t\);
- endowment level \(e\);
- discount factor \(\beta\);
- state-contingent asset payoff \(x_{t+1}\).
Stochastic Discount Factor (SDF)
We denote the stochastic discount factor (SDF) as: \[\begin{aligned} & m_{t+1} := \frac{u'(c_{t+1})}{u'(c_t)} \\ \implies & p_t = \BB{E}_t [m_{t+1} x_{t+1}] \end{aligned}\]
Intuitively, this factor serves as a risk-adjusted discount factor on individual level.
Applications
Denoting Returns
By the definition of asset return, it is the dollar amount of payoff if we were to invest in one dollar of the asset. Transcribing this into our pricing equation:
\[\begin{aligned}
1 &= \BB{E}_t [m_{t+1} R_{t+1}]
\end{aligned}\]
Risk-Free Assets
Suppose the risk-free rate is \(R^f\). Since this rate is not a random variable, we can further derive the pricing equation:
\[\begin{aligned}
1 &= \BB{E}_t [m_{t+1} R^f] = R^f \BB{E}_t [m_{t+1}]
\end{aligned}\] If we move back for one time period and use the law of iterated expectation:
\[\begin{aligned}
1 &= R^f \BB{E}_{t-1} [m_t] \\
\implies \BB{E}_{t-1} [m_t] &= \frac{1}{R^f} \\
\\
\BB{E} [m_t] &= \BB{E}[\BB{E}_{t-1} [m_t]] = \BB{E}[\frac{1}{R^f}] \\
&= \frac{1}{R^f}
\end{aligned}\] This links the risk-free rate to the stochastic discount factor!
Risk-Free Rate Determinants
Suppose our utility function: \[\begin{aligned}
u(c_t) &= \frac{1}{1-\gamma} c_t^{-\gamma}
\end{aligned}\] with \(\gamma > 0\).
If future consumption level is not random, then:
\[\begin{aligned}
m_{t+1} &= \left(\frac{c_{t+1}}{c_t} \right)^{-\gamma} \\
\implies m_t &= \beta \left(\frac{c_{t+1}}{c_t} \right)^{-\gamma}
\end{aligned}\]
Then we can denote the risk-free rate as:
\[\begin{aligned}
R^f &= \frac{1}{\beta} \left(\frac{c_{t+1}}{c_t} \right)^{\gamma}
\end{aligned}\]
The following components can shift risk-free rate. Assume risk-free rate is increasing, this can be caused by:
| Factor | Direction | Intuition |
|---|---|---|
| \(\beta\) | \(\downarrow\) | If investors are more impatient, the risk-free rate must be high enough for them to give up liquidity today. |
| \(c_{t+1}/c_t\) | \(\uparrow\) | A high consumption growth in the future deters savings today. This also shifts \(p_t\) down as the asset must be more lucrative to attract investment. |
| \(\gamma\) | \(\uparrow\) | A higher curvature will leads to a steeper decline to the first derivative. Then, investors are less prone to use asset to smooth consumption ceteris paribus. |
Risk Correction
We will derive and show that the stochastic discount factor is essentially a risk-corrected discount factor.
By the definition of covariance:
\[\begin{aligned}
\sigma_{m, x} &= \BB{E}[mx] - \BB{E}[m] \BB{E} [x]
\end{aligned}\]
Using it we derive:
\[\begin{aligned}
p &= \BB{E} [mx] = \BB{E} [m] \BB{E} [x] + \sigma(m, x) \\
&= \frac{\BB{E}[x]}{R^f} + \sigma(m, x)
\end{aligned}\]
Covariance of SDF and Payoff
On the covariance, assets whose payoff \(x\) is negatively correlated with the SDF (we will later show that together with convex utility, a higher SDF implies a smaller consumption growth) have a lower price.
We first further derive the risk-adjusted form of pricing formula:
\[\begin{aligned}
p &= \frac{\BB{E}[x]}{R^f} + \sigma_{m, x} \\
&= \frac{\BB{E}[x]}{R^f} + \sigma_{\beta \frac{u'(c_{t+1})}{u'(c_t)}, x} \\
&= \frac{\BB{E}[x]}{R^f} + \frac{\beta \sigma_{u'(c_{t+1}), x}}{u'(c_t)}
\end{aligned}\]
By the assumption of the utility function, \(u''<0\) holds. This implies that \(u'(c_1) > u'(c_2)\) whenever \(c_1 < c_2\).
Then, the covariance \(\sigma_{u'(c_{t+1}), x}\) is positive when the payoff of bad states (low consumption growth, higher \(u'\)) is high, vice versa.
On Risk Corrections
Risk-Neutral
If investors are risk-neutral in this economy, they only care about the expected consumption level. Hence, their utility function can be assumed to have a linear form:
\[\begin{aligned}
u(c_t) &= a + b c_t
\end{aligned}\]
Hence:
\[\begin{aligned}
u'(c_t) &= b = u'(c_{t+1})
\end{aligned}\]
Then, with \(m_t = \beta\) comes \(\sigma_{m, x} = 0\): there are no risk adjustments to asset pricing:
\[\begin{aligned}
p &= \frac{\BB{E}[x]}{R^f} + \sigma_{m, x} \\
&= \frac{\BB{E}[x]}{R^f}
\end{aligned}\]
Constant Consumption
It is obvious that \(u'(c_t) = u'(c_{t+1})\) when \(c_t = c_{t+1}\).
Systematic/Idiosyncratic Risks
Assume we have an risky asset:
- idiosyncratic risks are payoff risks uncorrelated with market conditions (in here the SDF): \(\sigma_{m, x} = 0\). These types of risks do not need to be adjusted by the market-wise SDF, as it is already implied by cash flow across states.
- systematic risks are payoff risks correlated with market conditions (consumptions). Only systematic risks generate risk correction through SDF.
Expected Return of Risky Assets
From our results above, we have:
\[\begin{aligned}
1 &= \BB{E}[mR^i] \\
&= \BB{E} [m] \BB{E} [R^i] + \sigma_{m, R^i} \\
&= \frac{\BB{E} [R^i]}{R^f} + \sigma_{m, R^i} \\
\\
\implies \BB{E} [R^i] - R^f &= - R^f \sigma_{m, R^i}
\end{aligned}\] Further expanding \(m\) and \(R^f\) yields:
\[\begin{aligned}
\BB{E} [R^i] - R^f &= - \left(\BB{E} [m]\right)^{-1} \sigma_{m, R^i} \\
&= - \left(\BB{E} [\beta \frac{u'(c_{t+1})}{u'(c_t)}]\right)^{-1} \sigma_{\beta \frac{u'(c_{t+1})}{u'(c_t)}, R^i} \\
&= - \frac{\sigma_{u'(c_{t+1}), R^i}}{\BB{E}[u'(c_{t+1})]}
\end{aligned}\]
Intuition:
- A higher excess return is required if \(\sigma_{u'(c_{t+1}), R^i}\) is negative.
By convexity of utility, lower \(u'\) indicates higher consumption. This means that the asset has a higher return when consumption growth is high (good state) and vice versa. This makes consumption even more volatile and has to compensate investors. - A lower excess return is required if \(\sigma_{u'(c_{t+1}), R^i}\) is positive.
Consider insurance products that pays when consumption growth is small (bad state) and do not pay vice versa. Since these products smooth consumption, we do not need additional incentive for investors to hold on to this asset.