Part II Financial Markets and Portfolio Theory
C- CAPM and Hansen–Jagannathan-Bounds
The consumption-based capital asset pricing model (C-CAPM) comes in many guises and is strongly related to the CAPM, if we assume that agents use the market portfolio to shift consumption between periods. Let’s take Equation (9.63) and remember that the stochastic discount factor is defined by
We then obtain the consumption-based CAPM equation
(9.70)
(9.71)
(9.72)
(9.73)
(9.74)
(9.75)
(9.76)
It is now immediately obvious why (9.69) is called the C-CAPM, because it relates the return on the stock S to the marginal utility of consumption at time t. What is its connection to the original CAPM?
To get a handle on this, let’s first ask what we get for the market portfolio. Replacing RS by RMP in (9.69) and rearranging yields
Using (9.70) back in (9.69), we obtain the already more familiar form
We now have to ask, what is the relation of marginal utility and the return of the market portfolio?
There is more than one way to obtain a suitable answer (see Cuthbertson and Nitzsche, 2004, sect.
13.2, for four different ways). We will take the short tour here. Let’s assume that the unknown utility function u(c) can be sufficiently approximated by the quadratic utility function
Now remember that the agent has to divide her initial wealth endowment w between time zero consumption c0 and the proportion invested in the market portfolio, making uncertain time t consumption available, measured in c0-units. Her budget constraint is thus
Calculating marginal utility of the time t consumption Ct under quadratic utility yields
with a = η − b, and b = w − c0. The important point is that the return on the market portfolio is an affine transformation of the marginal utility of consumption. Computing the covariances in (9.71) yields
Remember that the quotient of the two covariances in (9.75) is exactly the definition of the beta- coefficient βS in the original CAPM. We have thus from (9.71)
which is the capital asset pricing model.
We can even examine the relation of marginal utility and the market portfolio a little bit further.
Remember that the correlation coefficient between two arbitrary random variables X and Y is defined as
(9.77)
(9.78)
(9.79)
(9.80)
Correlation is a normalized measure of linear dependence of random variables. So let’s calculate the correlation between marginal utility u′(Ct) and the market portfolio return RMP,
Quick calculation 9.17 Confirm the first equality.
Fig. 9.2 Hansen–Jagannathan-bounds in the μ-σ-diagram – Upper leg is the capital market line Returns on the market portfolio are perfectly negative correlated with marginal utility. Negative correlation was of course to be expected because higher returns coincide with states of higher consumption and thus lower marginal utility of consumption, if the agent is risk averse. The perfect correlation is due to the affine relation of marginal utility and market portfolio returns, because the dependence structure is fully linear.
Quick calculation 9.18 Show that for X = a + bY the correlation of X and Y is ρ = 1.
The work of Hansen and Jagannathan (1991) revealed another close connection between the C- CAPM and the original CAPM. This one is particularly important, because it ultimately implies certain empirical predictions. The basic idea is actually straightforward. Rewriting the definition of the correlation coefficient (9.77) in a slightly different way yields
Using this relation, we can write the C-CAPM equation (9.69) in the following form
(9.81)
(9.82)
•
•
•
•
9.6
Note that the correlation coefficient is by definition |ρ| ≤ 1. Therefore, we can formulate an inequality, which is valid for every security S, and in alignment with our earlier notation reads
where again the standard deviation StD[ã] is the positive root of the variance. Figure 9.2 shows the Hansen–Jagannathan-bounds in the μ-σ-diagram. The upper leg is the mean-variance efficient frontier for risky assets (capital market line, CML), whereas the lower leg represents insurance contracts that are perfectly correlated with the stochastic discount factor. For the market portfolio, which is perfectly negative correlated with marginal consumption, and therefore with the stochastic discount factor, we obtain an upper theoretical limit for the Sharpe-ratio
to be observed in reality, if the respective SDF model is correct.
Quick calculation 9.19 Why are all returns on the CML perfectly negative correlated with the SDF?
The Equity Premium Puzzle
The equity premium puzzle of Mehra and Prescott (1985) is probably the most prominent puzzle in modern economics. Today it is really understood to consist of two parts, the equity premium puzzle and the risk-free rate puzzle. Each part of the puzzle generates empirical predictions, irreconcilable with the other part. In order to make empirical predictions, we have to specify the stochastic discount factor concretely. Here are the assumptions originally made by Mehra and Prescott:
Agents have utility functions of the HARA-type, in particular
with relative risk aversion γ ≥ 0.
Agents maximize their lifetime utility successively, according to a time separable von Neumann–Morgenstern-utility functional
depending solely on the consumption stream Ct for . Markets are complete.
There are no frictions like trading costs, etc.
(9.84)
(9.85)
(9.86)
(9.87) (9.83)
Of course the last two assumptions imply the existence of a representative agent (Constantinides, 1982). The first thing to do is to formulate a model for the historically observed consumption stream.
One particularly useful model is the AR(1)-process for the logarithmic consumption
where we followed the widely accepted convention in economics to represent the natural logarithm of a random variable by its lowercase letter. In (9.83), g is the growth rate of the consumption process and . It will come in handy to write the consumption growth model in a slightly different form,
where Δct = ct − ct−1. Note that (9.84) is a pure noise process, because is not related to the history of the consumption process. Using hyperbolic risk aversion, as assumed by Mehra and Prescott (1985), we can now specify the SDF in a very concrete form
Before we proceed, let’s summarize briefly the empirical findings of Mehra and Prescott. Without going into too much detail, they used various time series to calculate inflation adjusted annual returns for the Standard & Poor’s Composite Stock Price Index (S&P 500), per capita real consumption growth on non-durable commodities and services and an annual risk-free interest rate proxy from relatively riskless shortterm securities like US Treasury bills, in the period of 1889 to 1978.
According to this data, the S&P 500 realized an annual average growth rate of , with a standard deviation of . The average risk-free interest rate was r = 0.8% p.a. and thus, the average equity premium was . Consumption growth rate and standard deviation of the random error in (9.84) were estimated as and . What do these figures tell us about the agent’s risk aversion? First of all, the average Sharpe-ratio of the S&P 500 is roughly 37.36%. We can also assume that the Standard & Poor’s index is a good proxy for the market portfolio and thus it should be perfectly negative correlated with the SDF. We have thus by (9.82)
In order to complete our calculations we need another standard result from statistics. Consider again the log-normal distributed random variable Y = eX, with X ∼ N(μ, σ2). We already used that
. The variance of Y is given by
Now consider the ratio between the variance and the squared expectation value of such a random variable
(9.88)
(9.89)
(9.90)
(9.91)
(9.92)
(9.93)
(9.94)
for small variances. The ratio of standard deviation and expectation value is the positive root of (9.88).
According to (9.88), the right hand side of (9.86) is approximately , because by (9.85) we have . Using the estimate obtained from the data and rearranging yields
Is this alarmingly high? From experiments on gambles the relative risk aversion was expected to be in the range 3 ≤ γ ≤ 10. There is a neat self test you can try to determine your own personal relative risk aversion in Cuthbertson and Nitzsche (2004, p. 328). The bottom line is that if your relative risk aversion is too high, sticking to HARA-utility leads to implausible results (see Rabin’s paradox, Rabin, 2000). So the question is, is our , induced by the observed equity premium, too high? To answer this question, we need a second piece of information provided by the stochastic discount factor framework, that we have not yet extracted. This one leads to what is known as the risk-free rate puzzle.
I n (9.60) on page 167, we established the relation between the risk-free gross return and the expectation value of the SDF. For continuous compounding over one period of time, we thus obtain
Because we have now specified the concrete family of utility functions, and we have a stochastic model for the evolution of the consumption stream, we get another equation from (9.85)
Equating (9.90) and (9.91) one obtains for the risk-free interest rate
Using the estimates for γ, g, and , we have extracted from the data, we can compute a lower bound for the estimate of r,
This is way too high. Even if we take into account that the risk-free interest rate was averaged by (Mehra and Prescott, 1985). From their data we have , where the estimator is asymptotically normal, with standard deviation
There is no way this discrepancy could be explained by random fluctuations. The probability of accidently observing an average rate of or smaller, like the one in the data, whereas the true risk-free interest rate is indeed r = 12% is of order .
(9.95)
9.7
Quick calculation 9.20 Can you see how this probability was computed?
This means you would have to wait something like 1078 years on average, to see one such weird sample. The age of the entire universe is roughly 1.5 ã 1010 years. You can now appreciate how overwhelmingly improbable such an event would be.
Of course our analysis is based on certain assumptions regarding the utility structure, the consumption stream process, and so forth. It is perfectly possible that relaxing some of those restrictions might work in our favor. But there is still a long way to go and there is no guarantee that modifying the assumptions does not make things worse. Here is one example. We assumed that the market portfolio can be represented by the S&P 500 index and thus, the correlation between the SDF and the return on the index is ρ = − 1. Empirical findings indicate that the correlation could be substantially smaller, because the S&P 500 is a limited proxy of the real market portfolio. Assume that this correlation was estimated as . Where would it go in the equation? From (9.80) and (9.89) we would conclude that
Quick calculation 9.21 Confirm this result.
This would make things substantially worse. It has also been questioned, if data reaching more than 100 years into the past is appropriate, because economies and markets have undergone substantial changes due to two world wars and environmental developments in the law and social systems, globalization, technology, etc. Cochrane (2005, sect. 21.1) reports for the New York Stock Exchange index (NYSE) a post-WWII Sharpe-ratio of roughly one half. His empirical findings require a relative risk aversion of γ ≈ 50. The corresponding lower bound for the risk-free rate of return is r ≥ 37.5%. This is far beyond observation and would cause the economy to collapse because of a massive drop in consumption and investments.
The equity premium puzzle leaves us with two equivalent questions that cannot be answered simultaneously. We can either ask, how can a required moderate relative risk aversion, say in the range 0.5 ≤ γ ≤ 2.5 generate such a high equity premium as observed in the data, or we can accept a high γ, but are immediately faced with the problem of explaining far too low risk-free interest rates. A number of quite brilliant suggestions have been made since the seminal article of Mehra and Prescott (1985), to resolve the equity premium puzzle. We will discuss one of them in the sequel.
The Campbell–Cochrane-Model
The model of Campbell and Cochrane (1999) resolves the puzzles discussed so far essentially by extending the state space. Until now, we assumed that utility solely depends on the consumption stream. Campbell and Cochrane added another variable, representing a common consumption level all agents in the economy got used to in the recent past. This concept is called habit formation and it
(9.96)
(9.97)
(9.98)
(9.100) (9.99)
generally comes in two flavors. The first one is an individual habit formation, based on becoming adapted to a certain standard of living. Hence, not the absolute levels of consumption dominate the utility function, but the changes with respect to the individual habit level. The second version of habit formation is oriented on the average consumption level of all other agents in the economy. Thus, the stimulus for a given agent is exogenous and is called “keeping up with the Joneses” by Abel (1990).
This is the type used by Campbell and Cochrane (1999). In particular they suggested the following utility function
where the second state variable x represents a common consumption level. There is another useful quantity called the surplus consumption ratio
Obviously, the surplus consumption ratio increases with consumption and for Ct = Xt, we have St = 0, which is a very bad state. Let’s compute the relative risk aversion of an arbitrary agent with utility function (9.96) and keep in mind that the average consumption level Xt is exogenous for her
with .
Quick calculation 9.22 Confirm this result.
This is extremely enlightening, because we already get a glimpse of how this mechanism might work.
First of all, we may have a small γ, but the surplus ratio s is most likely very small, because it is a relative quantity. Thus, the relative risk aversion may come out large, even if γ is small, because it is divided by a very small number. Second, in times of recession, where St decreases and possibly tends to zero, the relative risk aversion becomes very large, and this is precisely what is observed in reality. The only problem is that the whole argument breaks down, if the surplus consumption ratio becomes negative. Campbell and Cochrane (1999) prevented this from happening by making suitable assumptions about the surplus consumption process.
In the following, we again stick to the economic tradition and represent the logarithm of random variables by their lowercase letters. Campbell and Cochrane (1999) also used the AR(1)-model (9.83) for logarithmic consumption ct = log Ct,
with . They also assumed that the logarithmic surplus consumption ratio process st = log St is a mean reverting process of the kind
(9.101)
(9.102)
•
•
•
(9.103)
(9.104)
That is, the logarithmic surplus process is driven by the same random error as the logarithmic consumption process (random consumption shocks). For 0 ≤ ϕ < 1, the process eventually drifts towards its mean reversion level , whereas λ(s) is called the sensitivity function by Campbell and Cochrane, to be determined in order to satisfy additional model assumptions. The immediate next step is to compute the stochastic discount factor in order to see, if there is an additional contribution, accounting for the large equity premium. A somewhat lengthy calculation yields
Using our log-normal trick (9.88), we obtain for the ratio of the conditional standard deviation and the conditional expectation of the SDF
Remember that (9.102) is the Sharpe-ratio of the market portfolio we expect to observe, conditional on the information . And indeed, there is an extra term , possibly accounting for the high equity premium observed. It all depends on how λ(s) is chosen. In order to determine a suitable form of λ(s), Campbell and Cochrane imposed three conditions:
The risk-free interest rate is constant.
Habit is predetermined at the steady state .
Habit moves non-negatively with consumption everywhere.
Surprisingly, these conditions are indeed sufficient to determine the sensitivity function
with , and
We will not go through the entire proof (see the original paper of Campbell and Cochrane, 1999), but let’s show that the first condition is indeed satisfied. The risk-free interest rate can be computed from the conditional expectation of the SDF (9.101). In particular for the continuous one period risk-free gross return one obtains
(9.105)
(9.106)
(9.107)
(9.108)
(9.109)
which is indeed constant.
Quick calculation 9.23 Verify the first and second equality in (9.105).
The big question is, can the large equity premium be explained by the Campbell– Cochrane-model, while maintaining a relatively small risk-free interest rate? In their original article, Campbell and Cochrane (1999) chose the model parameters as in Table 9.1. From these figures, we can calculate a steady state surplus consumption ratio of , and a steady state Sharpe-ratio of
Table 9.1 Parameters in the Campbell–Cochrane-model
Parameter Variable Value
Mean consumption growth rate (%) g 1.89
Standard deviation of consumption growth (%) 1.50
Risk-free interest rate (%) r 0.94
Surplus consumption persistence coefficient ϕ 0.87
Local utility curvature γ 2.00
matching more or less exactly the observed Sharpe-ratio of the post-WWII NYSE-index data.
But there is much more. In their subsequent paper, Campbell and Cochrane (2000) set up a toy economy with their habit formation model and simulated consumption streams, asset returns, and other quantities. The latter is possible because the price-dividend ratio can be obtained from a functional equation. To understand how this is done, first recall that for the gross return of an arbitrary asset
holds, and that the gross return on a dividend paying security1 V is
Multiplying both sides of (9.107) with , one obtains
(9.110)
(9.111)
(9.112)
(9.113)
9.8
Quick calculation 9.24 Confirm the last equality in (9.109).
Equation (9.109) is a functional equation for the price-dividend ratio. To make this more transparent, let’s write it in a slightly different way and call it the price-dividend function
where the logarithmic surplus consumption ratio st is the only state variable for the economy. The objective is now to find a particular function that obeys (9.110). To this end, Campbell and Cochrane assume that the logarithmic dividend process is driven by the same growth factor as consumption
with dt = log Dt, and that the random error is positively correlated with . Now, the conditional expectation can be computed and (9.110) is an equation in the unknown function . Campbell and Cochrane solved this functional equation numerically on a grid for the state variable st and afterwards interpolated the price– dividend function. Once they obtained this function, they were able to simulate all interesting quantities inside their toy economy.
Campbell and Cochrane (2000) simulated 100 000 months of time series data for their toy economy and used this artificial data to compare the performance of several asset pricing models. Here is what they surprisingly found: Asset pricing models, focusing on the return of a wealth or market portfolio, like the CAPM, perform much better than the C-CAPM pricing method, based on conventional HARA-utility. This is a remarkable result, because the artificial data was generated under the consumption-based habit formation model. Obviously, the CAPM has the edge over the C-CAPM, because its predictions are conditional on the filtration , whereas the utility function
gives rise to an unconditional stochastic discount factor
Campbell and Cochrane do not reject consumption-based asset pricing models in general, but they attribute the poor performance of the C-CAPM to the limitations of the simple parametric form of the utility function.
Further Reading
There are several comprehensive and accessible textbooks on financial economics like Cochrane
9.1
9.3
9.4
9.9
9.2
(2005), Cuthbertson and Nitzsche (2004), and Lengwiler (2004). The classical references on stock price bubbles are Blanchard (1979) and Froot and Obstfeld (1991). An additional approach to bubbles and crashes, based on herding and critical behavior of complex systems, can be found in Johansen et al. (2000) and Sornette (2003). For the classical paradoxes see Shiller (1981), Mehra and Prescott (1985), and also Weil (1992). To explain the equity premium puzzle, several strategies were suggested. An incomplete list covers generalizing the expected utility functional to disentangle intertemporal substitution and risk aversion (Epstein and Zin, 1989, 1991), habit formation (Constantinides, 1990), incorporating wealth into the utility function (Bakshi and Chen, 1996), incomplete markets (Constantinides and Duffie, 1996), and habit persistence (Campbell and Cochrane, 1999, 2000).
Problems
In adding up an infinite sum, one usually assumes the following two properties to hold
Show that using these properties, the surprising result
can be established.
The rational valuation formula for stochastic discount factor models is
How does this formula simplify, if the SDF can be assumed conditionally
uncorrelated, for k ≠ 0?
Consider an economy at t = 0 and t = T. Show that under CARA-utility, the stochastic discount factor is
if the usual time separable von Neumann–Morgenstern-utility functional is assumed.
The generalized Epstein–Zin-utility functional is based on the idea of a discounted certainty equivalent. One possible form is