In this section weassumethat market prices are generated by an individual decision problem. The question we are interested in is which utility function is compatible with the empirical findings in asset prices. Whether this assumption is plausible is left to the reader. Some justifications on this were given in the previous section.
Some authors say that assuming the utility market hypothesis for market aggregates is maybe more plausible than assuming it for individual investors. As early as 1956 John Hicks [Hic86, page 55] wrote:
. . . the preference hypothesis only acquires a prima facie plausibility when it is applied to a statistical average . . . to assume that an actual person, the Mr. Brown or Mr. Jones who lives round the corner, does in fact act in such a way does not deserve a moment’s consideration.
Hence, Hicks has already anticipated the psychologists’ critique that describing individual decisions by utility maximization is wrong. The mystery that remains is how the individual irrationalities are washed out at the level of the market. In the next chapter, we give an evolutionary argument for this, based on market selection.
From the evolutionary point of view market aggregates can be thought of as being derived from a rational utility function even though no individual ever has attempted to behave rationally.
We first have to distinguish between the implications that long-term and that short-term data of asset returns have for the utility function of the representative agent. Then we will suggest a synthesis of both. We will argue that in the long run, the utility function must have constant relative risk aversion, while in the short run it must have features of Prospect Theory.
That the utility of the representative investor must have CRRA in the long run was made pretty clear by Campbell and Viceira [CV02, page 24]:
The long run behavior of the economy suggests that relative risk aversion cannot depend strongly on wealth. Per capita consumption and wealth increased greatly over the past two centuries. Since financial risks are multiplicative, this means that the absolute scale of financial risks has also increased while the relative scale of financial risks is unchanged.
Interest rates and risk premia do not show any evidence of long-term trends in response to this long-term growth; this implies that investors are willing to pay almost the same relative costs to avoid given relative risks as they did when they were much poorer, which is possible only if relative risk aversion is almost independent of wealth.
Now supposing the utility function has CRRA, the question that remains is the magnitude of the risk aversion parameter. Let’s write the utility function asu.w/WD w1˛=.1˛/. An upper bound for˛can be found from the first order condition of utility maximization since, as we show next, the Sharpe ratio of any asset is bounded above by the volatility of the agent’s consumption growth. This derivation goes back to Hansen and Jagannathan [HJ91], hence the lower bound is called the Hansen and Jagannathan bound.
LetWD `=Rf be the likelihood ratio process divided by the risk-free rate. Then the no-arbitrage condition readsE.Rk/D1. By the definition of the correlation we can write:
1DE.Rk/DE./E.Rk/Ccorr.;Rk/./.Rk/;
hence
E.Rk/DRfcorr.;Rk/./
E./.Rk/:
Since the correlation is bounded between1andC1, we get the inequality:
ˇˇE.Rk/Rfˇˇ
.Rk/ ./
E./:
In the consumption based asset pricing model with expected utility, we have qkDu0.c0/1 1
1CıE
u0.c1/Ak
;
hence
D 1
1Cı u0.c1/ u0.c0/:
198 4 Two-Period Model: State-Preference Approach
Table 4.2
Hansen-Jagannathan bounds for alternative risk aversion based on annual data of the S&P 500 from 1973 to 2005
Risk aversion Consumption SDF
1 0.0165
2.5 0.0415
5 0.0844
8 0.1376
10 0.1743
18 0.3311
20 0.3730
30 0.5987
And in the case of CRRA we get:
D 1
1Cı c1
c0 ˛
; so that
ˇˇE.Rk/Rfˇˇ
.Rk/
1 1Cı
c1 c0
˛ :
Hence, taking data on the risk-free rate, the Sharpe ratio and the volatility of consumption growth we can estimate the relative risk aversion˛. In a recent paper Hens and Wửhrmann [HW07b] estimate the Sharpe ratio of the S&P 500 sampled from annual data since 1973 and compare it with the Hansen-Jagannathan bounds resulting for alternative risk aversions. The result is reported in Table4.2.
The actual Sharpe ratio in that data is about 0.328. Hence, a relative risk aversion of about 18 would explain the risk adjusted equity premium. On data with a higher frequency the risk aversion is higher. Some authors estimate numbers in the range of 30–40. It is typically claimed that numbers above 10 are too high to be reasonable values for risk aversion. Hence, the typical finding of numbers in the area of at least 18 is puzzling to many researchers. This puzzle is called theequity premium puzzle.
But why is a number above 18 considered to be too high? After all nobody has ever met the fictitious agent called representative investor and asked him about his degree of relative risk aversion. The idea is that the representative investor represents the individual investors and estimates of individuals’ risk aversion are feasible using questionnaire techniques as explained in Sect.2.2.4. However, we have already remarked there that such elicitations crucially depend on the assumptions on a person’s wealth level.
Let us take a look at a typical question that is used to determine the relative risk aversion:
Consider a fair lottery where you have a 50% chance of doubling your income, and a 50%
chance of losing a certain percentage, sayx% of your income. What is the highest lossx that you would be willing to incur to agree to taking part in this lottery?
The typical answer to this question is anxof about 23 %. Interpreting this answer based on a CRRA utility function we get an˛of 3.22: we set
0:5.2W/1˛ 1˛ C0:5
.1x/W1˛
1˛ D W1˛ 1˛: This gives21˛C.1x/1˛ D2or1˛D 2:22.
Hence, the typical answer to this question is far away from the value obtained from stock market data. There is a huge literature trying to bring down the alpha obtained from stock market data. Some authors say that in the optimization of the representative agent borrowing constraints are missing. Others say that maybe the utility function is not CRRA but includes aspects like habit formation (e.g.
[Abe90]). Yet others claim that consumption should be restricted to the consumption of stock holders. Finally, some researchers claim that one should calculate the equity premium based on expected stock returns that are typically smaller than their realizations. A recent book on these attempts was edited by Mehra [Meh06], the inventor of the equity premium puzzle.
There is another possible explanation: based on the observation that the back- ground wealth plays an important role in measuring any utility function, we notice that in the above derivation of˛we have implicitly assumed that the money at stake is the whole wealth of a person. However, in the question “only” the wholesalary is at stake. Now assume that the person’s background wealth is non-zero, let us say 50 % of his/her salary, then the degree of risk aversion is computed as follows:
0:5
.0:5C2/w1˛
1˛ C0:5
.0:5C.1x//w1˛
1˛ D
.0:5C1/w1˛
1˛ :
If we setx WD 23%, we obtain˛ D 21, and the alpha increases even more with higher background wealth (see [HW07b]). Please keep in mind that the background wealth should reflect the total wealth of our society since the representative investor whose consumption we used to explain market data needs to own everything we can think of (land, houses, factories, cars, etc.).
In the exercise book we do similar computations based on mean-variance and on Prospect Theory. The main message is unchanged: evaluating the degree of risk aversion from market data and from experimental data under the same assumption on background wealth the equity premium puzzle may not be that puzzling.
Now we turn to the short-run properties of the representative’s utility function as it can be estimated from daily data on stock market indices and their derivatives.
Again we use the first order condition for optimal investment decisions as a starting point:
`sDRf
u0.cs/
.1Cı/u0.c0/; sD1; : : : ;S:
200 4 Two-Period Model: State-Preference Approach
Fig. 4.14 Utility function of the representative investor estimated from daily return on the DAX for a typical trading day
Now we read this in the following way: we estimate the likelihood ratio process from observed stock market returns, which fix theps’s and from option price data which determines thes’s. Finally, we take the risk-free rate, the discount factor and the consumption growth as before. Following this approach Jackwerth [Jac00]
showed that the`is typically not a monotonic decreasing function, as is typically assumed, but instead “hump-shaped”. Detlefsen, Họrdle and Moro [DHM09] follow this approach and estimate the utility function on DAX-data (compare Fig.4.14and also [RH10]).
Note the convexity in the left part of the figure, which is reminiscent of Prospect Theory. If one were to include probability weighting, then the concavity for extreme losses which can be seen in Fig.4.14could be explained by the overweighting of extreme losses – compare the four fold pattern of risk from Sect.2.4.1.
There is another possible interpretation of the “empirical” utility function: we have seen that in the long run the representative agent is a CRRA maximizer while in the short run he might be better described by elements of Prospect Theory. We can therefore propose the following synthesis of the long-run and the short-run view:
define a utility function
u.c/WDu.c0/Cı
u.c1/.1h/Chv.c1c0/ Cı2
u.c2/.1h/Chv.c2c1/ C: : : ;
where we use a new parameter, the habit formation parameterh2.0; 1/. ForhD0 we get the standard expected discounted utility function that, e.g., may have CRRA.
For a positivehbelow 1 we can blend in the Prospect Theory value functionvwith reference point equal to last period’s consumption. We may also include discounting by which the marginal rates of substitution between today and any future period
poor rich
wealth utility
Fig. 4.15 A long-run CRRA utility with a short-run prospect-theory-overlay
are changed while those between two future periods remain as in the standard exponential discounting case (compare Sect.2.7and [BHS01]).
Graphically our suggested synthesis can be displayed as in Fig.4.15. Adding both terms we arrive at a utility function which is close to the empirical one from Fig.4.14.
It is interesting that there is also a completely different approach to the empirical pricing kernel puzzle: whereas we have so far assumed rational (or more precisely:
extrapolated) expectations, but behavioral preferences, one can alternatively assume biased expectations and estimatep. This approach has been developed by Hersh Shefrin [She08] and is an extension of the representative agent results discussed in Sect.4.6.2. In this way, it is possible to explain the observed pattern of the pricing kernel even within a standard CRRA utility framework.
It is difficult to discriminate empirically between both potential explanations for the “hump-shape” of the pricing kernel. The fact, however, that most trades on financial markets are based on heterogeneity of beliefs rather than on differences in risk preferences supports the explanation by Shefrin.
Finally, an explanation of the empirical pricing kernel puzzle based on incom- plete markets is presented in the exercise book