3.1 Geomentropic Preferences over Lotteries

A lottery is a list L=(S,S,p,w) where S is a finite state space, S is a partition of S, p:S→[0,1] assigns probabilities to partition sets, and w:S→R++ assigns payoffs to partition sets. The geometric mean of the lottery L is G(L)=∏σ∈Sw(σ)p(σ) and its entropy is N(L)=−∑σ∈Sp(σ)lnp(σ).^[8]

Let ⪰ be a preference ordering over lotteries. If for two lotteries L and L′, both L⪰L′ and L′⪰L then we write L∼L′. If for two lotteries L and L′, L⪰L′ but it is not the case that L′⪰L then we write L≻L′.

Our lottery index can be derived from preference axioms in straightforward fashion. The first two axioms, A1 and A2, are standard:

A1. (Completeness and Transitivity) For all lotteries L and L′, either L⪰L′ or L′⪰L. Furthermore, for all lotteries L, L′, and L′′, if L⪰L′ and L′⪰L′′ then L⪰L′′.

By a slight abuse of notation, let (S,S,1,w) denote the the zero-entropy lottery that assigns w to every state. Specifically, S=S is meant to suggest that the partition contains only one set: the entire state space, S. This lottery is analogous to a risk-free lottery in the mean-variance framework.

A2. (Continuity) For any lottery L, the sets {(S,S,1,w)⪰L} and {L⪰(S,S,1,w)} are closed.

A consequence of A1 and A2 is the existence of a zero-entropy equivalent, that is, G∗(L)∈R++ such that (S,S,1,G∗(L))∼L. A3 makes explicit the trade-off between geometric mean and entropy.

A3(a). (Geometric Mean Monotonicity) If N(L)=N(L′) and G(L)>G(L′) then L≻L′.

A3(b). (Entropy Monotonicity) If G(L)=G(L′) and N(L)<N(L′) then L≻L′.

Under A1–A3, lotteries can be ranked by their zero-entropy equivalents, as shown in the following proposition:

Now suppose G∗(L)>G∗(L′). Recall that G∗(L) and G∗(L′) are the geometric means of the risk-free lotteries K and K′ that are equally preferred to L and L′, respectively. Since K and K′ are risk-free, and G∗(L)>G∗(L′), by A3, L≻L′. QED

Although more flexibility is possible, a constant rate of substitution between risk and reward is a reasonable benchmark, and can be thought of as a first order approximation to actual preferences. It will also be seen to be general enough for many purposes.

A4. (Constant Marginal Rate of Substitution) There exists a constant λ>0 such that, for any two lotteries L and L′, if L∼L′ then lnG(L)−lnG(L′)=λ[N(L)−N(L′)].

We will refer to the index constructed in Proposition 2 as the geomentropic index and the preferences described by A1–A4 as geomentropic preferences. In contrast to expected utility preferences, geomentropic preferences do not imply an index that is linear in the probabilities of the various outcomes. This linearity is a consequence of A2 along with Neumann’s and Morgenstern’s independence axiom: L⪰L′ iff αL+(1−α)L”⪰αL′+(1−α)L” for all α∈[0,1] and all lotteries L”. The independence axiom has long had its critics,^[9] and we do not impose it here.

3.2 Geomentropic Preferences over Compound Lotteries

Let {Sk,Sk,pk,wk}k=1K, be a list of lotteries. The associated compound lottery with probability weights α={αk}k=1K is of the form {∪kSk,∪kSk,αp,w}, where αp assigns αkp(σ) to all σ∈Sk and w assigns wk(σ) to all σ∈Sk. This compound lottery can be interpreted as first randomly choosing a component lottery and then randomly generating the outcome of the chosen lottery. The geometric mean of the compound lottery L is G(L)=∏k∏σ∈Skwk(σ)αkpk(σ) and its entropy is defined as N(L)=−∑k∑σ∈Skαkpk(σ)lnαkpk(σ).

Thus the geomentropic index distributes lottery weights. However, it does not merge branches with the same outcomes by adding probabilities, nor does it divide one branch outcome into two or more by dividing the probabilities. Because it does not exhibit scalability over outcomes, it does not satisfy the reduction of compound lottery assumption implicit in the independence axiom. For example, an expected utility maximizer is indifferent between the compounding of two identical lotteries with probability weights α and (1−α)) and the same lottery with certainty. By contrast, the geomentropic agent prefers the simple lottery (S,S,w,p) to the compounding of two identical simple lotteries due to the higher entropy of the compound lottery (the probability-partitioning effect). For the non-expected utility maximizer, the two lotteries are not equivalent, a point made years ago by Mark Machina (1987). Geomentropic preferences assume a trade-off between the log of the lottery’s geometric mean and its entropy, instead of the reduction of compound lottery assumption inherent in the independence axiom of expected utility.

In summary, under geomentropic preferences, history matters in the sense that lottery presentation matters, with individuals perceiving subsequent algebraic manipulations involving outcomes as different lotteries. Scalability over outcomes implies non-unique geomentropic preferences. Harrison, Martinex-Correna, and Swarthout (2015) find that when choosing among many alternatives, the choices of experimental subjects do not support the reduction of compound lotteries assumption in his random lottery incentive mechanism. (See also Hillel (1973), Kahneman and Tversky (1979), Bernasconi and Loomes (1992), Miao and Zhong (2012), Abdellaoui, Klibanoff, Placido (2015)).

3.3 Geomentropic Preferences over Portfolios

A portfolio is a non-negative unit vector v=(v1,...,vK) where vk is interpreted as the fraction of the portfolio value W invested in the kth asset. The kth asset is a lottery (Sk,Sk,pk,wk). Given σ=(σ1,...,σK), where σk∈Sk for all k, let γ(σ) be the joint probability that sk∈σk for all k∈{1,...,K}. The geometric mean of the one-period payoff from a portfolio is

where W is the wealth invested and πk is the price of a unit of the kth asset. Portfolio risk is derived from the risks of the individual assets, as measured by the entropy of their marginal distributions, and their respective importance in the portfolio. Formally, define the marginal probabilities pk(σk) for the marginal distribution of asset k by

Define the entropy of the kth asset as

Then the entropy of the portfolio is defined as

Specifically, portfolio risk consists of the arithmetic average of individual asset uncertainty (Nk), weighted by their presence in the portfolio. Unlike the standard portfolio model based on mean-variance analysis, unsystematic risk does not go to zero as the number of assets increases. Rather, the portfolio risk converges to the mean asset risk.

It is common in the finance literature to assume the existence of a risk-free asset. The geomentropic analog is an asset that exhibits only the trivial partition, that is, the partition S that contains only the entire space S. Such an asset has entropy of zero and exhibits one outcome in all states of the world. Denote the price of a unit of the risk-free asset (cash) as πC. It is straightforward to show that as the share of the riskless asset increases, holding the proportions of the other assets constant, portfolio entropy falls. If only the risk-free asset is held in the portfolio, then portfolio entropy is zero.

3.4 Dynamic Geomentropic Preferences

Suppose the investor behaves as if the stochastic process governing the states of the world is ergodic, that is, there exists γ:×kSk→[0,1], such that ∑σ∈×kSkγ(σ)=1, γ(σ)>0 for all σ, and Limt→∞pασt=γ(σ) for all α and for all σ, where pασt are the transition probabilities from state α to σ after t periods.^[10] We assume the dynamic geomentropic investor acts as if investing to maximize relative to this ergodic distribution, and thus write the investor’s objective function as:

for the T-period horizon reinvestment problem when all available assets are risky. We assume that the appropriate entropy-metric for the portfolio, is the entropy of this ergodic distribution. Under these assumptions, dynamic geomentropic preferences when there is a risk-free asset (wC) can be written as

A lottery is a portfolio with one risky asset.

4.1 Geomentropic Risk Aversion

An expected utility maximizing agent who strictly prefers to receive the expected value of a lottery rather than the lottery itself is said to be risk averse. Analogously, we will define geomentropic risk aversion as a strict preference for receiving the geometric mean of a lottery rather than receiving the lottery itself. Given A3b, geomentropic agents exhibit geomentropic risk aversion. This follows directly from the entropy associated with a non-degenerate lottery.

4.2 Geomentropic Complexity Aversion

We will say that an agent is complexity averse if L′≻L whenever w(σ)=w′(σ) for all σ∈S and the partition S′ is coarser than the partition S. Then the definitions immediately imply the following theorem:

Note, by contrast, that expected utility maximizers are not inclined to complexity aversion. This is because of the scalability properties of expected utility. Thus compound lotteries are equally preferred to their expected utility equivalents, a property not shared by geomentropic preferences over lotteries.

Financially, investors with geomentropic preferences would rather hold a treasury bill than stock with an option that yields the same return.

4.3 Geomentropic Uncertainty Aversion

Geomentropic preferences exhibit a property that is analogous to the notion of uncertainty aversion introduced by Gilboa and Schmeidler (1989). Given two equally preferred lotteries, geomentropic individuals prefer a convex combination of the two lotteries over the chance of getting either one. The result relies on the distinction between taking a convex combination in which the weighting variable, call it ν, is a choice variable in the sense that its value is realized a priori and hence treated as non-random, and a true lottery in which one of the lotteries is received at random. In this latter case say the chance of getting the first lottery is γ, and as a random variable it induces some entropy. Uncertainty aversion means that any portfolio of the two lotteries, whatever the value of ν, is preferred to a lottery with those two lotteries as prizes, whatever the value of γ.

Subtracting terms on the right hand side with similar terms on the left hand side, we can rewrite the last inequality as (ν−γ)[ΣipilnXi+λΣipilnpi−ΣiqilnYi−λΣiqilnqi]>λ(γlnγ+(1−γ)ln(1−γ)).

Since the two lotteries are equally preferred, the terms in the square brackets sum to zero, and the last inequality becomes 0>λ(γlnγ+(1−γ)ln(1−γ)), which is always true because γ is bounded between 0 and 1. QED

4.4 Geomentropic Prudence

(p. 236 Gollier 2001) defines prudence, a concept due to Kimball (1992), by designating an agent as prudent if adding an uninsurable zero mean risk to his future wealth raises his optimal level of saving. In other words, an increase in risk for a given arithmetic mean causes the prudent agent to increase holdings of the riskless asset. The geomentropic analogue is for an increase in the entropy of the risky asset for a given geometric mean to cause an agent to increase holdings of the risk-free asset.

Consider a two-asset world where asset 1 is a risky asset with price π1 and asset 2 is a riskless asset with price πC. The payoffs associated with the two assets are w1(σ) and wC, respectively. The partition of the state space associated with the risk-free asset is the trivial partition containing only the state space and thus its entropy is N(L2)=−ln(1)=0. If v is the share of the risky asset in the portfolio then the geomentropic index of the portfolio (v,1−v) is

For any horizon length, T, the geomentropic index is concave in v. If real asset prices are such that there is an interior solution–-that is, if the optimal v is in the interior of (0,1)–-then the first order condition for the optimal choice of v is

Inspection of this first order condition for fixed asset prices and fixed distribution of returns reveals that a decrease in risk tolerance–-that is, an increase in λ–-must imply a decrease in {vWπ1w1(σ)+(1−v)WπCwC} and (assuming the mean return of the risky asset exceeds that of the riskless asset) a decrease in v, the fraction of the portfolio invested in the risky asset, and thus an increase in risk-free investment.

4.5 Fanning and the Allais Paradox with Geomentropic Preferences

To explain some empirical anomalies (including the Allais Paradox) for the expected utility model, Machina (1992) employs a three outcome lottery model with p=(p1,p2,p3) and corresponding fixed wealth outcomes w=(w1,w2,w3), where w3>w2>w1. In the probability triangle with p3 on the vertical axis, and p1 on the horizontal axis (with p2’s value implicitly determined by the choice of the other two probabilities within the triangle), under expected utlity sets of indifferrent lotteries lie on parallel straight lines with slopes equal to

Expected Utility indifference curves slope =

Given the linearity in probability assumptions under expected utility preferences (the result of the independence assumption in expected utility, see for example, (Section 1.3 Gollier 2001)), lotteries lying further to the northwest receive higher probability weights on the better outcome (p3 is larger and p1 is smaller, holding p2 constant), and hence, are preferred to all lotteries on indifference curves to the southeast.

Geomentropic indifference curves exhibit more complex patterns within the Machina probability triangle than expected utlity. Geomentropic indifference curves additionally depend on the log-odds ratios as follows:

Geomentropic indifference curves slope =

Geomentropic indifference curves tend to “fan out” (that is, their slopes increase as p3 increases, holding p2 constant) when lottery prizes are skewed to the right (that is, when (ln(w3)−ln(w2)>ln(w2)−ln(w1)). Unlike the expected utility indifference curves, geomentropic indifference curves may be negative for low values of p2 or high values of λ as they “fan out.”

In particular, this section shows that for values of λ that are consistent with empirical observation (roughly 3>λ>.4), geomentropic preferences are consistent with Allais paradox choices. This is in contrast to expected utility theory, which is inconsistent with such choices, regardless of the form of the utility function.

The Allais paradox considers choices between two paired lotteries (Machina, 1989), which can be appropriately regarded as choices between lotteries changing the wealth level of participant. (Recall that Latané’s maximum chance principle is essentially a wealth investment principle.) Given initial wealth W>0, consider four lotteries:

L1:

1.00 chance of $1,000,000 + W

L2:

0.10 chance of $5,000,000 + W

0.89 chance of $1,000,000 + W

0.01 chance of $0 + W

L3:

0.10 chance of $5,000,000 + W

0.90 chance of $0 + W

L4:

0.11 chance of $1,000,000 + W

0.89 chance of $0 + W

Most people claim to prefer L1 over L2 while preferring L3 over L4. However, whatever the utility function used, this is an inconsistent set of choices under the expected utility model because of the reduction of compound lotteries assumption (see Machina 1989, the diagram on the middle of the right hand column of p. 1629 and corresponding discussion).

By contrast, for feasible values of wealth W, and λ, geomentropic preferences are consistent with experimentally observed preferences of L1 over L2, and at the same time, preferences for L3 over L4. L1

is preferred to L2 if, upon factoring out ln(W) from both sides,

For geomentropic preferences, preference of L1 over L2 indicate the certain relative wealth term ln($1m/w+1) is preferred to the uncertain income 0.1ln($5m/W+1)+0.89ln($1m/W+1), discounted by the entropy associated with the uncertainty λ(0.1ln(0.1)+0.89ln(0.89)+0.01ln(0.01)). Simulations indicate that for all values of wealth W, L1 is preferred to L2 whenever λ is 0.34 or greater. This critical value of λ falls monotonically as W increases. For W=$1m, the critical value is λ=0.28; for W=$5m, the critical value is λ=0.13; and for W=$10m, λ=0.08. When λ is above these critical values L1 is always preferred to L2.

At the same time, L3 is preferred to L4 if

The upper area in Figure 1 indicates that for every wealth level, when λ>0.34, geomentropic perferences are consistent with typical Allais preferences (L1>L2 and L3>L4). Low levels of wealth are consistent with Allais choices, even for very low values of λ as indicated. The highest value of the lower bound for λ of 0.34 (achieved at a wealth level of about $80,000) monotonically declines as wealth increases. This establishes the existence of an open set of parameters for which Allais-style preferences are consistent with geomentropic preferences.

4.6 Insurance Decisions with Geomentropic Preferences

Figure 1:

Wealth-λ values consistent with Allais choices.

The results regarding fair pricing and insurance demand from expected utility maximization also derive from geomentropic preferences.

Specifically, let I∗ denote the amount of insurance an agent purchases at unit price s. The agent faces a loss of I with probability p; with probability (1−p) there is no loss. The agent chooses I∗ to maximize

The first order condition is

With actuarially fair insurance, so s=p, the first order condition implies I∗=I; thus the agent purchases full coverage.

When s>p, so insurance prices are not actuarially fair, the left side of the first order condition above is less than one so the right side must be less than one at the optimum. Inspection reveals this to be true only if I∗<I, so the agent purchases only partial insurance. This differs somewhat from standard expected utility, which can generate a demand for full insurance coverage, even with unfair prices, if the consumer is sufficiently risk adverse.

5.1 Equilibrium and Re-Investment Certainty Equivalence

Assume that all investors have the same marginal rate of substitution, λ. As explained in Section 2.4, and consistent with Latanés wealth reinvestment principle, we assume geomentropic investors behave as if maximizing over their time horizon, T, with respect to the ergodic distribution, γ.

Investors’ re-investment horizon has not been settled in the literature, though given Modigliani’s (1966) life cycle saving/investing model, values for an individual’s investment lifetime would plausibly be in the range of 20 to 40 years. The empirical sections below assume a 30-year horizon, but the linearity of key identifying restrictions implies that the estimates will be quite robust for the 20–40 year range. For long investment horizons, the value of a portfolio will grow close to its long-run geometric mean growth path with high probability. Hence, the compound growth of the risky portfolio over T periods less the λ-weighted entropy of the risky portfolio should roughly equal the compound growth of the risk-free portfolio. In particular, the index of risky and risk-free rates will satisfy the following equality because of re-investment certainty equivalence:

where wC is the risk-free return. The empirical sections below report estimates of λ using the certainty equivalencies of risky and risk-free portfolios. These estimates lie between 1 and 2, values that account for the equity premium puzzle.

5.2 Financial Market Equilibrium with Homogeneous Agents

A financial market is a triple, (x,W,ϕ), where x=(x1,...,xK) is the exogenous supply of each asset, W=(W1,...,WI) is the exogenous initial wealth of each investor, and ϕ(G,N) is the investors’ geomentropic preference index. A financial equilibrium is a pair (π,v) such that v maximizes the geomentropic index and the market clearing conditions hold for each asset: v∈argvmax{ϕ(Gp(v),Np(v))}, Σi=1IvkWiπk=xk for all risky assets k, and ∑i=1I(1−∑kvk)Wi/πC=xC for the risk-free asset. The first order conditions associated with the maximization of

set the partial derivatives of the objective function equal to zero: namely,

for all j∈{1,...,K}. An aggregate first order condition can be derived by multiplying the jth first order condition by vj, summing over j, and canceling the common wealth terms:

For the portfolio as a whole, the net portfolio returns (given on the right side of the previous equation) are the average risky returns above the risk free return, normalized by the weighted returns of the whole portfolio. Investors set portfolio returns equal to λ times portfolio entropy, λ∑kvkNk=λNp(v). As shown in the appendix, geomentropic preferences are concave in shares (that is, concave in v).

Substituting the market clearing conditions into the first order conditions, and simplifying, yields

where W=∑i=1IWi, the sum of wealth invested in the market by all the I investors.

The representative investor holds less of every risky asset in equilibrium given an exogenous increase in λ–that is, the greater the investors’ taste for certainty, the more risk-free asset they hold in their equilibrium portfolio (the lower the share of risky assets, ∑kvk). This is the content of the following theorem:

Designating the price of the risk-free asset as numeraire, as λ increases on the right side, the price of the jth asset, πj, must fall to maintain equilibrium. From the market clearing condition for that asset, Σi=1IvjWiπj=xj, a decrease in the price of the risky asset must be accompanied by a proportional decrease in the share of the asset. QED

For an individual investor, any portfolio satisfying the portfolio maximization condition

Figure 2:

Portfolio net returns and asset shares. (author’s calculation)

is equivalent, and different investors may choose different portfolios as long as the compounded, normalized excess returns to the risky assets in the portfolio equaled λ times that portfolio’s entropy. Portfolios with higher entropy must also have higher geometric returns.

To picture the equilibrium condition in eq. (26) geometrically, let the portfolio consist of three potential assets: a risk free asset, a riskier asset (with entropy N1), and a less risky asset (with entropy N2). The dotted simplex above the plane of shares of risky assets (v1,v2) gives the portfolio entropy as a function of those asset shares, assuming that λ=1. (Different values of λ merely rescales the slope of the plane, without changing the intuition.) The equilibrium condition in eq. (26) indicates that the asset prices (and, hence, returns) do not increase linearly as entropy increases (even if asset shares are held constant). The left-hand side of eq. (26) is not a linear function of those returns (prices) as shares are held constant.

6.1 Table 1: International Estimates Using Monthly Returns Data

This data comes from Marmi (2013), available for 16 developed economies. Again, the distinguishing feature of these data is that only monthly returns are available (where daily returns are available for US stock market data sets). As we have a relatively smaller time frame for the data in Table 1, all the entropy and standard deviations here are constructed from the last twelve months of market returns using only the monthly return data. This means that there is likely more noise in our uncertainty measures in Table 1 than there is in Tables 2 and 3. Nonetheless, the estimated λs are surprisingly similar to those estimated with the US data, available over a much longer time period.

The starting time for each data set varies in Table 1. The data for “developed” markets starts in 1988, and “emerging” markets (South Korea, India, Brazil, and China) have later starting dates (1992, 1993, 1995, and 1998, respectively), as reflected in the sample sizes. Marmi (2013) used the 1000 largest companies in the market for at least some of the estimates. Excluding Japan (whose estimated λ value is very imprecise, even among all the imprecision in Table 1), estimates for λ range generally between 1.4 and 2.0, except for Australia, Germany, and South Korea, which were a little bit lower (and hence, seem to discount information uncertainty a little bit less than other countries in this period).

Larger estimated entropy coefficients indicate greater ‘caution’ against market uncertainty for the respective country. Note that the estimated Chinese λ is virtually identical to that of the US (2.0345 vs. 2.0272) for this period, as are those λ values for Brazil (2.1906), Canada (1.8846), and India (1.9976). The most conservative economies in this analysis are Hong Kong (3.4881), Singapore (2.4874), and Sweden (2.6209)–relatively older, established market for their respective geographical regions.

6.2 Tables 2 and 3

The data came from the Fama/French factors available from the Wharton Business School. Data used in the first two tables was for the January 1, 1927 to July 31, 2013. The results in Table 2 are divided into two ‘columns’. The ‘Entropy R2 F-Stat’ on the left hand side are the results for entropy estimates (regressing Rm∗ on the market entropy, Nm, of results, calculated returns in the indicated window), the results under the headings ‘SD R2 F-Stat’ on the right hand side are for standard deviations (regressing Rp∗ on the standard deviations calculated from the respective market window). The data represent the returns from the value-weighted returns of all CRSP firms (Center for Research in Security Prices data base) incorporated in the US and listed on the NYSE, NASDAQ, or AMEX. It only reports information on ordinary common shares which either haven’t been or don’t need to be further defined (no firms incorporated outside the US, no trust components, no closed end funds).

The estimates in Table 2 and are OLS regressions without an intercept, as implied by the geomentropic model’s first order condition for portfolios (given in Section 5). The estimated value of λ using a 4-week market window (entropy and the standard deviation of market returns constructed from the last 20 days of trades) over the whole sample period from 1927 to 2013 is about 1.5, statistically significant at better than the one percent level (significance level is 0.0002). The overall fit using entropy as the dependent variable is not different statistically from the fit using the market standard deviation as the measure of risk/uncertainty: the R-square is the same on the left and right hand sides (0.0078 vs 0.0015), and the F-statistic for model fit are both close to significance (8.75 for the entropy model, and 1.56 for the standard deviation model).

Many things are apparent from the 4-week window models in the upper panel: very little of the variation in the normalized net returns is explained by either measure of uncertainty (one tenth of one percent at most), and entropy generally does as good a job at explaining the returns as does the standard deviation (SD). Indeed, in the post-war models (last three lines), the entropy R-square and F-statistics are much better than the standard deviations for the data based on monthly market returns.

In the estimates, we examine the overall period with and without information from World War II (WWII), and also examine just changes in the post war period to see whether risk tolerance has changed over time. The slight increase in λ suggests less tolerance for risk after the war, but when placed in the context of a shift in a spline function, the upward shift in λ towards less tolerance for uncertainty is statistically insignificant at the 10 percent level (it is significant at the 15 percent level). Perhaps the slight postwar increase in λ is associated with lingering concerns resulting from the stock market crash of the 1930s.

Similarly, the division of the post war period into pre- and post-ERISA (Employee Retirement Income Act) periods was to test whether the expansion of stock market participation after the enactment of ERISA in 1974, significantly changed the observed λ: either because of a change in perceived market transparency–which may cause λ to fall, or a change in the marginal investor’s tolerance for uncertainty–which may cause λ to fall or rise. ERISA encouraged corporate pension fund investments in stocks by requiring that pensions be appropriately funded, and prohibited plans from holding more than 10 percent of the corporation’s own stock. Institutional stock ownership also increased as individuals shifted from direct investment in equities to holding mutual fund shares and as public pension funds and nonprofit endowments abandoned their traditional policies against investments in equities. For example, public funds in California and 15 other states did not invest in any stocks until 1968. After ERISA, there was a dramatic increase in stock ownership by public entities (Bhide 1993). Using either the 4-week window estimates in the upper panel (where λ increases from about 1.52 to 1.61) or the 6-week estimates in the lower panel (where λ increases from 1.61 to 1.64), the estimated λ is relatively unchanged.

Results for the entropy coefficient indicate reasonable values for λ, from about 1.5 to 1.8 when estimating first order conditions using daily returns. These results were virtually unchanged when the model was estimated by first order autocorrelation regressions. (The estimated value of λ is obviously sensitive to our assumption that half the portfolio is in the risk-free asset: if the share of the risk-free asset fell to zero, the estimated value of λ tends to one.)

Overall, the results suggest reasonable approximations for portfolio λs given the first order conditions derived from our theoretical model, and, entropy measures of uncertainty fit the data as well as variance (standard deviation) measures of uncertainty.

6.3 Table 3: “Certainty Equivalencies” Over A Lifespan

Data on the annualized risky rates, risk free rates, as well as real disposal gross domestic product per capita are available from 1929. We assume, following the literature, that the ex post trends in the last 30 years of returns approximate the ex ante returns, and use 30-year lags to calculate the geometric mean of risky returns, the probability distribution of risky returns (we use a 6-bucket decomposition in Table 3) and entropy. (As a practical matter, we also employ the (log of) geometric mean for the riskfree rates over the prior 30 years. The results using the arithematic mean instead over the period yielded virtually the same results.) The certainty equivalency of the risky, and risk-free, portfolios condition allows for a robustness check on the data, as it implies a particular value for λ (here, p(σ) are derived from the observed aggregate returns of risky assets):

Or,

For example, by 1959, the previous 30 years of data indicated that the annual log of the geometric mean of the risky asset was 0.06308, and the annual log of the geometric mean of the risk-free asset was 0.012533. Hence, the implied λ consistent with the certainty equivalence condition is thirty times this difference (30(0.06308−0.012533) divided by the entropy value for this period, 1.69337, which equals 0.89550, as indicated in the middle column of estimates.

The data for remaining years shows consistent values of λ with the empirical findings in Tables 1 and 2, which is also consistent with Allais preferences. Overall, λ values are around 1.5, generally bounded between 1 and 2.

To compare our reconciliation of the risky and risk-free rates with the expected utility model, we also present implied relative risk aversion coefficients (RRA) using the same data (prior 30 years means for the risky and risk-free rates, whose differences represent the equity premium for risk) using the simple model of Gollier (2001, Section 5.2). The “equity premium” for 1959, the difference in risky and risk-free rates that compensates the investor for holding the risky assets, for example, is 0.109−0.011=0.098. Employing Gollier’s formula, we find the RRA factor is 100, an implausibly large value. Indeed for all years, these estimates of the RRA imply that an investor is so risk averse that she is willing to give up 29 percent of her wealth to avoid a gamble in which she receives a 30 percent wealth gain half the time, and a 30 percent wealth reduction half the time.

The four right hand columns suggest that as entropy rises, so does the equity premium. Indeed, a simple regression indicates as entropy goes up, so does the equity premium (a standard deviation increase in entropoy results in almost half a standard deviation increase in the equity premium), with an R-square of .18.