Hyperbolic absolute risk aversion
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finance Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline of f ...
,
economics Economics () is the social science that studies the production, distribution, and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics anal ...
, and
decision theory Decision theory (or the theory of choice; not to be confused with choice theory) is a branch of applied probability theory concerned with the theory of making decisions based on assigning probabilities to various factors and assigning numerical ...
, hyperbolic absolute risk aversion (HARA) (Chapter I of his Ph.D. dissertation; Chapter 5 in his ''Continuous-Time Finance'').Ljungqvist & Sargent, Recursive Macroeconomic Theory, MIT Press, Second Edition refers to a type of
risk aversion In economics and finance, risk aversion is the tendency of people to prefer outcomes with low uncertainty to those outcomes with high uncertainty, even if the average outcome of the latter is equal to or higher in monetary value than the more ...
that is particularly convenient to model mathematically and to obtain empirical predictions from. It refers specifically to a property of
von Neumann–Morgenstern utility function The expected utility hypothesis is a popular concept in economics that serves as a reference guide for decisions when the payoff is uncertain. The theory recommends which option rational individuals should choose in a complex situation, based on th ...
s, which are typically functions of final wealth (or some related variable), and which describe a decision-maker's degree of satisfaction with the outcome for wealth. The final outcome for wealth is affected both by
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s and by decisions. Decision-makers are assumed to make their decisions (such as, for example, portfolio allocations) so as to maximize the
expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
of the utility function. Notable special cases of HARA utility functions include the quadratic utility function, the exponential utility function, and the isoelastic utility function.


Definition

A utility function is said to exhibit hyperbolic absolute risk aversion if and only if the level of risk tolerance T(W)—the
reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another pol ...
of absolute risk aversion A(W)—is a linear function of wealth ''W'': :T(W) = \frac = \frac + \frac, where ''A''(''W'') is defined as –''U "''(''W'') / ''U'' '(''W''). A utility function ''U''(''W'') has this property, and thus is a HARA utility function, if and only if it has the form : U(W) = \frac \left(\frac + b\right)^ with restrictions on wealth and the parameters such that a>0 and b + \frac > 0. For a given parametrization, this restriction puts a lower bound on ''W'' if \gamma <1 and an upper bound on ''W'' if \gamma >1. For the limiting case as \gamma → 1, L'Hôpital's rule shows that the utility function becomes linear in wealth; and for the limiting case as \gamma goes to 0, the utility function becomes logarithmic: U(W) = \text(a W+b).


Decreasing, constant, and increasing absolute risk aversion

Absolute risk aversion is decreasing if A'(W) < 0 (equivalently ''T'' '(''W'') > 0), which occurs if and only if \gamma is finite and less than 1; this is considered the empirically plausible case, since it implies that an investor will put more funds into risky assets the more funds are available to invest. Constant absolute risk aversion occurs as \gamma goes to positive or negative infinity, and the particularly implausible case of increasing absolute risk aversion occurs if \gamma is greater than one and finite.


Decreasing, constant, and increasing relative risk aversion

Relative risk aversion In economics and finance, risk aversion is the tendency of people to prefer outcomes with low uncertainty to those outcomes with high uncertainty, even if the average outcome of the latter is equal to or higher in monetary value than the more ce ...
is defined as ''R''(''W'')= ''WA''(''W''); it is increasing if R'(W) > 0, decreasing if R'(W) < 0, and constant if R'(W) = 0. Thus relative risk aversion is increasing if ''b'' > 0 (for \gamma \ne 1), constant if ''b'' = 0, and decreasing if ''b'' < 0 (for - \infty < \gamma < 1).


Special cases

*Utility is linear (the
risk neutral In economics and finance, risk neutral preferences are preferences that are neither risk averse nor risk seeking. A risk neutral party's decisions are not affected by the degree of uncertainty in a set of outcomes, so a risk neutral party is indif ...
case) if \gamma = 1. *Utility is quadratic (an implausible though very mathematically tractable case, with increasing absolute risk aversion) if \gamma = 2. *The exponential utility function, which has constant absolute risk aversion, occurs if ''b'' = 1 and \gamma goes to negative infinity. *The power utility function occurs if \gamma < 1 and a = 1 - \gamma. :*The more
special case In logic, especially as applied in mathematics, concept is a special case or specialization of concept precisely if every instance of is also an instance of but not vice versa, or equivalently, if is a generalization of . A limiting case ...
of the
isoelastic utility In economics, the isoelastic function for utility, also known as the isoelastic utility function, or power utility function, is used to express utility in terms of consumption or some other economic variable that a decision-maker is concerned wit ...
function, with constant relative risk aversion, occurs if, further, ''b'' = 0. *The logarithmic utility function occurs for a=1 as \gamma goes to 0. :*The more special case of constant relative risk aversion equal to one — ''U''(''W'') = log(''W'') — occurs if, further, ''b'' = 0.


Behavioral predictions resulting from HARA utility


Static portfolios

If all investors have HARA utility functions with the same exponent, then in the presence of a risk-free asset a two-fund monetary separation theorem results: every investor holds the available risky assets in the same proportions as do all other investors, and investors differ from each other in their portfolio behavior only with regard to the fraction of their portfolios held in the risk-free asset rather than in the collection of risky assets. Moreover, if an investor has a HARA utility function and a risk-free asset is available, then the investor's demands for the risk-free asset and all risky assets are linear in initial wealth. In the
capital asset pricing model In finance, the capital asset pricing model (CAPM) is a model used to determine a theoretically appropriate required rate of return of an asset, to make decisions about adding assets to a well-diversified portfolio. The model takes into ac ...
, there exists a representative investor utility function depending on the individual investors' utility functions and wealth levels, independent of the assets available, if and only if all investors have HARA utility functions with the same exponent. The representative utility function depends on the distribution of wealth, and one can describe market behavior as if there were a single investor with the representative utility function. With a complete set of state-contingent securities, a sufficient condition for security prices in equilibrium to be independent of the distribution of initial wealth holdings is that all investors have HARA utility functions with identical exponent and identical rate of time preference between beginning-of-period and end-of-period consumption.


Dynamic portfolios in discrete time

In a discrete time dynamic portfolio optimization context, under HARA utility optimal portfolio choice involves partial myopia if there is a risk-free asset and there is serial independence of asset returns: to find the optimal current-period portfolio, one needs to know no future distributional information about the asset returns except the future risk-free returns. With asset returns that are
independently and identically distributed In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usual ...
through time and with a risk-free asset, risky asset proportions are independent of the investor's remaining lifetime.


Dynamic portfolios in continuous time

With asset returns whose evolution is described by
Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...
and which are independently and identically distributed through time, and with a risk-free asset, one can obtain an explicit solution for the demand for the unique optimal mutual fund, and that demand is linear in initial wealth.


References

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External links


Closed form solution for a consumption savings problem with HARA utility
Financial risk modeling Expected utility Utility function types