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Risk Aversion for Security Analysis and Portfolio Management MCOM sem 3 Delhi University

Risk Aversion for Security Analysis and Portfolio Management MCOM sem 3 Delhi University

In economics and finance, risk aversion is the behavior of humans (especially consumers and investors), when exposed to uncertainty, in the attempt to lower that uncertainty. It is the hesitation of a person to agree to a situation with an unknown payoff rather than another situation with a more parented, but possibly lower, expected payoff. Risk Aversion for Security Analysis and Portfolio Management MCOM sem 3 Delhi University.For example, a risk-averse investor might choose to put his or her money into a bank account with a low but guaranteed interest rate, rather than into a stock that may have high expected returns, but also involves a chance of losing value.

Risk Aversion for Security Analysis and Portfolio Management MCOM sem 3 Delhi University

Risk Aversion for Security Analysis and Portfolio Management MCOM sem 3 Delhi University

Example

A person is given the choice between two scenarios, one with a guaranteed payoff and one without. In the guaranteed scenario, the person receives $50. In the uncertain scenario, a coin is flipped to decide whether the person receives $100 or nothing. The expected payoff for both scenarios is $50, meaning that an individual who was insensitive to risk would not care whether they took the guaranteed payment or the gamble. However, individuals may have different risk attitudes. Risk Aversion for Security Analysis and Portfolio Management MCOM sem 3 Delhi University.

A person is said to be:

  • risk-averse (or risk-avoiding) – if he or she would accept a certain payment (certainty equivalent) of less than $50 (for example, $40), rather than taking the gamble and possibly receiving nothing.
  • risk-neutral – if he or she is indifferent between the bet and a certain $50 payment.
  • risk-loving (or risk-seeking) – if he or she would accept the bet even when the guaranteed payment is more than $50 (for example, $60).

The average payoff of the gamble, known as its expected value, is $50. The dollar amount that the individual would accept instead of the bet is called the certainty equivalent, and the difference between the expected value and the certainty equivalent is called the risk premium. For risk-averse individuals, it is positive, for risk-neutral persons it is zero, and for risk-loving individuals their risk premium is negative. Risk Aversion for Security Analysis and Portfolio Management MCOM sem 3 Delhi University.

Risk Aversion for Security Analysis and Portfolio Management MCOM sem 3 Delhi University

Utility of money

In expected utility theory, an agent has a utility function u(x) where x represents the value that he might receive in money or goods (in the above example x could be 0 or 100). Risk Aversion for Security Analysis and Portfolio Management MCOM sem 3 Delhi University.

Time does not come into this calculation, so inflation does not appear. (The utility function u(x) is defined only up topositive linear affine transformation – in other words a constant offset could be added to the value of u(x) for all x, and/or u(x) could be multiplied by a positive constant factor, without affecting the conclusions).

An agent possesses risk aversion if and only if the utility function is concave. For instance u(0) could be 0, u(100) might be 10, u(40) might be 5, and for comparison u(50) might be 6. Risk Aversion for Security Analysis and Portfolio Management MCOM sem 3 Delhi University.

The expected utility of the above bet (with a 50% chance of receiving 100 and a 50% chance of receiving 0) is,

E(u)=(u(0)+u(100))/2{\displaystyle E(u)=(u(0)+u(100))/2}E(u)=(u(0)+u(100))/2,

and if the person has the utility function with u(0)=0, u(40)=5, and u(100)=10 then the expected utility of the bet equals 5, which is the same as the known utility of the amount 40. Hence the certainty equivalent is 40.

The risk premium is ($50 minus $40)=$10, or in proportional terms

($50−$40)/$40{\displaystyle (\$50-\$40)/\$40}(\$50-\$40)/\$40

or 25% (where $50 is the expected value of the risky bet: (120+12100{\displaystyle {\tfrac {1}{2}}0+{\tfrac {1}{2}}100}\tfrac {1}{2} 0 + \tfrac{1}{2} 100). This risk premium means that the person would be willing to sacrifice as much as $10 in expected value in order to achieve perfect certainty about how much money will be received. In other words, the person would be indifferent between the bet and a guarantee of $40, and would prefer anything over $40 to the bet.

In the case of a wealthier individual, the risk of losing $100 would be less significant, and for such small amounts his utility function would be likely to be almost linear, for instance if u(0) = 0 and u(100) = 10, then u(40) might be 4.0001 and u(50) might be 5.0001. Risk Aversion for Security Analysis and Portfolio Management MCOM sem 3 Delhi University.

The utility function for perceived gains has two key properties: an upward slope, and concavity. (i) The upward slope implies that the person feels that more is better: a larger amount received yields greater utility, and for risky bets the person would prefer a bet which is first-order stochastically dominant over an alternative bet (that is, if the probability mass of the second bet is pushed to the right to form the first bet, then the first bet is preferred). (ii) The concavity of the utility function implies that the person is risk averse: a sure amount would always be preferred over a risky bet having the same expected value; moreover, for risky bets the person would prefer a bet which is a mean-preserving contraction of an alternative bet (that is, if some of the probability mass of the first bet is spread out without altering the mean to form the second bet, then the first bet is preferred). Risk Aversion for Security Analysis and Portfolio Management MCOM sem 3 Delhi University.

Measures of risk aversion under expected utility theory

There are multiple measures of the risk aversion expressed by a given utility function. Several functional forms often used for utility functions are expressed in terms of these measures. Risk Aversion for Security Analysis and Portfolio Management MCOM sem 3 Delhi University.

Absolute risk aversion

The higher the curvature of u(c){\displaystyle u(c)}u(c), the higher the risk aversion. However, since expected utility functions are not uniquely defined (are defined only up to affine transformations), a measure that stays constant with respect to these transformations is needed. One such measure is the Arrow–Pratt measure of absolute risk-aversion (ARA), after the economists Kenneth Arrow and John W. Pratt, also known as the coefficient of absolute risk aversion, defined as

A(c)=−u″(c)u′(c){\displaystyle A(c)=-{\frac {u”(c)}{u'(c)}}}A(c)=-\frac{u''(c)}{u'(c)}

where u′(c){\displaystyle u'(c)}{\displaystyle u'(c)} and u″(c){\displaystyle u”(c)}{\displaystyle u''(c)} denote the first and second derivatives with respect to c{\displaystyle c}c of u(c){\displaystyle u(c)}u(c).

Relative risk aversion

The Arrow-Pratt measure of relative risk-aversion (RRA) or coefficient of relative risk aversion is defined as

R(c)=cA(c)=−cu″(c)u′(c){\displaystyle R(c)=cA(c)={\frac {-cu”(c)}{u'(c)}}}R(c) = cA(c)=\frac{-cu''(c)}{u'(c)}.

Like for absolute risk aversion, the corresponding terms constant relative risk aversion (CRRA) and decreasing/increasing relative risk aversion (DRRA/IRRA) are used. This measure has the advantage that it is still a valid measure of risk aversion, even if the utility function changes from risk-averse to risk-loving as c varies, i.e. utility is not strictly convex/concave over all c. A constant RRA implies a decreasing ARA, but the reverse is not always true. As a specific example of constant relative risk aversion, the utility function u(c)=log⁡(c){\displaystyle u(c)=\log(c)}u(c) = \log(c) implies RRA = 1. Risk Aversion for Security Analysis and Portfolio Management MCOM sem 3 Delhi University.

In intertemporal choice problems, the elasticity of intertemporal substitution often cannot be disentangled from the coefficient of relative risk aversion. The isoelastic utility function

u(c)=c1−ρ−11−ρ{\display style u(c)={\frac {c^{1-\rho }-1}{1-\rho }}}u(c)={\frac {c^{{1-\rho }}-1}{1-\rho }}

exhibits constant relative risk aversion with R(c)=ρ{\displaystyle R(c)=\rho }R(c) = \rho and the elasticity of intertemporal substitution εu(c)=1/ρ{\displaystyle \varepsilon _{u(c)}=1/\rho }\varepsilon_{u(c)} = 1/\rho. When ρ=1,{\displaystyle \rho =1,}\rho =1, using l’Hôpital’s rule shows that this simplifies to the case of log utility, u(c) = log c, and the income effect and substitution effect on saving exactly offset. Risk Aversion for Security Analysis and Portfolio Management MCOM sem 3 Delhi University.

A time varying relative risk aversion can be considered.

Public understanding and risk in social activities

In the real world, many government agencies, e.g. Health and Safety Executive, are fundamentally risk-averse in their mandate. This often means that they demand (with the power of legal enforcement) that risks be minimized, even at the cost of losing the utility of the risky activity. It is important to consider the opportunity costwhen mitigating a risk; the cost of not taking the risky action. Writing laws focused on the risk without the balance of the utility may misrepresent society’s goals. The public understanding of risk, which influences political decisions, is an area which has recently been recognised as deserving focus. David Spiegelhalter is theWinton Professor of the Public Understanding of Risk at Cambridge University, a role he describes as “outreach”. Risk Aversion for Security Analysis and Portfolio Management MCOM sem 3 Delhi University.

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