hyperbolic absolute risk aversion( final presentation)

8/9/2019 Hyperbolic Absolute Risk Aversion( Final Presentation)

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Hyperbolic Absolute Risk

Aversion

Presented by: Nitasha Ahmed


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ABSOLUTE RISK

AVERSION• With the change in amount of wealth how muchthe investor’s preference changes (risk aversion)

• he higher the curvature of u(c)! the higher therisk aversion"


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• #$pected utility functions are not uni%uelyde&ned (are de&ned only up to afnetransormations)! a measure that staysconstant with respect to these transformations "'ne such measure is the Arrow Pratt measureo absolute risk-aversion (AA)

• n geometry! an a*ne transformation! a*ne mapor an a*nity (from the +atin! a*nis! ,connectedwith,) is a function between afne spaces which

preserves points, straight lines and planes.


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HYPERBOLIC ABSOLUTE

RISK AVERSION• -yperbolic absolute risk aversion is part of the

family of utility functions originally proposed by .ohnvon Neumann and 'skar /orgenstern in the late0123s"

• Also called linear risk tolerance

• t is the most general class of utility functions that

are usually used in practice (speci&cally! 4A(constant relative risk aversion)! 4AA (constantabsolute risk aversion)! and %uadratic utility alle$hibit -AA


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• A utility function e$hibits -AA if its absolute riskaversion is a hyperbolic function! namely


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Thus relative risk aversion is increasing ifb > 0 (for ), constant ifb = 0, and decreasing i


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• f the investors prefer more to less than the utilityfunction will re5ect their desires over the

restricted range of wealth" he most common%uadratic utility &nction is


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risk-aversion coefficient=ϒ


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• While utility functions have been mined by economiststo derive elegant and powerful models! there are

niggling details about them that should give us pause"• he &rst is that no single utility function seems to &t

aggregate human behavior very well"

• he second is that the utility functions that are easiestto work with! such as the %uadratic utility functions!

yield profoundly counter intuitive predictions about howhumans will react to risk"

• he third is that there are such wide di7erences acrossindividuals when it comes to risk aversion that &nding autility function to &t the representative investor or

individual seems like an e$ercise in futility"Notwithstanding these limitations! a working knowledgeof the basics of utility theory is a prere%uisite forsensible risk management"


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Examples

• nvestment in saving and retirement funds! stocks! insurance"

Static portolios

• f all investors have -AA utility functions with the same

e$ponent! then in the presence of a risk8free asset a two8fundmonetary separation theorem results" #very investor holdsthe available risky assets in the same proportions as do allother investors! and investors di7er from each other in theirportfolio behavior only with regard to the fraction of theirportfolios held in the risk8free asset rather than in the

collection of risky assets"

• /oreover! if an investor has a -AA utility function and a risk8free asset is available! then the investor9s demands for therisk8free asset and all risky assets are linear in initial wealth"

hyperbolic absolute risk aversion( final presentation)

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