4 utility - What Practitioners Need to Know . . . About...

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What Practitioners Need to Know . . . About Utility Mark Kiitzinan Windham Capital Manage- ment An important axiom of modern financial theory is that rational investors seek to maximize ex- pected utility. Many financial ana- lysts, however, find the concept of utility somewhat nebulous. This column discusses the origin of utility theory as well as its appli- cation within the context of finan- cial analysis. In his classic paper, "Exposition of a New Theory on the Measure- ment of Risk," first published in 1738, Daniel Bernoulli proposed the following: "the determination of the value of an item must not be based on its price, but rather on the utility it yields. The price of the item is dependent only on the thing itself and is equal for everyone; the utility, however, is dependent on the particular cir- cumstances of the person making the estimate. Thus there is no doubt that a gain of one thousand ducats is more significant to a pauper than to a rich man though both gain the same amount."^ Bernoulli's insight that the utility of a gain depends on one's wealth may seem rather obvious and perhaps even pedestrian, yet it has profound implications for the theory of risk. Figure A helps us to visualize Bernoulli's notion of utility, which economists today call diminishing marginal utility. The horizontal axis represents wealth, while the vertical axis rep- resents utility. The relation be- tween wealth and utility is mea- sured by the curved line. Utility clearly increases with wealth, be- cause the curve has a positive slope. The positive slope simply indicates that we prefer more Figure A Diminishing Marginal Lltihty Figure B Changes in Utility versus Changes in Wealth Wealth wealth to less wealth. This as- sumption seldom invites dispute. As wealth increases, the incre- ments to Utility become progres- sively smaller, as the concave shape of the curve reveals. This concavity indicates that we derive less and less satisfaction with each subsequent unit of incremental wealth. Technically, Bernoulli's notion of utility implies that its first derivative with respect to wealth is positive, while its sec- ond derivative is negative. A negative second derivative im- plies that we would experience greater disxiiiXiVf from a decline in wealth than the utility we would derive from an equal increase in wealth. This tradeoff is apparent in Figure B, which shows the changes in utility associated with an equal increase and decrease in wealth. According to Bernoulli, the pre- cise change in utility associated with a change in wealth equals the logarithm of the sum of initial wealth plus the increment to wealth, divided by initial wealth. For example, the increase in util- ity associated with an increase in Initial Wealth wealth from $100 dollars to $150 equals 0.405465, as follows: 0.405465 / = ln \ 100-H50 100 The next $50 increment to wealth, however, yields a smaller increment to utility: 0.287682 Risk Aversion From Bernoulli's assumption of diminishing marginal utility,
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This note was uploaded on 08/13/2010 for the course FINS 2624 R taught by Professor Yippie during the Three '10 term at University of New South Wales.

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4 utility - What Practitioners Need to Know . . . About...

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