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20b_risk_aversion

20b_risk_aversion - Uncertainty Risk Aversion and Expected...

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Uncertainty, Risk Aversion and Expected Utility If investors always evaluated risky ventures by taking expected payo ff s, uncer- tainty would not create many di culties. But there is tremendous evidence that uncertainty has a profound e ff ect on fi nancial markets. The fi rst empirical law of fi nancial markets is that over short periods of time (like an hour, or perhaps a day), asset returns have expectation zero. If there are no dividends in the intervening perioid, today’s price is the expectation of tomorrow’s price. But over long periods of time, there is compelling evidence that di ff erent assets have substantially di ff er- ent returns. From 1926 until today, stocks have averaged about 9.1% returns above in fl ation, while bonds have averaged only 2.3% above in fl ation. That is too big a di ff erence to attribute to luck. (This second fact is not inconsistent with the fi rst, since over short periods of time the small measurement error in observing the initial and fi nal prices can swamp any tiny trend.) One obvious factor is that stock returns are much riskier than bond returns. If investors were afraid of risk, then they would pay less for a riskier asset with the same expected payo ff . The return (expected payo ff divided by price) would then be higher for riskier assets. This will be the key idea in what follows. But how do we measure risk? And how do we put risk aversion into the model? 1 St Petersburgh Paradox and the Invention of Expected Utility Expected utility was invented by Daniel Bernoulli while he was living in St Peters- burgh on account of the followng puzzle. Suppose somebody told you he would continue fl ipping a fair coin until he got a tails, and then pay you 2 n +1 dollars, where n is the number of consecutive heads before the tail. (He is e ff ectively guaranteeing you at least $2.) How much would you pay for the privilige of receiving this lottery? Your expected revenue if you play is 1 2 2 + 1 4 4 + 1 8 8 + ... = 1 + 1 + 1 + ... = where we have computed that the chance of n consecutive heads is 1 / 2 n for the n heads in a row times 1 / 2 for the following tails which ends the streak, or 1 / 2 n +1 . Bernoulli noticed that nobody would actually pay very much money to play this game. In short, the price of the lottery is much lower than its expected payo ff , reminding us of the story of risky stocks.
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