FINC3017 Journal Articles
FINC3017 Journal Articles Elton, Gruber, Brown and Goetzmann (2003) Chapter 10 [Utility] Jefferson [22 pages] (http://ereserve.library.usyd.edu.au.ezproxy2.library.usyd.edu.au/fisher/EltonModern2003 Ch10.pdf) Modern Portfolio Theory and Investment Analysis Introduction to Utility Analysis • How to choose among investor’s opportunity set, or how to specify preference function • Prefer more to less, reduce set to efficient frontier • Riskless lending and borrowing, one preferred risky portfolio, regardless of preference function • Preference function determines combination of optimal risky portfolio, and risk-‐less asset • Characteristics of utility functions consistent with investor behaviour Preference Functions • Probability of each outcome multiplied by utility derived from each outcome o Utility (weighting) function, by proportion of each outcome, expected utility theorem o Maximise expected utility by maximising utility function outcome o Constants do not change the ranking of investments • Determine weighting function investors are implicitly using o Compare utilities for each investment under the same investor o Process of rational choice – eliminate some portfolios, reduce chance of bad decision-‐making Economic Properties of Utility Functions • Non-‐satiation, more wealth preferred to less wealth o First derivative positive • Risk aversion in relation to fair gamble o Risk-‐averse investor rejects fair gamble, disutility of loss is greater than utility of an equivalent gain o Implies negative second derivative o Risk neutral indifferent to fair gamble – zero second derivative • Exhibit utility functions in: (1) Utility of wealth space & (2) As indifference curves in mean-‐standard deviation space
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Absolute risk aversion ARA = A(W)
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Relative risk aversion RRA o Similar to ARA, except deals with percentages, instead of absolute dollar values Quadratic utility function
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o ARA and RRA both positive, reduce $ and % amount invested in risky assets as wealth increases o Mean-‐variance analysis, function leads to optimisation Log utility function (ln) o ARA is negative, RRA is 0
Investor Horizon • Utility functions generally based on investor choice in single-‐period horizon o In reality, multi-‐period choice problem • Certainty equivalents for a risky investment o Risk-‐free value that makes you indifferent between taking and not taking a fair gamble • Investors who have some risk tolerance may be willing to invest in risky assets over long horizons, but not short horizons o Willingness to invest in risky assets as investment horizon grows o Asset allocation can depend on how long they expect to keep the money invested
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Empirical Evidence on Alternative Preference Functions Consistency of assumptions on investor behaviour with observations of actual behaviour o Empirical evidence on simple choice situations, survey data on investor asset choices Prefer more to less is consistent Risk aversion consistent – insurance purchases, even though premiums are larger than expected loss Most individuals choose to gamble with the least risky alternatives – risk-‐seeking behaviour o Risk-‐averse individual may still gamble (e.g. lottery tickets) Asset holding and wealth to draw inference on ARA and RRA o Data does not exist on an investor’s wealth over time o Draw inferences from different investors at different wealth levels o First study by Blume and Friend – decreasing ARA, constant RRA o Second study by Cohn, Lewellen, Lease and Schlarbaum – decreasing ARA and RRA o Limitations, did not examine same investor at different wealth levels, but still suggestive of behaviour Appendix A – Expected Utility Theorem Axioms Preference ordering of certain outcomes o (1) Comparability – investor can state preference amongst all alternative certain outcomes o (2) Transitivity – consistent in ranking of outcomes, unless complex and cannot understand implications Rationality when ordering random prospects
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Absolute risk aversion ARA = A(W)
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Relative risk aversion RRA o Similar to ARA, except deals with percentages, instead of absolute dollar values Quadratic utility function
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o ARA and RRA both positive, reduce $ and % amount invested in risky assets as wealth increases o Mean-‐variance analysis, function leads to optimisation Log utility function (ln) o ARA is negative, RRA is 0
Investor Horizon • Utility functions generally based on investor choice in single-‐period horizon o In reality, multi-‐period choice problem • Certainty equivalents for a risky investment o Risk-‐free value that makes you indifferent between taking and not taking a fair gamble • Investors who have some risk tolerance may be willing to invest in risky assets over long horizons, but not short horizons o Willingness to invest in risky assets as investment horizon grows o Asset allocation can depend on how long they expect to keep the money invested
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Empirical Evidence on Alternative Preference Functions Consistency of assumptions on investor behaviour with observations of actual behaviour o Empirical evidence on simple choice situations, survey data on investor asset choices Prefer more to less is consistent Risk aversion consistent – insurance purchases, even though premiums are larger than expected loss Most individuals choose to gamble with the least risky alternatives – risk-‐seeking behaviour o Risk-‐averse individual may still gamble (e.g. lottery tickets) Asset holding and wealth to draw inference on ARA and RRA o Data does not exist on an investor’s wealth over time o Draw inferences from different investors at different wealth levels o First study by Blume and Friend – decreasing ARA, constant RRA o Second study by Cohn, Lewellen, Lease and Schlarbaum – decreasing ARA and RRA o Limitations, did not examine same investor at different wealth levels, but still suggestive of behaviour Appendix A – Expected Utility Theorem Axioms Preference ordering of certain outcomes o (1) Comparability – investor can state preference amongst all alternative certain outcomes o (2) Transitivity – consistent in ranking of outcomes, unless complex and cannot understand implications Rationality when ordering random prospects
