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Money

Money. Utility and wealth. Example. Consider a stock investment for 5000 which could increase or decrease by +/- 100 0 Let current wealth be C An investor has the choice of entering or not The probability of a ‘positive’ outcome is p W hat are the consequences and the de cisions?.

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Money

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  1. Money Utility and wealth

  2. Example • Consider a stock investment for 5000 which could increase or decrease by +/- 1000 • Let current wealth be C • An investor has the choice of entering or not • The probability of a ‘positive’ outcome is p • What are the consequences and the decisions?

  3. Utility • Utility is a function, or a numerical value which represents the value of a consequence to a decision maker. • It is sufficient, and convenient, to restrict our consideration to utilities between 0 and 1. • Utility of 1 can be seen as the best consequence and 0 as the worst. • NB these restrictions are ours, not given by ‘God’.

  4. Features of Utility for Money • The utility of every extra euro becomes less. • That is, an extra €100 is ‘worth’ less to a millionaire than it is to a poor student. • Equally, the loss of €100 may be acceptable to some rich folk, but a pain to normal mortals. • Utility increases with increasing wealth, and decreases as wealth moves to the left. • The restriction that wealth should be positive bounds the lower part of utility, whereas the first point bounds the right hand edge.

  5. Typical curves • 1 - exp(-cx) • 1 - exp(-ax)/2 - exp(-bx)/2 • These give the functional form of some curves satisfying our restrictions. (c,a = 1/2000 and b = 1/1000 in the plots) • NB the restrictions are ours; the utilities can be anything for any values, but often this would not make sense in practice.

  6. Diminishing Marginal Utility • These curves bring out the feature of money previously described and well recognised. • That is, extra wealth has diminishing marginal utility. • This is also true of other ‘pleasures.’ Like ice cream. • Formally, fix a. Then • if C1>C2, u(C1+a)-U(C1) < U(C2+a)-U(C2)

  7. Investment Example • Go back to the investment which could gain or lose 1000 euro. • Let the current wealth of the individual be 5000 euro. Use the second utility function. • What is the utility associated with each of the consequences? • For what value of probability of success, p, does the investment become attractive? • Repeat with current wealth equal to 8000.

  8. Risk Aversion • Note that a monetarily fair bet is one where the expected increase in wealth is zero. • A bet which has monetary expectation less than zero is termed subfair. • An investor is said to be ‘Risk averse’ if the only acceptable bets are those where the monetary expectation is positive.

  9. Risk Aversion and Utility • An investor will start to find a ‘bet’ attractive when the expected gain in utility becomes zero. • Risk aversion says that this can only occur if the monetary expectation is positive. • Show that these combined yield a utility curve as drawn. • That is, concavity of utility  risk averse

  10. Risk Prone • Risk prone behaviour, is the accepting of ‘bets’ or investments which have negative expected monetary value. • Playing of the lotto may be one such behaviour. • Sketch out the implications for utility. • Note that Friedman-Savage had a hypothesis about complex utilities such as this, but Lindley disagrees.

  11. Probability Premium • Let p be the probability value at which a gamble becomes monetarily fair. • Let P be the probability of success for which a gamble becomes acceptable (that is expected utility is zero). • Then P-p is termed the ‘probability premium.’ • This is the ‘extra odds’ a decision maker requires in order to accept ‘risk’ given their utility function.

  12. Insurance • Consider a person with assets C, a house valued at h, and an insurance which will cover loss of house costing m. • Let us suppose the house is destroyed with probability p, or else it survives to the end of the contract. • A householder wants to decide whether to take out insurance. What are the consequences?

  13. Insurance calculation • If C = 5000, and h = 5000 with p = 0.01, what is the maximum the decision maker will pay for insurance? • Use the second type of utility with a=1/20000 and b=1/2000. • What is the expected monetary value of the insurance contract?

  14. Risk Premium • The difference between the amount that the an individual will pay to insure a risk and the expected monetary loss associated with the risk is called the ‘Risk Premium.’ • Personal utility makes individuals want to transfer risk, and it is the risk premium that can be earned by insurance companies in accepting the risk.

  15. Portfolio Management • A portfolio is a collection of investments. • Different stocks give different returns, but also are associated with different risks. • If one uses expected monetary value to decide which asset to invest in, the only quantity of interest is expected return. • Expected utility matches the common wisdom of a balanced portfolio.

  16. Example • Returning to the stock. If 500 is invested, then +-100 can be returned. Let the p of favourable outcome be 0.51. • Let the decision maker have to consider how much to invest (in 250 euro units) in the stock (+-50 per unit). • Let the utility be given by the first type with c = 1/1000.

  17. Exercise • For this case; • Draw the table of decisions and consequences. • Calculate the table of utlities. • Calculate the expected utility for each decision. • Choose how much to invest. • Note the qualitative result that the investment strategy is to retain a mixture of assets.

  18. What about risk? • Some authors suggest that we should quantify the risk as well as the ‘expected’ utility. • This is indeed the case, and many of our simulation studies examine the possible as well as the probable. • However, a properly defined utility can ensure that the risk of ‘calamity’ has an appropriate negative weighting when it comes to our decision making paradigm.

  19. Multiple Criteria • As mentioned before, one may have to decide between, say, an ice cream or a beer. • There is an argument that says one cannot find an equivalency between ice cream and beer, so they cannot be reduced onto a single dimension. • However, the concept of ‘best’ is necessarily univariate, which may become clearer when looking at multiple criteria decision making.

  20. Exercise • A company has a utility function of the second form with a = 1/20 and b = 1/200, and x in millions. It has assets of €40M. • A risk of €40M is to be insured, with a probability of event = 0.05. • Calculate the minimum premium that will make this acceptable to the company (to the nearest €1M). • If a similar company, with same utility and assets is willing to take half the risk, what now is the minimum combined premium.

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