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Money

Utility and wealth

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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?

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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’.

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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.

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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.

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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)

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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.

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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.

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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

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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.

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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.

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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?

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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?

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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.

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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.

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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.

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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.

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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.

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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.

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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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