# Exercises - uncertainty, finance, time preferences (‘problem set’)

## Some questions from previous exams (somewhat easier questions)

Hedging bets?

Consider Betty, a UK resident working at a company that ships goods from Britain to France. In May 2016, an economist (Al) advises Betty that if the UK votes ‘leave’ in the June referendum, this may reduce trade with France. Al advises Betty to buy an asset (a ‘bet on leave’ with a bookmaker) that will pay off in the event that the UK votes for ‘leave’. The bookmaker offers odds that are seen as fair, and he only takes a small commission. What would justify the economist’s advice to buy this asset? What sort of preferences would Betty have to have to make this advice worth following?

Main points:

• If Leave passes she may lose her job or suffer reduced income.

• If she ‘bets on leave’ this loss would be counterbalanced by an income gain from the asset.

• On the other hand if leave does not pass she keeps her job, but loses the bet.

• Thus both the gains and losses are reduced by making this bet; i.e., the variance is reduced.

• If she is risk-averse she prefers to reduce the variance of her returns, holding the expected value the same. If she is substantially risk-averse, she is willing to sacrifice at least some amount of expected monetary value (i.e., the commission) to reduce the variance.

Note to Beem101 2020: this was part of a question on a previous exam. For the upcoming midterm, I would probably add an additional challenging element to such a question, e.g., asking you to formally specify her preferences in some way (concavity of value function, etc.) or consider the measurement of risk (certainty equivalent, Arrow-Pratt measures, etc.)

Risk aversion, choices

Define risk aversion formally and intuitively. Describe one choice that a risk neutral person might make that a risk-averse person would never make. Describe a particular measur} of risk-aversion that would allow us to rank individuals according to their level of risk aversion, considering the strengths and weaknesses of this measure.

A risk averse person always prefers the expected monetary value of a gamble to the gamble itself. They will never take ‘fair bets’ and will refuse even some gambles that have a positive expected value. Please see (and present and give intuition for) formal presentations as given above.

If utility is differentiable we can define risk aversion in terms of a diminishing marginal utility of income (or in general, concavity). More formal definitions, depictions, and intuition is given in this web book above.

The ‘coefficient of absolute’ risk aversion is one measure but it may not be constant within the range of an individual’s income; thus some normalisation or averaging would be required to make this comparison across individuals.

However, this depends what our purpose is in making this comparison across individuals – if we want to compare how risk averse they are given their current wealth, this may not be a problem. On the other hand, if we want to make a comparison (e.g., between men and women) to say something about genetic or culturl predisposition to risk-seeking, then the issue of ‘differing baseline incomes’ may be important.

‘Coefficient of relative risk aversion’ is another measure; it also may not be constant throughout the range of income (but that is at least more plausible). These measures, and the intuition for them, are discussed above.

Other measures include specific empirical elicitations/comparisons as those done in experiments, such as Holt and Laury discussed here. (You should briefly characterise it).

These are also arbitrary and may be sensitive to the experimental framing. In particular, there is some evidence (cite) that the Holt and Laury does not substantially predict real-world behavior. One possibility is that it is too complicated and analytical for most people to handle or to take seriously given low stakes. Another possibility is that the succession of choices presented by HL leads people to consider it in a way they would not naturally have done, to aim for an ‘arbitrary coherence’.

Risk aversion and diversification

Define ‘risk aversion’. Explain why an economist would advise a risk-averse investor to `diversify’ her investments. Would the advice be the same for any risk-averse investor, or would it vary depending on her level of risk-aversion? Does this depend on whether she can borrow or lend at the ‘risk-free’ rate? . Explain why or why not, referring to equations and diagrams as needed.

Risk aversion: The extent to which uncertainty of an outcome (holding the expected material or monetary value constant) implies an individual values it less. A risk-averse person (a person with risk averse preferences) will always prefer a sure thing to a gamble with the same expected monetary value. Equivalently, a risk averse person will always reject a fair gamble.

Neoclassical microeconomics concieves of and models this using an ‘outcome based’ (Von-Neuman Morgenstern) value function that increases at a diminishing rate, and an individual who tries to maximize the expected value of the outcome as measured by this utility function.

The greater the curvature (relative to the slope) of the VnM utility function, the more risk averse, at least by the popular ‘Arrow-Pratt’ measures.

An economist would advise a risk-averse investor to ‘diversify’ her investments, no matter how risk averse she is … as long as she is at least a little bit risk-averse, she will prefer to minimise the variance of the return (for a given expected return). As the returns of assets are not perfectly correlated, dividing the investment over ‘more coin flips’ implies a lower overall variance. (See discussion under ‘benefits of diversification’
as well as the discussion of the CAPM model).

This allows her to reduce the variance of her returns for a given expected return, or increase the expected return for a given variance. She can optimise along this margin by ‘optimally diversifying’, buying assets in proportion to their representation (relative value) in the market. She can then move to her desired point on the risk/return frontier, aka the ‘market line’, by either leveraging (borrowing) or putting some of her investment in a risk-free asset.

However (advanced point) if she cannot borrow/lend at the risk-free rate she cannot choose along the ‘market line’ and thus may not want to diversify quite as much; buying the ‘market basket’ may then be too-risky/too-safe relative to her preferences. (To fully answer this last part it will help to have read into the ‘CAPM’ model: see, e.g., the hypothes.is annotated Wikipedia entries on referred to above).

\end{solution}

Describe the ‘Allais paradox’, giving a specific example of a set of choices that illustrate this paradox. Explain why these choices are inconsistent with the standard theory of expected utility maximisation.

Please see lecture notes on Allais paradox

Allais paradox illustrated by a scenario such as

• Gamble A: an 89 percent chance of winning 1 million a 10 percent chance of winning £ 5 million, and a 1 pct chance of winning nothing.
• Gamble B: £ 1 million with certainty.

vs

• Gamble C: an 89 percent chance of winning nothing and an 11% chance of winning 1 million
• Gamble D: a 90 percent chance of winning nothing and a 10 percent chance of winning £ 5 million.

Many people choose B over A and choose D over C:

This contradicts Expected Utility theory:

• If $B \succ A$ then $EU(A) > EU(B)$
• $\rightarrow U(1m) > 0.89 \: U(1m) + 0.1 \: U(5m) + 0.01 \: U(0)$
• $\rightarrow$ $0.11 \: U(1m) > 0.1 U(5m) + 0.01 \: U(0)$
• If $D \succ C$ then $EU(D)>EU(C)$
• $\rightarrow 0.9 \: U(0) + 0.1 U(5m) > 0.89 \: U(0) + 0.11 \: U(1m)$
• Implying $0.1 \: U(5m) + 0.01 \: U(0) > 0.11 \: U(1m)$
• Contradicting the above!

## 3.13 From O-R

(Note: Suggested answers provided to Beem101 students, not to be posted on the web by request of O-R. Beem101 students can consult the Class Notebook, or the direct link HERE)

O-R Q2: “A parent”

A parent. A parent has two children, A and B. The parent has in hand only one gift. He is indifferent between giving the gift to either child but prefers to toss a fair coin to determine which child obtains the gift over giving it to either of the children. Explain why the parent’s preferences are not consistent with expected utility.

Hint below:

Note that expected utility requires the ‘independence’ property.

O-R Q7: “Additional lottery”

An individual faces the monetary lottery $p$. He is made the following offer. For each realization of the lottery another lottery will be executed according to which he will win an additional dollar with probability $\frac{1}{2}$ and lose a dollar with probability $\frac{1}{2}$.

Describe the lottery $q$ that he faces if he accepts the offer

Note: Here you are being asked to depict the lottery he faces in net including the lottery $p$, which may have any number of prizes, as well as the additional ‘coin flip’ lottery mentioned above.

and show that if he is strictly risk-averse he rejects the offer.

Note: In answering this question, you can assume that he is an ‘expected utility’ maximiser, and thus the continuity and independence axioms must hold (and by extension, monotonicity).

Note: I can probably improve the notation in the above video.

O-R Q8: “Casino”

An individual has wealth $w$ and has to choose an amount $x$, after which a lottery is conducted in which with probability $\alpha$ he gets $2x$ and with probability $1 − \alpha$ he loses $x$.

Show that the higher is $\alpha$ the higher is the amount $x$ he chooses.

O-R Q9: “Insurance”

Insurance. An individual has wealth $w$ and is afraid that an accident will occur with probability $p$ that will cause him a loss of $D$.

Please assume, of course, that this is indeed the probability that such an accident will occur.

The individual has to choose an amount, $x$, he will pay for insurance that will pay him $\lambda x$ (for some given $\lambda$) if the accident occurs.

Note that $\lambda$ will determine, in effect, the ‘price’ of the insurance, per unit of compensation in the event of an accident.

1. The insurer’s expected profit is $x − \lambda p x$

Because the individual paid $x$ and the insurer must compensate him $\lambda x$ with probability $p$

Assume that $\lambda$ makes this profit zero, so that $\lambda = 1/p$.

This is referred to as ‘actuarially fair insurance’. This should hold in a perfectly competitive insurance market if there are no moral hazard or asymmetric information issues, no transactions costs, etc.

Show that if the individual is risk-averse he optimally chooses $x = pD$ , so that he is fully insured: [implying that] his net wealth is the same whether or not he has an accident.

Exeter students: I cover this question at length in this recorded session

For a ‘state-space’ diagram presenting the insurance problem, please see Joon Song’s video here