Efficient markets and utility theory: questions

Three claims and the efficient markets hypothesis

Paper A style · Utility · Straightforward · 8 marks

Three employees of Harrowgate Asset Management make the following claims about shares listed on the London Stock Exchange.

Claim 1 (a technical analyst): "By studying charts of past share prices and trading volumes I can identify patterns that earn returns above those justified by risk."
Claim 2 (a fundamental analyst): "By reading company results and broker reports the moment they are published, I can pick shares that beat the market after allowing for risk."
Claim 3 (a director): "Because I sit on the board of a company in takeover talks that have not yet been announced, I could make abnormal profits by trading its shares."

(i)
Describe the weak, semi-strong and strong forms of the Efficient Markets Hypothesis.
3 marks
(ii)
For each claim, state the weakest form of the hypothesis which, if it held, would mean the claim is false. Give a brief reason in each case.
3 marks
(iii)
Explain one consequence for an investment manager if the semi-strong form holds.
2 marks

Properties of power utility

Paper A style · Utility · Straightforward · 8 marks

An investor's utility of wealth $w$ $> 0$ is $U(w)$ $= \dfrac{w^{1-\gamma}}{1-\gamma}$ , where $\gamma > 0$ and $\gamma \neq 1$ .

(i)
Show that the investor is non-satiated and risk averse.
2 marks
(ii)
Show that the coefficient of absolute risk aversion is $\gamma/w$ and the coefficient of relative risk aversion is $\gamma$ . Describe what each result means for the investor's behaviour as wealth increases.
3 marks
(iii)
For $\gamma = 2.5$ and current wealth £50,000, the investor is offered a gamble that adds or subtracts £5,000 with equal probability. Using the approximation that the risk premium is about $\tfrac12 A(w)\sigma^2$ , where $\sigma^2$ is the variance of the gamble, estimate the risk premium to the nearest £1.
2 marks
Your answer
(iv)
State the utility function obtained in the limit as $\gamma \to 1$ (after subtracting the constant $1/(1-\gamma)$ ).
1 mark

Certainty equivalent with square-root utility

Paper A style · Utility · Straightforward · 6 marks

An investor has utility of wealth $U(w)$ $= \sqrt{w}$ , where $w$ is in £000. She holds an investment that will leave her with wealth of either £196,000 or £64,000 in one year's time, each with probability 0.5.

(i)
Show that the investor is non-satiated and risk averse.
2 marks
(ii)
Calculate the certainty equivalent of the investment, to the nearest £1.
2 marks
Your answer
(iii)
Calculate the risk premium the investor attaches to the investment, to the nearest £1.
1 mark
Your answer
(iv)
Someone offers to buy the investment from her now in exchange for a guaranteed £125,000 in one year's time. Should she accept? Explain your answer.
1 mark

Abnormal returns after a profit warning

Paper A style · Utility · Standard · 9 marks

Kestrel Retail plc issued an unexpected profit warning before the market opened on day 0. A researcher models the share's expected daily return by the market model, with expected return $0.0002 + 1.2R_m$ on a day when the market return is $R_m$ . The observed daily returns were:

Day 0: share −6.10%, market −0.40%
Day 1: share −1.25%, market 0.30%
Day 2: share −0.90%, market −0.15%
Day 3: share −0.60%, market 0.10%

The abnormal return on a day is the observed return less the expected return.

(i)
Calculate the abnormal return on day 0, as a percentage to 2 decimal places.
2 marks
Your answer
(ii)
Calculate the cumulative abnormal return over days 1 to 3, as a percentage to 2 decimal places.
3 marks
Your answer
(iii)
Suppose that a study of many profit warnings finds similar negative abnormal returns persisting for several days after the announcement. Comment on what this suggests about the semi-strong form of the Efficient Markets Hypothesis.
2 marks
(iv)
Explain the joint hypothesis problem in testing market efficiency.
2 marks

Risk aversion under quadratic utility

Paper A style · Utility · Standard · 8 marks

An investor's utility of wealth $w$ (in £000) is $U(w)$ $= w - 0.004w^2$ .

(i)
Determine the largest level of wealth (in £000) up to which the investor is non-satiated.
2 marks
Your answer
(ii)
Calculate the investor's coefficient of absolute risk aversion when wealth is £50,000, to 5 decimal places (in units of per £000).
2 marks
Your answer
(iii)
Show that both absolute risk aversion $A(w)$ and relative risk aversion $R(w)$ increase with wealth over the range found in part (i).
2 marks
(iv)
Explain why increasing absolute risk aversion is considered an unrealistic property, and state one other drawback of quadratic utility.
2 marks

Log utility and the size of wealth

Paper A style · Utility · Standard · 8 marks

An investor has utility of wealth $U(w)$ $= \ln w$ . She may choose one of two investments, each lasting one year:

Investment X increases her wealth by £8,000 with probability 0.4 and reduces it by £4,000 with probability 0.6.
Investment Y increases her wealth by £500 with certainty.

Ignore interest and expenses.

(i)
Her current wealth is £20,000. Calculate the certainty equivalent of her wealth if she chooses Investment X, to the nearest £1, and state which investment she prefers.
3 marks
Your answer
(ii)
Repeat part (i) for an investor with the same utility function whose current wealth is £200,000. Give the certainty equivalent gain from Investment X (the certainty equivalent wealth minus £200,000), to the nearest £1, and state which investment this investor prefers.
3 marks
Your answer
(iii)
Explain the difference between your answers to parts (i) and (ii) in terms of the investor's absolute risk aversion.
2 marks

Exponential utility and the equity allocation

Paper A style · Utility · Standard · 10 marks

An investor with wealth £400,000 has utility of wealth $U(w)$ $= -e^{-aw}$ with $a$ $= 2 \times 10^{-5}$ per £. She will invest £ $x$ in an equity fund whose one-year return is normally distributed with mean 7% and standard deviation 18%, and the rest in a deposit account paying 3% a year with certainty. There are no expenses or taxes.

(i)
Show that if wealth at the end of the year is $W \sim N(m, s^2)$ , then $E[U(W)] = -\exp\left(-am + \tfrac12 a^2 s^2\right)$ , and hence that the investor ranks choices by $m - \tfrac12 a s^2$ .
3 marks
(ii)
Calculate the amount $x$ that maximises her expected utility, to the nearest £1.
3 marks
Your answer
(iii)
Calculate the certainty equivalent of her end-year wealth when she invests the amount found in part (ii), and hence the amount by which it exceeds the £412,000 she would have by holding only the deposit, to the nearest £1.
2 marks
Your answer
(iv)
State how the optimal amount in equities would change if her wealth were £800,000, and comment on whether this is realistic.
2 marks

Maximum and minimum premiums for cover

Paper A style · Utility · Standard · 8 marks

A self-employed van driver has total wealth of £60,000, which includes his van. Over the next year there is a probability of 0.08 that the van is written off, causing a loss of £15,000, and otherwise there is no loss. His utility of wealth is $U(w)$ $= \ln w$ .

An insurer whose utility of wealth is $U_I(w)$ $= -e^{-0.00001w}$ is willing to provide full cover against this loss. Ignore interest and expenses.

(i)
Calculate the maximum premium the driver would pay for full cover, to the nearest £1.
3 marks
Your answer
(ii)
Calculate the minimum premium the insurer would accept for providing the cover, to the nearest £1.
3 marks
Your answer
(iii)
Comment on your answers.
2 marks

Quadratic utility and two portfolios

Paper A style · Utility · Standard · 10 marks

An investor's utility of wealth $w$ (in £000) is $U(w)$ $= w - 0.01w^2$ . She must choose one of two portfolios. Her wealth in one year's time will have mean £30,000 and standard deviation £6,000 under Portfolio A, and mean £32,000 and standard deviation £10,000 under Portfolio B.

(i)
Show that for quadratic utility the expected utility of wealth $W$ depends only on the mean and variance of $W$ .
2 marks
(ii)
Calculate the expected utility of each portfolio and state which she prefers. Give the expected utility of Portfolio B as your numerical answer, to 2 decimal places.
3 marks
Your answer
(iii)
Keeping the mean of Portfolio B at £32,000, calculate the standard deviation of Portfolio B (in £, to the nearest £10) that would make her indifferent between the two portfolios.
3 marks
Your answer
(iv)
Comment on a weakness of using this utility function to compare these portfolios.
2 marks

Choosing an excess with exponential utility

Paper A style · Utility · Hard · 13 marks

A small business faces a property loss $X$ over the next year. With probability 0.8 there is no loss; with probability 0.2 a loss occurs whose size is uniformly distributed between £0 and £10,000. The owner's utility of wealth is $U(w)$ $= -e^{-0.0002w}$ . An insurer offers two policies, each priced at 130% of the expected claim payment:

Policy F pays the whole loss.
Policy E pays the loss in excess of £2,000 (the business pays the first £2,000 of any loss).

The owner may also buy no cover. Ignore interest and expenses.

(i)
Calculate the premium for Policy E, to the nearest £1. (You should also find that the premium for Policy F is £1,300.)
2 marks
Your answer
(ii)
Let $C$ be the total cost to the business over the year (premium plus any part of the loss it bears). Show that the owner prefers the option with the smallest value of $c = \dfrac{1}{0.0002} \ln E\left[e^{0.0002C}\right]$ , whatever the owner's initial wealth.
2 marks
(iii)
Calculate $c$ if the owner buys no cover, to the nearest £1.
3 marks
Your answer
(iv)
Calculate $c$ if the owner buys Policy E, to the nearest £1.
4 marks
Your answer
(v)
Recommend which option the owner should choose and comment on the result.
2 marks

State-dependent utility and optimal cover

Paper A style · Utility · Hard · 10 marks

A collector has wealth of £100,000, which includes a painting worth £40,000. Over the next year there is a probability of 0.1 that the painting is stolen. She can buy cover of any amount £ $C$ (with $0$ $\le C$ $\le 40{,}000$ ) at a premium of £0.10 per £1 of cover, paid at the start of the year; if the painting is stolen the insurer pays £ $C$ . Work in units of £000 and ignore interest.

Her utility of wealth $w$ (in £000) is $\ln w$ if the painting is not stolen and $k \ln w$ if it is stolen, where $k$ is a positive constant.

(i)
Show that her expected utility is maximised when $C$ satisfies $60 + 0.9C$ $= k(100 - 0.1C)$ , and deduce that with $k$ $= 1$ she buys full cover.
4 marks
(ii)
Calculate the optimal amount of cover when $k$ $= 0.8$ , to the nearest £1.
3 marks
Your answer
(iii)
Explain why she chooses less than full cover when $k$ $= 0.8$ , even though the premium is actuarially fair, and give one realistic example of a situation where utility may depend on the state of the world.
3 marks

Power utility with lognormal fund returns

Paper A style · Utility · Hard · 11 marks

An investor with utility of wealth $U(w)$ $= \dfrac{w^{1-\gamma}}{1-\gamma}$ (with $\gamma > 0$ , $\gamma \ne 1$ ) will invest all of her current wealth $w_0$ for one year in one of two funds. If $X$ is the accumulation factor of the chosen fund over the year (so her wealth becomes $w_0X$ ), then $\ln X \sim N(0.05, 0.10^2)$ for Fund A and $\ln X \sim N(0.085, 0.20^2)$ for Fund B.

(i)
Show that if $\ln X \sim N(\mu, \sigma^2)$ , the certainty equivalent of her end-year wealth is $w_0 \exp\left(\mu + \tfrac12(1-\gamma)\sigma^2\right)$ .
3 marks
(ii)
For $\gamma = 3$ , calculate the certainty equivalent rate of return of each fund (that is, $c/w_0 - 1$ ) and state which fund she prefers. Give the value for Fund B as your numerical answer, as a percentage to 2 decimal places.
3 marks
Your answer
(iii)
Calculate the value of $\gamma$ at which she would be indifferent between the two funds, to 4 decimal places.
3 marks
Your answer
(iv)
Explain why her choice of fund does not depend on her current wealth $w_0$ , and why this is consistent with the relative risk aversion of her utility function.
2 marks

Choosing a fund under three utility functions

Paper B style · Utility · Standard · 20 marks

An investor has £100,000 to invest for one year in exactly one of three funds, A, B and C. An economic scenario generator has produced 30 equally likely scenarios; the table gives each fund's one-year return in each scenario (0.0523 means 5.23%). Work in units of £000, so end-year wealth in a scenario is $100(1 + r)$ . Ignore expenses and taxes.

For the power utility function use $U(w)$ $= \dfrac{w^{1-\gamma}}{1-\gamma}$ for $\gamma \ne 1$ , and $U(w)$ $= \ln w$ when $\gamma = 1$ . The certainty equivalent $c$ of a fund satisfies $U(c)$ $= E[U(W)]$ .

ut-p13.csv: 30 rows, 4 columns

Show the data

Scenario	FundA	FundB	FundC
1	0.006	0.1142	-0.0281
2	0.0529	0.0909	0.2208
3	0.1116	0.1342	0.3999
4	0.0752	0.1788	0.2063
5	0.0549	0.1887	-0.1477
6	0.0995	-0.0266	-0.0475
7	0.0691	0.0364	0.6256
8	0.1232	0.1598	-0.4843
9	-0.0135	0.0859	0.4626
10	0.045	0.0676	-0.5285
11	0.0618	0.1848	-0.1986
12	0.013	0.0598	0.3273
13	0.0408	0.146	0.0087
14	-0.0023	-0.0683	-0.1597
15	-0.0139	0.2773	0.3412
16	0.0888	-0.0011	0.3813
17	-0.056	0.0278	-0.1008
18	-0.0223	0.362	0.1764
19	0.0234	0.2426	0.6778
20	-0.0002	0.1287	0.0925
21	0.0788	0.0225	0.1977
22	0.0659	0.0151	0.1762
23	0.0507	-0.1206	0.1942
24	0.1019	-0.0158	0.1687
25	0.0871	-0.2556	-0.018
26	0.086	0.1177	0.212
27	0.0552	0.0887	-0.1929
28	0.0414	-0.0579	0.02
29	0.0751	0.0793	-0.0754
30	0.0791	-0.1255	0.0746

(i)
A risk-neutral investor would choose the fund with the highest expected end-year wealth. Calculate the expected end-year wealth of each fund. State which fund the risk-neutral investor chooses, and give its expected end-year wealth (in £000, to 3 decimal places).
3 marks
Your answer
(ii)
For an investor with $U(w)$ $= \ln w$ , calculate the certainty equivalent of each fund. State which fund this investor chooses and give its certainty equivalent (in £000, to 3 decimal places).
4 marks
Your answer
(iii)
Repeat part (ii) for power utility with $\gamma = 5$ . Give the certainty equivalent of the chosen fund (in £000, to 3 decimal places).
4 marks
Your answer
(iv)
Use Goal Seek (or otherwise) to find the value of $\gamma$ , greater than 1, at which the investor is indifferent between Funds A and B. Give $\gamma$ to 2 decimal places.
5 marks
Your answer
(v)
Comment on your results.
4 marks