ECON 4070-001 Topics in Microeconomics

Spring 2021, Homework 7

Due 5pm, April 27

Complete the five questions below. Each question is worth 4 points and the overall presentation is worth 5 points. Total points is 25. Please prove your answers for maximum points. Please save your completed homework in a single pdf file and upload the file to Canvas by 5pm on April 27.

1. A rational investor is constructing a stock portfolio comprising of i = 1, …, n stocks from the NYSE, Assume that the annual returns from each stock are independently and identically distributed from a parent distribution with mean r, variance = 0.1225 and correlation 0 ≤ ρ ≤ 1.

(i) Assume that ρ = 1. Calculate the standard deviation of the sampling distribution of average returns from this portfolio for n = 50 stocks.

(ii) Very carefully show on a two dimensional diagram (with standard deviation on the vertical axis and number of stocks on the horizontal axis) how the standard deviation of the sampling distribution of average returns from this portfolio changes as you increase portfolio size from i = 1 to n = 50. Label this diagram Figure 1.

2. A rational investor is constructing a stock portfolio comprising of i = 1, …, n stocks from the NYSE, Assume that the annual returns from each stock are independently and identically distributed from a parent distribution with mean r, variance = 0.1225 and correlation 0 ≤ ρ ≤ 1.

(i) Assume that ρ = 0.3. Calculate the standard deviation of the sampling distribution of average returns from this portfolio for n = 50 stocks. Very carefully show on a two dimensional diagram how the standard deviation of the sampling distribution of average returns from this portfolio changes as you increase portfolio size from i = 1 to n =50. Label this diagram Figure 2.

(ii) Compare Figure 1 and Figure 2 and comment on how diversification (here, portfolio size) impacts stock portfolio risk.

3. Jeanne and Danny are risk-averse and both own residential homes worth $1,000,000. Danny’s home is brick construction and is located in the North Fruita desert. Jeanne’s home is wood construction and is located in the Sam Isabel National Forest. 2

(i) Assume that the insurance company cannot distinguish between consumers and will charge the same premium for fire insurance to all owners of $1,000,000 residential properties. Carefully explain which consumer, Jeanne or Danny, is more likely to buy fire insurance for their home?

(ii) Does your answer in Q3.(i) above imply a particular market failure problem for the insurance company and/or consumers of insurance that we have discussed in class? If so, carefully explain the source of the market failure.

4. Sixteen year-old persons have a probability of 0.9 of having a car accident that results in damage of $10,000. Twenty year-old persons have a probability of 0.1 of having a similar accident with damage of $10,000.

(i) An insurance company is unable to distinguish between individual sixteen and twenty year-old persons but assumes that they are equally represented in the population and are equally likely to purchase insurance. What is the actuarially fair premium?

(ii) Assume the utility for each consumer age group is U = lnW, where W is wealth, and each age group has a current wealth level of $20,000. Will both age groups purchase insurance at the fair premium?

5. Continue on this question with the same initial assumptions from question 4 above.

(i) Given your answer to Q4.(ii) above, what should the fair premium be? What is the utility for the typical individual in each age group?

(ii) Assume that the insurance company can price discriminate between sixteen and twenty year old car drivers. What are the insurance premiums for each age group and what is the utility for an individual in each age?