Questions to be discussed in this tutorial:

Question 1

Alice owns a house with a market price of 10 million dollars. She is an expected utility maximizer and her utility function is u(x) = ln(x) . During any given year she faces the following risk: with probability 1/3 her house will be completely destroyed by a tornado. She has an option to by any amount of insurance for 80 cents per 1 dollar of insurance. That is, if she wants to get $q in case tornado destroys her home, she needs to pay an insurance company $0.8q at the start of the year.

(a) Suppose she insures her house for q dollars. Calculate her final wealth in each possible state of the world (tornado and no tornado).

(b) Using your results from (a), set up her expected utility maximization problem and calculate the amount of insurance q* she will buy.

Answer:

(a) When she buys coverage her wealth in the good state (no tornado) is 10 − 0.8q and her wealth when tornado occurs is  0 − 0.8q + q = 0.2q

(b) She chooses the amount of insurance to maximize her expected utility:

2                           1

ln(10 − 0.8q) +   ln(0.2q)

3                          3

The First order condition is given by

2     −0.8       1  0.2 

1         1.6

=                                   

q    10 − 0.8q

10 − 0.8q = 1.6q

q =  = 4.1667

Note  the  price  of insurance  is  higher  than  the  expected  loss,  which  is  equal  to  10*0.33=3.3. Consequently, despite the fact that she is risk averse, Alice does not buy full insurance.

Question 2

For each of the following pairs of investments characterized by the expected return and st. deviation, state which would always be preferred by a rational investor:

(a) A has higher return for a given risk

Portfolio A: ER=18%, St. Dev= 20%

Portfolio B: ER=14%, St. Dev= 20%

(b) There is no clear winner: C has higher risk and higher return.

Portfolio C: ER=15%, St. Dev= 18%

Portfolio D: ER=13%, St. Dev= 8%

(c) F has lower risk for the same return.

Portfolio E: ER=14%, St. Dev= 16%

Portfolio F: ER=14%, St. Dev= 10%

Question 3

Plot the following risky portfolios on a graph with expected return on the vertical axis and standard deviation on the horizontal axis.

Which portfolios would a rational investor never invest in? Explain your answer.

Answer

A rational investor would not invest in A (dominated by B for example), D (dominated by E) and G (dominated by F).

Question 4

Consider the formula for the variance of the portfolio of two stocks discussed in class: P(2) = wA(2)A(2) + wB(2)B(2)  + 2wAwBpAB

Recall that wA  denotes the portfolio weight of stock A (proportion of funds invested in A) and wB         denotes the portfolio weight of stock B, with wA + wB  = 1 . The coefficient p , −1   p  1  measures the correlation between the returns of the stock A and B.

(a) Suppose the two returns are perfectly correlated: p = 1 .

(i) Derive the formula for the variance of the portfolio return in this case.

Answer

QP(2)  = wA(2)QA(2)  + wB(2)QB(2)  + 2wAwBQAQB  = (wAQA  + wBQB )2

(ii) Show that the standard deviation of the portfolio return is equal to the weighted average of the standard deviations of the two stocks.

Answer:

From (i) it follows that

Q  = w Q + w Q

(b) Consider now the case when the two returns have zero correlation: b = 0 . Derive the formulas for the variance and standard deviation of the portfolio return in this case.

Answer:

Q  = w Q  + w Q

QP  = (wA(2)QA(2)  + wB(2)QB(2))0.5

(c) Assuming that the expected returns are the same in both cases, would you rather invest in a          portfolio of two stocks which are perfectly correlated (i.e. b = 1 ) or the stocks that are not correlated at all ( b = 0 )? Hint: compare the standard deviations of the two portfolios.

Answer:

The standard deviation of the portfolio with zero correlation would be lower. The easiest way to see it is to take the derivative of the portfolio variance with respect to b :

9QP(2)   = 2w w Q Q  > 0

9b

This implies that the higher is b , the higher is the variance and standard deviation.

(d) Finally consider the case when the two stocks have a perfect negative correlation: b = − 1 . Show that in this case it is possible to construct a portfolio of these two stocks which completely eliminates risk, i.e. a portfolio with positive expected return and zero variance.

Answer:

QP(2)  = wA(2)QA(2)  + wB(2)QB(2)  − 2wAwBQAQB  = (wAQA  − wBQB )2