# Mean-variance efficient frontiers

1. (2 points). Mean-variance efficient frontiers. There are two risky assets A and B. The means

and standard deviations of the assets’ returns are given by

Mean �” Standard Deviation �”

A 5% 10%

B 8% 15%

a. Write the equations for the mean-variance efficient frontiers when the correlation between

the two assets is -1, 0, and 1. Graph the efficient frontiers with the standard deviation on

the x axis and the mean on the y axis using your preferred graphing software. Note that

the efficient frontiers are only the regions of the feasible sets that provide the maximum

mean return for any given standard deviation. You will lose 1 point if you do not distinguish

the efficient frontier from the set of feasible mean-standard deviation combinations.

b. Consider the case where the correlation between the two assets is zero. What is the

standard deviation of asset A above which no portfolio on the mean-variance efficient

frontier includes a positive amount of asset A in the portfolio? Show your work.

2. (2 points). Consumption CAPM. Assume utility is of the quadratic form �(�) = �� − +

,

�,

and a representative investor chooses a portfolio to solve the two-period utility maximization

problem

max�(�0) + 3��”�(�6

”

)

7

“8”

�.�. �0 + 3�”�6

”

7

“86

= �0

where the terms are defined as in class lecture 11. Assume all C’s are low enough that marginal

utility is positive. Let g = (C1-C0)/C0, the growth in consumption. Let �̃

0 be the return to a

portfolio that is uncorrelated with �A.

Show that, in equilibrium, the expected return to any portfolio, �̃

B, equals the sum of the

expected return of �̃

0 plus the beta from a regression of the portfolio return on consumption

growth times some constant Z:

�[�̃

B] = �[�̃

0] + �G

B�

Where

�G

B = ���(�̃

B, �A)

���(�A)

Useful facts: For any constants d, f, and h and random variables v and w

���(��A, �O) = ����(�A, �O) ��� ���(�A + �, �O + ℎ) = ���(�A, �O)

3. (2 points). Fundamental Theorem of Asset Pricing. Using the notation from class lecture 12,

prove that there can be no arbitrage opportunities if there exists a risk-neutral probability

measure on the set of fundamental securities.

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