standard deviations of the assets

  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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