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               Part 1: Capital Market Theory

   Brief Recap of Chapter 8 material
       Investors like more return with less risk.
       Investors are risk averse
        -   More risk-averse investors have steeper indifference curves
        -   Less risk-averse investors have flatter indifference curves
       The mean and variance of any asset can be plotted on the mean-standard deviation
        diagram. Investors would ideally like to be near the northwest corner of this
        diagram. But there are not enough opportunities for all investors to be there. One
        needs to look at how much we can reduce standard deviation (risk) while earning
        some return.
       Portfolios of two assets formed with different weights on the assets (e.g. 50%-
        50%, 60%-40%, 90%-10% etc.) can be plotted on the mean-standard deviation
       At a correlation of +1, i.e. if the two assets are perfectly correlated, then, all
        portfolios formed from the two assets will lie on a straight line in the mean-
        standard deviation diagram.
       As correlation keeps decreasing from +1 to –1, this line will start curving towards
        the left. This can be seen below:




                               -0.5                       0.0

       The reason for this leftward shift is that by combining the assets into a portfolio,
        we are able to achieve lower standard deviation than either of the assets by
        themselves. This is diversification.
       In general, for all the assets in the economy, we can plot all the optimal risk-
        return combinations on the mean-standard deviation. This line is called the
        efficient frontier for risky assets. The efficient frontier looks like the following:



       Investors locate along the efficient frontier according to their risk-aversion.
   This chapter introduces a risk-free asset along with the bunch of risky assets that we
    have so far considered. This will help us refine our understanding of the efficient
    frontier, and we can proceed to develop a theory.
   Risk-free asset: A risk-free asset is one whose returns are certain. To understand the
    characteristics of such an asset, let us consider the following example with two assets,
    one risky and one risk-free asset.
                        State     Probability       Risky Risk-free
                        Recession      0.2          -20%      6%
                        Normal         0.5           15%      6%
                        Boom           0.3           30%      6%
                                  Exp. Return        13%      6%

    -   As you can see from the table, the return on a risk-free asset does not depend
        upon the state of the economy, which is uncertain; the risky asset’s return does
        depend upon the uncertain state.
    -   Let’s calculate the variance and standard deviation of the risk-free asset:

                  State     Probability       Return Deviation Sq. Dev
                  Recession     0.2            6%      0%         0
                  Normal        0.5            6%      0%         0
                  Boom          0.3            6%      0%         0

        All the squared deviations are zero, which means the variance and standard
        deviation of the risk-free asset are zero.  2(Rf) = 0
    -   Let’s calculate the covariance of the risk-free asset and the risky asset:
                                                    Deviations    Product
                   State        Probability       Risky Risk-free of devns.
                   Recession        0.2           -33%       0%      0%
                   Normal           0.5             3%       0%      0%
                   Boom             0.3            18%       0%      0%

        All the products of deviations are zero, which means the covariance of any asset
        with the risk-free asset is zero  Cov(Ri,Rf)=0.
   Risk-free asset in a portfolio:
    - Let us understand a portfolio of one risk-free asset and one risky asset.
        Recall that for two assets, the expected return and variance of a portfolio are
        given by the following equations:
                       E(Rp) = w1E(R1)+w2E(R2), and
                       2(Rp) = w122(R1)+w222(R2)+2w1w2Cov(R1,R2)
        If the weight on the risk-free asset is wRF, then the weight on the risky asset is
        (1-wRF). Substituting these weights into the above formula,
                       E(Rp) = wRF(Rf)+(1-wRF)E(Ri), and
                       2(Rp) = wRF22(Rf)+(1-wRF)22(Ri)+2wRF(1-wRF)Cov(Ri,Rf)
        But we know that 2(Rf)=0 and Cov(Ri,Rf)=0, which means that:
                       2(Rp) = (1-wRF)22(Ri), and

                                               (Rp) = (1-wRF).(Ri)
-   If we form all possible portfolios of the risk-free and the risky asset, where will
    they be on the mean-standard deviation diagram? Let us see with the risk-free and
    risky asset above.
           We had: E(Ri) = 13%, Rf=6%, (Ri)= 17.50% (verify this for yourself!)
    Let’s form some possible portfolios of the risky and risk-free assets using the
    above formulas:
                                              wRF       (1-wRF)          E(Rp)    sigma(Rp)
                              0.0                         1.0            12.5%      17.5%
                              0.2                         0.8            11.2%      14.0%
                              0.4                         0.6             9.9%      10.5%
                              0.6                         0.4             8.6%       7.0%
                              0.8                         0.2             7.3%       3.5%
                              1.0                         0.0             6.0%       0.0%

    If we plot these portfolios as a mean-standard deviation diagram, we see the

                                                     Portfolio of risky and risk-free asset

                 Portfolio expected return

                                                 0.0%             5.0%         10.0%         15.0%   20.0%
                                                                    Portfolio standard deviation

    Note two points from this diagram:
    1) The risk-free asset itself plots on the vertical axis (because it has zero standard
       deviation). This is the little circle in the diagram.

        2) All portfolios formed with a risk-free asset and a risky asset (like the
            diamonds in the above figure) lie along a straight line in the mean-standard
            deviation diagram, extending from the risk-free asset to the risky asset.
   The risk-free asset changes the efficient frontier
    -   In the last chapter, we chose the portfolios that are the best possible risk-return
        combinations, from among the infinite possible portfolios of all risky assets.
    -   Now, we add a risk-free asset to all the risky assets.
    -   There are many possible portfolios that can be formed by investing a proportion
        of the portfolio in the risk-free asset, and the remaining proportion in a risky
        portfolio that is on the efficient frontier. See below:

                                              Portfolios like this were not feasible in the
                                              absence of the risk-free asset

                                                            Standard Deviation

        Note the following from the diagram:
        -   We can form more portfolios than before because we have a risk-free asset.
        -   Since we want to be as much to the northwest as possible, we will find it
            optimal to combine the risk-free asset with the highest possible point on the
            risky-asset efficient frontier.
        -   Geometrically, this means that we will draw a tangent from the risk-free asset
            to the efficient frontier, which is tangent at the risky portfolio M.
        -   This tangent now becomes the new efficient frontier, because all points with
            optimal risk-return combinations (farthest to the northwest) lie along this
        -   This tangent is called the Capital Market Line (CML).

    -   I took the GM, IBM, USX and Rf data from your first assignment, and came up
        with the risky-asset efficient frontier (for the three stocks) and the new efficient
        frontier in the presence of a risk-free asset. This diagram is shown in the next

    -   The little dark circle on this diagram is the minimum variance portfolio among all
        risky assets.
    -   The little diamond represents a portfolio that is like portfolio M in the above
        example. It is the tangency portfolio, the only portfolio that lies on the new
        efficient frontier and on the old efficient frontier.
    -   The line joining the risk-free asset and the tangency portfolio is the Capital
        Market Line (CML) in a world consisting solely of GM, IBM and USX.

                                            Efficient Frontier for GM, IBM and USX

        Portfolio expected return

                                        0.00       0.05     0.10        0.15    0.20   0.25

                                                           Standard deviation

    -   To give you a little flavor of the numbers, in this diagram, the tangency portfolio
        represents a portfolio of 85.1% in IBM, 28.9% in GM, and –14% in USX. (What
        does a negative proportion mean?)

   Investors preferences meet the Capital Market Line

    -   We know that investors want to be as far to the northwest as they can, based on
        their preferences. The best they can do is to be on the efficient frontier.
    -   In the presence of a risk-free asset, investors locate on the Capital Market Line,
        which plots all optimal risk-return combinations. They locate at different points
        along the CML, according to their risk aversion.

                                                                       Less risk-averse
                                                                       investor with a
                                                                       flatter indifference

                                                   More risk-averse
                                                   investor with a steeper
                                                   indifference curve

                                                               Standard Deviation

    -   So, we are saying that all investors choose portfolios along the CML. That is
        every investor chooses to invest in a portfolio of the risk-free asset and the
        tangency portfolio. This tangency portfolio must then, include all risky assets.
        Why? Because, if there is a risky asset that does not belong to the tangency
        portfolio, then no investor would want to own that asset, and it would have no
        value at all.
    -   This means that the tangency portfolio should be the market portfolio. This is a
        portfolio of all risky securities combined in market proportions. Market
        proportion for, say, Microsoft can be computed by dividing the total market value
        of Microsoft (no. of Microsoft shares outstanding multiplied by the price per
        share of Microsoft) by the total value of all risky assets.

   How do investors locate at different points along the CML?
    -   We now know that every investor will invest some portion of his portfolio in the
        risk-free asset and the remaining in the market portfolio. How do they decide
        these proportions and what do the proportions mean?

-   Consider a case where the expected return on the market is 12%, and the standard
    deviation of the market is 9%. The risk-free rate is 6%. There are two investors L
    and B. L wants an expected return on her portfolio of 8%, while B wants an
    expected return of 15%. How might they achieve these returns? Let us see.

    Case 1: Investor L
    L wants to combine the risk-free asset and the market portfolio to give her an
    expected return of 8%. She solves the following equation:
                   E(RL) = wRF(Rf)+(1-wRF)E(RM) = 8%
    Plugging in the numbers for Rf and E(RM), we have:
           wRF(6%)+(1-wRF)12% = 8%
                        12% - wRF(6%) =8%
                        wRF = (12%-8%)/6% = 4%/6% = 2/3
               and (1- wRF) = 1/3
    This means that out of every $100 of money invested, L will lend $66.67 to the
    risk-free borrower (like the U.S. govt.), and she will invest the remaining $33.33
    in the market portfolio.
           The standard deviation on her portfolio is given by:
                    (RL) = (1-wRF).(RM) =(1/3).9% = 3%

    Case 2: Investor B
    B wants to combine the risk-free asset and the market portfolio to give her an
    expected return of 15%. She solves the following equation:
                   E(RB) = wRF(Rf)+(1-wRF)E(RM) = 15%
    Plugging in the numbers for Rf and E(RM), we have:
           wRF(6%)+(1-wRF)12% = 15%
                        12% - wRF(6%) =15%
                        wRF = (12%-15%)/6% = -3%/6% = -1/2
               and (1- wRF) = 1-(-1/2) = 3/2
    This means that for every $100 of money invested, B will borrow $50 at the risk-
    free rate, and she will invest the entire $150 in the market portfolio. In our earlier

    language (Chapter 4), she will do a margin transaction. (In reality, of course, you
    and I cannot borrow at the risk-free rate. But here we are assuming everyone can
    borrow and lend at the risk-free rate. A minor, but valid, concern.
            The standard deviation on her portfolio is given by:
                    (RB) = (1-wRF).(RM) =(3/2).9% = 13.5%
-   Look at B and L on the following graph. Note that B is located between the risk-
    free asset and the market portfolio on the capital Market Line. L is located beyond
    the market portfolio on the CML. (Fill in the numbers yourself in class)
-   In fact, all portfolios beyond the market portfolio on the CML are obtained by
    borrowing at the risk free rate and investing in the market portfolio, while all
    portfolios between the risk-free asset and the market portfolio are obtained by
    lending some money at the risk-free rate, and investing the rest in the market




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