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PORTFOLIO SELECTION WITH A RISK-FREE ASSET

By Lucille Greene,2014-07-08 09:47
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Q: IF YOU CAN COMBINE THE RISK-FREE ASSET TOGETHER WITH ANY COMBINATION OF THE TWO RISKY ASSETS, WHICH COMBINATION OF THE TWO RISKY ASSETS DO YOU (AND ALL OTHER ...

    Chapter 6: Efficient Diversification

    General thought: Risk comes from different places. Some risk comes from common sources, like the economy. Other risk comes from sources unique to each asset. This means that some kinds of risk can be diversified.

    Return and Risk for a Portfolio

First, need to know how much you’ve invested in each asset (w) as a percentage

    of your total funds invested.

    Expected return on a portfolio

    E(r)w*E(r)w*E(r);w*E(r);... Pii?1122i

    In other words, portfolio expected return is always a weighted average of the

    expected returns of the assets within the portfolio.

    However, this is not true of portfolio standard deviation!

    ? Portfolio standard deviation depends on covariance / correlation between

    each pair of assets within the portfolio…how much the movements between

    each pair of assets offset each other.

    ? In general, portfolio standard deviation will be less than the weighted

    average of the standard deviations of the individual assets within the

    portfolio.

    1

    : Tells you how much any pair (two) stocks (i and j) move Covariance =i,j

    around together:

    Cov(r,r)Pr(s)[r(s)E(r)][r(s)E(r)] ?121,21122

    s

     Prob. 1 2

    .3 .2 .1 Great

    .5 .1 .2 OK

    .2 - .05 .4 Bad

.3*(.2.1)*(.1.21);.5*(.1.1)*(.2.21);.2*(.05.1)*(.4.21) 1,2

     = -.0033 + 0 + -.0057 = -.009

    Correlation between Two Assets

The correlation coefficient “standardizes” covariance – puts it into a form that tells

    you how much two assets actually move together. Correlation coefficients are

    scaled between 1 and +1:

    .009ij = = -.995 ij(.0866)(.1044)ij

    2

    Finally, we can use covariance to tell how risky the entire portfolio is …

    Variance of a portfolio

    NN2ww ??,Pijij

    ??11ij

    22222w;w;2ww If N=2, P1122121,2

    2222222w;w;w;2ww;2ww;2wwIf N=3, P121,2131,3232,3112233

    ? Last step: Take square root of variance to get standard deviation.

    3

    Why are combinations of two risky assets concave?

    ? Depends on extent of correlation between assets making up the portfolio …

    Assume that we have two assets (1 and 2), and the extreme cases of perfect

    ;1.01.0positive () and perfect negative () correlation: 1,21,2

    E(r)E(r)-Assume , 2121

     E(r) P

     All Asset 2

    1.0 1,2

     Various weightings

    ;1.0 of the two assets 1,2

     result in these

     portfolios.

    1.0 1,2

     All Asset 1

     P

Observations:

    ? Perfect positive correlation: No benefit from diversification

    (portfolio standard deviation = a weighted average of the parts)

    ? Perfect negative correlation: Zero-risk portfolio possible (specific weights

    given on p. 157, footnote 2)

    General idea: As you reduce correlation between pairs of assets, you reduce risk for a given level of portfolio return. Since most assets are not perfectly correlated, this means that various portfolio combinations of most two-asset portfolios will lie on a curve that curves to the left.

     4

    Portfolio Variance and Diversification with numbers

    22222w;w;2wwUsing : P1122121,212

    Suppose that

    22ww0.500.05 and 1212

    222.50(0.05);.50(0.05);2(.50)(.50).05.05? P1,2

     = .025 + (.025) 1,2

Result: Variance of portfolio is less than the variance of each individual asset (.05)

    as long as < 1 (i.e. not perfectly correlated). 1,2

    2Ex: If = 0, = .025 < var(Asset 1) or var(Asset 2) 1,2P

     5

    Optimal Risky Portfolios with two risky assets and a risk-free asset

    ? Given two risky assets, we know that various portfolios curve to the left in

    an expected return/standard deviation graph if they are less than perfectly

    correlated

    ? We also know that combining any risky asset (or portfolio) with a risk-free

    asset results in a straight line (the CAL) in the same graph.

    Q: If you can combine the risk-free asset together with any combination of the

    two risky assets, which combination of the two risky assets do you (and all

    other investors) choose?

    ?Answer: The tangency portfolio (P*) -- this portfolio has the steepest CAL.

     CAL (P*)

     E(r) A less risk averse investor

     chooses this mix of P* and the

     risk-free asset(is a borrower)

     Without a risk-free asset, a less risk

     P* averse investor chooses this portfolio

     Without a risk-free asset, a more risk

     averse investor might choose this portfolio

     A more risk averse investor would instead choose this

     mix of P* and the risk-free asset (is a lender)

     r F

     P

    ? Note that risk averse investors are better off mixing the risky portfolio P*

    with the risk-free asset (more risk-averse investors lend, less risk-averse

    investors borrow.) Without the risk-free asset, each investor type chooses a

    unique risky portfolio along the curve. With the risk-free asset, all investors

    choose the same risky portfolio P* in combination with the risk-free asset on

    the CAL. This results in higher Sharpe ratios for both investors (better risk

    premium for a given amt of risk).

     6

    How do we know that investors will always prefer the tangency portfolio (P*) in some combination with the risk-free asset?

    ? Investors are risk averse require a risk premium in exchange for

    risk. Investors use the Mean Variance Criterion to pick assets

    and portfolios:

Investment A is better than B if …

()?()ErErAB

    and and, one of these inequalities is strict

    AB

    Points on the CAL are always better than points on the curved portion, since investors can always get a better return with the same (or less) risk.

Q: Is there a formula for the weights of the optimal risky portfolio (P*) ?

    Answer: Equation (6.10) on p. 159 (do not need to know this for any test!)

Final Q: Once you know how to put together the optimal portfolio P, how do you

    allocate your money between P and r? F

Answer: Use the formula from Chapter 5:

    Err()Pf*y2AP

     7

    Portfolios Using Many Risky Assets

    ? Given a fixed number of risky assets, you can form lots of portfolios

    ? Some of these portfolios form the minimum-variance frontier

    ? Of the minimum-variance frontier portfolios, efficient portfolios offer:

     maximum return for a given amount of risk, and

     minimum risk for a given return.

?In general, points along the efficient frontier have to satisfy the two following

    conditions:

    Lowest subject to some target E(r) P

    and Highest E(r) subject to some target P

     Efficient Frontier

     E(r)

     Global minimum variance

     portfolio

     Minimum variance frontier

     P

     8

? All of these risky portfolios combine with the risk-free asset in straight lines …

     E(r) CAL (optimal risky portfolio) Efficient Frontier

     Some risky asset

     P*

     Global minimum variance

     portfolio

     r F

     P

    ? Result: One portfolio (P) dominates all of the other efficient portfolio on the

    efficient set

    - Investors who choose combinations of P and the risk-free asset get the

    highest return for a given level of risk, compared to all other risky

    portfolios

    - In other words, all investors choose from points along the CAL passing

    through portfolio P.

    ?Separation Property: Risky portfolio selection is separate from how funds

    are allocated between risky and risk-free assets

     9

    Portfolio Optimization in Practice

Typical asset classes:

    Asset Allocation Matrix:

    Investment Grade REITs Equity, Mortgage Cash Dow Industrial Balanced Corporate Bonds and Hybrid

    Money Market Government/ S&P 500 Growth Other Alternative Classes Funds Agency Bonds

    Treasury Bills NASDAQ 100 Treasury Bonds Income Natural Resources

    Certificate of High Yield Hard Assets: Commodities Russell 2000 Growth & Income Deposits (CDs) Corporate Bonds Precious Metals

    International Canadian Dollars S&P Utilities Municipal Bonds Hedge Funds Equities

    Mortgage backed Hybrid Fixed Income & Japanese Yens Wilshire 5000 Sector Weightings Securities Equity Strategies

    World Money International International Bonds Total Return Convertibles Market Funds Equities

    .......................................................................................................................................................

    ...........................

    Typically, do not use utility analysis to identify optimal portfolio (P*) (where you need to know the exact form of the utility function and the distribution of returns):

? Maximize return for a given level of risk

    - E.g. suppose your current portfolio is made up of 60% US stock, 40% US

    bonds. Assume the expected return of this portfolio is 12.4% with a

    standard deviation of 14.5%. Moving up to the efficient frontier

    (involving a more diversified portfolio) results in the same risk but an

    expected return of 14.7%.

     10

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