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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

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.

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: 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

.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

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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.

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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.

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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

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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).

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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

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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

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? 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

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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%.

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