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
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
：: Tells you how much any pair (two) stocks (i and j) move • Covariance =i,j
Prob. 1 2
.3 .2 .1 Great
.5 .1 .2 OK
.2 - .05 .4 Bad
= -.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
Finally, we can use covariance to tell how risky the entire portfolio is …
Variance of a portfolio
22222：？w：;w：;2ww： If N=2, P1122121,2
2222222：？w：;w：;w：;2ww：;2ww：;2ww：If N=3, P121,2131,3232,3112233
? Last step: Take square root of variance to get standard deviation.
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
，？;1.0，？；1.0positive () and perfect negative () correlation: 1,21,2
：，：E(r)，E(r)-Assume , 2121
All Asset 2
，？;1.0 of the two assets 1,2
result in these
All Asset 1
? 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.
Portfolio Variance and Diversification with numbers
22222：？w：;w：;2ww，：：Using : P1122121,212
22w？w？0.50：？：？0.05 and 1212
， = .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
2：，Ex: If = 0, = .025 < var(Asset 1) or var(Asset 2) 1,2P
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
? 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.
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)
? 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).
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
Investment A is better than B if …
and and, one of these inequalities is strict
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:
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
：Lowest subject to some target E(r) P
：and Highest E(r) subject to some target P
Global minimum variance
Minimum variance frontier
? 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
Global minimum variance
? Result: One portfolio (P) dominates all of the other efficient portfolio on the
- 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
- 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
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%.