2. Portfolios
Rate of Return
Scenario Stock fund Bond fund Portfolio squared deviation
Recession -7% 17% 5.0% 0.0016
Normal 12% 7% 9.5% 0.0000
Boom 28% -3% 12.5% 0.0012
Expected return 11.00% 7.00% 9.0%
Variance 0.0205 0.0067 0.0010
Standard Deviation 14.31% 8.16% 3.08%
The rate of return on the portfolio is a weighted average of
the returns on the stocks and bonds in the portfolio:
rP wB rB wS rS
5% 50% ( 7%) 50% (17%)
3. Portfolios
Rate of Return
Scenario Stock fund Bond fund Portfolio squared deviation
Recession -7% 17% 5.0% 0.0016
Normal 12% 7% 9.5% 0.0000
Boom 28% -3% 12.5% 0.0012
Expected return 11.00% 7.00% 9.0%
Variance 0.0205 0.0067 0.0010
Standard Deviation 14.31% 8.16% 3.08%
The expected rate of return on the portfolio is a weighted
average of the expected returns on the securities in the
portfolio.
E (rP ) wB E (rB ) wS E (rS ) 9% 50% (11%) 50% (7%)
4. Portfolios
Rate of Return
Scenario Stock fund Bond fund Portfolio squared deviation
Recession -7% 17% 5.0% 0.0016
Normal 12% 7% 9.5% 0.0000
Boom 28% -3% 12.5% 0.0012
Expected return 11.00% 7.00% 9.0%
Variance 0.0205 0.0067 0.0010
Standard Deviation 14.31% 8.16% 3.08%
The variance of the rate of return on the two risky assets
portfolio is
σP
2
(wB σ B ) 2 (wS σ S ) 2 2(wB σ B )(wS σ S )ρ BS
where BS is the correlation coefficient between the returns
on the stock and bond funds.
5. Portfolios
Rate of Return
Scenario Stock fund Bond fund Portfolio squared deviation
Recession -7% 17% 5.0% 0.0016
Normal 12% 7% 9.5% 0.0000
Boom 28% -3% 12.5% 0.0012
Expected return 11.00% 7.00% 9.0%
Variance 0.0205 0.0067 0.0010
Standard Deviation 14.31% 8.16% 3.08%
Observe the decrease in risk that diversification offers.
An equally weighted portfolio (50% in stocks and 50%
in bonds) has less risk than either stocks or bonds held
in isolation.
6. As long at the correlation coefficient is < 1, the
standard deviation of a portfolio of two securities is
less than the weighted average of the standard
deviations of the individual securities.
In the above case: SD of portfolio= 3.08%
Weighted average of SD = 14.31%*0.5 + 0.0816*0.5
= 0.07155 + 0.0408 = 0.11235 = 11.235%
This difference is due to the negative correlation
between the two securities.
11. return Portfolios with Various Correlations
100% Since any probable correlation of
= -1.0 stocks securities X and Y will range
between – 1.0 and + 1.0, the triangle
in the above figure specifies the
limits to diversification. The risk-
= 1.0 return curves for any correlations
= 0.2 within the limits of – 1.0 and + 1.0,
100% will fall within the triangle.
bonds
Relationship depends on correlation coefficient
-1.0 < < +1.0
If = +1.0, no risk reduction is possible
If = –1.0, complete risk reduction is possible
12. Portfolio Risk Depends on
Correlation between Assets
12
Investing wealth in more than one security reduces
portfolio risk.
This is attributed to diversification effect.
However, the extent of the benefits of portfolio
diversification depends on the correlation between returns
on securities.
When correlation coefficient of the returns on individual
securities is perfectly positive then there is no advantage of
diversification. The weighted standard deviation of returns
on individual securities is equal to the standard deviation
of the portfolio.
Diversification always reduces risk provided the
correlation coefficient is less than 1.
13. The Efficient Set for Many Securities
return
Individual Assets
P
Consider a world with many risky assets; we can still
identify the opportunity set of risk-return combinations
of various portfolios.
14. The Efficient Set for Many Securities
return
minimu
m
variance
portfolio
Individual Assets
P
The section of the opportunity set above the
minimum variance portfolio is the efficient frontier.
15. Investment Opportunity Set:
The n-Asset Case
15
An efficient portfolio is one that has the highest
expected returns for a given level of risk.
The efficient frontier is the frontier formed by the
set of efficient portfolios.
All other portfolios, which lie outside the efficient
frontier, are inefficient portfolios.
16. Efficient Portfolios of risky securities
16
An efficient
portfolio is one that
has the highest
expected returns for
a given level of risk.
The efficient frontier
is the frontier formed
by the set of efficient
portfolios. All other
portfolios, which lie
outside the efficient
frontier, are
inefficient
portfolios.
17. Diversification and Portfolio Risk
Diversification can substantially reduce the
variability of returns without an equivalent reduction
in expected returns.
This reduction in risk arises because worse than
expected returns from one asset are offset by better
than expected returns from another.
However, there is a minimum level of risk that
cannot be diversified away, and that is the systematic
portion.
18. RISK DIVERSIFICATION:
SYSTEMATIC AND UNSYSTEMATIC
RISK
18
When more and more securities are included in a
portfolio, the risk of individual securities in the
portfolio is reduced.
This risk totally vanishes when the number of
securities is very large.
But the risk represented by covariance remains.
Risk has two parts:
1. Diversifiable (unsystematic)
2. Non-diversifiable (systematic)
19. Systematic Risk
19
Systematic risk arises on account of the economy-wide
uncertainties and the tendency of individual securities
to move together with changes in the market.
This part of risk cannot be reduced through
diversification.
It is also known as market risk.
Investors are exposed to market risk even when they
hold well-diversified portfolios of securities.
Risk factors that affect a large number of assets
Includes such things as changes in GDP, inflation,
interest rates, etc.
21. Unsystematic Risk
21
Unsystematic risk arises from the unique
uncertainties of individual securities.
It is also called unique risk.
These uncertainties are diversifiable if a large
numbers of securities are combined to form well-
diversified portfolios.
Uncertainties of individual securities in a portfolio
cancel out each other.
Unsystematic risk can be totally reduced through
diversification.
24. Hence…Total Risk
Total risk = systematic risk + unsystematic risk
The standard deviation of returns is a measure of
total risk.
For well-diversified portfolios, unsystematic risk is
very small.
Consequently, the total risk for a diversified portfolio
is essentially equivalent to the systematic risk.
25. Since the systematic risk can’t be diversified, the
investor will require compensation for bearing this
risk.
Diversified portfolios with no unsystematic risk,
move with the market
26. Portfolio Risk and Number of Stocks, p355
In a large portfolio the variance terms are effectively
diversified away, but the covariance terms are not.
Diversifiable Risk;
Nonsystematic Risk;
Firm Specific Risk;
Unique Risk,
VARIANCE
Portfolio risk
Non diversifiable risk;
Systematic Risk;
Market Risk; COVAR
n
27. Optimal Portfolio with a Risk-Free
Asset
return 100%
stocks
rf
100%
bonds
In addition to stocks and bonds, consider a world that
also has risk-free securities like
T-bills.
28. Riskless Borrowing and Lending
return
100%
stocks
Balanced
fund
rf
100%
bonds
Now investors can allocate their money across the T-
bills and a balanced mutual fund.
