1
Week 2
Efficient
Diversification
2
Diversification and Portfolio Risk
• The risky portfolio of only one stock.
– Market (Systematic; Non-diversifiable) risk: Risk
factors common to the whole economy (Risk from
general economic conditions, such as business
cycle, inflation, interest rate, exchange rate, etc.).
– Unique (Unsystematic; Diversifiable) risk: Risk
that can be eliminated by diversification (Risk from
firm-specific factors, such as success in R&D,
management style, philosophy, etc.).
• Portfolio risk does fall with diversification, but the
power of diversification to reduce risk is limited
by common source of risk.
3
Asset Allocation with Two Risky Assets
• Covariance and Correlation
– Key determinant of portfolio risk is the extent
to which the returns on the two assets tend to
vary either in tandem or in opposition.
– How to measure the degree and direction of comovement of returns on two assets.
– Covariance:
• Cov(r1,r2) =  p(s)[r1(s) – E(r1)][r2(s) – E(r2)]
• A measure of the extent to which the returns
tend to vary each other: Positive – Vary with
each other; Negative – Vary inversely.
• It is difficult to interpret: How strong.
4
Asset Allocation with Two Risky Assets
• Covariance and Correlation (continued)
– Correlation:
• 12 = Cov(r1,r2)/[s1  s2].
• A standardized measure of the extent to which
the returns tend to vary each other.
• Easier to interpret how strong the relationship is.
• Range of correlations: -1.0 <  < 1.0
• If  = 1.0: Perfectly positively correlated.
• If  = -1.0: Perfectly negatively correlated.
• If  = 0: Unrelated (independent) to each other.
5
Two-Risky-Assets Portfolio: Return
• Rate of return on the portfolio [rp]
rp = w1r1 + w2r2,
where w1 = Proportion of funds in security 1
w2 = (1 - w1)= Proportion of funds in security 2
r1 = Rate of return of security 1
r2 = Rate of return of security 2
• Expected rate of return on the portfolio [E(rp)]
E(rp)= w1E(r1 ) + w2E(r2 ),
where E(r1 ) = Expected rate of return of security 1
E(r2 ) = Expected rate of return of security 2
6
Two-Risky-Assets Portfolio: Risk
• Variance of the rate of return on the portfolio [sp2]
sp2 = (w1s1)2 + (w2s2)2 + 2w1w2 Cov(r1, r2)
sp2 = (w1s1)2 + (w2s2)2 + 2w1w2 12s1s2,
where s12 = Variance of rate of return on security 1
s22 = Variance of rate of return on security 2
Cov(r1, r2) = Covariance between r1 and r2
12 = Correlation between r1 and r2
• Expected rate of return of the portfolio is simply a
weighted average of returns on two securities.
• However, standard deviation of the portfolio return is
NOT a weighted average of standard deviations of
returns on two securities unless 12 = 1.0.
7
Two-Risky-Assets Portfolio: Risk-Return Trade-off
•
•
•
•
Two risky assets: Bond Fund (B) & Stock Fund (S).
E(rB )=10%, E(rS )=17%, sB=12%, sS=25%, BS=0
100% in bonds (wB=1.0, wS=0): E(rp)= 10%; sp = 12%
50% in bonds and 50% in stocks (wB=0.5, wS=0.5)
E(rp)= 0.510% + 0.517% = 13.5%;
sp 2 = (0.5  12)2 + (0.5  25)2 + (2  0.5  0.5  0  12  25)
= 192.25; sp = 13.87%
By shifting from 100% in bonds to 50% in each of bonds and
stocks, the portfolio standard deviation is increased by only
1.87%, not by 6.5% to the average of the component standard
deviations [(25+12)/2 = 18.5%].
• 100% in stocks (wB=0, wS=1.0): E(rp)= 17%; sp = 25%
8
Two-Risky-Assets Portfolio: Risk-Return Trade-off
• Analyst can and must show investors the entire
investment opportunity set, which is the set of all
attainable combinations of risk and return offered by
portfolios formed using available assets in differing
proportion.
• wB
wS
E(rp)
sp
0.0
0.2
0.4
0.5
0.6
0.8
0.8127
1.0
1.0
0.8
0.6
0.5
0.4
0.2
0.1873
0.0
17.0
25.00
15.6
20.14
14.2
15.75
13.5
13.87
12.8
12.32
11.4
10.824
11.31 10.818
10.0
12.00
9
Investment Opportunity Set for Bond & Stock Funds
Portfolio Return
20
15
10
Stock (S)
Portfolio
Minimum
Variance
Portfolio (A)
5
Z
Bond (B)
Portfolio
0
0
5
10
15
Portfolio Risk
20
25
30
10
Mean-Variance Criterion
• Investors compare portfolios using a mean-variance
criterion; Prefer higher return and lower risk.
• Portfolio A dominates Portfolio B if all investors prefer
A over B: This will be the case if it has higher mean
return and lower variance – E(rA)  E(rB) and sA  sB.
• The choice among the dominant portfolios is not as
obvious, because higher expected return is accompanied
by higher risk (The choice will depend on individual’s
willingness to trade off risk against expected return).
– The Choice? – E(rA) > E(rB) and sA > sB.
• Low correlation aid diversification and high correlation
results in a reduced effect of diversification.
11
Diversification Benefits and Correlation
• If BS = 1.0: No Diversification Benefits.
– sp2 = (wBsB)2 + (wSsS)2; sp = wBsB + wSsS
– Portfolio standard deviation is simply a weighted average
of standard deviations of two securities.
– No inefficient portfolio; choice among portfolios depends
only on risk preference.
• If –1.0 < BS < 1.0: Some Diversification Benefits.
– When –1.0 < BS <0, Greater benefits than when 0< BS <
1.0.
• If BS = -1.0: Greatest Diversification Benefits.
– sp2 = (wBsB - wSsS)2; sp = ABS[wBsB - wSsS]
– If wBsB = wSsS, sp = 0.
– wB = [sS2 - BSsBsS]/[sB2 + sS2 - 2BSsBsS]
12
Portfolio Expected Return
Investment Opportunity Set: Varying Correlation
18.0
17.0
16.0
15.0
14.0
13.0
12.0
11.0
10.0
9.0
8.0
7.0
Stock
Fund
=-1.0
=1.0
=0
=0.5
=0.2
Bond
Fund
0.0
5.0
10.0
15.0
20.0
25.0
Portfolio Standard Deviation
30.0
13
Minimum Variance Portfolio
Bond
E(rB) = .10
sB = .12
sS = .25
Stock E(rS) = .17
sS2 - sBsSBS
wB =
BS = .2
sB2 + sS2 - 2sBsSBS
(.25)2 - (.25)(.12)(.2)
wB=
(.12)2 + (.25)2 - 2(.25)(.12)(.2)
wB= .8706; wS=.1294
14
Optimal Risky Portfolio with a Risk-Free Asset
• Two risky funds (Bond & Stock) + Risk-free Asset.
• E(rB )=10%; E(rS )=17%; sB=12%; sS=25%;
BS=0.2; and rf = 8%.
• Two possible CALs from rf.
– CAL 1: Portfolio A (Minimum variance portfolio).
• E(rA) = (.8706)(10)+(.1294)(17)=10.91%
sA = [(.870612)2+(.129425)2+(2.8706.129412
250.2)] = 11.54%
• SA = [E(rA)-rf]/sA = (10.91 – 8)/11.54 = 0.25
– CAL 2: Portfolio X (wB=0.65).
• E(rX) = 12.45%; sX = 12.83%
• SB = [E(rX)-rf]/sX = (12.45 – 8)/12.83 = 0.35
• Higher reward-to-variability ratio for CAL 2.
15
Optimal Risky Portfolio with a Risk-Free Asset
• The higher reward-to-variability of Portfolio X (CAL2)
means that combinations of Portfolio X and a risk-free
asset provide higher expected return for any level of
risk: Portfolio X dominates Portfolio A.
• The CAL, which is tangent with investment opportunity
set will provide the highest feasible reward-to-variability
ratio.
• The tangency portfolio (O) is the optimal risky portfolio.
• Investors will choose the optimal risky portfolio regardless of their degree of risk aversion.
• Investors differ only in their allocation of investment
funds between the optimal risky portfolio an the riskfree asset.
16
ALTERNATIVE CALs
CAL (O)
E(r)
CAL (X)
Portfolio O
(Optimal
Portfolio)
CAL (A)
Portfolio X
rf=8%
Portfolio A
(Minimum
Variance
Portfolio)
s
17
Efficient Diversification with Many Risky Assets
• Three Separate Steps for Asset Allocation
– Identify the best possible (most efficient) risk-return
combinations available from the set of risky assets.
– Determine the optimal risky portfolio of risky assets.
– Choose an appropriate complete portfolio based on
investor’s risk aversion by mixing the risk-free assets with
the optimal risky portfolio.
• Efficient Frontier of Risky Assets
– The optimal combinations with many risky assets result in
the lowest level of risk for a given return (or the highest
level of return for a given risk).
– The optimal trade-off is described as the efficient frontier;
These portfolios are dominant.
18
E(rp
)
The Efficient Frontier of Risky Assets
and Choice of Optimal Risky Portfolio
Optimal
CAL
Optimal Risky
Portfolio
rf
Global
Minimum
Variance
Portfolio
Efficient Frontier of
Risk Assets
Individual
Assets
sp
19
Efficient Diversification with Many Risky Assets
• Efficient Frontier of Risky Assets (continued)
– In real world, various constraints may preclude a
particular investor from choosing portfolios on the
efficient frontier: Portfolio managers can tailor an
efficient frontier to meet any particular objective.
• Choosing the optimal risky portfolio
– The CAL formed from the optimal risky portfolio will be
tangent to the efficient frontier of risky assets; This CAL
will dominates all other CALs.
• Preferred Complete Portfolio
– Separation theory: Portfolio choice can be separated into
two independent tasks – (1) determination of the optimal
risky portfolio, and (2) the personal choice of the best mix
of risky portfolio and risk-free asset.
20
Efficient Diversification with Many Risky Assets
• Preferred Complete Portfolio (continued)
– The best risky portfolio is the same for all investors
regardless of risk-aversion: More (Less) risk-averse
investor will invest more in risk-free (risky) asset.
– In real world, the optimal risky portfolio for different
clients also may vary because of portfolio constraints:
But, a few portfolios may be sufficient to serve the
demands of a wide range of investors.
– If the optimal risky portfolio is the same for all clients,
professional management is more efficient and less costly.
– If different managers use different input data to develop
different efficient frontiers, they will offer different
optimal portfolios; Thus, security analysis is important in
portfolio selection.
21
A Single-Factor Model
• Factor Models: Statistical models designed to estimate
two components (systematic and firm-specific) of risk
for a particular security or portfolio.
• Ri = E(Ri) + ßiM + ei (Single Factor Model)
Ri = Excess return on a security = (ri – rf)
E(Ri) = Expected excess return at the start of the holding period
M = Market or macroeconomic surprises during the holding
period
ßi = sensitivity of the security’s return to macroeconomic factor
ei = impact of unanticipated firm-specific events.
• One reasonable approach to measure the factor is to use
the rate of return on a broad index of securities (like the
S&P500) as a proxy for the common macro factor
22
A Single-Index Model
• Index Model: A model of stock returns using a market
index to represent common (systematic) risk factor –
Separates the realized rate of return on a security into
systematic and firm-specific components.
• Ri = ai + ßiRM + ei (Single-Index Model)
Ri = Risk premium (Excess return) on a security = (ri – rf)
ai = Stock’s excess return when market’s excess return is zero.
RM = Risk premium (Excess return) on a market index = (rM – rf)
ßiRM = Component of return due to movements in overall market
ei = Component of return due to unexpected firm-specific events
• Single-index model specifies the two sources of security
risk: Systematic and Firm-specific Risks.
23
Components of Risk
• Market or systematic risk ( ßiRM ): Risk attributable to
the security’s sensitivity to movements in overall market.
• Unsystematic or firm specific risk ( ei ): Risk not related
to the market index
• Total risk = Systematic Risk + Unsystematic Risk
si2 = bi2 sm2 + s2(ei)
where;
si2 = Variance of the excess return of a stock (total variance)
bi2 sm2 = Variance attributable to the uncertainty common to
the entire market
s2(ei) = Variance attributable to firm-specific risk factors
which are independent of market performance
24
Security Characteristic Line (SCL)
• SCL is a statistical representation of the single-index
model, that is, the plot of a security’s excess returns as a
function of the excess return of the market.
• Single-index model can be estimated by the regression of
the excess return of a security (Ri) on the excess return of
(RM).
• Regression line: E(RiRM) = ai + ßiRM
• The greater the slope of the regression, the greater the
security’s systematic risk, as well as its total variance.
• The deviations of actual returns from the regression line
measure the effects of firm-specific events.
25
Security Excess
Returns (Ri)
Graphical Representation of
Single-Index Model
. .. . . .. . .
. ..
. ..
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. .
a
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. ..
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.. . . .
i
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. . .
Security
Characteristic
Line
ßi
Market Excess
Returns (RM)
26
Examining Percentage of Variance
• One way to measure the relative importance of systematic
risk is to measure the ratio of systematic variance to total
variance (2)
2 = ßi2 s M2 / si2 = bi2 sM2 / [bi2 sM2 + s2(ei)]
where  = Correlation between Ri and RM.
• This measures the ratio of explained variance to total
variance, that is, the proportion of total variance that can
be attributed to market fluctuations.
• A large (small) absolute correlation means systematic
(firm-specific) variance dominates total variance.
• At extreme, perfect correlation (positive or negative)
means security returns are fully explained by market
return (No firm-specific effects).
27
Diversification and Single-Index Model
• What are the systematic and unsystematic variances of
portfolio which includes securities whose returns are given
by single-index model?
• The beta of the portfolio (bp) is the simple average of the
individual security betas; the systematic variance is bp2sM2.
• The systematic component of each security return, bi2sM2, is
perfectly correlated with the systematic part of any other
security’s return because of the common factor – No
diversification effects on systematic risk: Average beta of
component securities is relevant to portfolio systematic risk.
• Because firm-specific effects are independent of each other,
their risk effects are offsetting – Diversification effects on
unsystematic risk by holding many securities.
• For a well-diversified investor, only relevant risk of a
security is the systematic risk of the security measured by bi.