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Efficient Diversification
Chapter Seven
Overview
• Last week we discussed capital allocation decision between
risk‐free asset and risky portfolio.
• This week, we concentrate on the construction of the risky
portfolio
• Construction of risky portfolio involves;
• Asset allocation across broad assets classes and
• Security selection within each assets class.
• The objective of the above is to construct the optimal risky
portfolio – the portfolio that provides the best risk‐return
trade‐off
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Overview cont.
• Diversification is an essential element in this process
• So, we start with the benefits of diversification
• Then move to efficient diversification and consider
• Two risk assets case
• Add risk‐free asset
• Incorporate entire universe of risky securities
• We learn how diversification can reduce risk without
affecting expected return
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Diversification and portfolio risk
• Investing money in a diversified portfolio of risky assets can
reduce risk
• Investors can achieve this through Naïve diversification
• The risk you can reduce, and eliminate when you are well
diversified, is called unique risk/firm‐specific
risk/diversifiable risk/non‐systematic risk.
• The risk that remains even after extensive diversification is
called market risk/systematic risk/non‐diversifiable risk.
• Graph in next page demonstrates this.
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Portfolio risk as a function of number of stocks in the portfolio
Panel B: Some risk is systematic.
Panel A: All risk is firm specific.
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Efficient diversification
• Is there an efficient way to diversify and reduce risk?
• Can we construct a portfolio that provides the lowest possible risk
for any given level of expected return?
• We start with a portfolio of two risky assets
• Let’s consider a portfolio created using a debt fund and an
equity fund
• Let, D denotes debt fund and E denotes equity fund
wD = proportion invested in the debt fund
wE = proportion invested in the stock fund
E(rD)= expected rate of return on the debt fund
E(rE)= expected rate of return on the equity fund
𝜎 = standard deviation of debt fund
𝜎 = standard deviation of equity fund
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Expected return and risk of a two-asset portfolio
• From your BFF2140/BFB2140 knowledge …
• Portfolio expected return
– Weighted average of expected returns on the component
securities
E (rp )  w D E (rD )  wE E (rE )
• Portfolio risk
– Variance of the portfolio is a weighted sum of covariances, and
each weight is the product of the portfolio proportions of the
pair of assets
  w   w   2wD wE Cov  rD , rE 
2
p
2
D
2
D
2
E
2
E
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Computation of portfolio variance from the covariance matrix

2
p
 w D2 
2
D
 w E2 
8
2
E
 2 w D w E C o v  rD , rE

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Portfolios of two risky assets: Covariance
• Covariance of returns on debt and equity
𝐷
–
–
–
𝐷𝐸 𝐷 𝐸
= Correlation coefficient of returns
= Standard deviation of bond returns
= Standard deviation of equity returns
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Portfolios of two risky assets: Correlation coefficient
• Range of values for correlation coefficient
‐1.0 > r > +1.0
• If r = 1.0, the two securities are perfectly positively correlated
• If r = ‐ 1.0, the two securities are perfectly negatively correlated
• If r = 0, the two securities are not correlated
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Portfolios of two risky assets: Correlation coefficient
• When ρDE = 1;
• There is no diversification benefit
2
• The right hand side of  p equation is a perfect square
• Therefore, portfolio variance equation simplifies to:

2
p
 (W
D D WE E )
2
• Therefore;
 P  wE E  wD D
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Correlation coefficient cont.
• When ρDE = ‐1, portfolio variance equation simplifies to

2
p
 (W
D D WE E )
2
• Therefore;
 P  wE E  wD D
• When ρDE = ‐1, a perfect hedge is possible
• i.e. Standard deviation of the portfolio can be reduced to zero
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Correlation coefficient cont.
• Let’s see how we can reduce portfolio risk to zero when
ρDE = ‐1
• From previous slide, when ρDE = ‐1,  P  wE E  wD D
• Setting left hand side of this equation to zero and solving
; 0 w  w 
for
E
wE 
E
D
D
D 
 1  wD
E
• Solving the same equation for
wD 
D
E
D 
;
 1  wE
E
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Portfolios of two risky assets: Example — 50%/50% split
Expected Return: E (rp )  w D E (rD )  wE E (rE )
 .50  8%  .50 13%  10.5%
Variance:  p2  wD2  D2  wE2 E2  2wD wE Cov  rD , rE 
 .502 122  .502  202  2  .5  .5  72  172
 P  172  13.23%
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Weights in individual assets and portfolio expected return
• Expected return is
proportional to weights
of individual assets
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• 𝑊
0; 𝑊
1
• 𝑊
1; 𝑊
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Weights in individual assets and portfolio standard deviation
• In addition to weights of
individual assets,
correlation between
assets influences the
standard deviation
• Lower the correlation,
lower the standard
deviation.
• STD for 50%/50% split
if 𝜌 ,
1
• STD for 50%/50% split if
𝜌 ,
1
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Portfolio expected return as a function of standard deviation
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•
Each line gives portfolio
opportunity set under a particular
correlation.
•
When ρDE = 1, both return and risk
increase when you move from D
to E.
•
When, ρDE = 0.30, risk is lower
for any expected return than
when ρDE = 1.
•
When ρDE = 0, risk is reduced further.
•
When ρDE = ‐1, risk can be eliminated.
•
Benefits of diversification increases
as correlation decreases
•
Perfectly hedged portfolio or
zero risk portfolio
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Minimum variance portfolio
• Minimum‐variance portfolio has a standard deviation
smaller than that of either of the individual component
assets
• Portfolio composed of the risky assets that has the
smallest standard deviation (the portfolio with least risk)
• Illustrates the power of diversification to limit risk
• Formula for minimum variance portfolio:
 22  Cov ( r1 , r2 )
W1  2
 1   22  2 Cov ( r1 , r2 )
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Assets allocation between risk-free asset and optimal
risky portfolio
7-19
• This is a revisit to last week’s lecture on capital allocation
• We now consider asset allocation across equity fund, debt
fund and risk‐free asset
• Stocks fund and bond fund represent risky‐assets portfolio
• Assume risk‐free rate is 5%.
• The following graph illustrates this
• Note that we use the opportunity set when ρDE = 0.30
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Opportunity set of the debt and equity funds and two
feasible CALs
•Graph has two possible CALs
•First CAL is drawn through
optimal Portfolio A (i.e.
minimum variance portfolio)
•Second CAL is drawn through
optimal Portfolio B
•Objective of capital allocation
is to maximize the Sharpe ratio
Sp 
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E r p   r f

p
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Opportunity set of the debt and equity funds and two
feasible CALs cont.
𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜𝐴
𝐸 𝑟
8.9%
11.45%
𝜎
8.9 5
0.34
𝑆
11.45
𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜𝐵
9.5%
𝐸 𝑟
11.70%
𝜎
.
0.38
𝑆
.
• CAL B has a higher Sharpe
ratio than CAL A.
• But why stop at CAL B?
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Debt and equity funds with the optimal risky portfolio
• Tangency portfolio labelled P
is the optimal risky portfolio
with risk‐free asset.
• It has the highest Sharpe
ratio.
𝐸 𝑟
11%
𝜎
14.2%
𝑆
𝑆
𝑆
22
𝐸 𝑟
𝑟
𝜎
11% 5%
14.2%
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Determination of the optimal complete portfolio
• This depends on the investor’s
risk averse coefficient.
• Optimal allocation to P for an
investor with A 4
• From last week’s lecture;
𝑟
𝐸 𝑟
𝑦
𝐴𝜎
11% 5%
𝑦
0.7439
4
14.2%
• Investor will invest 74.39% in
portfolio P and 25.61% in T‐bills.
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Proportions of the optimal complete portfolio
Note: Optimum risky portfolio has 𝑊
0.60 𝑎𝑛𝑑 𝑊
0.40
• 𝐶𝑜𝑚𝑝𝑙𝑒𝑡𝑒 𝑃𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜
11% 𝑦 0.7439
𝐸 𝑟
𝜎
14.2%
𝑟
5%
𝐸 𝑟
0.7439
9.46%
𝑦 𝐸 𝑟
1 𝑦
𝑟
11%
0.2561 5%
𝜎
.7439
𝑆
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14.2%
9.46% 5%
10.56%
10.56%
0.42
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Markowitz portfolio optimisation model
We can generalise portfolio construction problem to many risky
assets and risk‐free asset
•
•
•
Determine the risk‐return
opportunities available
• Minimum‐variance frontier of
risky assets
All portfolios that lie on the
minimum‐variance frontier from the
global minimum‐variance portfolio
and upward provide the best risk‐
return combinations
Efficient frontier of risky assets is the
portion of the frontier that lies above
the global minimum‐variance
portfolio
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Markowitz portfolio optimization model cont.
• Search for the CAL with the highest reward‐to‐variability
ratio
• Individual investor chooses the appropriate mix between
the optimal risky portfolio P and T‐bills
• Everyone invests in P, regardless of their degree of risk
aversion
• More risk averse investors put less in P
• Less risk averse investors put more in P
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Efficient frontier of risky assets with the optimal CAL
• What does an investor do
if he/she picks a complete
portfolio that lies between
rf and P in CAL?
• What does an investor do
if he/she picks a complete
portfolio that lies above P
in CAL?
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Markowitz portfolio optimization model cont.
• Capital allocation and the separation property
• Portfolio choice problem may be separated into two
independent tasks
• Determination of the optimal risky portfolio is purely
technical
• Allocation of the complete portfolio to risk‐free versus the
risky portfolio depends on personal preference
• The separation theorem separates the investing and
financing decisions.
– All investors will invest in the same optimal risky portfolio and
adjust the risk level of the portfolio by either lending (investing in
T‐bills, i.e., lending to the government) or borrowing (buying
risky securities on margin).
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Capital allocation lines with various portfolios from the
efficient set
• This graph looks similar to
the graph in slide 20.
• Only difference is that the
investor combines risk‐
free asset with optimum
portfolios in the efficient
frontier.
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