Asset Allocation

Risk and Return

Diversification
Efficient Portfolios
Capital Market Line

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 1. Investment Preferences
under uncertainty
• Use Risk and Return to represent investment preferences
under uncertainty
− Risk and return go hand in hand with each other
• The objective is to pick the investments that have
the best risk/return trade-off.

• We measure “return” using the expected return : E(R)

we measure “risk” using the standard deviation of returns : σR

How do we put them into a single objective function?

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 Investment Preferences
under uncertainty (cont’d)
Use Utility function to model investment preferences :

+ -
U ( E(R) , σ 2) = E(R ) - k σA2

k = Scale factor = 0.5 if decimals

0.005 if %
A = Risk Aversion parameter
The higher “A”, the more risk averse an investor is..
.i.e as RA increases ►U decreases faster as σ2 goes up

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 Investment Preferences
under uncertainty (cont’d)
Example
Assume a risk-free asset has a return of 4%, and a mutual fund
has an expected return of 10% and a standard deviation of 16%.
Question
which will you prefer if your degree of risk aversion is 4 using the
utility function U = E(R) - .5 *A*σ2 ?

U = .10 - .5(4) (.162) = .0488 for Mutual fund

U = .04 for the risk free asset

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 Investment Preferences
under uncertainty (cont’d)
Risk Averse Investor ► A > 0
Given an equal increase of risk, the investor will require an
increasing increments of expected return
E (R) 3
Risk Return Trade offs
2
For given preference A=4
1 Indifference Curve
σ
Δ Δ
Does the investor need to diversify and why?
Yes, because the investor is risk averse

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 Investment Preferences
under uncertainty (cont’d)
Risk Neutral Investor ► A = 0
Given an equal increase of risk, the investor will require a
constant expected return

E (R)
1 2 3

σΔ Δ

Does the investor need to diversify and why?

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 Investment Preferences
under uncertainty (cont’d)
Risk Lover Investor ► A < 0
Given an equal increase of risk, the investor will require
a decreasing expected return

E (R) 1
2
3

σ
Δ Δ
Does the investor need to diversify and why?

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 2. Return
• Dollar return vs. Percentage return
0 1
Stock
P0 =$10 D1 =$1.5
P1 =$15
Dollar Return =D1 + (P1 – P0)
=$1.5 + ($15-$10) = $6.5
Holding Period Return-HPR =$1.5/$10+($15-$10)/$10=65%
(Percentage return)
Dividend Yield Capital Gain Yield

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 3. Predicting Risk-Return : One Stock
A. Ex-ante Method (Prospective)

States Probability R =HPR

Recession .30 0%
Normal .40 10 Expected return?
Boom 20
.30
∑=1

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Predicting Risk-Return: One Stock (cont’d)

Assumption about Returns

σ = risk

-σ +σ

E (R) - σ E (R) E(R) +σ Rate of return (%)

68% of chances

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 Predicting Risk-Return: One Stock (cont’d)
Example
E (R)= Weighted average return
=  [ Rt * Probt ]
= 0 x .3 + .1 x .4 + .2 x .3 = .1

Variance =  {Probt *[ Rt - E(R)] 2 }

= .3 (0-.1)2 +.4 (.1-.1)2 + .3 (.2-.1)2 = .00600625
Standard deviation is the square root of variance
σ = √.00600625 = .0775

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Predicting Risk-Return: One Stock (cont’d)

Interpretation

-σ=-7.75% +σ=7.75%

10% - 7.75% 10% 10% + 7.75%

2/3 of the time

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 Predicting Risk-Return: One Stock (cont’d)

B. Ex-post method: Compute past Returns (R1, R2, R3…….RT )

E(R)= = ∑ R
T
∑ ( Rt - R )2
VAR (R ) = (1/T-1)

σ?

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Predicting Risk-Return: One Stock (cont’d)
Example
Year 1 2 3 4
Return(%) 15% 0 5%20%

E(R) = (.15 + 0 + .05 +.20 )/4 = .10

VAR(R) = (.15 - .1) +(0-.1) +(.05-.10) +(.2-.1) = .00833

2 2 2 2

(4-1)
σ = √.00833 = .0913

Predict the return = (.10-.0913, .10+.0913)

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 Average Annual Returns: 1948-1992
Av returnSt. deviation
• Cnd com stocks 12.58% 16.82%
• Bonds 7.04 10.02
• Treasury Bills 6.19 4.26
• Inflation 4.68

Important Principle

Higher the Return ►Higher the Risk

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 4. Predicting Risk-Return :
Portfolio of Stocks
Suppose we have two stocks with the following information
Stock E (R) 
A 10% 7.75%
B 18.5 16.6
Expected Return of the Portfolio?

WA= % of the initial wealth invested in Stock A

WB= % of the initial wealth invested in Stock B

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 Predicting Risk-Return :
Portfolio of Stocks (cont’d)
The expected return of the portfolio is the weighted average
expected returns of the individual stocks in the portfolio

E(RP) = WA E(RA) + WB E (R B )
WA + WB = 1

or

E(RP) = WA E(RA) + (1-WA) E (R B )

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 Predicting Risk-Return :
Portfolio of Stocks (cont’d)
Var() = )

Var() = WA2 VAR() +WB2 VAR() + 2WAWB COV(, )

Weighted average of Co-movements

individual risks If > 0
If < 0

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 Predicting Risk-Return :
Portfolio of Stocks (cont’d)
Key is Covariance
Return
B

Portfolio

Cov < 0 Time

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 Predicting Risk-Return :
Portfolio of Stocks (cont’d)
Estimation of covariance

COV(R A,RB) = i Probi (R Ai -E(R A)) (R Bi -E(R B))

= t (1/T-1) (R A t - R A) (R Bt - R B)

= σ A σ B Corr(RA,RB),

-
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 Predicting Risk-Return :
Portfolio of Stocks (cont’d)
Example
State Prob RA RB
Boom .4 .10 .35
Normal .3 .00 .20
Recession .3 .20 -.05
E(R A)=.10, E(R B)=.185
COV (RA,RB)= .4 ( .1 - .1 ) ( .35 - .185 )
.3 ( .0 - .1 ) ( .20 - .185 )
.3 ( .2 - .1 ) ( -.05 - .185)
= -.0075

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 5. Covariance Variance Matrix
WA WB
Stock A Stock B
Stock A 1 Cov (A,A) 2
WA WAWB Cov(A,B)
= W Aσ
2 2
A

Stock B 3 4 Cov (B,B)

WBWA Cov (B,A) = W2B σ2B
WB

# terms? 4 terms = 2 Var + 2 Cov

σP2 = W2A σ 2A+ W2B σ 2B + 2 W AW B Cov (RA ,RB)

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Covariance Variance Matrix (cont’d)

Var Cov Cov

Cov Var Cov

Cov Cov Var

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 Correlation Matrix
Example
Stock Wi Var (Ri) std (Ri)
A 0.3 0.16 0.40
B 0.6 0.25 0.50
C 0.1 0.09 0.30
A B C
Correlation Matrix
1.0 …A…
0.2 1.0
B…
0.4 0.5
C 1.0
Var (Rp) = .32 x.16+.62 x.25+.12 x.09
+ 2 x .3 x .6 x (.4x.5x.2) (A,B)
+ 2 x .3 x .1 x (.4x.3x.4) (A,C)
+ 2 x .6 x .1 x (.5x.3x.5) (B,C)
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 6. Diversification
• Diversification is a portfolio management strategy
that mixes a wide variety of assets within a portfolio.
• Why?
The goal is to reduce risk (Min σP)

There is a famous quote :

Don’t put all eggs in the same basket,
because if the basket drops….?

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 Diversification (cont’d)
Assess the impact of diversification upon risk
Sk Var Cov # of terms
1 1 1
2 2 2 4
3 3 6 9
4 4 12 16
N N N2-N N2

∞ 0 Average Cov

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 Diversification (cont’d)
σ P 2 = ΣNi W2i σ 2i+ ΣNiΣNj W iW j Cov (Ri ,R j)
N terms (N2 – N) terms
Assume Wi =1/N
All stocks have the same var, each = σ2
All cov are the same, each = cov
σP2 = N (1/N2 * σ2) + (N2-N) (1/N2 * cov)
σP2 = (1/N) σ + (1-1/N) cov
2

As N goes to infinity ►Left over with σP2 = cov

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 Diversification (cont’d)
σ
Unsystematic risk or
firm specific risk
( Mgmt Skills, winning or losing
contracts,Labor strike…)

Well Cov called systematic risk

Diversified
(inflation, Taxation, interest rate…)
Portfolio

σ =Total risk= Systematic Risk + Unsystematic Risk

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 Diversification (cont’d)
Example
Based on data for 1982-2010, we find that
σGE = 6.49%
and σIBM= 8.10%
The correlation between GE and IBM is 0.377
Variance of an equally weighted portfolio
=0.52x.06492+0.52x.0812 + 2x.5x.5x(.0649x.081x0.377)
=.00368
and σ=.0607 or 6.07%
The portfolio is less risky than GE or IBM
► This is called the benefit of diversification
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 Diversification (cont’d)

Conditions for Diversification

Key to diversification is Cov

But - ∞ ≤ Cov ≤ + ∞
use Correlation because it is more convenient to assess the
portfolio decisions
Corr(RA,RB),

-1 +1

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 Diversification (cont’d)
Case 1 Corr(RA,RB) = 1
E(RP) = W A E(R A) + W B E (R B )
σP2 = W2A σ 2A+ W2B σ 2B+2 W AW B (σA*σ B* Corr(RA,RB))
a2 + b2 + 2ab
σ2P= (a + b) 2= (WAσA+WBσB)2
σP = (WAσA+WBσB)
Feasible set? WA WB E(RP) σP
1 0
Plot the feasible set

0 1
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 Diversification (cont’d)
Don’t gain from
Long Short diversification
WA +WB =1
WB>1 WA<0
E (RP)
B WA=0,WB=1
WA=1/4, WB=3/4
WA=WB=1/2
WA=3/4, WB=1/4
WA=1, WB=0
A Asset allocation

σP=0 σP
P WA>1 WB<0
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 Diversification (cont’d)
Case 2 Corr(RA,RB) = - 1

E(RP) = W A E(R A) + W B E (R B )
σP2 = W 2A σ 2A+ W2B σ 2B + 2 WA WB (σ A *σ A * Corr(RA,RB) )
a 2
+ b2 - 2ab
σP = (a – b) 2
2

σ P = (WA σA – WB σB)
Feasible set & plot

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 Diversification (cont’d)

E (R ) B

P
Zero risk A
portfolio
WA?
σ
WB? 0
Asset allocation
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 Diversification (cont’d)

Question:
What is the asset allocation between A and B to achieve zero
risk portfolio?
Method 1 set σ P = 0 and solve for WAand WB
WA σA – WB σB = 0
Solving 2 equations 2
WA + WB = 1 unknowns
W*A = σB
σ A +σB

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 Diversification (cont’d)

Method 2 Take the first derivative of σ 2P with respect to

WA, set it equal to zero and solve for WA
σP2 = W2Aσ2A + W2Bσ2B + 2WAWBcov

WB=1-WA
∂σ2P = 0 and solve for WA
∂WA 2
σ B - cov
W*A = For any Corr
σ2A+σ2B-2cov
General formula for any MVP

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 Diversification (cont’d)

Example
σA=10% σB=20% Corr(RA,RB) = -1

WA = .67
σP=0
WB = .33

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 Diversification (cont’d)
Case 3 Corr(RA,RB) = 0
E(RP) = W A E(R) A + W B E (R B )
σ P 2 = W2A σ 2A+W2B σ 2B+[2 W A W B (σA *σB* Corr(RA,RB) )]
0
E(R )
MVP

σP

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 Diversification (cont’d)

Solving MVP using general formula

σ2 B
W*A =
σ2A+σ2B

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 Diversification (cont’d)
Example
σA=10% σB=20% Corr(RA,RB) = 0
WA = .8
σP= .0894
WB= .2

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 Diversification (cont’d)
Review of the 3 cases
E(R)
Corr =-1

Corr =1
Corr =0

σ
As Corr goes down ►σ decreases

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 Diversification (cont’d)

If add Ford ►Assume Corr = 1 /If add JJ instead ►Corr(GM,JJ) <1

σ
GM GM,Ford if Corr=1
σ
Gain from diversification
GM,JJ if Corr < 1

1 2 ? Securities

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 7. Efficient Portfolios
An efficient portfolio is the one

Max Return given Risk

or
Min Risk given Return

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 Efficient Portfolios (cont’d)
Case 1 Two Risky Assets

Using Markowitz method

σ P 2 = W 2A σ 2A+ W2B σ 2B+ 2 WAWB Cov (RA,RB)
W A E(R A) + (1-W A) E (R B ) = k (given)

• Decision variables? WA? WB?

• Inputs? σ 2 ? σ2 ? Cov ? E (R ) ? E(R ) ?
A B A B

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 Efficient Portfolios (cont’d)
Efficient Frontier
MVP B
E(Rp)
3 B
2

MVP
1

Next? Solve for the asset allocation σp

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 Efficient Portfolios (cont’d)
Separation in Portfolio Managment
Step 1 Derive the feasible set
WA? WB?
Step 2 Determine the efficient set

Step 3 Investor will choose according to

yA? yB?
his risk-return preferences

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 Efficient Portfolios (cont’d)
Example Asset A Asset B
E (R ) 8% 13%
σ 12 20
Corr .3
Questions
1. What is the asset allocation if an investor requires 10%?
10% = y 8% + (1- y) 13% ► y = .6 (1-y) = .4
2. What is the risk level of the portfolio?
σP2 = .62 x.122+ .42x.22+ 2 x .6 x .4 x (.12 x .20 x .3)=.015
σP = √.01512 =.123

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 Efficient Portfolios (cont’d)
E(RP)
B
.10
Asset allocation
MVP yA= .60 yB=.4

σP
.123

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 Efficient Portfolios (cont’d)

Case 2 Two Risky Assets and One Risk Free Asset

σP2 = WA2 σA2 + WB2 σB2+ 2 WAW B Cov(RA,RB)
WA E(RA) + WB E (RB ) + (1-WA-WB) RF = K

Portfolio of Risky Assets Risk Free Asset

Asset allocation decision is concerned with how we allocate the

initial wealth between
the portfolio of risky assets and the risk free asset

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 Efficient Portfolios (cont’d)
Step 1 Efficient set CAL(M)
B
E (R)
M
MVP CAL(MVP)
C CAL(C)
A CAL(A)
RF

σP
Start with (RF,A) Slope= E(RA)-RF
σA
Compared with (RF,C) Slope= E(RC)-RF Reward to risk
σC
…till highest slope = E(RM) - RF
σM 50
 Efficient Portfolios (cont’d)
• Slope of the Capital Allocation Line (CAL)
– measures the excess return being earned per unit of risk
– This “reward-to-risk ratio” is also called the Sharpe ratio.

• M is the optimal portfolio of risky assets which dominates

all other risky portfolios

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 Efficient Portfolios (cont’d)

• We solved the weights for the risky assets A and B in M

W*A=

W*B = 1 – W*A

Equation 7.13-Page 216 in textbook

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 Efficient Portfolios (cont’d)
E (R )
y Efficient set
M

(1-y)
RF

σP

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 Efficient Portfolios (cont’d)

For any efficient portfolio P on the efficient set

E(RP) = y E(RM) + (1-y) RF (1)

σ P 2 = y 2 σ 2M + (1-y) 2 σ 2F + 2 y(1-y) cov (M,F)

σ P 2 = y 2 σ 2M

σ P =y σ M (2)

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 Efficient Portfolios (cont’d)

Substitute (1) into (2) and rearrange

E(RP) = RF + (E(RM) – RF) Equation of

σM *σP
Capital
Quantity of Market Line
CAL(M) Market
total risk or CML
Risk
Premium

Risk Premium for P

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 Efficient Portfolios (cont’d)
Step 2 Investor’s Decisions
CML
M
lender
borrower
y>1
Rf Unlevered
(1-y) < 0
0 < (1-y)≤ 1 y=1
0≤ y < 1

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 Efficient Portfolios (cont’d)
Example
Assume the following assets:
Asset A Asset B Rf
E (R ) 8% 13% 5%
σ 12 20
Corr(A,B) .3

% in M .4 .6

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 Efficient Portfolios (cont’d)
Questions
1) What is expected return given the required σ P = 10%?

E(RM) = .4 x .08 + .6 x .13 = .11

σM2 =.4 2x .12 2+ .62x .22+ 2 x.4x.6x.12x.2x.3
σM = .142

Given σP = .10 ► E (RP) = .092 or 9.2%

Max expected return

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 Efficient Portfolios (cont’d)
2) Asset allocation ?
y (1-y)

M RF
9.2% = y * 11% + ( 1 – y ) 5% y=.70 (1-y) =.3

σP = y σM ►.10 = y * .142 ► y =.70 and (1-y) =.3

E(RP) = .7 * 11% + .3 * 5% = 9.2%

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 Efficient Portfolios (cont’d)

If initial wealth = $1,000

M = $700 RF=$300

Asset A $700 Asset B = $700 x .6 = $420

x.4=$280

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 Efficient Portfolios (cont’d)

3) Asset allocation given A= 2?

Max U = E(RP) – k *A*σP2
y
y E(RM)+(1-y)RF y2σ2M

U = yE(RM)+(1-y)RF - ½ * A*y2σ2M
∂ U/∂ y = E(RM)-RF-A* y *σ2M = 0
.11−.05
¿
2 𝑥 .142
2 ¿1.48
Given W = $1,000 ►borrow 48% of $1,000 to raise $480 and
invest $1,480 in M

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