Portfolio Theory I
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Transcript of Portfolio Theory I
1
Portfolio Theory
Slide 2
Premise of Portfolio Theory
• Any asset can go up or down depending on the market conditions.
• When these assets are put together to form a portfolio, their interaction reduces overall volatility which then contributes to the stability of the portfolio.
• Any portfolio is designed with returns in mind. One can then choose an expected return and then seek to minimize the risk.
Slide 3
Assumptions of Portfolio Theory • Asset returns are normal distributed. • Correlations between the asset returns (not raw
prices) are fixed and constant over a period of time.
• Investors seek to maximize their overall return. • All players in the market are rational and risk
averse • Common information is available to all players in
the market • Many other assumptions …….
Slide 4
Topics
• Two risky assets - with specified expected return - global minimum variance portfolio
• One risky asset & one risk-free asset
• Two risky assets & one risk-free asset
• Matrix Formulation
Slide 5
Motivation
• Suppose investment in a certain stock offers an expected return of 12% while the expected return on a certain bond is only 8%. Would you put all your money in the stock ?
• If not, what is the optimal combination of these two assets ?
Slide 6
Data Set for Demonstration
• The following data will be repeatedly used to illustrate the techniques :
μA μB σA2 σB
2 σA σB σAB ρAB 0.175 0.055 0.067 0.013 0.258 0.115 -0.004875 -0.164
Slide 7
Portfolio with 2 Risky Assets
• Suppose a percentage of wA of the capital is invested in security A and wB in security B.
• Thus wA + wB = 1 with wA , wB ≥ 0
• Rate of return rp = wArA + wBrB
• Expected return of portfolio
E(rp) = wAE(rA) + wBE(rB)
Slide 8
Portfolio with 2 Risky Assets (cont’d)
• Variance of rate of return of portfolio
σp2 = var(rp)
= var(wArA + wBrB)
= wA2var(rA) + wB
2var(rB) + 2wAwBcov(rA, rB)
= wA2σA
2 + wB2 σB
2 + 2wAwB ρAB σAσB
where ρAB is the correlation coefficient between rA and rB.
Portfolio with 2 Risky Assets – Example 1
• Consider a portfolio with wA = wB = 0.5.
– μP = (0.5)(0.175) + (0.5)(0.055) = 0.115
– σp2 = (0.5)2(0.067) + (0.5)2(0.013) + 2(0.5)(0.5)(-0.004875)
= 0.01751
σp = 0.1323
The portfolio expected return is the average of the expected returns of assets A and B The portfolio s.d. is less than the average of the asset s.d. This shows diversification reduces risk.
Slide 9
Portfolio with 2 RiskyAssets – Example 2
• Consider a long-short portfolio with wA = 1.5 and wB = -0.5.
– μP = E(rp) = (1.5)(0.175) + (-0.5)(0.055) = 0.235
– σp2 = (1.5)2(0.067) + (-0.5)2(0.013) + 2(1.5)(-0.5)(-0.004875)
= 0.1604
σp = 0.4005
This portfolio has a higher expected return and a higher s.d. than both asset A and asset B.
Slide 10
Slide 11
Shape of Portfolio Frontier : ρ ≠ ±1
( )
( ) ( )( ) ( ) ( )
2 2 2 2 2
22 2 2
2 2 2 2 2
2
1 2 1
2 2 2
P A A B B A B A B
P A A B B A B A B
A A A B A A A B
A B A B A B A B A B
w w w
w w w w
w w w w
w w
µ µ µ µ µ µ
σ σ σ ρσ σ
σ σ ρσ σ
σ σ ρσ σ σ ρσ σ σ
= + = − +
= + +
= + − + −
= + − + − + +
Portfolio lies on the quadratic curve joining A and B.
Slide 12
Shape of Portfolio Frontier : ρ = 1
2 2 2 2 2 2
2 ( )
P A A B B
P A A B B A B A B A A B B
P A A B B
w ww w w w w ww w
µ µ µ
σ σ σ σ σ σ σσ σ σ
= +
= + + = +⇒ = +
Portfolio lies on the line joining A and B.
Slide 13
Shape of Portfolio Frontier : ρ = -1
The portfolio is riskless when :
( )
( )( )
( )
( )
22 2 2 2 2
2
0
1
µ µ µ µ
σ σ σ σ σ σ σ
σ σ σ σ σ σ
σσ σ σ
σ σ
σσ σ σ
σ σ
− +
= + − = −
⇒ − +
++
++
≤ ≤
≤ ≤
=
= = −
− += −
A B A B
P A A B B A B A B A A B B
P A A B B A B A B
BA B A B A
A B
BA B A B A
A B
P w
w w w w w w
w w w
w w
w w
if
if
, 1BA B A
A B
w w wσσ σ
= = −+
Slide 14
Portfolio Frontier : ρ = -1 Example
The portfolio is riskless when :
115
373115
373
0
1
0.12
0.373
0.373
µ µ
σσ
σ
+
⇒
≤ ≤
≤ ≤
=
− + −
=A A
A A
BA
B
B
P
P
w
w w
w w
if
if
115 258, 373 373
= =A Bw w
μA μB σA σB 0.175 0.055 0.258 0.115
Slide 15
Portfolio Frontier : ρ = -1 Example (cont’d)
The portfolio is riskless when :
258
373258
373
0
1
0.12
0.373
0.373
µ µ
σσ
σ
+
⇒
≤ ≤
≤ ≤
= −
− + −
=A
A A
A
A B
B
B
P
P
w
w w
w w
if
if
258 115, 373 373
= =A Bw w
μA μB σA σB 0.055 0.175 0.115 0.258
Shape of Portfolio Frontiers for Various ρ
Efficient Portfolio & Efficient Frontier
• Portfolios with the highest expected return for a given level of risk are called efficient portfolios.
• The curve passing through a collection of efficient portfolios is called the efficient frontier (EF).
Slide 17
Efficient Portfolios & EF with 2 Risky Assets : ρ ≠ ±1
Slide 18
Efficient portfolios in green Inefficient portfolios in red
Global Minimum Variance Portfolio
Slide 19
This portfolio has the smallest variance among all efficient portfolios
Global Minimum Variance Portfolio
• To find this portfolio, one has to solve the constrained minimization problem
• It can be shown that
Slide 20
2 2 2 2 2
, 2
s.t. 1
minA B
P A A B B A B ABw w
A B
w w w w
w w
σ σ σ σ= + +
+ =
2min
2 2
min min
2
1
B ABA
A B AB
B A
w
w w
σ σσ σ σ
−=
+ −
= −
Global Minimum Variance Portfolio - Example
• Using the data, we have
The expected return, variance and s.d. of this portfolio are
Slide 21
min min0.013 ( 0.004875) 0.1992 0.80080.067 0.013 2( 0.004875)A Bw w− −
= = ⇒ =+ − −
2 2 2
(0.1992)(0.175) (0.8008)(0.055) 0.0789
(0.1992) (0.067) (0.8008) (0.013) 2(0.1992)(0.8008)( 0.004875) 0.00944
0.00944 0.09716
P
P
P
µ
σ
σ
= + =
= ++ − =
= =