Fi8000
Optimal
Risky Portfolios
Milind Shrikhande
Investment Strategies
☺Lending
vs. Borrowing (risk-free asset)
☺
Lending: a positive proportion is invested in
the risk-free asset (cash outflow in the
present: CF0 < 0, and cash inflow in the
future: CF1 > 0)
☺
Borrowing: a negative proportion is invested
in the risk-free asset (cash inflow in the
present: CF0 > 0, and cash outflow in the
future: CF1 < 0)
Lending vs. Borrowing
E(R) 8.0%
A
A
6.0%
Lend
B
4.0%
2.0%
rf
Borrow
C
rf
0.0%
0.0%
1.0%
2.0%
3.0%
4.0%
STD(R)
Investment Strategies
☺A Long
vs. Short position in the risky asset
☺
Long:
A positive proportion is invested in the risky
asset (cash outflow in the present: CF0 < 0,
and cash inflow in the future: CF1 > 0)
☺
Short:
A negative proportion is invested in the risky
asset (cash inflow in the present: CF0 > 0,
and cash outflow in the future: CF1 < 0)
Long vs. Short
E(R)
Long A and Short B
Long A and Long B
A
Short A and Long B
B
Investment Strategies
☺ Passive
risk reduction:
The risk of the portfolio is reduced if we invest a larger
proportion in the risk-free asset relative to the risky one
☺ The
perfect hedge:
The risk of asset A is offset (can be reduced to zero) by
forming a portfolio with a risky asset B, such that ρAB=(-1)
☺ Diversification:
The risk is reduced if we form a portfolio of at least two
risky assets A and B, such that ρAB<(+1)
The risk is reduced if we add more risky assets to our
portfolio, such that ρij<(+1)
One Risky Fund and one Risk-free
Asset: Passive Risk Reduction
E(R) 8.0%
Reduction in
portfolio risk
4.0%
2.0%
rf
A
A
6.0%
B Increase of
portfolio Risk
C
rf
0.0%
0.0%
1.0%
2.0%
3.0%
4.0%
STD(R)
Two Risky Assets with ρAB=(-1):
The Perfect Hedge
E(R)
Minimum
Variance is zero
Pmin
B
A
The Perfect Hedge – an Example
What is the minimum variance portfolio if we
assume that
μA=10%; μB=5%; σA=12%; σB=6% and ρAB=(1)?
   A B  AB
wA  2
2
 A   B  2 A B  AB
2
B
(6%)  (12%)(6%)(1)
1


2
2
(12%)  (6%)  2(12%)(6%)(1) 3
2
The Perfect Hedge – Continued
What is the expected return μmin and the standard
deviation of the return σmin of that portfolio?
min  wA  A  (1  wA )  B
1
2
2
 10%   5%  6 %
3
3
3
 min  wA2 A2  (1  wA ) 2  B2  2wA (1  wA ) A B  AB
2
2
1 2
1
2
    (12%) 2     (6%) 2  2   12%  6%  (1)
3 3
3
3
 0%
Diversification: the Correlation
Coefficient and the Frontier
E(R)
A
ρAB=(-1)
-1<ρAB<1
B
ρAB=+1
Diversification: the Number of
Risky assets and the Frontier
E(R)
A
C
B
Diversification: the Number of
Risky assets and the Frontier
E(R)
A
C
B
Capital Allocation:
n Risky Assets
State all the possible investments – how
many possible investments are there?
Assuming you can use the Mean-Variance
(M-V) rule, which investments are M-V
efficient?
Present your results in the μ-σ (mean –
standard-deviation) plane.
The Expected Return and the Variance of
the Return of the Portfolio
wi = the proportion invested in the risky asset i (i=1,…n)
p = the portfolio of n risky assets (wi invested in asset i)
Rp = the return of portfolio p
μp = the expected return of portfolio p
σ2p = the variance of the return of portfolio p
n
R p  w1  R1  w2  R2  ...  wn  Rn   wi  Ri
i 1
n
E ( R p )   p   wi  i
i 1
n
n
V ( R p )   p2   wi  w j   ij
i 1 j 1
The Set of Possible Portfolios
in the μ-σ Plane
E(R)
The Frontier
i
The Set of Efficient Portfolios
in the μ-σ Plane
E(R)
The Efficient Frontier
i
Capital Allocation:
n Risky Assets
The investment opportunity set:
{all the portfolios {w1, … wn} where Σwi=1}
The Mean-Variance (M-V or μ-σ )
efficient investment set:
{only portfolios on the efficient frontier}
The case of n Risky Assets:
Finding a Portfolio on the Frontier
Optimization:
Find the minimum variance portfolio for a
given expected return
Constraints:
A given expected return;
The budget constraint.
The case of n Risky Assets:
Finding a Portfolio on the Frontier
n
Min
{ w1 ,... wn }
n
 w  w
i
i 1 j 1
n
S .t.
w 
i
i 1
n
i
w 1
i 1
i
j
  ij
 p
Capital Allocation: n Risky
Assets and a Risk-free Asset
State all the possible investments – how
many possible investments are there?
Assuming you can use the Mean-Variance
(M-V) rule, which investments are M-V
efficient?
Present your results in the μ-σ (mean –
standard-deviation) plane.
The Expected Return and the Variance of
the Return of the Possible Portfolios
wi = the proportion invested in the risky asset i (i=1,…n)
p = the portfolio of n risky assets (wi invested in asset i)
Rp = the return of portfolio p
μp = the expected return of portfolio p
σ2p = the variance of the return of portfolio p
n
R p  w0 rf  w1 R1  w2 R2  ...  wn Rn  w0 rf   wi Ri
n
E ( R p )   p  w0 rf   wi i
n
i 1
n
V ( R p )   p2  0   wi w j ij
i 1 j 1
i 1
The Set of Possible Portfolios
in the μ-σ Plane (only n risky assets)
E(R)
The Frontier
i
The Set of Possible Portfolios
in the μ-σ Plane (risk free asset included)
E(R)
The Frontier
i
rf
n Risky Assets and a Risk-free
Asset: The Separation Theorem
The process of finding the set of MeanVariance efficient portfolios can be separated
into two stages:
1.
Find the Mean Variance efficient frontier
for the risky assets
2.
Find the Capital Allocation Line with the
highest reward to risk ratio (slope) - CML
The Set of Efficient Portfolios
in the μ-σ Plane
μ
The Capital Market Line:
m
i
rf
μp= rf + [(μm-rf) / σm]·σp
The Separation Theorem:
Consequences
The asset allocation process of the risk-averse
investors can be separated into two stages:
1.Decide on the optimal portfolio of risky assets m
(the stage of risky security selection is identical for all the
investors)
2.Decide on the optimal allocation of funds between
the risky portfolio m and the risk-free asset rf –
choice of portfolio on the CML (the asset allocation stage is
personal, and it depends on the risk preferences of
the investor)
Capital Allocation: n Risky
Assets and a Risk-free Asset
The investment opportunity set:
{all the portfolios {w0, w1, … wn} where Σwi=1}
The Mean-Variance (M-V or μ-σ )
efficient investment set:
{all the portfolios on the Capital Market Line CML}
n Risky Assets and One Risk-free
Asset: Finding a Portfolio on the Frontier
Optimization:
Find the minimum variance portfolio for a
given expected return
Constraints:
A given expected return;
The budget constraint.
n Risky Assets and One Risk-free
Asset: Finding the Market Portfolio
n
Min
{ w1 ,... wn }
 w w 
i 1 j 1
i
j
ij


wi i  1- wi  rf   p

i 1
 i 1 
n
S .t.
n
n
n Risky Assets and One Risk-free
Asset: Finding the Market Portfolio
Solve the following system of equasions and
find the proportions {w1 ,...wn } invested in the risky assets
w1 11  w2 12  ...  wn 1n  1  rf
w1 21  w2 22  ...  wn 2 n   2  rf
...
w1 n1  w2 n 2  ...  wn nn   n  rf
Scale the proportions: zi 
wi
for i  1,...n
n
w
j 1
j
and m  {z1 , z2 ,...zn } is the market portfolio.
A Numeric Example
Find the market portfolio if there are only two risky assets,
A and B, and a risk-free asset rf.
μA=10%; μB=5%; σA=12%; σB=6%; ρAB=(-0.5) and rf=4%
To find the proportions {wA , wB } invested in assets A and B
use the system of equations for two risky assets:
wA AA  wB AB   A  rf
wA BA  wB BB   B  rf
Using our data we get two equations:
wA (12%) 2
 wB (12%)(6%)(0.5)  10%  4%
wA (12%)(6%)(0.5)  wB (6%) 2
 5%  4%
Example Continued
If we solve the two equations:
wA (12%) 2
 wB (12%)(6%)(0.5)  10%  4%
wA (12%)(6%)(0.5)  wB (6%) 2
 5%  4%
we get the proportions {wA , wB }  {0.06481, 0.09259}.
Now we have to scale the proportions
wA
0.06481
zA 

 0.41176 and z B  0.58824.
wA  wB 0.06481  0.09259
and m  {z A , z B }  {0.41176, 0.58824} is the market portfolio.
Example Continued
The expected return of the market portfolio is
m  z A  A  zB  B
 0.41176 10%  0.58824  5%  7.06%
The standard deviation of the return of the market portfolio is
 m  z A2 A2  z B2 B2  2 z A z B A B  AB
 (0.41176) 2 (12%) 2  (0.58824) 2 (6%) 2
 2  0.41176  0.58824 12%  6%  (-0.5)
 4.41%
Practice Problems
BKM Ch. 8: 1-7, 11-14
Mathematics of Portfolio Theory:
Read and practice parts 11-13.