Fi8000
Capital Allocation and
Efficient Portfolios
Milind Shrikhande
Today
☺ Portfolio
Theory
☺
The Mean-Variance Criterion
☺
The Normal Distribution
☺
Capital Allocation
☺
The Mathematics of Portfolio Theory
The Mean-Variance Criterion
(M-V or μ-σ criterion)
Let A and B be two (risky) assets. All riskaverse investors prefer asset A to B if
{ μA ≥ μB
{ μA > μB
and
or if
and
σA < σB }
σA ≤ σB }
* Note that these rules apply only when we assume that the
distribution of returns is normal.
The Mean-Variance Criterion
(M-V or μ-σ criterion)
☺
E(R) = μR
☺
The Normal Distribution of Returns
Pr(R)
68%
95%
μ - 2σ
μ-σ
μ
μ +σ
μ +2σ
R
The Normal Distribution of Returns
Pr(Return)
σR: Risk
0
μR: Reward
R=Return
The Normal Distribution
Higher Reward (Expected Return)
Pr(Return)
μB
<
μA
R=Return
The Normal Distribution
Lower Risk (Standard Deviation)
Pr(Return)
A
B
μA= μB
σA < σB
R=Return
Capital Allocation - Outline
☺n
mutually exclusive assets (i.e. one can
only invest in one asset but not in a
portfolio)
☺ One risky asset and one risk-free asset
☺ n risky assets and one risk-free asset (the
risky investments are mutually exclusive)
☺ Two risky assets
☺ n risky assets
☺ n risky assets and one risk-free asset
Capital Allocation - Data
There are three (risky) assets and one risk-free
asset in the market. The risk-free rate is rf = 1%,
and the distribution of the returns of the risky
assets is normal with the following parameters
Asset
A
B
C
Expected Return
5.6%
4.2%
1.7%
Standard Deviation
of the Return
2.5%
5.0%
2.1%
Capital Allocation:
n mutually exclusive assets
State all the possible investments.
Assuming you can use the Mean-Variance
(M-V) rule, which investments are M-V
efficient (i.e. which assets can not be thrown
out of the set of desirable investments by a
risk-averse investor who uses the M-V rule)?
Present your results on the μ-σ (mean –
standard-deviation) plane.
The Expected Return and
the STD of the Return (μ-σ plane)
E(R) 8.0%
A
6.0%
B
4.0%
C
2.0%
rf
0.0%
0.0%
2.0%
4.0%
6.0%
8.0%
STD(R)
The Mean-Variance Criterion
(M-V or μ-σ criterion)
☺
E(R)
☺
Capital Allocation:
n mutually exclusive assets
The investment opportunity set:
{rf, A, B, C}
The Mean-Variance (M-V or μ-σ ) efficient
investment set:
{rf, A, C}
Note that investment B is not in the efficient set since investment
A dominates it (one dominant investment is enough).
Capital Allocation:
One Risky Asset (A) and One 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 on the μ-σ (mean –
standard-deviation) plane.
The Expected Return and STD of the
Return of the Portfolio
α = the proportion invested in the risky asset A
p = the portfolio with α invested in the risky asset A
and (1- α) invested in the risk-free asset rf
Rp = the return of portfolio p
μp = the expected return of portfolio p
σ p = the standard deviation of the return of portfolio p
Rp = α·RA + (1-α)·rf
μp = E[ α·RA + (1-α)·rf ] = α·μA + (1-α)·rf
σ2p = V[ α·RA + (1-α)·rf ] = (α·σA)2
Or
σp = α·σA
Capital Allocation:
One Risky Asset and One Risk-free Asset
The investment opportunity set:
{all portfolios with proportion α invested in A
and (1-α) invested in the risk-free asset rf}
The Mean-Variance (M-V or μ-σ )
efficient investment set:
{all the portfolios in the opportunity set}
The Capital Allocation Line
 E ( RA )  rf
E ( R p )  rf  
 STD ( RA )
or
  A  rf
 p  rf  
 A

  p


  STD ( R p )

The Expected Return and
the STD of the Return (μ-σ plane)
E(R) 8.0%
A
A
6.0%
B
4.0%
2.0%
rf
C
rf
0.0%
0.0%
1.0%
2.0%
3.0%
4.0%
STD(R)
The Capital Allocation Line (CAL):
Four Basic Investment Strategies
E(R) 8.0%
A
6.0%
P2
BA
4.0%
C
2.0%
P1
rf
rf
0.0%
0.0%
1.0%
2.0%
3.0%
4.0%
STD(R)
Portfolios on the CAL
Portfolio
α
E(Rp) = μp
Std(Rp) = σp
rf
0
1.00%
0.00%
P1
0.25
2.15%
0.625%
A
1
5.60%
2.50%
P2
1.5
7.90%
3.75%
Capital Allocation: n Mutually
Exclusive Risky Asset and One 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 on the μ-σ (mean –
standard-deviation) plane.
The Expected Return and
the STD of the Return (μ-σ plane)
E(R) 8.0%
A
6.0%
B
4.0%
2.0%
rf
0.0%
0.0%
C
2.0%
4.0%
6.0%
8.0%
STD(R)
Capital Allocation:
One Risky Asset and One Risk-free Asset
The investment opportunity set:
{all the portfolios with proportion α invested in the
risky asset j and (1-α) invested in the risk-free asset,
(j = A or B or C)}
The Mean-Variance (M-V or μ-σ ) efficient
investment set:
{all the portfolios with proportion α invested in the
risky asset A and (1-α) invested in the risk-free asset
– (why A?)}
Capital Allocation:
Two 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 on the μ-σ (mean –
standard-deviation) plane.
The Expected Return and STD of
the Return of the Portfolio
wA = the proportion invested in the risky asset A
wB = (1-wA) = the proportion invested in the risky asset B
p = the portfolio with wA invested in the risky asset A and
(1-wA) invested in the risky asset B
Rp = the return of portfolio p
μp = the expected return of portfolio p
σ p = the standard deviation of the return of portfolio p
Rp =
wA·RA + (1-wA)·RB
μp = E[ wA·RA + (1-wA)·RB ]
σ2p= V[ wA·RA + (1-wA)·RB ]
Two Risky Assets:
The Investment Opportunity Set
E(Rp)
A
B
Two Risky Assets:
The M-V Efficient Set (Frontier)
E(Rp)
A
B
Two Mutually Exclusive Risky Assets:
The M-V Efficient Set
E(R)
A
B
Two Risky Assets:
The M-V Efficient Set (Frontier)
E(R)
P
A
B
Capital Allocation:
Two Risky Assets
The investment opportunity set:
{all the portfolios on the frontier: with
proportion wA invested in the risky asset A and
(1-wA) invested in the risky asset B}
The Mean-Variance (M-V or μ-σ )
efficient investment set:
{all the portfolios on the efficient frontier}
Two Risky Assets:
The M-V Efficient Set (Frontier)
E(R)
A
P2
Pmin
B
P3
P1
Portfolios on the Efficient Frontier
wA = the proportion invested in the risky asset A
wB = (1-wA) = the proportion invested in the risky asset B
What is the value of wA for each on of the portfolios
indicated on the graph? - Assume that
μA=10%; μB=5%; σA=12%; σ B=6%; ρAB=(-0.5).
What is the investment strategy that each portfolio
represents?
How can you find the minimum variance portfolio?
What is the expected return and the std of return of
that portfolio?
Portfolios on the Frontier
E(Rp) = μp Std(Rp) = σp
Portfolio
wA
P1
1.3
11.50%
16.57%
A
1
10.00%
12.00%
P2
0.35
6.75%
4.06%
Pmin
?
?
?
B
0
5.00%
6.00%
P3
-0.5
2.50%
13.08%
The Minimum Variance Portfolio
The variance of a portfolio on the frontier
(2 risky assets, A and B) is
V ( R p )   p2  wA2   A2  wB2   B2  2 wA wB A B  AB
If you differentiate this expression with respect to wA
and set the derivative equal to zero,
you will get the minimum variance portfolio:
 B2   A B  AB
wA  2
2
 A   B  2 A B  AB
and
wB  1  wA
The Minimum Variance Portfolio
The minimum variance portfolio in our case is:
 B2   A B  AB
wA  2
 A   B2  2 A B  AB
(6%)  12%  6%  (0.5)

 0.2857
2
2
(12%)  (6%)  2 12%  6%  ( 0.5)
2
Therefore,
min  6.43%
and
 min  3.93%
Investment Strategies
☺Lending
vs. Borrowing (bonds)
☺Long vs. Short position (stocks)
☺Passive
risk reduction
☺Diversification
☺
☺
The number of risky assets in the portfolio
The correlation between the returns of the assets
☺A perfect
hedge
Practice Problems
BKM Ch. 7: 1-6, 8, 9, 13, 20, 22, 23
BKM Ch. 8: 1-7
Mathematics of Portfolio Theory:
Read and practice parts 4-10.