Techniques for Calculating the Efficient Frontier
Weerachart Kilenthong
RIPED, UTCC
c
⃝Kilenthong
2017
Tee (Riped)
Introduction
1 / 43
Two Fund Theorem
The Two-Fund Theorem states that we can reach any point on the
m-v frontier using only two efficient portfolios.
Formally, let Xa and Xb be m-v portfolios with mean return Ra and
Rb ̸= Ra , respectively. Hence,
1
2
A combination αXa + (1 − α)Xb is an m-v portfolio.
An m-v portfolio is a combination of (Xa , Xb ).
Proof: Consider a portfolio Xz = αXa + (1 − α)Xb , where Xa and Xb
are m-v portfolios.
We need to show that this Xz is a solution to m-v problem, i.e., we
can write
(
)
(
) −1
1
−1
1 R̄ A
Xz = Σ
(1)
Rz
where Rz = αRa + (1 − α)Rb is the mean return of portfolio Xz .
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Introduction
2 / 43
Two Fund Theorem: Proof continued
We now have
Xz
= αXa + (1 − α)Xb
(
)
(
) −1
1
−1
1 R̄ A
= αΣ
Ra
(
)
(
)
1
+ (1 − α)Σ−1 1 R̄ A−1
Rb
(
)
(
) −1
1
−1
1 R̄ A
= Σ
.
αRa + (1 − α)Rb
In addition, we know that
Rz
= R̄T Xz = R̄T (αXa + (1 − α)Xb )
= αR̄T Xa + (1 − α)R̄T Xb = αRa + (1 − α)Rb
Therefore, we can write
Xz
Tee (Riped)
−1
= Σ
(
1 R̄
Introduction
)
−1
A
(
1
Rz
)
.
(2)
3 / 43
Two Fund Theorem: Proof continued
We just showed that Xz is an m-v portfolio.
We now can prove the second statement (the opposite direction). In
fact, we just reverse the earlier proof.
Suppose that Xz be an m-v portfolio, i.e., we can write
(
)
(
) −1
1
−1
1 R̄ A
Xz = Σ
.
(3)
Rz
For any pair Ra and Rb , we can find a real number α such that
−Rb
.
αRa + (1 − α)Rb = Rz . Inf act, we set α = RRza −R
b
We can now rewrite the above equation as
(
)
(
) −1
1
−1
1 R̄ A
Xz = Σ
αRa + (1 − α)Rb
(
)
(
) −1
1
−1
1 R̄ A
= αΣ
Ra
(
)
(
)
1
+ (1 − α)Σ−1 1 R̄ A−1
Rb
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Introduction
4 / 43
Two Fund Theorem
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Introduction
5 / 43
Tangency Portfolio with Risk-Free Asset
With the risk-free asset, the optimal portfolio selection problem
becomes
min XT ΣX
X
(4)
subject to
(
)
XT R̄ − Rf 1 = Rt − Rf .
(5)
where Rf is the risk free rate. Note that total weight on risky assets
does not need to be equal to 1. If 1T X > 1, the investor must
borrow, and vice versa.
The Lagrangian now is
(
))
(
L = XT ΣX + µ Rt − Rf − XT R̄ − Rf 1
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Introduction
(6)
6 / 43
Tangency Portfolio with Risk-Free Asset
The optimal condition or FOC with respect to X is
(
)
(
)
µ
2ΣX − µ R̄ − Rf 1 = 0 ⇒ X = Σ−1 R̄ − Rf 1
2
(7)
Similarly to the above problem we first solve for µ:
) µ(
)T
(
)
(
R̄ − Rf 1 Σ−1 R̄ − Rf 1
Rt − Rf = XT R̄ − Rf 1 =
2
Thus,
µ
Rt − Rf
=(
)T
(
)
2
R̄ − Rf 1 Σ−1 R̄ − Rf 1
(8)
Thus, we get
)
(
(Rt − Rf )Σ−1 R̄ − Rf 1
X= (
)T
(
)
R̄ − Rf 1 Σ−1 R̄ − Rf 1
Tee (Riped)
Introduction
7 / 43
Tangency Portfolio with Risk-Free Asset
The tangency portfolio Xt is such that
1T X t = 1
(9)
Using this condition with the optimal portfolio, we get
(
)T
(
)
R̄ − Rf 1 Σ−1 R̄ − Rf 1
(
)
Rt − Rf =
1T Σ−1 R̄ − Rf 1
(10)
(
)
Σ−1 R̄ − Rf 1
)
Xt = T −1 (
R̄ − Rf 1
1 Σ
(11)
Hence,
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Introduction
8 / 43
Efficient Frontier
Finding an efficient frontier when
I
I
I
I
short sales are allowed and riskless borrowing and lending is possible,
short sales are allowed but riskless borrowing and lending is not
possible,
short sales are not allowed but riskless borrowing and lending is
possible,
neither short sales nor riskless borrowing and lending are possible.
Roles of a riskless asset (or riskless lending and borrowing).
Roles of short sales.
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Introduction
9 / 43
Efficient Frontier with Riskless Lending and Borrowing
Let riskless be RF . Let RA be the return of portfolio of risky assets.
The average return of the combination between A and riskless asset is
R̄C = (1 − X )RF + X R̄A
Variance is
√
σC = (1 − X )2 σF2 + X 2 σA2 + 2X (1 − X )σA σF ρFA = X σA
(12)
(13)
Hence,
R̄C = RF +
Tee (Riped)
R̄A − RF
σC
σA
Introduction
(14)
10 / 43
Efficient Frontier with Riskless Lending and Borrowing
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Introduction
11 / 43
Efficient Frontier with Riskless Lending and Borrowing
Tee (Riped)
Introduction
12 / 43
Efficient Frontier with Riskless Lending but not Borrowing
Tee (Riped)
Introduction
13 / 43
Efficient Frontier with Riskless Lending and Borrowing at
Different Rates
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Introduction
14 / 43
An Example with Bonds and Stocks
Returns and variances of bonds and stocks are
R̄S = 12.5%, σS = 14.9%ρS,B = 0.45, R̄B = 6%σB = 4.8%
(15)
Now assume that the investor and borrow and lend at 5 %. Then we
can identify tangent portfolio T. The new efficient frontier is now a
straight line.
R̄P = 5 + 0.50σP
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Introduction
(16)
15 / 43
Efficient Frontier with Bonds and Stocks
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Introduction
16 / 43
An Example with Bonds and Stocks (Con’t)
From the graph, R̄T = 13.54% and σT = 16.95%
We can find the corresponding portfolio weight XS for the tangent
portfolio.
13.54 = XS (2.5) + (1 − XS )6 ⇒ XS = 116%, XB = −16%
Tee (Riped)
Introduction
(17)
17 / 43
Efficient Frontier with Bonds and Stocks
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Introduction
18 / 43
Short Sales and Riskless implies Maximum Slope
The existence of riskless asset implies that the efficient frontier in the
mean-standard deviation space is the line between the riskless asset
and the portfolio of risky assets that gives the maximum slope of
the line.
The efficient frontier is the line passing through RF and B.
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Introduction
19 / 43
Optimal Portfolio Problem with Short Sales and Riskless
Asset
Mathematically, we can find the efficient frontier by solving the
following problem.
The problem is to find a portfolio of risky assets P, whose mean
return and standard deviation are R̄P and σP respectively, that
maximize the slope
max
Xi
R̄P − RF
σP
(18)
subject to
N
∑
Xi = 1
(19)
i=1
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Introduction
20 / 43
Mean and Standard Deviation
The mean return of portfolio P is given by
R̄P =
N
∑
Xi R̄i
(20)
i=1
where R̄i is the mean return of asset i.
The standard deviation of portfolio P is given by
1

2
N ∑
N
N
∑
∑
2 2


Xi Xj σij
σP =
Xi σi +
i=1
Tee (Riped)
(21)
i=1 j̸=i
Introduction
21 / 43
Solving for Optimal Portfolio
The problem can be written as
[
max
Xi
|
N
∑
i=1
− 1
] N
2
N
N
(
) ∑ 2 2 ∑∑
Xi R̄i − RF 
Xi σi +
Xi Xj σij 
{z
F1 (X)
}|
i=1
i=1 j̸=i
{z
(22)
}
F2 (X)
Solving this: using first order conditions (FOCs): differentiating the
objective function with respect to a choice variable and take it equal
to zero.
The FOC w.r.t. Xk is
∂F1 × F2
∂F2
∂F1
= 0 =⇒ F1
+ F2
=0
∂Xk
∂Xk
∂Xk
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Introduction
(23)
22 / 43
Solving for Optimal Portfolio: More Details
Each derivative is
∂F1
= R̄k − RF
∂Xk
and
− 3 


2
N
N N
∑
∂F2
1 ∑ 2 2 ∑ ∑
=−
Xi σi +
Xi Xj σij  2Xk σk2 + 2
Xj σjk 
∂Xk
2
i=1
Tee (Riped)
i=1 j̸=i
Introduction
j̸=k
23 / 43
Solving for Optimal Portfolio: More Details
Putting these together:

− 3
2
( ) ∑
N
N ∑
N
∑
1 
2 2

0 = R̄P − RF −
Xi σi +
Xi Xj σij
2
i=1
i=1 j̸=i


∑
× 2Xk σk2 + 2
Xj σjk 
[
]
j̸=k
− 1

2
N ∑
N
N
∑
[
] ∑
2
2
Xi Xj σij 
Xi σi +
+ R̄k − RF 
i=1
[
]

i=1 j̸=i
= − R̄P − RF σP−3 Xk σk2 +
∑

]
[
Xj σjk  + R̄k − RF σP−1
j̸=k
Tee (Riped)
Introduction
24 / 43
Solving for Optimal Portfolio: More Details
Multiplying this equation by σP and rearranging the terms give


∑
[
]
R̄P − RF 
2
 + R̄k − RF
0 = −
X
σ
X
σ
+
j
k
jk
k
σ2
j̸=k
| {zP }
λP
which can be rewritten in a compact form as, for each asset k,
∑
λP Xj σjk
R̄k − RF = λP Xk σk2 +
| {z }
| {z }
Zk
R̄k − RF
= Zk σk2 +
j̸=k
∑
Zj σjk
Zj
(24)
j̸=k
Tee (Riped)
Introduction
25 / 43
System of Simultaneous Equations
After collecting all FOC for every asset together, we will end up with
a system of simultaneous equations:
R̄1 − RF
= Z1 σ12 + Z2 σ12 + Z3 σ13 + . . . + ZN σ1N
R̄2 − RF
= Z1 σ12 + Z2 σ22 + Z3 σ23 + . . . + ZN σ2N
..
.
R̄N − RF
2
= Z1 σ1N + Z2 σ12 + Z3 σ1N + . . . + ZN σN
Tee (Riped)
Introduction
26 / 43
System of Simultaneous Equations
This system of simultaneous equations can be written in a matrix
form as
R̄ − RF 1 = Σ × Z
where


R̄1



R̄ =  ...  , 1 = 
R̄N
 2
σ1 σ12 . . .
 σ12 σ 2 . . .
2

Σ =  .
..
..
 ..
.
.

σ1N
Tee (Riped)
σ2N
...
Introduction


1
..  , Z = 

. 
1

σ1N
σ2N 

.. 
. 

Z1
..  ,
. 
ZN
2
σN
27 / 43
Recovering the Optimal Portfolio
In principle, we will be able to solve for Zi using several methods,
e.g.,
1
inverse matrix
(
)
Z = Σ−1 R̄ − RF 1
2
(25)
repetitive substitution (see example)
The solution of this mathematical problem is Zi but what we really
want is Xi . How can we get Xi ?
From Zi = λP Xi , we can show that
∑
∑
Zi = λ P
Xi = λP
(26)
i
i
Hence we can recover the optimal portfolio X from Z using
Xk =
Tee (Riped)
Zk
Zk
=∑
λP
i Zi
Introduction
(27)
28 / 43
Example
Suppose there are three risky assets, say CP (asset 1), Centrals (asset
2), and PTT (asset 3).
Using past information, we can calculate mean returns and variance
covariance matrix of these assets as

14
R̄ =  8  ,
20


6×6
0.5 × 6 × 3 0.2 × 6 × 15
3×3
0.4 × 3 × 15 
Σ =  0.5 × 6 × 3
0.2 × 6 × 15 0.4 × 3 × 15
15 × 15

Suppose that the riskless rate is 5 %.
Tee (Riped)
Introduction
29 / 43
Example
Hence, a system of equations for this problem is
9 = 36Z1 + 9Z2 + 18Z3
3 = 9Z1 + 9Z2 + 18Z3
15 = 18Z1 + 18Z2 + 225Z3
The solution is

Z = 
Tee (Riped)
Introduction
14
63
1
63
3
63


30 / 43
Example
We can then find portfolio weight X as
X1 =
14
1
3
, X2 = , X3 =
18
18
18
The mean return of the optimal portfolio is
R̄P =
1
3
14
× 14 +
×8+
× 20 = 14.67%
18
18
18
The variance of the optimal portfolio is
σP2 = XT ΣX = 33.83%
Tee (Riped)
Introduction
31 / 43
Example: Solution
The slope of the efficient frontier is equal to 1.66.
Tee (Riped)
Introduction
32 / 43
Short Sales without Riskless
We will now consider a case where short sales are allowed but there is
no riskless asset.
We can use the same technique as before but with an assumed rate
R̃F . Different assumed rates will lead to different efficient portfolios.
Tee (Riped)
Introduction
33 / 43
Example Continued.
Suppose that the riskless rate is now R̃F = 2.
The system of equations now becomes
12 = 36Z1 + 9Z2 + 18Z3
6 = 9Z1 + 9Z2 + 18Z3
18 = 18Z1 + 18Z2 + 225Z3
whose solution is
Z1 =
42
72
6
7
12
1
, Z2 =
, Z3 =
and X1 = , X2 = , X3 =
189
189
189
20
20
20
and
R̄P = 10.7, σP2 = 13.70
Tee (Riped)
Introduction
34 / 43
Example Continued
In principle, we need to find only two of them (Two Fund Theorem).
That is, any combination of these two portfolios (which themselves
are assets) is on the efficient frontier.
For example: put 50 − 50 weight. We can show that σP2 = 21.859.
Then we can find the covariance between the two portfolios: using
σP2 = X12 σ12 + X22 σ22 + 2X1 X2 σ12
This leads to σ12 = 19.95.
With the information of expected returns, variances, and covariance
between the two portfolios, we can trace out the whole frontier.
Tee (Riped)
Introduction
35 / 43
Riskless but No Short Sales
We will now consider a case where short sales are not allowed but
there is a riskless asset.
In principle, an efficient portfolio problem is a constrained
maximization problem. In this case, we can write
max
Xi
R̄P − RF
σP
(28)
Xi
= 1
(29)
Xi
≥ 0, ∀i
(30)
subject to
∑
i
where the last one represents the no short-sales constraint.
Tee (Riped)
Introduction
36 / 43
Riskless but No Short Sales: Example
Consider again the example with three assets and risk-free rate
RF = 5%.
Recall that the efficient portfolio in this case is
X1 =
14
1
3
, X2 = , X3 =
18
18
18
Remember that this solution is solved under an assumption that short
sales are allowed.
What if we now impose the no short-sales constraint, should we get a
different answer?
Tee (Riped)
Introduction
37 / 43
Example II
Suppose there are three risky assets, say CP (asset 1), Centrals (asset
2), and PTT (asset 3).
Using past information, we can calculate mean returns and variance
covariance matrix of these assets as

14
R̄ =  8  ,
20


6×6
0.5 × 6 × 10 0.2 × 6 × 15
10 × 10
0.4 × 10 × 15 
Σ =  0.5 × 6 × 10
0.2 × 6 × 15 0.4 × 10 × 15
15 × 15

Suppose that the riskless rate is 5 %.
Tee (Riped)
Introduction
38 / 43
Example
Hence, a system of equations for this problem is
9 = 36Z1 + 30Z2 + 18Z3
3 = 30Z1 + 100Z2 + 60Z3
15 = 18Z1 + 60Z2 + 225Z3
The solution is


0.3000
Z =  −0.1019 
0.0698
Tee (Riped)
Introduction
39 / 43
Example
We can then find portfolio weight X as
X1 = 1.1197, X2 = −0.3803, X3 = 0.2607
But we do not allow for a negative weight! What should we do?
Tee (Riped)
Introduction
40 / 43
More General Efficient Portfolio Problem
This problem started from the seminal work by Markowitz (1959).
∑
∑∑
min
Xi2 σi2 +
Xi Xj σij
(31)
X
i
i
subject to
∑
∑
Xi
j̸=i
= 1
(32)
≥ R̄P
(33)
≥ 0, ∀i
(34)
≥ D
(35)
i
Xi R̄i
i
∑
Xi
Xi di
i
where the last constraint is the so called dividend requirement
constraint.
The role of a riskless asset is to simplify the objective function as a
slope.
Tee (Riped)
Introduction
41 / 43
CAPM: A first look
From the tangency portfolio, we have
σt2 = XT ΣX =
Rt − Rf
(
T
−1
1 Σ
R̄ −
Rf 1
)
(36)
Hence we can write
(
) Rt − Rf
1T Σ−1 R̄ − Rf 1 =
σt2
(37)
Similarly, covariance between the tangency portfolio and all assets is
given by
Covt = ΣX =
Tee (Riped)
R̄ − Rf 1
)
(
R̄ − Rf 1
1T Σ−1
Introduction
(38)
42 / 43
CAPM: A first look
Combining the last two equations lead to CAPM
R̄ − Rf 1 =
Covt
(Rt − Rf )
σt2
(39)
We now set the ratio between covariance and variance as β, e.g.,
1 ,Rt )
, where Rt is a return process of the tangency portfolio
βi = Cov (R
σt2
(a random variable).
We now have a CAPM with the tangency portfolio as a market
portfolio:
R̄ − Rf 1 = βt (Rt − Rf )
(40)
where βt = (βi )ni=1 .
Tee (Riped)
Introduction
43 / 43