Financial Economics
Week 3
Introduction
Filippo Massari
University of Bologna
23 Feb 2024
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The efficient frontier with many assets
Suppose there are N risky assets (where N can be arbitrarily large)
A portfolio of assets is a vector (we shall define vectors as column
vector and indicate row vectors by the transpose symbol X′ )
X′ = [X1 , X2 , ..., XN ]
where the X s are the fraction of wealth invested in each asset
If short sales are allowed, some of the X s can be negative (of course,
not all of them, since the sum must be 1)
Each asset i has an expected return R̄i
The vector of expected returns is
R′ = [R̄1 , R̄2 , ..., R̄N ]
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Many assets
The matrix of variances and covariances (for brevity, covariance
matrix) is

 2
σ1 σ12 σ13 ... σ1N
 σ21 σ22 σ23 ... σ2N 


2

Σ=
 σ31 σ32 σ3 ... σ3N 
 ...
...
... ... ... 
σN1 σN2 σN3 ... σN2
With N assets, the variance-covariance matrix has N 2 entries.
N (N + 1)
However, σ12 = σ21 etc. so the independent entries are
.
2
The inputs for the portfolio selection process (in addition to the
N (N + 1)
investor’s preferences over risk) are
variances and
2
covariances plus the N expected returns
With 500 assets, the inputs are about 125,000!
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Many assets
We assume that all assets are risky
In matrix notation, the expected return of a portfolio is
R̄P = X′ R
The variance of the portfolio rate of return is
σP2 = X′ ΣX
One can easily verify that in the two asset case one re-obtains the
formulas of Lecture 2
Note : the covariance matrix is symmetric and positive defined, so
X′ ΣX is a quadratic form
In our analysis we shall assume that Σ has full rank (is not singular,
has a determinant different from 0)
The economic interpretation is that there are no risk free assets and
no asset is perfectly correlated with a combination of the others
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Many assets: a simple case
In general the variance-covariance matrix can be very complicated
A simple special case is that all variances are equal and also that all
covariances are equal (but different from the variances)
In this case the covariance matrix becomes

 2
σ
ρσ2 ... ρσ2
ρσ2 σ2 ... ρσ2 

Σ = 
 ...
... ... ... 
ρσ2 ρσ2 ... ρσ2


1 ρ ... ρ
 ρ 1 ... ρ 

= σ2 
... ... ... ...
ρ ρ ... ρ
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Many assets: a simple case
Assume, in addition, that
1 1
1
X =
, , ...,
N N
N
′
That is, the same fraction of wealth is invested in each of the N assets
In this case the variance of the portfolio becomes
σP2
2
1 2 2
1
σ + N (N − 1)
ρσ2
N
N
N −1
σ2
ρσ2
+
N
N
= N
=
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Many assets: a simple case
σP2 =
σ2
+
N
N −1
N
ρσ2
The first term is the contribution to the portfolio risk of the riskiness
of the individual assets (idiosyncratic risk – also called diversifiable
risk, firm-specific risk etc.), the second term is the contribution to
the portfolio variance of the various assets being correlated
(nondiversifiable risk – also called market risk, systematic risk
etc.)
As N increases, the first term vanishes and the second term converges
to ρσ2
That is, idiosyncratic risk can be eliminated by diversification, but
nondiversifiable risk cannot
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Many assets: a simple case
The figure shows how the risk of the portfolio decreases as the
number of securities included increases using average variance and
covariance of the relevant markets.
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The efficient frontier with many assets
One way to calculate the efficient frontier is to solve the problem
= X′ ΣX
s.t. X′ R = R̄P (a given constant)
X′ 1 = 1
min σP2
where 1 is a vector of ones, for different values of Rp (Merton, JFQA,
1972, Roll, JFE 1977)
An alternative but equivalent method is to solve the problem
1
1
max t R̄P − σP2 = tX′ R− X′ ΣX
2
2
′
s.t. X 1 = 1
for different values of t (Szego 1974)
The parameter t is called tolerance for reasons that will become clear
when we study the investor’s preferences
For each value of R̄P (or t) we find one point of the efficient frontier;
by varying R̄P (or t) we can trace out the entire frontier
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The efficient frontier with many assets
We start from the second method
Since we have a constrained maximisation problem, we set up the
Lagrangian
1
L = tX′ R− X′ ΣX+λ(1 − X′ 1)
2
where λ is a Lagrange multiplier
First order conditions for a maximum are (the derivative of a
quadratic form is similar to the derivative of ax 2 )
tR−ΣX−λ1 = 0
X′ 1 = 1
The solution is
X =Σ−1 (tR−λ1)
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The efficient frontier with many assets
There are two parameters in the solution
However, one is pinned down by the adding up condition
X′ 1 = 1′ X = 1
We can get rid of the redundant parameter by noting that
λ
−1
X =tΣ
R− 1
t
Denoting
λ
= c we have
t
Σ−1 (R−c1)
X = ′ −1
1 Σ (R−c1)
By varying c we can now trace out the entire frontier
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The efficient frontier with many assets
Using the constraint 1′ X = 1 to eliminate λ we get
B −1
1 −1
−1
X = Σ 1 + t Σ R− Σ 1
C
C
where
C = 1′ Σ −1 1
and
B = R′ Σ −1 1
are scalars
Clearly, X is linear in t
Note that we are allowing for short sales and hence some of the X s
can be negative
Define also
A = R′ Σ−1 R and D = AC − B 2
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The efficient frontier with many assets
Given the solution we have found, we can calculate
RP
= R′ X
B2
B
+ t (A −
)
=
C
C
1
=
(B + Dt )
C
and
σP2 = X′ ΣX =
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Dt 2 + 1
C
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The efficient frontier with many assets
From these conditions we easily obtain
σP2 =
A − 2BRP + CRP2
D
This is the equation of a parabola
In terms of the standard deviation
s
A − 2BRP + CRP2
σP =
D
which is the equation of a hyperbola
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The efficient frontier with many assets
Consider now the other procedure, which is equivalent to
= X′ ΣX
s.t. X′ R = R̄P
X′ 1 = 1
max −σP2
The Lagrangian is
1
L = − X′ ΣX+µ(R̄P − X′ R)+λ(1 − X′ 1)
2
where now we have two Lagrange multipliers, λ and µ
But this is equivalent to the Lagrangian we have used before setting
µ = t (they differ only by a constant term µR̄P )
In fact µ will depend on R̄P and so must vary arbitrarily, just as t does
This shows that the two methods are equivalent
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The two mutual funds theorem
We can now prove one of the most elegant results of portfolio theory:
the two mutual funds theorem (also known as the two funds
theorem, the mutual funds theorem, or the separation theorem)
The theorem says that the efficient frontier can be generated by
combining any two portfolios on the frontier, just as we combined
single assets in Lecture 1
This implies that all an investor has to do is to calculate two efficient
portfolios; any other efficient portfolio then is a combination of these
two (possibly with negative weights for one portfolio)
Thus the problem of finding the optimal portfolio becomes the
problem of finding the optimal weights
We shall see later that when there is a risk free asset the two mutual
funds theorem has even striker implications
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The two mutual funds theorem
Proof: The trick is to note that X is linear in t
B −1
1 −1
−1
Σ 1 + t Σ R− Σ 1
X =
C
C
= M + tN
Take two different values of t, t1 ̸= t2 , (say t1 > t2 ) and consider the
corresponding efficient portfolios
X1 = M + t1 N
X2 = M + t2 N
Solve this system for M and N
N =
M =
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X1 − X2
t1 − t2
t 1 X2 − t 2 X1
t1 − t2
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The two mutual funds theorem
Given the solution for M and N
N =
M =
X1 − X2
t1 − t2
t 1 X2 − t 2 X1
t1 − t2
we can express any efficient portfolio X as a combination of X1 and
X2
(t1 − t )X2 −(t2 − t )X1
X = M + tN =
t1 − t2
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Risk-free asset
Now suppose that there is a risk free asset (if there were many they
should all have the same return, otherwise – if they are really risk-free
– there would be arbitrage opportunities)
In the past, the prototypical example of risk-free assets would have
been a government bond
Today it is apparent that even government bonds can be risky for
many countries
Nevertheless, one can still think of government bonds of such
countries as US or Germany as risk free
With a risk free asset, the covariance matrix is singular (there will be
one row and one column of 0s) so we cannot apply the above analysis
However, it turns out that the analysis is in fact even simpler
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Two risky assets and one risk-free asset
Recall that we can combine not only individual assets but also
portfolios of assets
In particular, we can combine any portfolio on the risk-return frontier
with the risk free asset
As we have shown, the expected return/standard deviation for any
portfolio which is the combination of a risky asset or portfolio and the
risk free asset must lie on the straight line connecting the risky
portfolio to the risk free asset
Going short on the risk free asset is equivalent to borrowing at the
risk free interest rate (something that individuals and even many
governments cannot in fact do).
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Two risky assets and one risk-free asset
Clearly, there will be one efficient risky portfolio, the tangency
portfolio.
The efficient frontier will be a mixture of the risk-free asset and the
tangency portfolio
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The separation theorem
From the figure above we can draw a simple conclusion
Because the efficient frontier is a mixture of the risk free asset and
the optimal risky portfolio, risk preferences do not determine which
risky portfolio is chosen, but only the proportion of the risk-free asset
and the efficient risky portfolio
This conclusion, first noticed by Nobel laureate James Tobin (REStat
1958), is called the separation theorem (or sometimes, more
modestly, the separation property)
It may be seen as a special case of the two mutual funds theorem and
implies that the selection of the optimal portfolio can be divided into
two steps:
finding the optimal risky portfolio
finding the share of wealth to be invested in the risky portfolio
The investor’s preferences come into play only at the second stage
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The separation theorem
The optimal risky portfolio will of course depend on the investor’s
assessment of the expected return, the variance of returns and the
covariance of returns of the risky assets
However, it does not depend on the investor’s preferences over risk
and return
This implies that if you have many different clients, with different
attitudes towards risk, you will recommend different combinations of
the risk free asset and the optimal risky portfolio
However, you will recommend the same risky portfolio to all of
your clients, irrespective of their risk attitude
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The Capital Allocation Line (CAL)
The capital allocation line (CAL) is the line that joins the risk free
asset and the optimal risky asset
It must be tangent to the efficient frontier
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The Sharpe ratio
The Sharpe ratio is the slope of the capital allocation line
It is given by
R̄P − RF
Sharpe ratio =
σP
Risk premium
=
Volatility
where R̄P is the expected rate of return of the optimal risky portfolio
and σP is its standard deviation
The numerator of the Sharpe ratio can be interpreted as a risk
premium (the excess return which is obtained by holding a risky
portfolio); the denominator as a measure of volatility of the portfolio
which yields excess returns
This reward to volatility measure was first proposed by William
Sharpe, who shared with Markowitz the Nobel prize in 1990
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The optimal risky portfolio
The optimal risky portfolio is the one with the highest Sharpe ratio
To find it, one must
maximise θ =
R̄P − RF
σP
among all feasible portfolios of risky assets
Clearly, the solution will necessarily lie on the efficient frontier
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The optimal risky portfolio
Recalling the expressions for R̄P and σP , our problem is to
maximise θ =
=
R̄P − RF
σP
X1 (R̄1 − RF ) + X2 (R̄2 − RF )
q
X12 σ12 + X22 σ22 + 2X1 X2 σ12
We need not impose the constraint X1 + X2 = 1 as the objective
function is homogeneous of degree 0 in the X s
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The optimal risky portfolio
The first order conditions are
dθ
dX1
dθ
dX2
= 0,
= 0
which are equivalent to
1
2X1 σ12 + 2X2 σ12
(R̄1 − RF ) σP − (R̄P − RF )
2
σP
1
2X2 σ22 + 2X1 σ12
(R̄2 − RF ) σP − (R̄P − RF )
2
σP
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= 0,
= 0
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The optimal risky portfolio
Rearranging
R̄P − RF
2
X
σ
+
X
σ
= 0,
1
2
12
1
σP2
R̄P − RF
X2 σ22 + X1 σ12 = 0
(R̄1 − RF ) −
2
σP
(R̄1 − RF ) −
Define new variables
Z1 =
R̄P − RF
X1
σP2
Z2 =
R̄P − RF
X2
σP2
This trick works because all we actually need to find are the shares
X1
X1 +X2 , which are the same for the X s and the Z s
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The optimal risky portfolio
With this substitution, the system becomes
Z1 σ12 + Z2 σ12 = (R̄1 − RF )
Z1 σ12 + Z2 σ22 = (R̄2 − RF )
The solution is
Z1 =
Z2 =
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(R̄1 − RF ) σ22 − (R̄2 − RF ) σ12
2
σ12 σ22 − σ12
(R̄2 − RF ) σ12 − (R̄1 − RF ) σ12
2
σ12 σ22 − σ12
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The optimal risky portfolio
With the Z s, we can then obtain the X s as follows:
X1 =
(R̄1 − RF ) σ22 − (R̄2 − RF ) σ12
(R̄1 − RF ) (σ22 − σ12 ) + (R̄2 − RF ) (σ12 − σ12 )
and X2 = 1 − X1
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Optimal risky portfolio with many assets
However large the number of assets, the efficient frontier in the
(σP , RP ) space is a hyperbola
Above the minimum variance portfolio, the frontier is necessarily
concave (the geometric arguments used in the two asset case
continue to hold with no change in the N asset case)
Thus, the optimal risky portfolio is again the one that maximises the
Sharpe ratio
θ =
=
=
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R̄P − RF
σP
XR − RF
√
X′ ΣX
′
X (R−RF 1)
√
X′ ΣX
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Optimal risky portfolio with many assets
The first order condition is (again we need not worry about the
constraint X′ 1 = 1)
(R−RF 1)
√
X′ ΣX −
or
(R−RF 1) −
Filippo Massari (University of Bologna)
1 X′ (R−RF 1) 2ΣX
√
=0
2
X′ ΣX
X′ (R−RF 1)
ΣX = 0
X′ ΣX
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Optimal risky portfolio with many assets
Denoting
X ′ ( R − RF 1 )
X′ ΣX
(a scalar) and λX = Z,the system reduces to
λ=
ΣZ = (R−RF 1)
The solution is
Z = Σ − 1 ( R − RF 1 )
The X s can now be easily recovered as
X =
=
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1
Z
1′ Z
Σ−1 (R−RF 1)
1 ′ Σ − 1 ( R − RF 1 )
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Optimal risky portfolio: no short sales
The solution we have found typically involves Xi ̸= 0; however, Xi can
be either positive or negative
A negative value of Xi means that you should go short on asset i
If short sales are prohibited, one must add the constraint X ≥ 0 in the
optimisation problem considered above
A closed form solution is no longer available, although all the main
qualitative results we have obtained (namely, the shape of the frontier
and the separation property) continue to hold
However, the relationship between the efficient frontier with and
without short sales is not as simple as in the two assets case
With no short sales, not only does the frontier not extend indefinitely:
it also lies everywhere below the frontier with short sales
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Borrowing constraints
If the investor cannot borrow at the risk free rate, the separation
theorem no longer holds
The efficient frontier becomes
itbpF 2.2477in2.4785in0inFigure
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Borrowing constraints
If the investor can only borrow at an interest rate higher the risk free
rate, the efficient frontier will be
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Borrowing constraints
With borrowing constraints, we have a three mutual funds theorem
That is, the efficient frontier can be generated by combing three
different portfolios: the risk free asset (which has a different rate of
return depending on whether you go short or long), portfolio G, and
portfolio H
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