MVS with N risky assets and a risk–free
asset
The weight of the risk–free asset is
xf = 1 − 10N x,
and the expected portfolio return is
µp = xf rf + x0µ = rf + x0(µ − 1N rf ).
Thus, the MVS–problem with a risk–free asset becomes
1
min x0Σx
x 2
subject to µp = rf + x0(µ − 1N rf )
Note that we can drop the restriction 10N x = 1 on
the weights of the risky assets, because the risk–free
asset makes up the residual.
1
MVS with N risky assets and a risk–free
asset
The Lagrange function is
1 0
L = x Σx + λ [µp − rf − x0(µ − 1N rf )] .
2
The solution to this problem is
xM V S (µp)
µp − r f
−1
(µ − 1N rf )
=
Σ
0
−1
(µ − 1N rf ) Σ (µ − 1N rf )
"
#
µp − r r
−1
=
(µ − 1N rf ).
Σ
2
a − 2brf + crf
2
MVS with N risky assets and a risk–free
asset
Thus, provided rf 6= b/c, minimum variance portfolios are combinations of the risk–free asset and a
portfolio of risky assets M with weight vector
xM =
1
Σ−1(µ − 1N rf ),
b − rf c
(1)
the “tangency portfolio”.
To see that (i) M ∈ M V S, and (ii) that M
is indeed the tangency portfolio, left–multiply (1) by
x0M Σ to get
2
σM
x0M (µ − 1N rf ) µM − rf
=
=
.
b − rf c
b − rf c
(2)
3
MVS with N risky assets and a risk–free
asset
The mean of M is obtained by left–multiplying (1)
by µ0, i.e.,
µM =
a − brf
bµM − a
⇔ rf =
.
b − crf
cµM − b
(3)
Substituting (the equality on the right–hand side
of) (3) into (2), we observe
2
σM
=
M −a
µM − bµ
cµ −b
M
M −a
b − c bµ
cµ −b
M
=
(cµ2M − 2bµM + a)/(cµM − b)
(bcµM − b2 − bcµM + ac)/(cµM − b)
=
cµ2M − 2bµM + a
,
d
which is the equation of the MVS, so M ∈ M V S.
4
MVS with N risky assets and a risk–free
asset
For (ii), note that the right–hand side equality in (3)
implies that rf is the mean of the zero–beta portfolio
with respect to M .
Thus, a tangency to the hyperbola at M intersects
the return axis at rf , or, portfolio M is at the tangency
point of the hyperbola and a line passing through rf .
That is, M is the tangency portfolio.
5
MVS with N risky assets and a risk–free
asset
The tangency portfolio is on the upper (lower)
branch of the hyperbola if rf < (>)b/c = µGM V P ,
i.e., we either have
µM > µGM V P > rf
or
µM < µGM V P < rf .
To prove this claim, just evaluate the product
=
(µM − µGM V P ) (µGM V P − rf )
a − brf b b − crf
−
b − crf c
c
=
(ac − cbrf − b2 + bcrf )(b − crf )
c2(b − crf )
=
d
> 0.
c2
6
MVS with N risky assets and a risk–free
asset
In the economically more relevant case, where rf <
b/c, efficient portfolios are combinations of a long
position in portfolio M and lending or borrowing at
the risk–free rate.
In the case where rf > b/c, efficient portfolios are
generated by short (or zero) positions in the tangency
portfolio (which is not efficient) and risk–free lending.
The efficient set is above the hyperbola.
The first analysis of these situations appeared in
Robert C. Merton (1972): An Analytic Derivation of
the Efficient Portfolio Frontier, Journal of Financial
and Quantitative Analysis, 7: 1851–1872.
7
Example
Consider a two–asset example with
µ=
2
5
,
Σ=
4 2
2 6
.
The MVS–constants are given by a = 4.2, b = 0.9,
c = 0.3, and d = ac − b2 = 0.45.
Thus, µGM V P = b/c = 0.9/0.3 = 3.
8
Example
Assume rf = 1 < 3 = µGM V P .
Then
xM =
Σ
−1
(µ − rf 12)
=
b − crf
−0.1667
1.1667
,
with µM = 5.5 and σM = 2.7386, and
θM
µM − r f
=
= 1.6432.
σM
(4)
Thus, the MVS is
µp(M V S) = 1 ± σp(M V S)1.6432.
9
Example
case 1: 1 = rf < µGMVP = b/c = 3
10
8
6
4
tangency portfolio M
2
0
−2
−4
−6
−8
0
1
2
3
4
5
6
10
Example
Assume rf = 4 > 3 = µGM V P .
Then
xM =
Σ
−1
(µ − rf 12)
=
b − crf
2.3333
−1.3333
,
with µM = −2 and σM = 4.4721, and
θM
µM − r f
=
= −1.3416.
σM
(5)
Thus, the MVS is
µp(M V S) = 4 ± σp(M V S)(−1.3416).
The efficient set is above the hyperbola.
11
Example
case 2: 4 = rf > µGMVP = b/c = 3
12
10
8
6
4
2
tangency portfolio M
0
−2
−4
0
1
2
3
4
5
6
12
Example
Now consider the case rf = 3 = µGM V P .
Then,
10N Σ−1(µ − 1N rf ) = b − rf c = 0,
so that the portfolio of risky assets is a zero–investment
portfolio, i.e., a portfolio with zero net value, created
by buying and shorting equal amounts of securities.
13
Example
Holding this portfolio on a scale γ > 0 results in a
portfolio mean
µQ = γx0M µ = (µ − rf 12)0Σ−1µ
b
= γ(a − brf ) = γ a − b
c
= (γ/c)(ac − b2) = γd/c > 0,
so efficient portfolios are combinations of a full investment in the risk–free asset and a long position in the
zero–investment portfolio.
14
Example
To figure out the MVS, we compute the variance,
2
σQ
= γ 2(µ − rf 12)0Σ−1ΣΣ−1(µ − rf 12)
= γ 2(µ − rf 12)0Σ−1(µ − rf 12)
= γ 2(a − 2brf + crf2 )
2
2
2b
b
= γ2 a −
+
c
c
=
γ2
d,
c
or
σQ = γ
r
d
⇔ γ(σQ) = σQ
c
r
c
.
d
15
Example
Substituting γ(σQ) into µQ = γd/c, we find
µQ =
r
cd
σQ =
dc
r
d
σQ ,
c
and combining a full investment in the risky asset with
the zero–investment portfolio Q produces the MVS
µp = r f ±
r
d
σQ = µGM V P ±
c
r
d
σp ,
c
so that the MVS is given by the asymptotes of the
hyperbola which describes the risky assets–only MVS.
There is no tangency portfolio.
16
Example
In our example,
xQ =
4 2
2 6
−1 2−3
5−3
=
−0.5
0.5
.
The MVS is
µp = r f ±
r
d
σp = 3 ±
c
r
0.45
σp = 3 ± 1.225σp.
0.3
17
Example
case 3: 3 = r = µ
f
GMVP
= b/c = 3
20
15
10
5
0
−5
−10
0
2
4
6
8
10
12
18