The paper presents a mean-variance frontier based on dynamic frictionless investment strategies in continuous time. The result applies to a finite number of risky assets whose price process is given by multivariate geometric Brownian motion with deterministically varying coefficients. The derivation is based on the solution for the frontier in discrete time. Using the same multiperiod framework as Li and Ng (2000), I provide an alternative derivation and an alternative formulation of the solution. It allows for a nice asymptotic formulation of the efficient hyperbola and its underlying efficient processes that applies in continuous time.

Bekker, Paul A.

English

Research Papers in Economics

A Mean-Variance Frontier in Discrete and Continuous Time
Paul A. Bekker
Department of Economics
University of Groningen
2004
Department of Economics, University of Groningen, P.O.Box 800, 9700 AV Groningen, The Netherlands.
Email: P.A.Bekker@eco.rug.nlAbstract
The paper presents a mean-variance frontier based on dynamic frictionless investment strate-
gies in continuous time. The result applies to a ﬁnite number of risky assets whose price process
is given by multivariate geometric Brownian motion with deterministically varying coeﬃcients.
The derivation is based on the solution for the frontier in discrete time. Using the same mul-
tiperiod framework as Li and Ng (2000), I provide an alternative derivation and an alternative
formulation of the solution. It allows for a nice asymptotic formulation of the eﬃcient hyperbola
and its underlying eﬃcient processes that applies in continuous time.
Keywords: multiperiod mean-variance frontier, discrete time, continuous time1 Introduction
Dynamic mean-variance optimal solutions have been used, mainly in a univariate context, to
study asset allocation and derivative pricing by, among many others, Richardson (1989), Duﬃe
and Richardson (1991), Sch¨ al (1994), Schweizer (1995), Bajeux-Besnainou and Portait (2002).
In particular Li and Ng (2000) generalized Markowitz (1952, 1959). They provided an analytical
expression for the mean-variance frontier along with optimal portfolio policy for a multivariate
multiperiod portfolio selection problem where the returns of risky assets are assumed to be
independent over time. The solution was hard to ﬁnd due to problems of nonseparability in
the sense of dynamic programming. Their solution scheme was to embed the original problem
into a tractable auxiliary separable problem and to investigate the relationship between the
solution sets. Leippold et al. (2004) use this approach to cover portfolios consisting of both
assets and liabilities. They emphasize the use of orthogonal projections and provide a ﬁnancial
interpretation of the desired optimal policies.
Here I start by considering the same problem and derive the solution in a more direct way.
Similar to the standard single-period problem the mean-variance frontier is fully described by
three of the ﬁrst two moments of two eﬃcient strategies. As the moments of the returns of
the risky assets are not path dependent, the strategy that minimizes the expected squared
return is easily found. Its solution does not depend on the number of periods. So the problem
reduces to ﬁnding a single moment of another eﬃcient strategy. If there is no riskless asset
present this problem is nontrivial. Still, relatively simple steps lead to its solution. The overall
solution describes the three moments in terms of the mean-variance parameters that apply in
the separate periods.
In a next step the limits of the relevant expressions are found as the number of periods
increases, while the variances and expectations in the separate periods are shrunk at the same
time. The underlying price process converges to geometric Brownian motion, with deterministi-
cally varying coeﬃcients, and the mean-variance solution converges to the solution in continuous
time. It generalizes the continuous-time eﬃcient frontier of Zhou and Li (2000) to the case where
a riskless asset need not be present. In a separate paper the solution is used to model the current
1yield curve1. The implied mean-variance parameters, which are found by ﬁtting yield curves,
are shown to be closely related to the business cycle and to the equity risk premium.
2 The model and the solution in discrete time
Consider a ﬁnite number of assets whose price vector at time ti is given by Sti, i = 0,1,...,n.
Let the vector of gross returns over the separate periods be given by θti = Diag(Sti−1)−1Sti,
i = 1,...,n. The returns are assumed to be independent over time with ﬁnite ﬁrst and second
moments. Consider frictionless trading and self-ﬁnancing strategies. That is, if equity positions
at time ti are given in the vector φti, and the portfolio value is given by Vti = ι0φti,where ι is a
vector of ones, then Vti = ι0φti = θ0
tiφti−1, i = 1,...,n. The returns of the strategies are given
by Rtn = Vtn/Vto. Consider strategies with ﬁnite ﬁrst and second moments. They form a linear
space. Consequently, standard one-period mean-variance analysis and its results (cf. Cochrane
2001) apply to the dynamic framework as well.
2.1 The frontier
Let the ﬁrst two moments be denoted by m = E(Rtn), s2 = E(R2
tn) and let the variance be
given by σ2 = Var(Rtn). The eﬃcient frontier is usually formulated in terms of the Global
Minimum Variance (GMV) portfolio. Let the strategy that minimizes σ2 have moments m
GMV
and σ
GMV2 and let Γ be a third parameter, then the frontier for eﬃcient strategies, where σ2 is
minimal conditional on m, can be formulated as
σ2 = σ
GMV2 +

m − m
GMV
Γ
2
. (1)
Another formulation of the frontier is in terms of the Least Squared Return (LSR) strategy with
moments m
LSR and s
LSR2, that minimizes s2 . It is given by
s2 = s
LSR2 +

m − m
LSR
F
2
. (2)
1Based on approximate bond-replicating strategies that hedge against constant claims at varying maturities and
by using appropriate risk premia for the residual risk, I arrive at a model for the current yield curve without making
assumptions about the dynamic evolution of yield curves.
2The relations between the two sets of parameters are easily found by using the optimality of
GMV and LSR in (2) and (1), respectively:
m
GMV =
m
LSR
1 − F2,
σ
GMV2 = s
LSR2 −
m
LSR2
1 − F2, (3)
Γ2 =
F2
1 − F2.
Another useful eﬃcient strategy is found by minimizing σ2/m2 and will be referred to as the
Minimum Risk (MR) strategy. Its expected return is given by
m
MR = m
LSR +
s
LSR2
mLSRF2. (4)
2.2 The solution
Deﬁne the ﬁrst two moments of the returns mti ≡ E(θti+1) and Ωti ≡ E(θti+1θ0
ti+1), which is
assumed to be nonsingular, and let single period mean-variance parameters, for i = 0,...,n−1,
be given by
slsr
ti
2
=
1
ι0Ω−1
ti ι
, mlsr
ti =
m0
tiΩ−1
ti ι
ι0Ω−1
ti ι
, f2
ti = m0
ti
 
Ω−1
ti −
Ω−1
ti ιι0Ω−1
ti
ι0Ω−1
ti ι
!
mti. (5)
The solution for the multi-period mean-variance frontier parameters of frontier (2), which will
be derived in the next subsection, is given by
m
LSR =
n−1 Y
i=0
mlsr
ti , (6)
s
LSR2 =
n−1 Y
i=0
slsr
ti
2
, (7)
F2 =
n−1 X
j=0

f2
tj
n−1 Y
i=j+1
mlsr
ti
2
slsr
ti
2

, (8)
where
Qn−1
i=n mlsr
ti
2
/slsr
ti
2
= 1. For n = 1 the multi-period solution coincides with the one-period
solution.
32.3 The derivation
Describing the LSR strategy and its moments is easy. As the returns are independent the
minimization of s2 amounts to minimization in each period separately, i.e.
E(R2
tn) =
n Y
i=1
E



 
θ0
tiφti−1
ι0φti−1
!2



is minimized by
φ
LSR
ti =
Ω−1
ti ι
ι0Ω−1
ti ι
V
LSR
ti , (9)
i = 0,...,n − 1. Consequently,
m
LSR =
n Y
i=1
E
 
θ0
tiΩ−1
ti−1ι
ι0Ω−1
ti−1ι
!
=
n Y
i=1
m0
ti−1Ω−1
ti−1ι
ι0Ω−1
ti−1ι
=
n−1 Y
i=0
mlsr
ti ,
s
LSR2 =
n Y
i=1
E



 
θ0
tiΩ−1
ti−1ι
ι0Ω−1
ti−1ι
!2

 =
n Y
i=1
1
ι0Ω−1
ti−1ι
=
n−1 Y
i=0
slsr
ti
2
,
which amounts to (6) and (7).
To derive (8) consider the strategy that minimizes E{(Vtn − 1)2}. As EVtn = Vtom and
VarVtn = V 2
toσ2, it follows that E{(Vtn−1)2} = (Vtom−1)2+V 2
toσ2, which is minimized for ﬁxed
m and σ2 by Vto = m/s2. For this optimal value of Vto the minimum of E{V tn−1)2)} = σ2/s2
is found by investing Vto in the MR strategy that minimizes both σ2/m2 and σ2/s2. Therefore,
the portfolio value at time ti will be indicated as V
MR
ti . Cochrane (2001) refers to its return as
the constant-mimicking return.
Alternatively, the initial value V
MR
to = m
MR/s
MR2 can be found as follows. Due to the
optimality of the portfolio, V
MR
tn − 1 is orthogonal to any other portfolio, in particular to R
LSR
tn ,
i.e.
E

R
LSR
tn (V
MR
tn − 1)
	
= 0. (10)
Similarly, due to its optimality, R
LSR
tn is orthogonal to any portfolio with initial value equal to
0, in particular to V
MR
tn /V
MR
to − R
LSR
tn ,
E
(
R
LSR
tn
 
V
MR
tn
V
MR
to
− R
LSR
tn
!)
= 0. (11)
4Combining (10) and (11) the initial value is found as
V
MR
to =
m
LSR
sLSR2. (12)
Consequently, the expected portfolio value at maturation is given by
E(V
MR
tn ) =
m
MRm
LSR
sLSR2 . (13)
Due to (4) the problem is solved once this expectation is expressed in the mean-variance pa-
rameters of the separate periods.
To achieve this goal the equity positions φ
MR
ti of the portfolio have to be found. To start
with to, φ
MR
to satisﬁes ι0φ
MR
to = V
MR
to as given by (12). Now, the steps that led to this result can
be repeated with ι replaced by another, arbitrary vector ρ, say. That is, consider the LSR-ρ
portfolio that minimizes E{(V
LSR−ρ
tn )2} under the restiction ρ0φ
LSR−ρ
to = 1. Similar to (6) and
(7) the ﬁrst two moments of this portfolio are found as
E(V
LSR−ρ
tn ) =
m0
toΩ−1
to ρ
ρ0Ω−1
to ρ
n−1 Y
i=1
mlsr
ti , (14)
E{(V
LSR−ρ
tn )2} =
1
ρ0Ω−1
to ρ
n−1 Y
i=1
slsr
ti
2
. (15)
Furthermore, (10) holds if R
LSR
tn is replaced by R
LSR−ρ
tn and similar to (11),
E
(
R
LSR−ρ
tn
 
V
MR
tn
ρ0φ
MR
to
− R
LSR−ρ
tn
!)
= 0 (16)
must hold, and so, similar to (12),
ρ0φ
MR
to = ρ0Ω−1
to mto
n−1 Y
i=1
mlsr
ti
slsr
ti
2.
As ρ is arbitrary, the initial equity positions of the MR portfolio are given by
φ
MR
to = Ω−1
to mto
n−1 Y
i=1
mlsr
ti
slsr
ti
2. (17)
5To derive the equity positions for ti, i = 1,...,n − 1, ﬁrst consider the LSR-i and MR-i
portfolios that are deﬁned similar to the LSR and MR portfolios with the diﬀerence that the
former portfolios start at ti instead of to. Similar to (9) and (17) the initial equity positions are
given by
φ
LSR−i
ti =
Ω−1
ti ι
ι0Ω−1
ti ι
, φ
MR−i
ti = Ω−1
ti mti
n−1 Y
j=i+1
mlsr
tj
slsr
tj
2. (18)
The positions of the MR portfolio at time ti will be diﬀerent in general from φ
MR−i
ti due to the
fact that the initial value of the MR-i portfolio is not restricted, whereas φ
MR
ti should satisfy
ι0φ
MR
ti = V
MR
ti .
So, the MR portfolio for tj, j = i,...,n, can be deﬁned as the solution to the problem of
minimizing E(Vtn − 1)2 conditional on Vti = V
MR
ti . As Vtn = V
MR−i
tn +(Vtn −V
MR−i
tn ) and due to
the orthogonality of V
MR−i
tn −1 to any other portfolio, in particular to Vtn−V
MR−i
tn , the deﬁnition
of the MR portfolio for tj, j = i,...,n amounts to minimizing
E(Vtn − 1)2 = E(V
MR−i
tn − 1)2 + E(Vtn − V
MR−i
tn )2,
conditional on Vti = V
MR
ti . Then, clearly, Vti − V
MR−i
ti should be invested in the LSR-i strategy,
i.e. V
MR
tj − V
MR−i
tj = V
LSR−i
tj (V
MR
ti − V
MR−i
ti ) for j = i,...,n. As a result we ﬁnd the equity
positions of the MR portfolio for all ti equal to
φ
MR
ti = φ
MR−i
ti + φ
LSR−i
ti (V
MR
ti − V
MR−i
ti )
= Ω−1
ti mti
n−1 Y
j=i+1
mlsr
tj
slsr
tj
2 +
Ω−1
ti ι
ι0Ω−1
ti ι

V
MR
ti −
n−1 Y
j=i
mlsr
tj
slsr
tj
2

,
= vti
n−1 Y
j=i+1
mlsr
tj
slsr
tj
2 + wtiV
MR
ti , (19)
where (18) has been used for the second equality and
vti ≡
 
Ω−1
ti −
Ω−1
ti ιι0Ω−1
ti
ι0Ω−1
ti ι
!
mti, (20)
wti ≡
Ω−1
ti ι
ι0Ω−1
ti ι
. (21)
6Finally, (19) can be used to describe the MR portfolio values recursively:
V
MR
ti+1 = θ0
ti+1vti
n−1 Y
j=i+1
mlsr
tj
slsr
tj
2 + θ0
ti+1wtiV
MR
ti .
Consequently, the ﬁnal portfolio value is given by
V
MR
tn =
n−1 Y
i=0

θ0
ti+1wti
mlsr
ti
slsr
ti
2

 +
n X
j=1

θ0
tivtj−1
n−1 Y
i=j
θ0
ti+1wti
mlsr
ti
slsr
ti
2

.
As the returns are independent and E(θ0
ti+1vti) = f2
ti and E(θ0
ti+1wti) = mlsr
ti , as in (5), the
expectation of the ﬁnal portfolio value is given by
E(V
MR
tn ) =
n−1 Y
i=0
mlsr
ti
2
slsr
ti
2 +
n X
j=1

f2
tj−1
n−1 Y
i=j
mlsr
ti
2
slsr
ti
2

.
Using (4) and (13) the result (8) follows.
3 Mean-variance eﬃciency in continuous time
In order to formulate the eﬃcient mean-variance frontier in continuous time, along with two
eﬃcient strategies, consider the returns θt+∆t over a period ∆t, where ∆t = (T − to)/n and
T = tn is ﬁxed as ∆t → 0. Let the ﬁrst two moments of the returns be given by
Ωt = mtm0
t + Σt∆t and mt = ι + µt∆t,
and the limit process by
dSt = Diag(St)(µtdt + Σ
1/2
t dWt),
where Wt is multivariate Brownian motion and µt and Σt vary deterministically over time and
Σt is assumed to be nonsingular for all t ∈ [t0,T]. Finally, deﬁne instantaneous mean-variance
parameters
α2
t =
1
ι0Σ−1
t ι
, βt =
µ0
tΣ−1
t ι
ι0Σ−1
t ι
, γ2
t = µ0
t
 
Σ−1
ti −
Σ−1
t ιι0Σ−1
t
ι0Σ−1
t ι
!
µt, (22)
7which are assumed to be bounded and integrable over [t0,T].
After some algebraic manipulations I ﬁnd
m0
tΩ−1
t ι =
1 + βt∆t
α2
t∆t(1 + γ2
t ∆t) + (1 + βt∆t)2,
ι0Ω−1
t ι =
1 + γ2
t ∆t
α2
t∆t(1 + γ2
t ∆t) + (1 + βt∆t)2,
m0
tΩ−1
t mt =
α2
tγ2
t (∆t)2 + (1 + βt∆t)2
α2
t∆t(1 + γ2
t ∆t) + (1 + βt∆t)2.
Thus, the parameters (5) can be expressed as
slsr
t
2
= 1 + (2βt + α2
t − γ2
t )∆t + o(∆t),
mlsr
t = 1 + (βt − γ2
t )∆t + o(∆t),
f2
t = γ2
t ∆t + o(∆t),
and the the frontier parameters (6), (7) and (8) are given in continuous time by
sLSR2
= exp
(Z T
to
(2βt + α2
t − γ2
t )dt
)
, (23)
mLSR = exp
(Z T
to
(βt − γ2
t )dt
)
, (24)
F2 =
Z T
to
γ2
t exp
(
−
Z T
t
(α2
u + γ2
u)du
)
dt. (25)
Notice that, although Σt was assumed to be nonsingular in the derivation, the solution allows αt
to be arbitrary close to 0. Due to continuity the expressions where αt is replaced by 0 represent
the parameters that describe the frontier in the presence of a riskless asset.
If αt, βt and γt are constant, the parameters (23), (24) and (25) equal
sLSR2
= exp
n
(2β + α2 − γ2)(T − to)
o
,
mLSR = exp
n
(β − γ2)(T − to)
o
,
F2 =
γ2
α2 + γ2
h
1 − exp
n
−(α2 + γ2)(T − t0)
oi
,
respectively. For example, the problem of ﬁnding the minimum variance for a portfolio value
8satisfying E(VT) = 1, which is given by
σ
MR2
mMR2 =
s
LSR2
mMRmLSR − 1,
converges to the ﬁnite value α2/γ2 as T → ∞.
To describe the LSR and MR processes, for which the discrete versions have been given in
(9) and (19), respectively, consider the limits of (20) and (21):
vt →
 
Σ−1
t −
Σ−1
t ιι0Σ−1
t
ι0Σ−1
t ι
!
µt,
wt →
Σ−1
t ι
ι0Σ−1
t ι
−
 
Σ−1
t −
Σ−1
t ιι0Σ−1
t
ι0Σ−1
t ι
!
µt.
Consequently, the following limits are found:
µ0
tvt → γ2
t ,
µ0
twt → βt − γ2
t ,
and
v0
tΣ−1
t vt → γ2
t ,
v0
tΣ−1
t wt → −γ2
t ,
w0
tΣ−1
t wt → α2
t + γ2
t .
Based on these moments the processes are given by
dV
LSR
t = V
LSR
t

(βt − γ2
t )dt +
q
α2
t + γ2
t dW
(1)
t

,
dV
MR
t = V
MR
t

(βt − γ2
t )dt +
q
α2
t + γ2
t dW
(1)
t

+ (γ2
t dt + γtdW
(2)
t )exp
(
−
Z T
t
(βt + α2
t)dt
)
,
where W
(1)
t and W
(2)
t are Brownian motions with deterministically varying correlation equal to
−
r
γ2
t
α2
t+γ2
t
. Notice the correlation equals −1 if αt = 0, t ∈ [tto,T]. In that case the MR strategy
reduces to investment in a riskless asset.
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A mean-variance frontier in discrete and continuous time

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