Transport and Telecommunication
Vol.7, No 1, 2006
MULTIPERIOD PROJECT PORTFOLIO INVESTMENT STRATEGY Gundars Berzins, Nicholas A. Nechval, Edgars K. Vasermanis Mathematical Statistics Department, University of Latvia Raina Blvd. 19, LV-1050, Riga, Latvia E-mail:
[email protected] This paper considers an analytical optimal solution to the mean-variance formulation of the problem of optimisation of multiperiod investment strategy for project portfolio. Specifically, analytical optimal multiperiod investment strategy for project portfolio is derived. An efficient algorithm is proposed in order to maximize the expected value of the terminal wealth under constraint that the variance of the terminal wealth is not greater than a pre-assigned risk level or to minimize the variance of the terminal wealth under constraint that the expected terminal wealth is not smaller than a pre-assigned level. A numerical example is given.
Keywords: project portfolio, multiperiod investment strategy, optimisation
1. INTRODUCTION Investment of project portfolio is to seek the best allocation of wealth among a basket of projects. Project portfolio investment is the periodic activity involved in investing a portfolio of projects, that meets an organization’s stated objectives without exceeding available resources or violating other constraints. Some of the issues that have to be addressed in this process are the organization’s objectives and priorities, financial benefits, intangible benefits, availability of resources, and risk level of the project portfolio. Investment strategy for a set of proposed projects is a crucial decision in many organizations, which must make informed decisions on investment, where the appropriate distribution of investment is complex, due to varying levels of risk, resource requirements, and interaction among the proposed projects. The mean-variance formulation by Markowitz [1-2] provides a fundamental basis for project portfolio investment in a single period. Smith [3], Mossin [4], Merton [5], Samuelson [6], Fama [7], Hakansson [8], Elton and Gruber [9], Winkler and Barry [10], Francis [11], Dumas and Luciano [12], Grauer and Hakansson [13], and Pliska [14] have studied the problem of multiperiod portfolio investment. The literature in multiperiod portfolio investment has been dominated by the results of maximizing expected utility functions of the terminal wealth and/or multiperiod consumption. Specially, investment situations where the utility functions are of power form, logarithm function, exponential function, or quadratic form have been extensively investigated in the literature. To our knowledge, no analytical or efficient numerical method for finding the optimal strategy for the mean-variance formulation of the problem of multiperiod project portfolio investment has been reported in the literature. In this sense, the concept of the Markowitz’s mean-variance formulation has not been fully utilized in multiperiod project portfolio investment. This paper considers an analytical optimal solution to the mean-variance formulation of the problem of multiperiod project portfolio investment. Specifically, analytical optimal multiperiod investment strategy for project portfolio is derived. An efficient algorithm is proposed in order to maximize the expected value of the terminal wealth under constraint that the variance of the terminal wealth is not greater than a pre-assigned risk level or to minimize the variance of the terminal wealth under constraint that the expected terminal wealth is not smaller than a pre-assigned level.
2. PROBLEM STATEMENT We consider a portfolio with (m+1) risky projects, with random rates of returns. Let w0 be an initial wealth of an investor at time 0. The investor can allocate his wealth among the (m+1) projects. The wealth can be reallocated among the (m+1) projects at the beginning of each of the following (T−1) consecutive time periods. The rates of return of the risky projects at time period τ within the 60
Proceedings of the 5th International Conference RelStat’05
Part 1
planning horizon are denoted by a vector rτ=[rτ(0), rτ(1), …, rτ(m)]′, where rτ(j) is the random return for project j at time period τ . It is assumed in this paper that vectors rτ, τ = 0, 1, ..., T−1, are statistically independent and return rτ has a known mean E{rτ}=[E{rτ(0)}, E{rτ(1)}, …, E{rτ(m)}]′ and a known covariance
σ τ (00) Cov{rτ } = σ τ (om )
σ τ (0m)
. σ τ ( mm)
(1)
Let wτ be the wealth of the investor at the beginning of the τth period, and let sτ(j), j∈{1, …, m}, be the amount invested in the jth risky project at the beginning of the τth time period. The amount investigated in the 0th risky project at the beginning of the τth time period is equal to wτ −
m
∑ sτ ( j ) .
(2)
j =1
An investor is seeking a best multiperiod investment strategy, sτ=[sτ(0), sτ(1), …, sτ(m)]′ for τ = 0, 1, 2, . . . , T −1, such that either (i) the expected value of the terminal wealth wT , E{wT}, is maximized if the variance of the terminal wealth, Var{wT}, is not greater than a pre-assigned risk level v•, or (ii) the variance of the terminal wealth, Var{wT}, is minimized if the expected terminal wealth, E{wT}, is not smaller than a pre-assigned level e•. Mathematically, a mean-variance formulation for multiperiod project portfolio investment can be posed as one of the following two forms: (i) Maximize E{wT} subject to Var{wT}≤v•, wτ +1 =
rτ ( j ) sτ ( j ) + wτ − j =1 m
∑
m
j =1
∑ sτ ( j ) rτ (0) = rτ (0) wτ + Rτ′ sτ ,
τ=0, 1, 2, …, T−1,
(3)
sτ ( j ) rτ ( 0) = rτ (0) wτ + Rτ′ sτ , τ=0, 1, 2, …, T−1, j =1
(4)
and (ii) Minimize Var{wT} subject to E{wT}≥e•, wτ +1 =
rτ ( j ) sτ ( j ) + wτ − j =1 m
∑
m
∑
where Rτ = [rτ (1) − rτ (0) , rτ ( 2) − rτ (0) , ..., rτ ( m) − rτ (0) ]′.
(5) 61
Transport and Telecommunication
Vol.7, No 1, 2006
Notice that E{rτ rτ′ } = Cov{rτ } + E{rτ }E{rτ′ }.
(6)
It is assumed in this paper that E{rτ rτ′ } is positive definite for all time periods, that is, E{rτ rτ′ } >0, ∀τ=0(1)T−1. Formulation (i) or (ii) enables an investor to specify a risk level he can afford when he is seeking to maximize his expected terminal wealth or specify an expected terminal wealth he would like to achieve when he is seeking to minimize the corresponding risk. A strategy of multiperiod project portfolio investment is an investment sequence, ST=[s0, s1, s2, …, sT-1],
(7)
where sτ=[sτ(1), sτ (2), …, sτ (m)]′, ∀τ=0(1)T−1,
(8)
More specifically, ST is a feedback policy and sτ maps the wealth at the beginning of theτth period, wτ , into a project portfolio decision in the τth period, i.e., sτ ≡ sτ(wτ). A multiperiod project portfolio investment strategy, S T∗ , is said to be efficient if there exists no other multiperiod portfolio policy, ST, such that E{wT;ST}≥E{wT; ST∗ } and Var{wT;ST}≤Var{wT; ST∗ } with at least one equality strictly. By varying the value of v• in (i) or the value of e• in (ii), we can generate the set of efficient multiperiod project portfolio investment strategies.
3. OPTIMAL MULTIPERIOD INVESTMENT STRATEGY Theorem 1. The optimal strategy for multiperiod project portfolio investment for problems (i) and (ii) is specified by the following analytical expression:
sτ∗ = −E −1{Rτ Rτ′ }E{rτ (0) Rτ }wτ gw0 + b1 (2ch) −1 T −1 a1t + 2 t =τ +1a2t
∏
−1 E {Rτ Rτ′ }E{Rτ }, τ=0(1)T−2,
(9)
sT∗ −1 = − E −1{R T −1R ′T −1}E{rT −1( 0 ) R T −1}wT −1 gw + b (2ch) −1 −1 E {R T −1R ′T −1}E{R T −1}, + 0 1 2
(10)
where b1 , when (i) is solved, 2 • 2 c(v − qw0 ) h= b12 , when (ii) is solved, • 2c[e − (a1 + gb1 ) w0 ]
(11)
a1τ = E{rτ ( 0) } − E{Rτ′ }E −1{Rτ Rτ′ }E{rτ ( 0) Rτ },
(12)
a 2τ = E{rτ2(0 ) } − E{rτ ( 0) Rτ′ }E −1{Rτ Rτ′ }E{rτ (0 ) Rτ },
(13)
62
Proceedings of the 5th International Conference RelStat’05
Part 1
bτ = E{Rτ′ }E −1{Rτ Rτ′ }E{Rτ },
(14)
−1
T −1 T −1 b1τ = bτ a1t 2 a 2t , t =τ +1 t =τ +1
∏
∏
(15)
with T −1
∏
T −1
∏ a2t = 1,
a1t =
t =T
(16)
t =T
T −1
∏ a1τ ,
a1 =
(17)
τ =0
a2 =
T −1
∏ a2τ ,
(18)
τ =0
b1 =
T −1 T −1
a1t b1τ , τ = 0 t =τ +1
∑ ∏
(19)
c=
b1 − b12 , 2
(20)
g=
a1b1 , c
(21)
q = a 2 − a12 − cg 2 .
(22)
In this case, the expected value and the variance of the terminal wealth are given by
E{wT } = (a1 + gb1 ) w0 +
b12 , 2ch
(23)
and
Var{wT } =
b12 4ch
2
+ qw02 .
(24)
Proof. The proof is carried out by means of the method of Lagrange multipliers, optimality principle, and the method of mathematical induction. It is omitted here and will appear elsewhere. ˛
Thus, the optimal multiperiod investment strategy for project portfolio consists of two terms and exhibits a decomposition property between the investor’s risk attitude and his current wealth. The ∗
second term in sτ is dependent on the investor’s risk attitude and is independent of his current wealth. ∗
It can be calculated off-line before the real investment process starts. The first term in sτ is dependent on the current wealth and is independent of the investor’s risk attitude. It is calculated on-line at every time period when the current wealth is observed.
63
Transport and Telecommunication
Vol.7, No 1, 2006
4. NUMERICAL EXAMPLE Consider the case of a stationary multiperiod process with T = 4. An investor has one unit of wealth at the very beginning of the planning horizon, i.e., w0 = 1 The investor is trying to find the best allocation of his wealth among three risky projects, 0, 1, and 2 in order to maximize E{w4} while keeping his risk not exceeding 1; that is, v• = 1. The expected returns for risky projects, 0, 1, and 2 are E{rτ(0)}=1.189, E{rτ(1)}=1.314, and E{rτ(2)}=1.335, τ = 0, 1, 2, 3. The covariance of rτ=[rτ(0), rτ(1), rτ(2)]′ is 0.0099 0.0114 0.0120 Cov{rτ } = 0.0114 0.0535 0.0508 , τ= 0, 1, 2, 3. 0.0120 0.0508 0.0864
(25)
Thus,
E{Rτ } = E{[rτ (1) − rτ ( 0) , rτ ( 2) − rτ ( 0) ]′} =[0.125, 0.146]′, rτ2( 0) − 2rτ ( 0) rτ (1) + rτ2(1) ′ E{Rτ Rτ } = E 2 rτ ( 0) − rτ ( 0) rτ (1) − rτ ( 0) rτ ( 2) + rτ (1) rτ ( 2)
(26)
rτ2( 0) − rτ ( 0) rτ (1) − rτ ( 0) rτ ( 2) + rτ (1) rτ ( 2) rτ2( 0) − 2rτ ( 0) rτ ( 2) + rτ2( 2)
0.056225 0.05555 = , 0.05555 0.093616
(27)
E{rτ (0) Rτ′ } = [E{rτ (0) rτ (1) } − E{rτ2(0) }, E{rτ (0) rτ ( 2) } − E{rτ2(0) }] =[0.150125, 0.175694], τ=0,1,2,3.
(28)
Furthermore, we have bτ = 0.290972,
(29)
a1τ = 0.839340049,
(30)
a2τ = 1.003433478, τ = 0, 1, 2, 3,
(31)
a1 = 0.496308579,
(32)
a2 = 1.013804806,
(33)
b1 = 0.36969,
(34)
c = 0.048174261,
(35)
g = 3.808681367,
(36)
and q = 0.068664187.
(37)
From (11), the corresponding h = 0.87266436.
(38)
The associated optimal strategy for multiperiod project portfolio investment is given, using equations (9) and (10), as follows: 64
Proceedings of the 5th International Conference RelStat’05
1.97189131 3.960204 s ∗0 = − w0 + , 0.706668067 1.394898
Part 1
(39)
1.97189131 4.734435 s1∗ = − w1 + , 0.706668067 1.667604
(40)
5.660031 1.97189131 s ∗2 = − w2 + , 1.993626 0.706668067
(41)
1.97189131 6.766584 s ∗3 = − w3 + . 0.706668067 2.383385
(42)
The investment in the 0th project at period τ is equal to sτ (0) = wτ −
m
∑ sτ ( j ) ,
∀τ=0(1)3.
(43)
j =1
Since it is known that w0=1, we obtain from (39) a percent investment (on w0) in three risky projects, 0, 1, and 2, at period τ=0 as follows: s0(0) = 97.323%, s0(1) = 1.988%, s0(2) = 0.689%. The corresponding expected terminal wealth and the risk level are given by E{x4} = 3.529827827 and Var{x4} = 1, respectively, using equations (23) and (24).
5. CONCLUSION The Markowitz mean-variance approach has been extended in this paper to multiperiod project portfolio investment problems. The derived analytical optimal multiperiod project portfolio investment strategy provides investors with the best strategy to follow in a dynamic investment environment.
References [1] Markowitz H.M. Portfolio Selection: Efficient Diversification of Investment. New York: John Wiley & Sons, 1959. [2] Markowitz H.M. Mean-Variance Analysis in Portfolio Choice and Capital Markets.Cambridge, MA: Basil Blackwell, 1989. [3] Smith K.V. A Transition Model for Portfolio Revision, Journal of Finance, 22, 1967, pp.425-439. [4] Mossin J. Optimal Multiperiod Portfolio Policies, Journal of Business, 41, 1968, pp.215-229. [5] Merton R.C. Continuous-Time Finance. Cambridge, MA: Basil Blackwell, 1990. [6] Samuelson P.A. Lifetime Portfolio Selection by Dynamic Stochastic Programming, The Review of Economics and Statistics, 50, 1969, pp.239-246. [7] Fama E.F. Multiperiod Consumption-Investment Decisions, American Economic Review, 60, 1970, pp.163-174. [8] Hakansson N.H. Multi-Period Mean-Variance Analysis: Toward a General Theory of Portfolio Choice, Journal of Finance, 26, 1971, pp.857-884. [9] Elton E.J. and Gruber M. J. Finance as a Dynamic Process. Englewood Cliffs, NJ: Prentice Hall, 1975. [10] Winkler R.L. and Barry C. B. A Bayesian Model for Portfolio Selection and Revision, Journal of Finance, 30, 1975, pp.179-192. [11] Francis J.C. Investments: Analysis and Management. New York: McGraw-Hill, 1976. [12] Dumas B. and Luciano E. An Exact Solution to a Dynamic Portfolio Choice Problem under Transaction Costs, Journal of Finance, XLVI, 1991, pp.577-595. [13] Grauer R.R. and Hakansson N. H. On the Use of Mean-Variance and Quadratic Approximations in Implementing Dynamic Investment Strategies: a Comparison of Returns and Investment Policies, Management Science, 39, 1993, pp.856-871. [14] Pliska S.R. Introduction to Mathematical Finance. Malden, MA: Basil Blackwell, 1997.
65
