4.3 Portfolio Properties
4.3.1 Per-stage Return Statistics
As stated in Section 4.2, the random vector corresponding to the N stage individual asset returns is Gaussian with mean µrN, defined in Eq. 4.3, and covariance matrix ΣrN, defined in Eq. 4.6. In static asset allocation, the portfolio weight vector, wN, is determined by the information available at time t0, and therefore the cumulative return on the portfolio, rN, is a scalar Gaussian random variable with mean wTµrN and variance wTΣrNw. In a two-stage problem, the per-stage portfolio returns are denoted by r1 = wT (x1− x0) and r2 = wT (x2− x1). Each of these returns is Gaussian, with distributions parametrized as follows:
r1 ∼ N wTΠx0, wTΨw
r2 ∼ N wTΠΠ1x0, wT Ψ + ΠΨΠT w
Furthermore, the covariance between r1 and r2 is given by:
cov [r1, r2] = E [r1r2] − E [r1] E [r2]
= Eh
wT(x1− x0) (x2− x1)Twi
− EwT (x1− x0) E wT(x2− x1)
= wTEh
(Πx0+ 1) (Π (Π1x0+ 1) + 2)Ti
w − wTΠx0xT0ΠT1ΠTw
= wT Πx0xT0ΠT1ΠT + ΨΠT w − wTΠx0xT0ΠT1ΠTw
= wTΨΠTw. (4.10)
Thus, in static portfolio choice, the inter-stage return covariance is a simple quadratic form of the portfolio weight vector. The following example compares the per-stage portfolio return statistics between the Beta and MVO portfolios.
4.3. PORTFOLIO PROPERTIES 87 Example 4.3.
Consider again the system of two assets from Ex. 4.2. The Beta portfolio that also satisfies the budget constraint of wT1 = 1, corresponds to a total portfolio return standard deviation of σ0 = 0.37. Table 4.1 displays the per-stage return statistics for all three strategies when executed at this risk level, and reveals two interesting characteristics of static portfolios.
First, for the two strategies with no budget constraint, there is a direct relationship between the total expected return and the degree of negative correlation between the inter-stage portfolio returns. Since the total return, rN, is defined as the sum of the per-stage log-returns, the total variance is computed as:
var [rN] = var [r1] + var [r2] + 2cov [r1, r2] .
Thus, higher negative correlation implies that the per-stage variances can increase, while the total variance remains constant.
The second property of static portfolios revealed in Table 4.1 is that the increased per-stage portfolio return variance is realized through the use of leverage. As the portfolio weights for the MVO solution with the budget constraint indicate (Table 4.1, line 3), the investor enters into a long position in asset 2 using 310% of the initial wealth, which is, in turn, financed by a short position in asset 1 using 210% of the initial wealth. The presence of an explicit risk-free asset is not required since the net position is unlevered. However, while this trading scheme is able to outperform the Beta strategy, it is the MVO solution without the budget constraint that achieves the highest expected return of all three strategies, due its use of both high negative inter-stage return correlation and net leverage.
As Ex. 4.3 illustrates, both the Beta and MVO portfolios exhibit negative inter-stage
Trading Budget Portfolio Weights Stage 1, r1 Stage 2, r2 Correlation Total, rT Strategy Constraint Asset 1 Asset 2 Total Mean Std Mean Std ρ =cov(rσ 1,r2)
1σ2 Mean Std
Leverage (σ1) (σ2) (σT)
Beta YES -1.50 0.50 1.00 0.14 0.30 0.10 0.33 -0.29 0.24 0.37
MVO NO -0.87 3.45 2.58 0.35 0.30 0.24 0.37 -0.41 0.59 0.37
MVO YES -2.10 3.10 1.00 0.30 0.33 0.21 0.37 -0.46 0.50 0.37
Table 4.1. Second-order statistics for static trading strategies in a two-stage example with total risk budget σ0= 0.37.
portfolio return correlations. This naturally raises the question as to whether negative correlation alone explains the increase in expected return for a given level of standard deviation realized by the MVO solution over the Beta solution. Furthermore, one might wonder whether the MVO solution corresponds to the portfolio weight vector that yields the largest negative correlation between the first and second stage returns over all possible vector directions. These questions are explored next.
Example 4.4.
Consider a first-order cointegrated VAR process with the following parameters:
Π1 =
0.7878 0.0707 0.2634 0.9122
, Ψ = I2, φ = 0.
Figure 4-3(a) depicts the direction of the vectors {α, α⊥, β, β⊥} and the two-stage MVO portfolio weight vector w∗2 centered at an arbitrary initial condition of x0 =
1.75 4.3
T
. In order to explore the relationship between the direction of the portfolio policy and the degree of negative correlation achieved between the inter-stage portfolio returns, a set of portfolios are determined by rotating the portfolio weight vector over a range of angles θ = [0, π] from the Beta direction. Figure 4-3(b) displays the correlation coefficient between the returns for the first and second stages as a function of the rotation angle. Also highlighted is the correlation level achieved by the MVO solution and by four vectors corresponding to the {α, α⊥, β, β⊥} subspaces. While the MVO solution is able to realize a larger degree of negative correlation than the Beta solution, it is not the direction with maximum negative correlation. This in turn suggests that negative correlation is only part of the story, and must be considered alongside individual asset and total portfolio leverage.
In addition to demonstrating that the MVO solution does not maximize the negative relation between the inter-stage returns, the preceding example also revealed that the cor-relation coefficient between the inter-stage portfolio returns is not negative for all portfolio vector directions. In order to understand this result, first represent an arbitrary portfolio weight vector w using the non-orthogonal basis {α⊥, β}, as follows:
w = c1β + c2α⊥, (4.11)
4.3. PORTFOLIO PROPERTIES 89
1 1.5 2 2.5
3.5 4 4.5 5
x 1
x2
x0
!
!
*
wMVO
!
!
(a) Geometry of system in Ex. 4.4. The vectors {α, α⊥, β, β⊥} and the MVO op-timal portfolio weight vector are depicted centered at an arbitrary initial condition of x0= (1.75 4.3)T.
0 30 60 90 120 150 180
-0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0
Rotation Angle
Correlation Coefficient Betaperpportfolio
MVO portfolio Alphaperpportfolio
Beta portfolio Alpha portfolio
(b) Correlation coefficient between first and second stage returns as a function of the port-folio vector direction. While the MVO solu-tion realizes a larger degree of negative cor-relation than the Beta solution, it does not correspond to the portfolio direction which maximizes the negative correlation.
Figure 4-3. Correlation coefficient between first and second stage returns as a function of the portfolio vector direction.
where the constants c1, c2 ∈ R are computed according to the oblique projections:
c1 = αTβ−1
αTw, c2 = βT⊥α⊥
−1
βT⊥w.
Next, substitute Eq. 4.11 into Eq. 4.10, the covariance between the first and second stage returns, which yields:
cov [r1, r2] = c21βTΨβαTβ + c22αT⊥ΨβαTα⊥+ c1c2βTΨβαTα⊥+ c1c2αT⊥ΨβαTβ.
Due to the fact that αTα⊥ = 0, the second and third terms are equivalently zero. The first term, which corresponds to a portfolio purely in the β direction, is always negative, as established in Thm. 4.1 below.
Theorem 4.1.
Consider a two-stage asset allocation problem in which the log-price process x ∈ Rp is assumed to evolve according to a first-order cointegrated VAR system. The matrix Π1 contains exactly p − 1 unit eigenvalues, so that the matrix Π can be factored as the outer product of two p−dimensional vectors, as Π = αβT. When the static portfolio weight
vector is chosen according to w ∝ β, the covariance (equivalently the correlation coefficient) between the portfolio returns for the first and second stages is negative.
The proof of Thm. 4.1 is given in Appendix 4.A. Thus, the covariance between the first and second stage returns is positive when the following condition is met:
c1βTΨβ < −c2αT⊥Ψβ → wTΨβ < 0,
which follows using the fact that αTβ < 0, as established in the proof of Thm. 4.1.
As a final point of interest, consider the two portfolio weight vectors corresponding to zero inter-stage return covariance. These can be directly determined through examination of Eq. 4.10, which can be factored as follows:
cov [r1, r2] = wT (Ψβ) αTw.
Hence, choosing w ∝ (Ψβ)⊥ or w ∝ α⊥ produces a correlation coefficient of zero. In Ex.
4.4, the covariance matrix of the input was chosen as Ψ = I2, and therefore the two zero crossings of the correlation coefficient function occur at w = {β⊥, α⊥}. The zero crossing at w = α⊥ can be best understood by examining the total system response for a cointegrated VAR process defined in Eq. 3.16. The portfolio value (i.e., log price) at any given point is given by the scalar process v[n] = wTx[n]. Therefore, when w = α⊥, the second term of x[n] corresponding to the AWSS random process drops out, leaving the portfolio value to be defined by a nonstationary integrated process. This in turn implies that the log-return process is an i.i.d. random process, with per-stage returns that are uncorrelated.