The persistence property of ˆβn(0)+ in (5.17) holds asymptotically. However, it is impossible
to obtain infinite samples in reality. In this section, we carry out simulation studies to investigate how LFn( ˆβ
(0+)
n ) is close to LFn( ˜β
(0+)
5.4.1
Simulation Methodology
In this simulation study, we first design a joint distribution of a large number of stocks. The number of stocks d is designed to be an increasing function of the number of simulated scenarios n. More specifically, we let d = d(n) = bnαc with α > 1 and b·c is the floor
function of any real number. Then, these stocks construct an equally weighted stock- market index. An equally weighted stock-market index is used since it is easy to construct and analyze. Given the true joint distribution of stock returns and the index return, an analytical form of LFn(β) can be obtained. The performance of our index tracking method with ˆβ(0)n + can be investigated by comparing the gap between LFn( ˆβ
(0)+
n ) and the minimized
true risk LFn( ˜β
(0)+
n ).
For simplicity, from time 0 to time 1 we assume stock returns, as well as the return on the money market account (0-th asset), follow Sharpe’s single-index model ([109]) which is given by
r = a + bRM + ε, (5.22)
where r = (r0, r1, . . . , rd(n))0 is the vector of asset returns, a = (a0, a1, . . . , ad(n))0 and
b = (b0, b1, . . . , bd(n))0 are vectors of constant coefficients. We assume that RM is a market
portfolio return which follows a normal distribution with mean µRM and variance σ2
RM. The vector of random noises ε = (ε0, ε1, . . . , εd(n))0 follows a multivariate normal distribution
(MVN) with mean 0 and covariance matrix Dε, which is denoted by MVN(0, Dε). The
matrix Dε is diagonal with positive diagonal elements σε2i for i = 0, 1, . . . , d(n). Hence, r follows MVN a + (µRM)b, σR2Mbb
0+ D ε.
Let e0 = (0, 1, . . . , 1)0 be a (1 + d(n))-column vector. Then the return of an equally-
weighted index R, which consists of r1, r2, . . . , rd(n), is given by
R = e 0 0r d(n), then Y = e 0 0X d(n), (5.23)
Given initial wealth W(0)− and dollar amounts for each asset x(0) −
i or equivalently β (0)−
i
for i = 0, 1, . . . , d(n) at time 0, we have LFn(β) = E h
Y − β0X2|β(0)−i= Eh Y − β0X2i . Hence, the true risk (or true tracking error) of β is given by
LFn(β) = E h Y − β0X2i = (−1, β0)E " Y X ! (Y0, X0) # −1 β ! = (−1, β0) E[Y Y 0] E[Y X0] E[XY0] E[XX0] ! −1 β !
= β0E[XX0]β − 2E[Y X0]β + E[Y Y0], (5.24)
where
E[Y Y0] = V ar(Y ) + E[Y ]2 = 1 d(n)2
e00ΣXe0+ µ2Y,
E[XY0] = Cov(X, Y ) + E[X]E[Y ]0 = 1
d(n)ΣXe0+ µXµY, E[XX0] = Cov(X, X) + E[X]E[X]0 = ΣX + µXµ0X,
and ΣX = σR2Mbb
0 + D
ε, µY = d(n)1 µ0Xe0, µX = e + a + (µRM)b. Since ΣX is positive definite, so is E[XX0]. Hence, given any fixed cn, we can efficiently obtain the optimal
solution ˜β(0)n + defined in (5.10)-(5.14) via quadratic programming solvers.
Given n and the parameters in (5.22), we can simulate samples of (RM s , εs)
n
s=0, and
then generate an in-sample dataset TSim =(Y
s, Xs,0, Xs,1, . . . , Xs,d(n))0 : s = 1, 2, . . . , n
according to (5.22) and (5.23). Based on TSim, the one-period index tracking strategy
ˆ
βn(0)+ at time (0)+ is given by (5.17) subject to (5.11)-(5.14). The actual risk LFn( ˆβ
(0)+
n )
can be computed by plugging ˆβn(0)+ into (5.24).
The performance of our tracking strategy ˆβn(0)+ in finite samples can be investigated
by repeating the simulation S times. Based on sufficiently many repetitions, we can con- struct a confidence interval of LFn( ˆβ
(0)+
n ) to evaluate how stable LFn( ˆβ
(0)+
between the averaged LFn( ˆβ
(0)+
n ) and LFn( ˜β
(0)+
n ) shows on average how close LFn( ˆβ
(0)+
n ) is
to LFn( ˜β
(0)+
n ).
5.4.2
An Implementation of the Simulation Study
In order to simulate samples, we let α = 1.25, and let µRM and σ2
RM be the sample mean and sample variance of the Russell 3000 index weekly returns described in Section5.5. For i = 1, 2, . . . , d(n), coefficients ai, bi, and σε2i are ordinary least square (OLS) estimators of regressing the i-th Russell 3000 component weekly return on the Russell 3000 index return. Parameters a0, b0, and σε20 are OLS estimators of regressing weekly interest rates against the Russell 3000 index return. All these parameter estimators are obtained from data in the recovery environment described in Section 5.5.
Based on TSim, we construct a tracking portfolio at time 0 given β(0)−
0 = 1 and β
(0)− i = 0
for i = 1, . . . , d(n). In this implementation, we assume the proportional rate of transaction cost is around the middle of the range [0.31%, 2.35%] described in Section 5.1, and let θ = 1%. Further, let the transaction cost limit γ be 1%. We increase the value of n from 100 to 200, and then 450 to evaluate how the tracking strategy behaves as n grows. Results are summarized in Figures 5.1-5.3 respectively. Each of Figures 5.1-5.3 shows the true risk (solid line) LFn( ˜β
(0)+
n ) i.e. true mean square error (MSE), the average of actual
risks LFn( ˆβ
(0)+
n )’s (dash-dot line) and corresponding 90% confidence band of LFn( ˆβ
(0)+
n )’s
(dashed lines) which are obtained from 30 repetitions of simulations.
In all Figures 5.1-5.3, cn varies from −1 to 3, and the minimized true risk LFn( ˜β
(0)+
n )
decreases as cn gets larger, which results from enlarged feasible sets. Once cn is sufficiently
large, the true risk does not change. This is because cn is large enough to disqualify the
constraint (5.7). Moreover, the minimized true risk is uniformly smaller than the actual risk LFn( ˆβ
(0)+
n ), which is true by definition.
Figures 5.1-5.3 show that as n increases the confidence band gets narrower, and the averaged actual risk approaches to the true risk. This verifies the persistence result
−10 −0.5 0 0.5 1 1.5 2 2.5 3 1 2 3 4 5 6 7 8x 10 −4 Cn TE
Actual MSE vs True MSE, sample size = 100 5% confidence band 95% confidence band Mean
True MSE minimun of mean
Figure 5.1: Minimized True Risk vs. Actual Risk: n = 100, #stock=316.
in Theorem 5.1. Moreover, for any fixed n, the confidence band gets wider as cn in-
creases. This verifies the result (5.20) in Theorem5.1, which says that the upper bound of PFn n LFn( ˆβn) − LFn(β ∗ n) ≥ δ o
becomes larger as supβ∈B(n)||β||1 gets bigger.
All Figures5.1-5.3indicate a tradeoff between the magnitude of the actual risk LFn( ˆβ
(0)+
n )
and its stableness. When cnis small, the averaged actual risk is close to the minimized true
risk LFn( ˜β
(0)+
n ) and the confidence band is narrow, but the minimized true risk is relatively
large. When cn is large, the minimized true risk is small, but the averaged actual risk
deviates from the minimized true risk due to larger estimation errors, which is suggested by wider confidence intervals. In practice, it is necessary to choose a cn to implement the
tracking strategy. In this chapter, we follow the suggestion in [69, p.221] and choose the value of cn which leads to the minimum of the averaged actual risk.