|
Article Excerpt
1. Introduction
It has been observed in the literature that the in-sample estimate of the variance of an efficient portfolio constructed using estimated inputs significantly understates the portfolio's true (out-of-sample) variance. (1) This downward bias, commonly referred to as in-sample optimism, increases with the number of assets used to construct mean-variance efficient portfolios. We demonstrate that in-sample optimism can be significantly large in certain cases, identify why that happens, and develop a jackknife-type estimator that is not subject to that bias. Because our focus is in assessing the risk associated with an efficient portfolio constructed using estimated inputs, we examine the global minimum variance and minimum benchmark tracking-error variance portfolios. We refer to such portfolios as sample minimum-risk portfolios (SMRPs) in this paper.
It might appear that we can avoid in-sample optimism by using the Bayesian approach because it would take into account the uncertainty associated with estimates of means and covariances of returns used in the optimization procedure. We find that with standard diffuse priors commonly used in empirical works, the variance of a minimum-risk portfolio computed using the predictive distribution is, on average, substantially below its true variance. The inadequacy of standard diffuse priors and the need for modifying them is discussed by Jacquier et al. (2005) in a related context. Another closely related work is Tu and Zhou (2009), which incorporates economic objectives into the Bayesian priors to address the parameter uncertainty problem in portfolio choice.
1.1. Relation to Literature on Testing Mean-Variance Efficiency of a Portfolio
There is a large literature on testing the mean-variance efficiency of a given portfolio. (2) These tests examine whether the distance between a given benchmark portfolio and a particular efficient portfolio constructed using sample moments is zero after allowing for sampling errors. Although these tests also involve constructing efficient portfolios based on estimated covariance matrices, the sampling theory associated with these tests differs in important ways from that associated with the estimate of the variance of the sample minimum-risk portfolio developed in this paper.
To see why, consider forming the global minimum-variance portfolio based on S, an estimate of the unknown covariance matrix, (SIGMA). The vector of portfolio weights is given by
[w.sub.s] = [S.sup.-1]1/1'[S.sup.-1]1. (1)
We use 1 to denote the column vector of ones throughout the paper. The variance of this portfolio's return is
[w'.sub.s][SIGMA][w.sub.s] = 1'[S.sup.-1][SIGMA][S.sup.-1]1/[(1'[S.sup.-1]1).sup.2]. (2)
Because [SIGMA] is unknown, the variance of [w.sub.s] is also unknown and has to be estimated.
In MacKinlay (1987) and Gibbons et al. (1989), the test statistic--which provides a measure of the distance between a given benchmark portfolio and a particular sample-efficient portfolio--also involves the unknown covariance matrix of returns, [SIGMA] However, when the unknown [SIGMA] is replaced with the sample covariance matrix S, the test statistic still has a known finite-sample distribution when returns are independent and identically distributed (i.i.d.) multivariate Normal. In contrast, if we replace the unknown [SIGMA] with its estimate S we just get the in-sample variance,
[w'.sub.s]S[w.sub.s] = 1/1'[S.sup.-1]1/ (3)
This does not help because the in-sample variance provides a downward-biased estimate of the population variance.
1.2. Estimating the Conditional Variance of a Given Sample Minimum-Risk Portfolio
When returns are not i.i.d. but instead exhibit persistence in their second moments, as in the data, we suggest two approaches. (3) The first approach is based on the dynamic conditional correlation model of Engle (2002). We find that empirically this approach provides an accurate estimate of risks in the SMRPs constructed using one- and three-factor models. However, in our sample, SMRPs constructed using one-and three factor models have a significantly higher risk when compared to SMRPs constructed using the sample covariance matrix. This suggests that three factors are insufficient to account for the correlation among returns, and further work is needed to model what a dynamic three-factor model misses in the covariance structure of asset returns. In the second approach, we develop a modification of the jackknife-type estimator that weights more recent observations more heavily than those in the distant past.
Kan and Smith (2008) derive the exact distribution of the sample mean-variance efficient frontier of returns when there are no portfolio weight constraints and returns are drawn from an i.i.d. multivariate Normal distribution. In contrast, our focus is limited to global minimum variance and global minimum tracking-error portfolios. However, we do not require returns to be normally distributed or independent over time, and allow for the presence of portfolio weight constraints that are commonly encountered in practice. We focus on the global minimum-variance portfolio and minimum tracking-error portfolio for two reasons. First, minimizing tracking-error variance is widely used for fund managers. Second, it is well known in the literature that sample mean is a very poor estimate of the corresponding population mean return for portfolio optimization purposes.
In the next section, we develop some theoretical results that help understand why in-sample optimism occurs. We also demonstrate that in-sample optimism will continue to exist even when the Bayesian approach is used to construct efficient portfolios. We develop a jackknife-type estimator of the out-of-sample variance in [section]3 that is valid when returns are i.i.d. over time. In [section]4, we propose two methods for estimating the conditional out-of-sample variance when returns do not satisfy the i.i.d. assumption and empirically evaluate these methods in [section]5. Section 6 concludes. The e-companion provides technical derivations. (4)
2. Relation Between In-Sample and Out-of-Sample Variances of Sample Minimum-Risk Portfolios
We use the following notation [R.sub.t] is the N x 1 vector of date I returns (or returns in excess of some benchmark return when the objective is to minimize the tracking-error variance) on N primitive assets; [SIGMA] Cov([R.sub.t]) (may depend on the information set at date T); and S is an estimate of [SIGMA] based on T observations on returns, Our analysis applies to both portfolio variance minimization and tracking-error variance minimization. For brevity, we will use the term "return" to mean both raw return and excess return in excess of the benchmark, and the term "portfolio variance minimization" to mean both portfolio variance minimization and tracking-error variance minimization in our analysis.
We term [w.sub.s] in (1) as the sample minimum-risk portfolio. Let [w.sub.p] be the population minimum-risk portfolio constructed using [SIGMA] instead of S. Then [w'.sub.s]S[w.sub.s] is the in-sample variance of [w.sub.s], [w'.sub.s][SIGMA][w.sub.s] is its out-of-sample variance, and [w'.sub.p] [SIGMA][w.sub.p] is the variance of the population minimum-risk portfolio.
The following proposition characterizes the relation among the expected in-sample and out-of-sample variances of the sample minimum-risk portfolio and the variance of population counterpart.
PROPOSITION 1. When S is an unbiased estimator of [SIGMA], the in-sample estimate of the variance of [w.sub.s] will on average be strictly smaller than the variance of the true (i.e., population) global minimum-variance portfolio, which on average will be strictly smaller than the out-of-sample variance of [w.sub.s]. That is,
E([w'.sub.s]S[w.sub.s]) < [w'.sub.p][SIGMA][w.sub.p] < E([w'.sub.s][SIGMA][w.sub.s]). (4)
Under the i.i.d. normality assumption and when S is the sample covariance matrix, we have the following proposition. (5)
PROPOSITION 2. Denote [[sigma].sub.p.sup.2] = [w'.sub.p][SIGMA][w.sub.p], [[
.[sigma]].sub.p.sup.2] = [w'.sub.s] S[w.sub.s], and [~.[[sigma]].sub.p.sup.2] = [w'.sub.s][SIGMA][w.sub.s]. Under the i.i.d. normality assumption on [R.sub.t], we have
[[
.[sigma].sub.p.sup.2] = u[[sigma].sub.p.sup.2]/T - 1, [[~.[sigma].sub.p.sup.2] = (1 + [u.sub.1]/[u.sub.2]) [[sigma].sub.p.sup.2], (5)
where u ~ [[chi].sub.T - N'.sup.2], [u.sub.1] ~ [[chi].sub.N - 1.sup.2], [u.sub.2] ~ [[chi].sub.T - N + 1'.sup.2], and they are independent of each other. It follows that
E[[
.[sigma].sub.p.sup.2]] = T - N/T - 1 [[sigma].sub.p.sup.2], E[[~.[sigma].sub.p.sup.2] = T - 2/T - N - 1 [[sigma].sub.p.sup.2], if T > N + 1, (6)
and an unbiased estimator of E[[~.[sigma].sub.p.sup.2]] is given by
(T -1) (T - 2)/(T -N) (T - N - 1) [[
.[sigma].sub.p.sup.2]. (7)
In addition to giving an unbiased estimator of the out-of-sample variance of [w.sub.s], this proposition gives the exact joint distribution of ([
.[[sigma]].sub.p.sup.2], [~.[[sigma]].sub.p.sup.2]). Also, it shows that the two are independent even though both are functions of S.
The following results indicate that the "in-sample optimism" problem is present even in the Bayesian decision-making framework.
PROPOSITION 3. Let [w.sub.sB] be the portfolio that minimizes var (w'[R.sub.T + 1]| [R.sub.1], ..., [R.sub.T]). Assume that the investor has diffuse prior about ([mu], [SIGMA]), as given by
p([mu], [SIGMA]) [varies] [|[SIGMA]|.sup.-(N + 1)/2].
p([mu], [SIGMA]) [varies] |[[SIGMA].sup.-(N + 1)/2]
Then, under the assumption that [R.sub.t], t = 1, ..., T have an i.i.d. Normal distribution, we have
[w.sub.sB] = [w.sub.s] (8)
var([w'.sub.sB][R.sub.T + 1] | [R.sub.1], ..., [R.sub.T]) = (T - 1) (T + 1)/T (T - N - 2) [[[
.[sigma].sub.p.sup.2]]. (9)
The adjustment to [[
.[sigma]].sub.p.sup.2] on the right-hand side of (9) is less than the correct adjustment in (7); hence, the in-sample optimism problem still exists. (6)
3. Jackknife-Type Estimator of the Out-of-Sample Variance with i.i.d. Data
3.1. Portfolio Holding Period Equals the Return Observation Interval
For expositional convenience, we first consider the special case where the investor estimates covariance matrices and recomputes optimal portfolio weights at the end of...
|
