Portfolio selection: from under-diversification to concentration

Abstract

The two opposing investment strategies, diversification and concentration, have often been directly compared. While there is much less dispute regarding Markowitz’s approach as the benchmark for diversification, the precise meaning of concentration in portfolio selection remains unclear. This paper offers a novel definition of concentration, along with an extreme value theory-based estimator for its implementation. When overlaying the performances derived from diversification (in Markowitz’s sense) and concentration (in our definition), we find an implied risk threshold, at which the two polar investment strategies reconcile—diversification has a higher expected return in situations where risk is below the threshold, while concentration becomes the preferred strategy when the risk exceeds the threshold. Different from the conventional concave shape, the estimated frontier resembles the shape of a seagull, which is piecewise concave. Further, taking the equity premium puzzle as an example, we demonstrate how the family of frontiers nested inbetween the estimated curves can provide new perspectives for research involving market portfolios.

Appendix A: Derivations of the statistical methods

1.1 A.1: The mean-variance optimization

This section provides the general setup of Markowitz’s mean-variance optimization. Let $\varvec{E}$ be the vector of expected asset returns in the stock pool, $\varvec{\mathrm {V}}$ be the covariance matrix of the returns, and $\varvec{w}$ be the vector of weights indicating the fraction of portfolio wealth held in each asset. Assuming that short sales are permitted, the constrained minimization problem is as follows:

$$\begin{aligned} \begin{aligned}&\underset{\varvec{w}}{\min }&\frac{1}{2} \varvec{w}^T \varvec{\mathrm {V}} \varvec{w}\\&\text {subject to}&\mu = \varvec{w}^T \varvec{E} \text { and }1 = \varvec{w}^T \varvec{1}, \end{aligned} \end{aligned}$$

where $\mu$ denotes the target expected return of the portfolio, and $\varvec{1}$ denotes a vector of ones. The analytical solution to this problem is derived following Merton (1972), which we will not expand on here.

1.2 A.2: The DR method

We elaborate on the DR method first introduced by Liu (2017), which effectively reduces the number of stocks, while still preserves the variance in the market. Suppose that there are N assets with asset prices $S^{(1)},S^{(2)},...,S^{(N)}$ in the market. Based on the multivariate Black-Scholes model, the asset price processes $\left\{ S_t^{(h)}\right\}$ for $h = 1,2,...,N$ solves the stochastic differential equation

$$\begin{aligned} \frac{dS_t^{(h)}}{S_t^{(h)}} = r_t dt + \sum _{l=1}^N \sigma _{hl} dB_t^{(l)}, \quad S_0^{(h)} = 1, \end{aligned}$$

(1)

where $B_t^{(1)},B_t^{(2)},...,B_t^{(N)}$ follow the independent standard Brownian motions, $r_t$ is the short rate of interest, and $[\sigma _{hl}]$ is the matrix capturing the correlation among the assets. Then, the solution to Equation (1) is

$$\begin{aligned} S_t^{(h)} = \exp \left[ \left( \int _0^t r_s ds - \frac{t}{2}\sum _{l=1}^N \sigma ^2_{hl}\right) +\sum _{l=1}^N \sigma _{hl}B_t^{(l)}\right] ,\quad h=1,2,...,N. \end{aligned}$$

(2)

Let $t_0=0$, $t_1 = \Delta ,...,t_m=m\Delta$ be the time steps with equal space $\Delta$, and suppose that the continuous forward rate is constant within each period. We denote $f_j$ as the annualized continuous forward rate for period $\left( t_{j-1},t_j\right)$ such that

$$\begin{aligned} f_j = \frac{1}{\Delta }\int _{t_{j-1}}^{t_j} r_s ds,\quad j=1,2,...,m. \end{aligned}$$

Then, we have

$$\begin{aligned} \exp \left( \Delta \left( f_1+f_2+...+f_j\right) \right) = \exp \left( \int _0^{t_j}r_s ds\right) ,\quad j=1,2,...,m. \end{aligned}$$

(3)

For $j=1,2,...,m$, let $A_j^{(h)}$ be the accumulation factor of the $h^{th}$ index for the period $(t_{j-1},t_j)$, that is,

$$\begin{aligned} A_j^{(h)} = \frac{S_{j\Delta }^{(h)}}{S_{(j-1)\Delta }^{(h)}}. \end{aligned}$$

(4)

Combining Equation (2) to (4), we get

$$\begin{aligned} A_j^{(h)} = \exp \left[ \left( f_j-\frac{1}{2}\sum _{l=1}^N\sigma ^2_{hl}\right) \Delta +\sum _{l=1}^N \sigma _{hl} \sqrt{\Delta }Z_j^{(l)}\right] , \end{aligned}$$

(5)

where

$$\begin{aligned} Z_j^{(l)} = \frac{B_{j\Delta }^{(l)}-B_{(j-1)\Delta }^{(l)}}{\sqrt{\Delta }}. \end{aligned}$$

By the property of Brownian motion, we know that $Z_1^{(l)},Z_2^{(l)},...,Z_m^{(l)}$ are independent random variables with a standard normal distribution. From Equation (5), we derive the continuous return for the period $(t_{j-1},t_j)$

$$\begin{aligned} R_j^{(h)} = \ln \left( A_j^{(h)}\right) = \left( f_j-\frac{1}{2}\sum _{l=1}^N \sigma _{hl}^2\right) \Delta + \sum _{l=1}^N \sigma _{hl} \sqrt{\Delta } Z_j^{(l)}. \end{aligned}$$

The mean and covariance matrix of the returns are given by

$$\begin{aligned} \mathrm {E}\left[ R_j^{(h)}\right] = \left( f_j-\frac{1}{2}\sum _{l=1}^N \sigma _{hl}^2\right) \Delta \end{aligned}$$

and

$$\begin{aligned} \mathrm {Cov}\left( R_j^{(h)},R_j^{(s)}\right)&=\mathrm {E}\left[ \left( R_j^{(h)}-\mathrm {E}\left[ R_j^{(h)}\right] \right) \left( R_j^{(s)}-\mathrm {E}\left[ R_j^{(s)}\right] \right) \right] \\&=\mathrm {E}\left[ \left( \sum _{l=1}^N \sigma _{hl} \sqrt{\Delta }Z_j^{(l)}\right) \left( \sum _{l=1}^N \sigma _{sl} \sqrt{\Delta }Z_j^{(l)}\right) \right] \\&=\sum _{l=1}^N \sigma _{hl}\sigma _{sl}\Delta , \quad h,s=1,2,...,N. \end{aligned}$$

Let $\Sigma$ be the covariance matrix of the annualized continuous returns of the N stocks and

$$\begin{aligned} {\mathbf {A}}= \begin{bmatrix} \sigma _{11} &{} \sigma _{12} &{}... &{} \sigma _{1N} \\ \sigma _{21} &{} \sigma _{22} &{}... &{} \sigma _{2N} \\ \vdots &{} \vdots &{}\ddots &{} \vdots \\ \sigma _{N1} &{} \sigma _{N2} &{}... &{} \sigma _{NN} \\ \end{bmatrix} \end{aligned}$$

be the Cholesky decomposition of $\Sigma$ such that

$$\begin{aligned} \mathbf {AA}^\intercal = \Sigma , \end{aligned}$$

where ${\mathbf {A}}^\intercal$ is the transpose of ${\mathbf {A}}$. Then, the variance contribution, also known as the explained variance (e.g., Kent 1983), of the first $N_{DR}$ assets with the highest Sharpe ratios can be defined as

$$\begin{aligned} \frac{\Vert A_1 \Vert ^2+...+\Vert A_{N_{DR}} \Vert ^2}{\Vert A_1 \Vert ^2+...+\Vert A_N \Vert ^2} \end{aligned}$$

where $A_i$ is the $i^{th}$ column of ${\mathbf {A}}$. In this paper, the reduced dimensionality $N_{DR}$ is the minimum number of assets needed to reach the $95\%$ explained variance, and the dimensionality reduction is achieved when $N_{DR}<<N$.

1.3 A.3: The EVT method

In this section, we discuss the EVT method in further details. We take one stock market, say the China market, as an example and exclude all non-positive returns since our concern is the right-tail return. Let $X_{1},X_{2},...,X_{n}$ denote the observations of returns in one group, say G1. We consider these n returns as i.i.d. observations from some distribution function F. Let $X_{1,n} \le X_{2,n} \le ... \le X_{n,n}$ be the associated order returns, so that $X_{n,n}$ denotes the maximum return in G1. Then, according to Mises (1954) and Jenkinson (1955), if the maximum $X_{n,n}$, suitably centered and scaled, converges to a non-degenerate random variable, then there exist sequences $\{a_n\}$ $(a_n > 0)$ and ${b_n}$ $(b_n \in {\mathbb {R}})$ such that

$$\begin{aligned} \lim _{n \rightarrow \infty } {\mathbb {P}} \left\{ \frac{X_{n,n}-b_n}{a_n} \le x\right\} = G_\gamma (x), \end{aligned}$$

(6)

where

$$\begin{aligned} G_\gamma (x) := \exp \left( -\left( 1+\gamma x\right) ^{-1/\gamma } \right) \end{aligned}$$

for some $\gamma \in {\mathbb {R}}$, with x such that $1+\gamma x > 0$. That is, F is in the domain of attraction of some extreme value distribution function $G_\gamma$ and $\gamma$ is the extreme-value index. By taking logarithms, Equation (6) can be written as

$$\begin{aligned} \lim _{q \rightarrow \infty } q\left( 1-F\left( a_qx+b_q\right) \right) = -\log G_\gamma (x) = \left( 1+\gamma x\right) ^{-1/\gamma },\quad G_\gamma (x)>0, \end{aligned}$$

where $q \in {\mathbb {R}}^+$ and $a_q$ and $b_q$ are defined by interpolation. We take $b_q = U(q)$ with

$$\begin{aligned} U(q) := \left( \frac{1}{1-F}\right) ^{-1}(q) = F^{-1}\left( 1-\frac{1}{q}\right) ,\quad q>1, \end{aligned}$$

where $-1$ denotes the left-continuous inverse.

We then estimate $\gamma$, $a_q$ and $b_q$ as follows. Let, for $1 \le k < n$,

$$\begin{aligned} M_n^{(p)} := \frac{1}{k} \sum _{i=0}^{k-1} \left( \log X_{n-i,n} - \log X_{n-k,n}\right) ^p,\quad p=1,2. \end{aligned}$$

We use the moment estimators for $\gamma \in {\mathbb {R}}$ introduced by Dekkers et al. (1989):

$$\begin{aligned} \hat{\gamma } := M_n^{(1)} + 1 - \frac{1}{2}\left( 1-\frac{\left( M_n^{(1)}\right) ^2}{M_n^{(2)}}\right) ^{-1}. \end{aligned}$$

Specifically, we first test that $\gamma$ exists for all groups according to Dietrich et al. (2002). Next, we plot $\hat{\gamma }$ as a function of k, which is the number of upper order statistics used for estimation minus 1. Then, we determine the first stable region in k of the estimate from the moment estimator plot. Namely, we try to identify a set of consecutive values of k where the estimated values do not fluctuate much, so that the procedure is insensitive to the choice of k in such a region. For the moment estimator in G1 for the China market, as illustrated in Fig. 9, such a stable region runs from around $k = 30$ to $k = 200$.

Fig. 9

Moment estimator versus k for G1 of the China Market

Next, we define the following estimators for $a_n/k$ and $b_n/k$:

$$\begin{aligned} \hat{a} := \hat{a}_{n/k}:= {\left\{ \begin{array}{ll} X_{n-k,n}M_n^{(1)}(1-\hat{\gamma }) &{} \text {if }\hat{\gamma }<0 \\ X_{n-k,n}M_n^{(1)} &{}\text {otherwise,} \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} \hat{b}:=\hat{b}_{n/k}:=X_{n-k,n}. \end{aligned}$$

Then, our goal is to estimate the right endpoint

$$\begin{aligned} x^* := \sup \left\{ x|F(x)<1\right\} \end{aligned}$$

of the distribution function F, that is, the ultimate return of G1 based on the observed returns. When estimating the endpoint, we assume that $\gamma < 0$. Next, it can be shown that Equation (6) is equivalent to

$$\begin{aligned} \lim _{q \rightarrow \infty } \frac{U(qx)-U(q)}{a(q)}=\frac{x^\gamma -1}{\gamma },\quad x>0. \end{aligned}$$

As t gets large, we can write

$$\begin{aligned} U(qx) \approx U(q) + a(q)\frac{x^\gamma -1}{\gamma }. \end{aligned}$$

Because $\gamma <0$ this yields, for large x and setting $q=n/k$,

$$\begin{aligned} x^* \approx U\left( \frac{n}{k}\right) -a\left( \frac{n}{k}\right) \frac{1}{\gamma }. \end{aligned}$$

Therefore, $x^*$ can be estimated as

$$\begin{aligned} \hat{x}^* := \hat{b} -\frac{\hat{a}}{\hat{\gamma }}, \end{aligned}$$

where $\hat{\gamma } <0$, and $\hat{x}^*:=\infty$ otherwise. The endpoint estimate of G1 for the China is shown in Fig. 10, and the selected estimate is the dotted horizontal line.

Fig. 10

Endpoint estimators versus k for G1 of the China Market