On the use of conditional expectation in portfolio selection problems

Abstract

In this paper, we examine the use of conditional expectation, either to reduce the dimensionality of large-scale portfolio problems or to propose alternative reward–risk performance measures. In particular, we focus on two financial problems. In the first part, we discuss and examine correlation measures (based on a conditional expectation) used to approximate the returns in large-scale portfolio problems. Then, we compare the impact of alternative return approximation methodologies on the ex-post wealth of a classic portfolio strategy. In this context, we show that correlation measures that use the conditional expectation perform better than the classic measures do. Moreover, the correlation measure typically used for returns in the domain of attraction of a stable law works better than the classic Pearson correlation does. In the second part, we propose new performance measures based on a conditional expectation that take into account the heavy tails of the return distributions. Then, we examine portfolio strategies based on optimizing the proposed performance measures. In particular, we compare the ex-post wealth obtained from applying the portfolio strategies, which use alternative performance measures based on a conditional expectation. In doing so, we propose an alternative use of conditional expectation in various portfolio problems.

Appendices

Appendix A

Proof of Theorem 1

To prove Theorem 1, we need to prove the following proposition. $\square$

Proposition

Let $W=(X,Y,Z)$ be an n-dimensional elliptically distributed vector $Ell(\mu ,\Sigma )$, where X, Y, and Z are p-dimensional, q-dimensional, and $n-p-q$ dimensional vectors ($n>p>q\ge 1$), respectively, with dispersion matrix:¹¹

$$\begin{aligned} \Sigma =((\Sigma _{X},\Sigma _{XY},\Sigma _{XZ}),(\Sigma _{YX},\Sigma _{Y} ,\Sigma _{YZ}),(\Sigma _{ZY},\Sigma _{XY},\Sigma _{Z})) \end{aligned}$$

and mean $\mu =(\mu _{X},\mu _{Y},\mu _{Z})$, such that Y and Z are uncorrelated. Moreover, assume that $W=\mu +AG$, where A is a continuous positive random variable, which is independent of the Gaussian vector G, which has a null mean and variance–covariance matrix $\Sigma$. Then

$$\begin{aligned} E(X|Y,Z)=E(X|Y)+E(X|Z)-\mu _{X}\sim Ell(\mu _{X},\Sigma _{XY} \Sigma _{K}^{-1} \Sigma _{YX}+\Sigma _{XZ} \Sigma _{K}^{-1} \Sigma _{ZX}).\nonumber \\ \end{aligned}$$

(23)

Proof

Let $K=(Y,Z)$. Then, $\Sigma _{K}=\left( \left( \Sigma _{Y},0\right) ,\left( 0,\Sigma _{Z}\right) \right)$, $\mu _K=(\mu _Y,\mu _Z)'$ because Y and Z are uncorrelated (i.e., $\Sigma _{YZ}=0$). This implies that:

$$\begin{aligned} \Sigma =\left[ \begin{array}{ll} \Sigma _{X} &{} \Sigma _{XK}\\ \Sigma _{KX} &{} \Sigma _{K} \end{array} \right] \end{aligned}$$

and

$$\begin{aligned} \Sigma _{K}^{-1}=\left[ \begin{array}{ll} \Sigma _{Y}^{-1} &{} 0\\ 0 &{} \Sigma _{Z}^{-1} \end{array} \right] , \end{aligned}$$

where $\Sigma _{XK}=\left[ \Sigma _{XY},\Sigma _{XZ}\right]$. According to Corollary 1 proposed by Ortobelli and Lando (2015), we know that:

$$\begin{aligned} E(X|K)=\mu _X+\Sigma _{XK}\Sigma _{K}^{-1}(K-\mu _{K})\sim Ell(\mu _{X},\Sigma _{XK}\Sigma _{K}^{-1}\Sigma _{KX}), \end{aligned}$$

from which we get

$$\begin{aligned} E(X|Y,Z)=\mu _{X}+\Sigma _{XY}\Sigma _{Y}^{-1}(Y-\mu _{Y})+\Sigma _{XZ} \Sigma _{Z}^{-1}(Z-\mu _{Z})=E(X|Y)+E(X|Z)-\mu _{X}. \end{aligned}$$

Thus, the thesis holds. $\square$

The proof of Theorem 1 is a logical consequence of the above proposition. As a matter of fact, we obtain the proof with two factors if we assume Y and Z from the previous Proposition are two uncorrelated one-dimensional factors. The general proof follows by induction.

Appendix B

This appendex provides practical examples of the proposed performance measures TOK and JTOK using five assets, namely Apple Inc. (AAPL), Boeing Company (BA), JPMorgan Chase & Co. (JPM), Coca-Cola Company (KO), and 3M Company (MMM), over the period from January, 2015 to August, 2016.

Let us consider the TOK and JTOK performance measures for the uniform portfolio (i.e., $x=[0.2,0.2,0.2,0.2,0.2]'$). First, for the TOK ratio, we approximate the $\sigma$-algebras $\mathfrak {I}_{L}(x)$ and $\mathfrak {I}_{P}(x)$ generated by the portfolio losses and by the portfolio profits respectively. Thus, for the $\sigma$-algebras $\mathfrak {I}_{L}(x)$ and $\mathfrak {I}_{P}(x)$ we consider the following partitions $\{A_1,A_2,A_3\}$ and $\{B_1,B_2,B_3\}$, respectively, as follows:

$$\begin{aligned} A_{1}&=\left\{ x^{\prime }z\le F_{x^{\prime }z}^{-1} (0.1)\right\} ,&B_{1}&=\left\{ x^{\prime }z> F_{x^{\prime }z}^{-1} (0.9)\right\} , \\ A_{2}&=\left\{ F_{x^{\prime }z}^{-1}( 0.1)<x^{\prime }z\le F_{x^{\prime }z}^{-1}( 0.2) \right\} ,&B_{2}&=\left\{ F_{x^{\prime }z}^{-1}( 0.8) < x^{\prime }z\le F_{x^{\prime }z}^{-1}( 0.9) \right\} ,\\ A_{3}&=\left\{ x^{\prime }z>F_{x^{\prime }z}^{-1}(0.2) \right\} .&B_{3}&=\left\{ x^{\prime }z \le F_{x^{\prime }z}^{-1}(0.8) \right\} . \end{aligned}$$

Second, for the JTOK ratio, we approximate the $\sigma$-algebras $\mathfrak {I}_{Loss}$ and $\mathfrak {I}_{G}$ generated by the joint losses and gains respectively. On the one hand, for $\mathfrak {I}_{Loss}$, we calculate the function g that counts the total number of assets that are jointly lossing more than their $VaR_{0.2}(z_i)$, i.e., $g(w)=\sum _{i=1}^5 {\mathbf {1}}_{L_{i}^{(0.2)}}(w)$, where ${\mathbf {1}}_{L_{i}^{(0.2)}}$ is the indicator function of $L_{i}^{(0.2)}=\{z_i\le F_{z_i}^{-1}(0.2)\}$, and $F_{z_i}^{-1}(p)=\text{ inf }\{v: F_{z_i}(v)>p\}$. On the other hand, for $\mathfrak {I}_{G}$, we compute the function f, that counts the total number of assets with returns greater than 0.8 percentile, as $f(w)=\sum _{i=1}^5 {\mathbf {1}}_{G_{i}^{(0.8)}}(w)$, where $G_{i}^{(0.8)}=\{z_i\ge F_{z_i}^{-1}(0.8)\}$. Table 8 reports the monthly gross returns and functions f and g of the five assets.

Table 8 Monthly gross return of the five assets, over the period January, 2015 to August, 2016, and functions g and f

Thus, for $\mathfrak {I}_{Loss}$ and $\mathfrak {I}_{G}$, we consider the following partitions $\{A_{1,g},A_{2,g},A_{3,g}\}$ and $\{B_{1,f},B_{2,f},B_{3,f}\}$, respectively, as follows:

$$\begin{aligned} A_{1,g}&=\left\{ w: g(w)> F_{g}^{-1} (0.9)\right\} ,&B_{1,f}&=\left\{ w: f(w) > F_{f}^{-1} (0.9)\right\} ,\\ A_{2,g}&=\left\{ w: F_{g}^{-1}( 0.8)<g(w)\le F_{g}^{-1}( 0.9) \right\} ,&B_{2,f}&=\left\{ w: F_{f}^{-1}( 0.8) < f(w) \le F_{f}^{-1}( 0.9) \right\} , \\ A_{3,g}&=\left\{ w: g(w)\le F_{g}^{-1} (0.8)\right\} .&B_{3,f}&=\left\{ w: f(w)\le F_{f}^{-1}(0.8) \right\} . \end{aligned}$$

Table 9 reports the uniform portfolio and the partitions used for the TOK and JTOK performance measures (1 when $x'z$ belongs to its proper partition, 0 otherwise).

Table 9 Uniform portfolio $x'z$ and the partitions used for the TOK and JTOK performance measures

Third, we calculate the quantities $E(x'z|\mathfrak {I}_{L}(x))$ and $E(x'z|\mathfrak {I}_{P}(x))$ for the TOK performance measure and $E(x'z|\mathfrak {I}_{Loss})$ and $E(x'z|\mathfrak {I}_{G})$ for the JTOK ratio. To this end, we use the following consistent estimator (see Ortobelli et al. 2017):

$$\begin{aligned} E(x'z|\mathfrak {I}_{k})= \sum _{j=1}^{3}\left( \frac{1}{\eta _{A_j}}\sum _{x'z\in A_j} x'z\right) {\mathbf {1}}_{A_j}, \end{aligned}$$

where $\eta _{A_j}$ are the numbers of elements of $A_j$. In this case, $A_j$ is either $A_j, B_j, A_{j,g}$, or $B_{j,f}$. Accordingly, $\mathfrak {I}_{k}$ is either $\mathfrak {I}_{L}(x), \mathfrak {I}_{P}(x)$, $\mathfrak {I}_{Loss}$, or $\mathfrak {I}_{G}$. Our results based on this estimator are reported in Table 10.

Table 10 Conditional expectation of $x'z$ given different $\sigma$-algebras over the period January, 2015–August, 2016

Fourth, we caculate the portfolio net returns every month as follows:

• $y_{P}=E(x^{\prime }z|\mathfrak {I}_{P}(x))-1$ and $y_{L}=E(x^{\prime }z |\mathfrak {I}_{L} (x))-1$ for the TOK performance.

• $y_{G}=E(x^{\prime }z|\mathfrak {I}_{G})-1$ and $y_{Loss}=E(x^{\prime }z |\mathfrak {I}_{Loss})-1$ for the JTOK performance.

Fifth, we compute the cumulative product as follows:

• $W_{P,20}=\Pi _{t=1}^{20} (1+y_{P})_{t}=1.1063$ and $W_{L,20}= \Pi _{t=1}^{20} (1-y_{L} )_{t}=0.8880$ for the TOK performance.

• $W_{G,20}=\Pi _{t=1}^{20} (1+y_{G})_{t}=1.1076$ and $W_{Loss,20}= \Pi _{t=1}^{20} (1-y_{Loss} )_{t}=0.8883$ for the JTOK performance.

Finally, we calcule the ratio between $W_{P,20}$ and $W_{L,20}$ and ratio between $W_{G,20}$ and $W_{Loss,20}$ which give respctively $TOK= 1.2457$ and $JTOK= 1.2468$.