Robust trade-off portfolio selection

Abstract

Robust portfolio selection explicitly incorporates a model of parameter uncertainty in the problem formulation, and optimizes for the worst-case scenario. We consider robust mean–variance portfolio selection involving a trade-off between the worst-case utility and the worst-case regret, or the largest difference between the best utility achievable under the model and that achieved by a given portfolio. While optimizing for the worst-case utility may yield an overly pessimistic portfolio, optimizing for the worst-case regret may result in a complete loss of robustness; we theoretically demonstrate this point. Robust trade-off portfolio compromises these two extremes, enabling more informative selections. We show that, under a widely used ellipsoidal uncertainty model, the entire optimal trade-off curve can be found via solving a series of semidefinite programs (SDPs). We then extend the model to handle the union of finitely many ellipsoids, and show that trade-off analysis under this quite general uncertainty model also reduces to a series of SDPs. For more general uncertainties, we propose an iterative algorithm based on the cutting-set method. Under the finite-union-of-ellipsoids model, this algorithm offers an alternative to the SDP in exploring the optimal trade-off curve. We illustrate the promises of the trade-off portfolios by using historical stock returns data.

Appendices

Appendix 1: Proofs

1.1 Appendix 1.1: Proof of Theorem 1

1.1.1 Part (a)

With the introduction of an auxiliary variable t, maximization of $U_{\text {wc}}(w)$ (5) can be written

$$\begin{aligned} \begin{array}{ll} {\text{ maximize }} &{} t \\ {\text{subject to }} &{} {\mathbf {1}}^Tw = 1 \\ &{} U(w,\mu ,\varSigma ) \ge t, \quad \forall (\mu ,\varSigma ) \in {\mathcal {U}} . \end{array} \end{aligned}$$

Then the Lagrangian of this concave program is

$$\begin{aligned} L(w,t,\lambda )&= t + \int _{{\mathcal {U}}}(U(w,\mu ,\varSigma )-t)\lambda (d(\mu ,\varSigma )) \\&= \left( 1 - \int _{{\mathcal {U}}}\lambda (d(\mu ,\varSigma ))\right) t + \int _{{\mathcal {U}}}U(w,\mu ,\varSigma )\lambda (d(\mu ,\varSigma )), \end{aligned}$$

where $\lambda$ is a measure defined on the Borel $\sigma$-field of ${\mathcal {U}}$. Then the Lagrangian dual function of problem (5) is

$$\begin{aligned} g(\lambda )&= \sup _{t,w:{\mathbf {1}}^Tw=1} L(w,t,\lambda ) \\&= \sup _{t,w:{\mathbf {1}}^Tw=1} \big (1 - \lambda ({\mathcal {U}})\big ) t + \int _{{\mathcal {U}}}U(w,\mu ,\varSigma )\lambda (d(\mu ,\varSigma )) \\&= {\left\{ \begin{array}{ll} \sup _{t,w:{\mathbf {1}}^Tw=1} \int _{{\mathcal {U}}}U(w,\mu ,\varSigma )\lambda (d(\mu ,\varSigma )), &\quad \lambda ({\mathcal {U}}) = 1, \\ +\infty , &\quad{} {\text {otherwise}}, \end{array}\right. } \end{aligned}$$

i.e., $g(\lambda )$ is finite if and only if $\lambda$ is a probability measure on ${\mathcal {U}}$. Note

$$\begin{aligned} \sup _{w:{\mathbf {1}}^Tw=1} \int _{{\mathcal {U}}}U(w,\mu ,\varSigma )\lambda (d(\mu ,\varSigma ))&= \sup _{w:{\mathbf {1}}^Tw=1} \mathbf {E}_{\lambda } \big ( w^T\mu - (\gamma /2)w^T\varSigma w \big ) \\&= \sup _{w:{\mathbf {1}}^Tw=1} U(w,\mu _{\lambda },\varSigma _{\lambda }) \\&= \bar{U}(\mu _{\lambda },\varSigma _{\lambda }), \end{aligned}$$

where $\bar{U}(\mu ,\varSigma )=\sup _{z:{\mathbf {1}}^Tz=1}U(z,\mu ,\varSigma )$; $\mathbf {E}_{\lambda }(\cdot )$ denotes the expectation with respect to probability measure $\lambda$, $\mu _{\lambda } = \mathbf {E}_{\lambda }[\mu ]$, and $\varSigma _{\lambda } = \mathbf {E}_{\lambda }[\varSigma ]$.

Thus the dual optimization problem of (5) is to find $\lambda$ for which $(\mu _{\lambda }, \varSigma _{\lambda })$ solves

$$\begin{aligned} \inf _{(\mu _{\lambda },\varSigma _{\lambda }) \in {\mathcal {U}}} \bar{U}({\tilde{\mu }}, {\tilde{\varSigma }}) = \inf _{(\mu _{\lambda },\varSigma _{\lambda }) \in {\mathcal {U}}} \sup _{{\mathbf {1}}^Tw=1} U(w, \mu _{\lambda }, \varSigma _{\lambda }) = \sup _{{\mathbf {1}}^Tw=1} \inf _{(\mu _{\lambda },\varSigma _{\lambda }) \in {\mathcal {U}}} U(w, \mu _{\lambda }, \varSigma _{\lambda } ), \end{aligned}$$

(33)

where the last equation is due to the strong duality. The latter holds because Slater’s condition for the primal (5) is satisfied: $\inf _{(\mu ,\varSigma )\in {\mathcal {U}}} U(w,\mu ,\varSigma ) > -\infty$ since ${\mathcal {U}}={\mathcal {M}}\times \mathcal {S}$ is compact by the assumption; note that the last optimization problem is precisely of the same form as problem (5).

Under the given separable uncertainty model, it is easy to verify that the optimal tuple $(w^{\star },\mu _{\lambda ^{\star }}, \varSigma _{\lambda ^{\star }})$ satisfies

$$\begin{aligned} \mu _{\lambda ^{\star }} = {\bar{\mu }} - P\frac{P^Tw^{\star }}{\Vert P^Tw^{\star }\Vert }, \end{aligned}$$

which implies that under the optimal probability measure $\lambda ^{\star }$, $\mu _{\lambda ^{\star }}=\mathbf {E}_{\lambda ^{\star }}[\mu ]$ is a point on the boundary of ${\mathcal {M}}$. This can only occur when $\lambda ^{\star }$ is a product measure of a Dirac measure that puts all the mass at a boundary point of ${\mathcal {M}}$ and any probability measure on $\mathcal {S}$.

1.1.2 Part (b)

Similarly to part (a), write the problem of minimizing $R_{\text {wc}}(w)$ (6) as

$$\begin{aligned} \begin{array}{ll} {\text{ minimize }} &{} t \\ {\text{subject to }} &{} {\mathbf {1}}^Tw = 1 \\ &{} \bar{U}(\mu ,\varSigma ) - U(w,\mu ,\varSigma ) \le t, \quad \forall (\mu ,\varSigma ) \in {\mathcal {U}}, \end{array} \end{aligned}$$

(P)

whose Lagrangian is

$$\begin{aligned} L(w,t,\lambda )&= \left( 1 - \int _{{\mathcal {U}}}\lambda (d(\mu ,\varSigma ))\right) t + \int _{{\mathcal {U}}}\big ( \bar{U}(\mu ,\varSigma )-U(w,\mu ,\varSigma )\big )\lambda (d(\mu ,\varSigma )) \end{aligned}$$

for $w\in {\mathbb {R}}^n$, $t\in {\mathbb {R}}$, and $\lambda$ a measure on the Borel $\sigma$-field of  ${\mathcal {U}}$. The associated dual optimization problem is to maximize

$$\begin{aligned} \begin{aligned} g(\lambda )&= \inf _{w:{\mathbf {1}}^Tw=1} \int _{{\mathcal {U}}}\big ( \bar{U}(\mu ,\varSigma )-U(w,\mu ,\varSigma )\big )\lambda (d(\mu ,\varSigma )) \\&= \inf _{w:{\mathbf {1}}^Tw=1} \mathbf {E}_{\lambda } \big ( \bar{U}(\mu ,\varSigma )-U(w,\mu ,\varSigma )\big ) \\&= \mathbf {E}_{\lambda } \big ( \bar{U}(\mu ,\varSigma ) \big ) - \sup _{w:{\mathbf {1}}^Tw=1} U(w,\mathbf {E}_{\lambda }(\mu ),\mathbf {E}_{\lambda }(\varSigma ) ) \\&= \mathbf {E}_{\lambda } \big ( \bar{U}(\mu ,\varSigma ) \big ) - \bar{U}(\mu _{\lambda },\varSigma _{\lambda } ) \ge 0, \end{aligned} \end{aligned}$$

(D)

for a probability measure $\lambda$ on ${\mathcal {U}}$. The last inequality is due to the convexity of $\bar{U}$ in $(\mu , \varSigma )$ and Jensen’s inequality.

Slater’s condition is satisfied for (P) since $\sup _{(\mu ,\varSigma )\in {\mathcal {U}}}\{\bar{U}(\mu ,\varSigma ) - U(w,\mu ,\varSigma )\} < \infty$, which in turn is because of the assumed compactness of ${\mathcal {U}}={\mathcal {M}}\times {\mathcal {S}}$. Thus the strong duality holds, which translates

$$\begin{aligned} g(\lambda ^{\star })&= \mathbf {E}_{\lambda ^{\star }}[\bar{U}(\mu ,\varSigma )] - \bar{U}(\mu _{\lambda ^{\star }}, \varSigma _{\lambda ^{\star }}) \\&= \mathbf {E}_{\lambda ^{\star }}[\bar{U}(\mu ,\varSigma )] - U(w^{\star }, \mathbf {E}_{\lambda ^{\star }}[\mu ], \mathbf {E}_{\lambda ^{\star }}[\varSigma ])\\&= \mathbf {E}_{\lambda ^{\star }}[\bar{U}(\mu ,\varSigma ) - U(w^{\star }, \mu , \varSigma )] \\&= \max _{(\mu ,\varSigma )\in {\mathcal {U}}}\{\bar{U}(\mu ,\varSigma ) - U(w^{\star }, \mu , \varSigma )\} = R_{\text {wc}}(w^{\star }) = t^{\star } , \end{aligned}$$

for the optimal tuple $(w^{\star }, t^{\star }, \lambda ^{\star })$. The fourth inequality shows that the opitimal $\lambda ^{\star }$ is a degenerate probability measure that puts all the masses on the maximizers of $\bar{U}(\mu , \varSigma ) - U(w^{\star },\mu ,\varSigma )$, where $w^{\star }=\text {arg max}_{w:{\mathbf {1}}^Tw=1} U(w, \mu _{\lambda ^{\star }}, \varSigma _{\lambda ^{\star }})$.

Hence we now consider the problem

$$\begin{aligned} \max _{\varSigma \in {\mathcal {S}}}\max _{\mu \in {\mathcal {M}}} \{\bar{U}(\mu , \varSigma ) - U(w^{\star },\mu ,\varSigma )\} . \end{aligned}$$

(R)

Recall ${\mathcal {M}}=\{{\bar{\mu }} + Pu: \Vert u\Vert _2 \le 1\}$ and that $\bar{U}(\mu ,\varSigma )$ attains a closed form:

$$\begin{aligned} \bar{U}(\mu ,\varSigma ) = \alpha (\varSigma ) + b(\varSigma )^T\mu + \mu ^T B(\varSigma ) \mu , \end{aligned}$$

where

$$\begin{aligned} \alpha (\varSigma )&= -\frac{\gamma }{2({\mathbf {1}}^T\varSigma ^{-1}{\mathbf {1}})}, \quad b(\varSigma ) = \frac{1}{{\mathbf {1}}^T\varSigma ^{-1}{\mathbf {1}}}\varSigma ^{-1}{\mathbf {1}},\\ B(\varSigma )&= \frac{1}{2\gamma }\left( \varSigma ^{-1}- \frac{1}{{\mathbf {1}}^T\varSigma ^{-1}{\mathbf {1}}}\varSigma ^{-1}{\mathbf {1}}{\mathbf {1}}^T\varSigma ^{-1}\right) \succeq 0 . \end{aligned}$$

Also since $w^{\star }$ maximizes $U(w, \mu _{\lambda ^{\star }}, \varSigma _{\lambda ^{\star }})$, it follows that

$$\begin{aligned} w^{\star } = b(\varSigma _{\lambda ^{\star }}) + 2B(\varSigma _{\lambda ^{\star }})\mu _{\lambda ^{\star }} . \end{aligned}$$

(see Sect. 3.2).

If $\mathcal {S}$ is singleton, i.e., ${\mathcal {S}}=\{{\bar{\varSigma }}\}$, then

$$\begin{aligned} g(\lambda )&= \mathbf {E}_{\lambda }[\bar{U}(\mu ,\varSigma )] - \bar{U}(\mu _\lambda , {\bar{\varSigma }}) \\&= \mathbf {E}_{\lambda }[\alpha (\varSigma ) + b(\varSigma )^T\mu + \mu ^T B(\varSigma )\mu ] - \alpha ({\bar{\varSigma }}) - b({\bar{\varSigma }})^T\mu _{\lambda } - \mu _{\lambda }^T B({\bar{\varSigma }})\mu _{\lambda }\\&= \mathbf {Tr}\left( B({\bar{\varSigma }}) (\mathbf {E}_{\lambda }[\mu \mu ^T] - \mu _{\lambda }\mu _{\lambda }^T) \right) = \mathbf {Tr}( B({\bar{\varSigma }}) {\mathbf {cov}}_{\lambda }[\mu ] )\\&= \mathbf {Tr}( B({\bar{\varSigma }})P{\mathbf {cov}}_{\pi }[u] P^T ) = \mathbf {Tr}(P^T B({\bar{\varSigma }})P\mathbf {E}_{\pi }[uu^T]) - \mathbf {Tr}(P^T B({\bar{\varSigma }})P\mathbf {E}_{\pi }[u]\mathbf {E}_{\pi }[u]^T)\\&= \mathbf {E}_{\pi }[u^TP^T B({\bar{\varSigma }})Pu] - \Vert \mathbf {E}_{\pi }[B^{1/2}({\bar{\varSigma }})Pu]\Vert _2^2\\&\le \Vert B^{1/2}({\bar{\varSigma }})P u_{\max }\Vert _2^2 , \end{aligned}$$

since $\mathbf {E}_{\lambda }[f(\varSigma )]=f({\bar{\varSigma }})$ for any function f of $\varSigma$; $\pi$ is the probability measure on $\{u: \Vert u\Vert \le 1\}$ induced from $\lambda$ by the transformation $\mu ={\bar{\mu }} + Pu$, and $u_{\max }$ solves the following optimization problem

$$\begin{aligned} \begin{array}{ll} {\text {maximize}} &{} \Vert B^{1/2}({\bar{\varSigma }})P u\Vert _2 \\ {\text {subject to}} &{} \Vert u\Vert \le 1 . \end{array} \end{aligned}$$

(34)

Note that $-u_{\max }$ also solves (34). The inequality in the last line holds with equality if and only if $\pi$ puts equal masses on $u_{\max }$ and $-u_{\max }$. Thus $\lambda ^{\star }$ is a degenerate probability measure that puts equal masses on $({\bar{\mu }}\pm Pu_{\max }, {\bar{\varSigma }})$, yielding $\mu _{\lambda ^{\star }}=\mathbf {E}_{\lambda ^{\star }}[\mu ]={\bar{\mu }}$. Trivially $\varSigma _{\lambda ^{\star }} = {\bar{\varSigma }}$. Therefore, we have

$$\begin{aligned} w^{\star } = b({\bar{\varSigma }}) + 2B({\bar{\varSigma }}){\bar{\mu }} = w_{\text {nom}} \end{aligned}$$

as claimed.

For general $\mathcal {S}$, if the worst-case regret criterion

$$\begin{aligned} \max _{\varSigma \in {\mathcal {S}}}\max _{\mu \in {\mathcal {M}}} \{\bar{U}(\mu , \varSigma ) - U(w,\mu ,\varSigma )\} = R_{\text {wc}}(w) \end{aligned}$$

(6 reformulated)

has a unique maximizer $\varSigma ^{\star }(w)$ for each weight vector w, then problem (R) has a unique maximizer $\varSigma ^{\star }=\varSigma ^{\star }(w^{\star })$ and the optimal probability measure $\lambda ^{\star }$ puts all the masses on $\varSigma ^{\star }$ in $\mathcal {S}$. Thus it suffices to consider probability measures $\lambda$ on ${\mathcal {M}}\times \{\varSigma ^{\star }\}$ in maximizing $g(\lambda )$ in (D). In particular, for these $\lambda$ there holds $\mathbf {E}_{\lambda }[f(\varSigma )]=f(\varSigma ^{\star })$ for any function f. Thus by the same argument as the singleton case above, $\lambda ^{\star }$ is a degenerate probability measure that puts equal masses on $({\bar{\mu }}\pm Pu_{\max }^{\star }, \varSigma ^{\star })$, where $u_{\max }^{\star }$ is a solution to problem (34) with ${\bar{\varSigma }}$ replaced by $\varSigma ^{\star }$. This yields $\mu _{\lambda ^{\star }}=\mathbf {E}_{\lambda ^{\star }}[\mu ]={\bar{\mu }}$ and $w^{\star } = \beta (\varSigma ^{\star }) + \varGamma (\varSigma ^{\star }){\bar{\mu }}$, as desired.

1.2 Appendix 1.2: Proof of Theorem 2

Assume that $\lambda _1 = \cdots = \lambda _r > \lambda _{r+1}$ for some $r \ge 1$, and let $V_1$ denote the first r columns of the V. The Lagrangian of (18) is

$$\begin{aligned} L(u,\lambda ) = -u^T(\lambda I-P^TBP)u + u^T\nu + \lambda , \quad \lambda \ge 0. \end{aligned}$$

The dual function is thus

$$\begin{aligned} g(\lambda )&= \sup _{u} L(u,\lambda ) \nonumber \\&= {\left\{ \begin{array}{ll} \lambda + (1/4)\nu ^T(\lambda I - P^TBP)^{\dagger }\nu , &{} \lambda I - P^TBP \succeq 0, ~ \nu \in {\mathcal {R}}(\lambda I - P^TBP),\\ \infty , &{} {\text {otherwise}} \end{array}\right. }\nonumber \\&= {\left\{ \begin{array}{ll} \lambda + (1/4)\nu ^T(\lambda I - P^TBP)^{-1}\nu , &{} \lambda > \lambda _1, \\ \lambda _1 + (1/4)\nu ^T(\lambda _1 I - P^TBP)^{\dagger }\nu , &{} \lambda = \lambda _1,~ V_1^T\nu =0, \\ \infty , &{} {\text {otherwise}} \end{array}\right. }\nonumber \\&= {\left\{ \begin{array}{ll} \lambda + (1/4)\sum _{i=1}^n {\bar{\nu }}_i^2/(\lambda -\lambda _i), &{} \lambda \ge \lambda _1, \\ \infty , &{} {\text {otherwise}}, \end{array}\right. } \end{aligned}$$

(35)

interpreting $g(\lambda _1)=\infty$ if $V_1^T\nu \ne 0$. Note ${\bar{\nu }}_1 = \cdots = {\bar{\nu }}_r = 0$ if $V_1^T\nu = 0$. The dual optimization problem is to minimize $g(\lambda )$, which is equivalent to the SDP (19) in that $t^{\star } = g(\lambda ^{\star })$.

For $\lambda > \lambda _1$, $g(\lambda )$ is strictly convex and also strictly increasing for sufficiently large $\lambda$, thus $\lambda ^{\star } > \lambda _1$ if the equation

$$\begin{aligned} g'(\lambda ) = 1 - \frac{1}{4}\sum _{i=1}^n\frac{{\bar{\nu }}_i^2}{(\lambda -\lambda _i)^2} = 0 \end{aligned}$$

(36)

has a root on $\lambda > \lambda _1$, otherwise $g'(\lambda ) \ge 0$ and $\lambda ^{\star }=\lambda _1$.

From (35), $u^{\star } = (1/2)(\lambda ^{\star } I -P^TBP)^{\dagger }\nu + v$, where $v \in {\mathcal {N}}(\lambda ^{\star } I - P^TBP)$. If $\lambda ^{\star } > \lambda _1$, $v=0$ and $u^{\star } = (1/2)(\lambda ^{\star } I -P^TBP)^{-1}\nu = (1/2)V{{\,\mathrm{diag}\,}}{(1/(\lambda ^{\star }-\lambda _1),\ldots ,1/(\lambda ^{\star }-\lambda _n))}V^T\nu$. Thus $\Vert u^{\star }\Vert _2^2 = (1/4)\sum _{i=1}^n {\bar{\nu }}_i^2/(\lambda ^{\star }-\lambda _i)^2 = 1$ due to (36), which is also confirmed by the Kuhn–Karush–Tucker (KKT) condition $\lambda ^{\star }(\Vert u^{\star }\Vert _2^2-1) = 0$ for (18). If $\lambda ^{\star } = \lambda _1$, then $\lambda _1v=P^TBPv$ hence $v=kv_{\max }$ for some scalar k. This also implies that ${\bar{\nu }}_1 = \cdots = {\bar{\nu }}_r = 0$. From the KKT condition, $\Vert u^{\star }\Vert _2^2 = (1/4)\sum _{i=r+1}^n{\bar{\nu }}_i^2/(\lambda _1-\lambda _i)^2 + k^2=1$, yielding $k=\pm \sqrt{1-(1/4)\sum _{i=r+1}^n{\bar{\nu }}_i^2/(\lambda _1-\lambda _i)^2}$. The latter is well-defined since $g'(\lambda _1) \ge 0$.

Appendix 2: Performance metrics

We first show how the wealth varies under the periodic rebalancing strategy described above and describe some descriptive statistics used to assess its performance. A dollar invested to asset i at the beginning of the $j-1$st period grows to $d^{(j-1)}_i = \prod ^{N_{\mathrm {estim}}+jL}_{t=N_{\mathrm {estim}}+1+(j-1)L}(1+r_{it})$. The initial portfolio $w^{(j-1)}$ held at the beginning of the $j-1$st period is therefore changed to the portfolio $\tilde{w}^{(j-1)}$ with weights

$$\begin{aligned} {{\tilde{w}}}^{(j-1)}_i = \frac{d^{(j-1)}_iw^{(j-1)}_i}{\sum ^n_{i=1}d^{(j-1)}_iw_t^{(j-1)}}, \qquad i = 1,\ldots ,n. \end{aligned}$$

For the first period, we take ${{\tilde{w}}}^{(0)}=0$, i.e., the initial holdings of the assets are zero.

At the start date of the jth period, asset weights are changed from ${{\tilde{w}}}^{(j-1)}$ to $w^{(j)}$. The turnover ratio, i.e., the relative amount of the purchase and sale of securities in a fund’s portfolio, associated with the change is computed as

$$\begin{aligned} {\mathrm {TO}}_j = \sum ^n_{i=1}\left| w^{(j)}_i-\tilde{w}^{(j-1)}_i\right| . \end{aligned}$$

The percentage of the portfolio that is bought and sold to exchange for other stocks is $100{\mathrm {TO}}_j\%$.

Portfolio rebalancing incurs transaction costs. Let the cost to buy or sell one share of stock i be $\eta _i$. The transaction cost ${\mathrm {TC}}_j$ due to the rebalancing at the beginning of the jth period is given by

$$\begin{aligned} {\mathrm {TC}}_j = W_{N_{\mathrm {estim}}+jL}\sum ^n_{i=1}\eta _i\left| w^{(j)}_i-\tilde{w}^{(j)}_i\right| . \end{aligned}$$

Then the (normalized) wealth grows according to the recursion

$$\begin{aligned} W_t = \left\{ \begin{array}{ll} W_{t-1}(1+\sum ^n_{i=1}w_{it}r_{it}), &{} t \not \in \{N_{\mathrm {estim}}+jL \;|\; j=1,\ldots ,K\}, \\ W_{t-1}(1+\sum ^n_{i=1}w_{it}r_{it})-{\mathrm {TC}}_j, &{} t = N_{\mathrm {estim}}+jL, \end{array} \right. \end{aligned}$$

for $t = N_{\mathrm {estim}}, \ldots , N_{\mathrm {estim}}+KL$, with $W_{N_{\mathrm {estim}}} = 1$.

We are interested in several performance metrics which include the annualized return, risk, and Sharpe ratio (SR) of the normalized wealth time series. (The annualized Sharpe ratio is the ratio of the annualized excess expected return of relative to the risk-free return $\mu _{\mathrm {rf}}$) Another performance metric of interest is the maximum drawdown, defined as

$$\begin{aligned} {\mathrm {MDD}} = \sup _{t=N_{\mathrm {estim}},\ldots ,N_{\mathrm {trading}}+KL}{\mathrm {MDD}}_t, \quad {\text {where}} \quad {\mathrm {MDD}}_t = \frac{W_t}{\sup _{s \le t} W_s}, \end{aligned}$$

i.e., the largest drawdown of the wealth experienced by the trading strategy over the entire trading periods.

Appendix 3: Portfolio rebalancing data basics

See Tables 3, 4 and 5.

Table 3 S&P 500 component stocks and their performance over the investment periods considered in the numerical study

Table 4 Market factors

Table 5 Investment periods