The effect of regularization in portfolio selection problems

Abstract

Portfolio selection problems have been thoroughly studied under the risk-and-return paradigm introduced by Markowitz. However, the usefulness of this approach has been hindered by some practical considerations that have resulted in poorly diversified portfolios, or, solutions that are extremely sensitive to parameter estimation errors. In this work, we use sampling methods to cope with this issue and compare the merits of two approaches: a sample average approximation approach and a performance-based regularization (PBR) method that appeared recently in the literature. We extend PBR by incorporating three different risk metrics—integrated chance-constraints, quantile deviation, and absolute semi-deviation—and deriving the corresponding regularization formulas. Additionally, a numerical comparison using index-based portfolios is presented using historic data that includes the subprime crisis.

A Proofs

Proof

(Lemma 1) We have

$$\begin{aligned} \mathrm {SV}[ {\mathsf {z}}]&= \frac{1}{n-1} \sum _{i=1}^{n} \left( z_{i} - \frac{1}{n}\sum _{j=1}^{n} z_{j}\right) ^{2} \\&= \frac{1}{n-1} \left\{ \sum _{i=1}^{n} z_{i}^{2} - \frac{2}{n} \sum _{i=1}^{n} \sum _{j=1}^{n} z_{i}z_{j} + n \cdot \frac{1}{n^{2}} \left( \sum _{j=1}^{n} z_{j} \right) ^{2}\right\} \\&= \frac{1}{n-1} \left\{ z^{T}I_{n}z - \frac{2}{n} \sum _{i=1}^{n} \sum _{j=1}^{n} z_{i}z_{j} + \frac{1}{n} \sum _{i=1}^{n} \sum _{j=1}^{n} z_{i}z_{j} \right\} \\&= \frac{1}{n-1} \left\{ z^{T}I_{n}z - \frac{1}{n} z^{T}1_{n}1_{n}^{T}z \right\} \\&= z^{T} \left\{ \frac{1}{n-1}[I_{n} - n^{-1}1_{n}1_{n}^{T}] \right\} z \\&= z^{T}\Omega _{n}z. \end{aligned}$$

$\square$

Proof

(Proposition 1) From Definition 6, given that the $Y_{i}$ are independent and identically distributed we have

$$\begin{aligned} \mathrm {Var}\left[ \widehat{\mathrm {ICC}}(w;{\mathsf {Y}}) \right]&= \frac{1}{n^{2}} \sum _{i=1}^{n} \mathrm {Var}\big [(h_{i} - w^{T}X_{i})^{+}\big ] \\&= \frac{1}{n^{2}} \cdot n\mathrm {Var}\big [(h - w^{T}X)^{+}\big ] \\&= \frac{1}{n} \mathrm {Var}\big [(h - w^{T}X)^{+}\big ]. \end{aligned}$$

$\square$

Proof

(Proposition 2) Let ${\mathsf {Y}} = (Y_{1}, \ldots , Y_{n})$ with $Y_{i} := -X_{i}$. Then the sample average of ${\mathsf {Y}}$ verifies $\hat{\mu _{{\mathsf {Y}}}} = -{\hat{\mu }}$ and we have that $Y_{i}$ are i.i.d. and they have finite second moment. Rewriting the expression for $\widehat{\mathrm {ASD}}_{n}[w; {\mathsf {X}}]$ in terms of ${\mathsf {Y}}$ we have

$$\begin{aligned} \widehat{\mathrm {ASD}}(w; {\mathsf {X}})&= \frac{1}{n} \sum _{i=1}^{n} \big (w^{T}X_{i} - w^{T}{\hat{\mu }} \big )^{+} \\&= \frac{1}{n} \sum _{i=1}^{n} \big (w^{T}\widehat{\mu _{{\mathsf {Y}}}} - w^{T}Y_{i}\big )^{+} \\&= \widehat{\mathrm {ICC}}\big (w,w^{T}\widehat{\mu _{{\mathsf {Y}}}};{\mathsf {Y}}\big ). \end{aligned}$$

Thus, by Corollary 1

$$\begin{aligned} \mathrm {SV}\Big [\widehat{\mathrm {ASD}}_{n}(w; {\mathsf {X}}) \Big ] = \frac{1}{n} {\tilde{z}}^{T}\Omega _{n}{\tilde{z}}, \end{aligned}$$

where $\Omega _{n}$ is as above, ${\tilde{z}} = ({\tilde{z}}_{1}, \ldots , {\tilde{z}}_{n})$ and ${\tilde{z}}_{i} = \big (w^{T}\widehat{\mu _{{\mathsf {Y}}}} - w^{T}Y_{i}\big )^{+}$. Rewriting the result back in terms of ${\mathsf {X}}$ gives the desired result. $\square$

Proof

(Lemma 2) The expression to be minimized in Problem (10), which we refer to as $F(\eta )$, is piecewise linear with breaking points at $Z_{1}, \ldots , Z_{n}$. For $m \in \{-\lceil n\alpha \rceil +1 , \ldots , \lceil n\alpha \rceil +n - 1\}$ we define

$$\begin{aligned} \Delta (m) := F(Z_{(\lceil n\alpha \rceil + m + 1)}) - F(Z_{(\lceil n\alpha \rceil + m)}), \end{aligned}$$

where $Y_{(i)}$ is the ith order statistic. Note that

$$\begin{aligned} \Delta (m)&= \frac{1}{n} \Bigg \{ \sum _{i=1}^{n} (1-\alpha ) \Big [ (Z_{(\lceil n\alpha \rceil + m + 1)} - Z_{i})^{+} - (Z_{(\lceil n\alpha \rceil + m)} - Z_{i})^{+} \Big ] \\&\quad + \alpha \Big [ (Z_{i} - Z_{(\lceil n\alpha \rceil + m + 1)})^{+} - (Z_{i} - Z_{(\lceil n\alpha \rceil + m)})^{+} \Big ] \Bigg \} \\&= \frac{1}{n} \Bigg \{ (1-\alpha ) \big (\lceil n\alpha \rceil + m \big ) \big (Z_{(\lceil n\alpha \rceil + m + 1)} - Z_{(\lceil n\alpha \rceil + m)} \big ) \\&\quad + \alpha (n - \lceil n\alpha \rceil - m) \big (Z_{(\lceil n\alpha \rceil + m)} - Z_{(\lceil n\alpha \rceil + m + 1)}\big ) \Bigg \} \\&= \frac{1}{n} \big (Z_{(\lceil n\alpha \rceil + m + 1)} - Z_{(\lceil n\alpha \rceil + m)} \big ) \Big [\lceil n\alpha \rceil - n\alpha + m \Big ] \\&= \frac{1}{n} \big (Z_{(\lceil n\alpha \rceil + m + 1)} - Z_{(\lceil n\alpha \rceil + m)} \big ) [p + m]. \end{aligned}$$

From the definition, $0 \le p < 1$. Thus $\Delta (m) < 0$ for $m \le -1$ and $\Delta (m) > 0$ for $m > 0$. If $p > 0$, then $\Delta (0) > 0$ and thus $\eta ^{*} = Z_{(\lceil n\alpha \rceil - 1)}$ is unique. If $p = 0$, then $\Delta (0) = 0$, i.e.,

$$\begin{aligned} F(Z_{(\lceil n\alpha \rceil + 1)}) = F(Z_{(\lceil n\alpha \rceil )}). \end{aligned}$$

Since $F(\cdot )$ is piecewise linear, then its minimum value is $F(Z_{\lceil n\alpha \rceil })$ and $\eta =Z_{(\rceil n\alpha \rceil )}$ is one of its minimizers, which concludes the proof. $\square$

Proof

(Proposition 3) Using the notation above, Lemma 2 implies that

$$\begin{aligned} \widehat{\mathrm {QDEV}}_{\alpha }(w; {\mathsf {X}})&= (\epsilon _{1} + \epsilon _{2}) \frac{1}{n} \sum _{i=1}^{n} (1-\alpha )\big (\eta (p) - Z_{i} \big )^{+} + \alpha \big (Z_{i} - \eta (p)\big )^{+} \\&= \frac{1}{n} \sum _{i=1}^{n} \epsilon _{1}\big (\eta (p) - Z_{i} \big )^{+} + \epsilon _{2} \big (Z_{i} - \eta (p)\big )^{+}, \end{aligned}$$

where $\eta (p)$ can be $Z_{(\lceil n\alpha \rceil )}$ or $Z_{(\lceil n\alpha \rceil - 1)}$ depending on the value of p. The result follows from the assumption that the $Z_i$ observations are i.i.d. $\square$