# Model Risk in Portfolio Optimization

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## Abstract

**:**

## 1. Introduction

**μ**and the covariance matrix Σ. The standard approach replaces true parameters by estimates ignoring any model uncertainty. In the literature, it has been shown that this standard approach has serious drawbacks in mean-variance optimization, see e.g., Michaud [2].

**μ**and Σ are introduced. In [18] the problem is studied under uncertainty in

**μ**only. In [19] the mean-variance problem with linear constraints is considered under uncertainty in both

**μ**and Σ. Ter Horst et al. [20] analyze the same loss function as in [17,18,19] under the assumption of known covariance matrix Σ and considering only estimation of

**μ**. A uniform rescaling of the optimal portfolio weights is proposed to account for model uncertainty. In Kan and Zhou [21] a closed-form expression for the same loss function as in [17,18,19] is derived in the unconstrained mean-variance problem under uncertainty in

**μ**and Σ. In Jobson and Korkie [22] the asymptotic distributions of the means, variances and optimal portfolio weights are derived in a constrained problem. Mori [23] obtains analytical results on the optimality of certain estimators in the constrained mean-variance problem. General linear equality constraints are considered under uncertainty in

**μ**and Σ. Unlike these papers we use a more general class of models for the data than the multivariate Gaussian assumption. Moreover, we consider a modification of the loss function used in [17,18,19].

**μ**and Σ is considered under elliptical asset returns. The considerations in our paper are in the spirit of [29,30] but relate to finite number of assets and sample size.

**μ**. From the analytical results derived in Kan and Zhou [21], it follows that the uncertainty in the covariance matrix also has a large impact on the optimal portfolio when the number of assets is large compared to the sample size. Therefore, in this paper, we consider uncertainty in both

**μ**and Σ.

**Organization of the paper**. In Section 2 we introduce our measure of model risk for general portfolio optimization problems. Using this measure we prove in Proposition 1 the following result: under the assumption of known constraints and unbiased estimators, optimal portfolios are on average negatively affected by model risk. In Section 3 we consider mean-variance optimization assuming we have observations drawn from a normal variance mixture model. We derive in Theorem 2 an analytical formula for our measure of model uncertainty. This provides a good description of the effects of model uncertainty. Moreover, we show that this measure of model risk attains its lowest value in the Gaussian case. In Section 4 we introduce an estimator for the covariance matrix with adjusted eigenvalues. In Theorem 4 we derive a relationship between our measure of model uncertainty and the adjusted eigenvalues. Using this, we derive in Theorem 5 a rule which reduces the effects of model uncertainty under the assumption of known eigenvectors. Even if these are unknown, we show numerically in Section 5 that the rule provides significant improvements to the portfolio optimization problem.

## 2. General Definitions

**Example 1.**

- (i)
- Mean-variance optimization: the investor is assumed to maximize$${U}^{\mathrm{mv}}\left(\mathit{w}\right)={\mathit{\mu}}^{\prime}w-\kappa {\mathit{w}}^{\prime}\Sigma \mathit{w}$$$${\mathit{w}}^{*}({U}^{\mathrm{mv}},{\mathbb{R}}^{n})=\frac{1}{2\kappa}{\Sigma}^{-1}\mathit{\mu}$$In the case of linear constraints we set $\mathcal{A}=\{\mathit{w}\in {\mathbb{R}}^{n}\phantom{\rule{0.222222em}{0ex}}|\phantom{\rule{0.222222em}{0ex}}{A}^{\prime}\mathit{w}=\mathbb{a}\}$, where $A\in {\mathbb{R}}^{n\times k}$ is a matrix of rank k and $\mathit{a}\in {\mathbb{R}}^{k}$. The solution of Equation (1) can be calculated using the method of Lagrange and is given by$${\mathit{w}}^{*}({U}^{\mathrm{mv}},\mathcal{A})=\frac{1}{2\kappa}\left({\Sigma}^{-1}-{\Sigma}^{-1}A{\left({A}^{\prime}{\Sigma}^{-1}A\right)}^{-1}{A}^{\prime}{\Sigma}^{-1}\right)\mathit{\mu}+{\Sigma}^{-1}A{\left({A}^{\prime}{\Sigma}^{-1}A\right)}^{-1}\mathit{a}$$
- (ii)
- Minimum variance: the investor is assumed to maximize the following utility function$${U}^{\mathrm{v}}\left(\mathit{w}\right)=-{\mathit{w}}^{\prime}\Sigma \mathit{w}$$Consider the following linear equality constraints $\mathcal{A}=\{\mathit{w}\in {\mathbb{R}}^{n}\phantom{\rule{0.222222em}{0ex}}|\phantom{\rule{0.222222em}{0ex}}{A}^{\prime}\mathit{w}=\mathit{a}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\mathrm{and}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}{\mathit{w}}^{\prime}\mathit{\mu}={\mu}_{0}\}$, where $A\in {\mathbb{R}}^{n\times k}$ is a matrix of rank k, $\mathit{a}\in {\mathbb{R}}^{k}$ and ${\mu}_{0}\in \mathbb{R}$. In this problem we are looking for the portfolio with minimal variance satisfying linear constraints and having expected return equal to ${\mu}_{0}$. The optimal portfolio can be calculated using the method of Lagrange and is given by$${\mathit{w}}^{*}({U}^{\mathrm{v}},\mathcal{A})={\Sigma}^{-1}\tilde{A}{\left({\tilde{A}}^{\prime}{\Sigma}^{-1}\tilde{A}\right)}^{-1}\tilde{\mathit{a}}$$
**μ**. - (iii)
- Equal contributions to risk: the investor is assumed to maximize$${U}^{\mathrm{ecr}}\left(\mathit{w}\right)=-\sum _{\begin{array}{c}i,j=1,\dots ,n\\ i<j\end{array}}{\left({\mathrm{RC}}_{i}\left(\mathit{w}\right)-{\mathrm{RC}}_{j}\left(\mathit{w}\right)\right)}^{2}$$
- (iv)
- Maximum diversification: the investor is assumed to maximize$${U}^{\mathrm{md}}\left(\mathit{w}\right)=\frac{{\sum}_{i=1}^{n}{w}_{i}\sqrt{{\Sigma}_{ii}}}{\sqrt{{\mathit{w}}^{\prime}\Sigma \mathit{w}}}$$

**μ**and Σ are not observable. Therefore, the investor cannot directly maximize ${U}^{\mathrm{mv}}$, ${U}^{\mathrm{v}}$, ${U}^{\mathrm{ecr}}$ or ${U}^{\mathrm{md}}$, because neither U nor $\mathcal{A}$ are explicitly known.

**Remark 1.**

**R**, except for certain integrability conditions specified below.

**R**we define the utility estimator by

**R**, i.e.,

**Remark 2.**

**Remark 3.**

**Proposition 1.**

**Remark 4.**

## 3. Analysis of the Loss Function in the Mean-Variance Case

- (i)
- ${\left({\mathit{Z}}_{i}\right)}_{i=1,\dots ,m}\stackrel{\mathrm{i}.\mathrm{i}.\mathrm{d}.}{\sim}\mathcal{N}({0}_{n},{I}_{n})$ for ${0}_{n}$ and ${I}_{n}$ being the n-dimensional zero vector and identity matrix, respectively;
- (ii)
- W is an almost surely positive random variable, independent of ${\left({\mathit{Z}}_{i}\right)}_{i=1,\dots ,m}$, satisfying $E\left[w\right]=1$; and
- (iii)
- ${\Sigma}^{\frac{1}{2}}\in {\mathbb{R}}^{n\times n}$ is a non-singular matrix such that ${\Sigma}^{\frac{1}{2}}{\left({\Sigma}^{\frac{1}{2}}\right)}^{\prime}=\Sigma $.

**Remark 5.**

- (i)
- ${\left({\mathit{R}}_{i}\right|W)}_{i=1,\dots ,m}\stackrel{\mathrm{i}.\mathrm{i}.\mathrm{d}.}{\sim}\mathcal{N}(\mathit{\mu},W\Sigma )$,
- (ii)
- $E\left[{\mathit{R}}_{i}\right]=\mathit{\mu}$, and $\mathrm{Cov}\left({\mathit{R}}_{i}\right)=E\left[W\right]\Sigma =\Sigma $ for all $i=1,\dots ,m$.

**Remark 6.**

**μ**and covariance matrix Σ. In order to obtain a more general class of models one introduces a random time change. Let T be a non-decreasing stochastic process with $T\left(0\right)=0$, almost surely. This process is called subordinator and represents a stochastic relationship between the calendar time and the pace of the market. Based on this time change one considers the model

**B**and T are independent processes, then we have for the logarithmic return

**B**we obtain model assumption (4) for fixed $t\ge 1$. For more details on subordinated models see e.g., De Giovanni et al. [40].

**Example 2.**

- (i)
- Multivariate Gauss: set $W=1$.
- (ii)
- 2-points mixture: let W be discrete and take positive values ${x}_{1}$ and ${x}_{2}$ with probabilities $p\in (0,1)$ and $1-p$. The condition $E\left[W\right]=1$ implies ${x}_{1}\in (0,{p}^{-1})$ and ${x}_{2}=\frac{1-p{x}_{1}}{1-p}$. This is a regime switching model with 2 variance regimes characterized by ${x}_{1}$ and ${x}_{2}$.
- (iii)
- Multivariate Student-t: let $W\sim \frac{\nu -2}{\nu}\mathrm{Ig}\left(\frac{\nu}{2},\frac{\nu}{2}\right)$, where $\nu >2$ and $\mathrm{Ig}(\xb7,\xb7)$ denotes the inverse gamma distribution. The scaling is chosen so that $E\left[W\right]=1$. Then, for all $i=1,\dots ,m$, the random vector ${\mathit{R}}_{i}$ has a multivariate Student-t distribution with ν degrees of freedom, mean vector
**μ**and covariance matrix Σ.

**μ**and Σ we consider the sample estimators

**Remark 7.**

- (i)
- Conditional on W we have$$\widehat{\mathit{\mu}}|W\sim \mathcal{N}\left(\mathit{\mu},\frac{W}{m}\Sigma \right)\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\mathrm{and}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}(m-1)\widehat{\Sigma}|W\sim \mathcal{W}(\Sigma ,m-1)$$
- (ii)
- For $m>n$ the sample covariance matrix is almost surely positive definite and ${\widehat{\Sigma}}^{-1}$ exists almost surely. See Appendix 2 for more details.
- (iii)
- For any $\mathit{w}\in {\mathbb{R}}^{n}$ we have $E\left[\widehat{\mathit{\mu}}\right]=\mathit{\mu}$, $E\left[\widehat{\Sigma}\right]=\Sigma $, and $E\left[{\widehat{U}}^{\mathrm{mv}}\left(\mathit{w}\right)\right]={U}^{\mathrm{mv}}\left(\mathit{w}\right)$. This means that $\widehat{\mathit{\mu}}$, $\widehat{\Sigma}$ and ${\widehat{U}}^{\mathrm{mv}}$ are unbiased estimators for
**μ**, Σ and ${U}^{\mathrm{mv}}$, respectively.

**Theorem 2.**

**Remark 8.**

- (i)
- Jensen’s inequality implies $E\left[{W}^{-2}\right]\ge E{\left[{W}^{-1}\right]}^{2}\ge E\left[{W}^{-1}\right]$. Hence for $m>n+4$$$\begin{array}{cc}\hfill L({U}^{\mathrm{mv}},{\widehat{U}}^{\mathrm{mv}},{\mathbb{R}}^{n})& \ge \frac{1}{4\kappa}\left(\beta (m-1,n)E\left[{W}^{-2}\right]-E\left[{W}^{-1}\right]\right){\mathit{\mu}}^{\prime}{\Sigma}^{-1}\mathit{\mu}\hfill \\ & \ge \frac{1}{4\kappa}\left(E\left[{W}^{-2}\right]-E\left[{W}^{-1}\right]\right){\mathit{\mu}}^{\prime}{\Sigma}^{-1}\mathit{\mu}\ge 0\hfill \end{array}$$
- (ii)
- If we fix the number of assets n and let $m\to \infty $ we obtain$$L({U}^{\mathrm{mv}},{\widehat{U}}^{\mathrm{mv}},{\mathbb{R}}^{n})\u27f6\frac{1}{4\kappa}\left(E\left[{W}^{-2}\right]-E\left[{W}^{-1}\right]\right){\mathit{\mu}}^{\prime}{\Sigma}^{-1}\mathit{\mu}\ge 0$$In particular, in the multivariate Gaussian, case we have $L({U}^{\mathrm{mv}},{\widehat{U}}^{\mathrm{mv}},{\mathbb{R}}^{n})\to 0$ as $m\to \infty $, see Example 2(i).
- (iii)
- For $n,m\to \infty $ and $n/m\to c\in (0,1)$ we get$$\begin{array}{cc}\hfill L({U}^{\mathrm{mv}},{\widehat{U}}^{\mathrm{mv}},{\mathbb{R}}^{n})& \u27f6\frac{1}{4\kappa}\frac{1}{(1-c)}\{c+\frac{c}{{(1-c)}^{2}}E\left[{W}^{-1}\right]\hfill \\ & \phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}+\left(\frac{1}{{(1-c)}^{2}}E\left[{W}^{-2}\right]-E\left[{W}^{-1}\right]\right){\mathit{\mu}}^{\prime}{\Sigma}^{-1}\mathit{\mu}\}\hfill \end{array}$$This asymptotic analysis corresponds to large number of risky assets n and comparable sample size m. This limit is considered in El Karoui [29,30] to obtain asymptotic results on model uncertainty in a problem with constraints. The same limit is also considered in Ledoit and Wolf [35] to study asymptotic properties of the spectrum of $\widehat{\Sigma}$. In this paper, we have an explicit characterization of model uncertainty for finite n and m. Note that for $c\to {0}^{+}$ we obtain the same limit as in (ii), and for $c\to {1}^{-}$ we have $L({U}^{\mathrm{mv}},{\widehat{U}}^{\mathrm{mv}},{\mathbb{R}}^{n})\to \infty $.
- (iv)
- Note that the loss function depends on the unobservable distribution of W, true mean vector
**μ**and true covariance matrix Σ.

**Corollary 1.**

- (i)
- multivariate Gauss:$$\begin{array}{cc}\hfill {L}^{\mathrm{Gauss}}({U}^{\mathrm{mv}},{\widehat{U}}^{\mathrm{mv}},{\mathbb{R}}^{n})& =\frac{1}{4\kappa}\alpha (m-1,n)\{\frac{n}{m}\left(1+\beta (m-1,n)\right)\hfill \\ & \phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}+\left(\beta (m-1,n)-1\right){\mathit{\mu}}^{\prime}{\Sigma}^{-1}\mathit{\mu}\}\hfill \end{array}$$
- (ii)
- 2-points mixture:$$\begin{array}{cc}\hfill {L}^{2\mathrm{pm}}({U}^{\mathrm{mv}},{\widehat{U}}^{\mathrm{mv}},{\mathbb{R}}^{n})& =\frac{1}{4\kappa}\alpha (m-1,n)\{\frac{n}{m}-\left(\frac{p}{{x}_{1}}+\frac{{(1-p)}^{2}}{1-p{x}_{1}}\right){\mathit{\mu}}^{\prime}{\Sigma}^{-1}\mathit{\mu}\hfill \\ & \phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}+\frac{n}{m}\beta (m-1,n)\left(\frac{p}{{x}_{1}}+\frac{{(1-p)}^{2}}{1-p{x}_{1}}\right)\hfill \\ & \phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}+\beta (m-1,n)\left(\frac{p}{{x}_{1}^{2}}+\frac{{(1-p)}^{3}}{{(1-p{x}_{1})}^{2}}\right){\mathit{\mu}}^{\prime}{\Sigma}^{-1}\mathit{\mu}\}\hfill \end{array}$$
- (iii)
- multivariate Student-t:$$\begin{array}{cc}\hfill {L}^{\mathrm{t}}({U}^{\mathrm{mv}},{\widehat{U}}^{\mathrm{mv}},{\mathbb{R}}^{n})& =\frac{1}{4\kappa}\alpha (m-1,n)\{\frac{n}{m}\left(1+\beta (m-1,n)\frac{\nu}{\nu -2}\right)\hfill \\ & \phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}+\frac{\nu}{\nu -2}\left(\beta (m-1,n)\frac{2+\nu}{\nu -2}-1\right){\mathit{\mu}}^{\prime}{\Sigma}^{-1}\mathit{\mu}\}\hfill \end{array}$$

**Remark 9.**

- (i)
- In the 2-points mixture model we have ${L}^{2\mathrm{pm}}({U}^{\mathrm{mv}},{\widehat{U}}^{\mathrm{mv}},{\mathbb{R}}^{n})\to \infty $ for ${x}_{1}\to {p}^{-1}$ or ${x}_{1}\to 0$, and ${L}^{2\mathrm{pm}}({U}^{\mathrm{mv}},{\widehat{U}}^{\mathrm{mv}},{\mathbb{R}}^{n})\to {L}^{\mathrm{Gauss}}({U}^{\mathrm{mv}},{\widehat{U}}^{\mathrm{mv}},{\mathbb{R}}^{n})$ for ${x}_{1}\to 1$. The more we increase the difference between the two variance regimes (i.e., by considering ${x}_{1}\to {p}^{-1}$ or ${x}_{1}\to 0$), the greater the average negative effect of model risk. If we decrease the difference (by considering ${x}_{1}\to 1$), then the loss function becomes closer to the Gaussian case.
- (ii)
- In the multivariate Student-t model we have ${L}^{\mathrm{t}}({U}^{\mathrm{mv}},{\widehat{U}}^{\mathrm{mv}},{\mathbb{R}}^{n})\to \infty $ for $\nu \to {2}^{+}$, and ${L}^{\mathrm{t}}({U}^{\mathrm{mv}},{\widehat{U}}^{\mathrm{mv}},{\mathbb{R}}^{n})\to {L}^{\mathrm{Gauss}}({U}^{\mathrm{mv}},{\widehat{U}}^{\mathrm{mv}},{\mathbb{R}}^{n})$ for $\nu \to \infty $. This means, by making the marginals “less normal” (i.e., by considering $\nu \to {2}^{+}$), we increase the average negative effect of model risk.

**Corollary 2.**

## 4. Adjusting for Model Risk

**Remark 10.**

**R**in the following results, unless explicitly specified.

**μ**and Σ. We assume that $\widehat{\Sigma}$ is positive definite almost surely. We want to analyze a class of estimators for the covariance matrix Σ obtained by individually rescaling the eigenvalues of $\widehat{\Sigma}$. An asymptotic analysis of this approach is proposed in Ledoit and Wolf [35]. In [35] this is done to improve the covariance matrix estimation without specific considerations on mean-variance optimization. Menchero et al. [38] analyze empirically spectral properties of $\widehat{\Sigma}$ in the context of mean-variance optimization. The spectral decomposition of the symmetric positive definite covariance matrix Σ is given by

- (i)
- $\Lambda =\mathrm{diag}({\lambda}_{1},\dots ,{\lambda}_{n})$ is the diagonal matrix of eigenvalues (also called eigenvariances in this context) of Σ, which satisfy $0<{\lambda}_{1}\le \dots \le {\lambda}_{n}$; and
- (ii)
- T is an orthogonal matrix with columns ${v}_{1}={({T}_{11},\dots ,{T}_{n1})}^{\prime},\dots ,{v}_{n}={({T}_{n1},\dots ,{T}_{nn})}^{\prime}$ being the eigenvectors (also called eigenportfolios) of Σ corresponding to ${\lambda}_{1},\dots ,{\lambda}_{n}$, respectively.

**Remark 11.**

**Proposition 3.**

**Remark 12.**

- (i)
- ${\widehat{\Sigma}}_{\overline{c}}=\overline{c}\widehat{\Sigma}$ is a simple proportional scaling type estimator. The unconstrained mean-variance portfolio constructed using ${\widehat{\Sigma}}_{\overline{c}}$ is given by$${\mathit{w}}^{*}({\widehat{U}}_{\overline{c}}^{\mathrm{mv}},{\mathbb{R}}^{n})=\frac{1}{2\overline{c}\kappa}{\widehat{\Sigma}}^{-1}\widehat{\mathit{\mu}}$$
- (ii)
- In general, adjustment factors (7) are unobservable since they depend on the unknown distribution of W. Note that, by Remark 8(i) we have$$\overline{c}\ge \beta (m-1,n)$$

**Theorem 4.**

- (i)
- $\widehat{\Lambda}=\mathrm{diag}({\widehat{\lambda}}_{1},\dots ,{\widehat{\lambda}}_{n})$ independent of $\widehat{\mathit{\mu}}$,
- (ii)
- $0<{\widehat{\lambda}}_{1}\le \dots \le {\widehat{\lambda}}_{n}$ almost surely and integrable, and
- (iii)
- $E\left[\widehat{\mathit{\mu}}\right]=\mathit{\mu}$.

**Remark 13.**

**Theorem 5.**

## 5. Case Studies

#### 5.1. Simulated Observations

**μ**and Σ. We consider the following four problems, see Example 1.

- (i)
- Unconstrained mean-variance: $U={U}^{\mathrm{mv}}$ and $\mathcal{A}={\mathbb{R}}^{n}$. We set $\kappa =1$ for the risk aversion parameter.
- (ii)
- Long only fully invested minimum variance: $U={U}^{\mathrm{v}}$ and $\mathcal{A}=\{\mathit{w}\in {\mathbb{R}}^{n}\phantom{\rule{0.222222em}{0ex}}|\phantom{\rule{0.222222em}{0ex}}{\mathbf{1}}^{\prime}\mathit{w}=1\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\text{and}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}{w}_{i}\ge 0\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4.pt}{0ex}}i=1,\dots ,n\}$.
- (iii)
- Long only fully invested equal contributions to risk: $U={U}^{\mathrm{ecr}}$ and $\mathcal{A}$ as in (ii).
- (iv)
- Long only fully invested maximum diversification: $U={U}^{\mathrm{md}}$ and $\mathcal{A}$ as in (ii).

**μ**and Σ equal to the sample estimates computed from the dataset of Section 5.2. We simulate independently N samples of m return observations from the three distributional models of Example 2. In the 2-points mixture case we set ${x}_{1}=5$ and $p=0.1$; in the multivariate Student-t case we set $\nu =3$.

**Figure 1.**${L}^{\mathrm{Gauss}}({U}^{\mathrm{mv}},{\widehat{U}}^{\mathrm{mv}},{\mathbb{R}}^{n})$, ${L}^{2\mathrm{pm}}({U}^{\mathrm{mv}},{\widehat{U}}^{\mathrm{mv}},{\mathbb{R}}^{n})$ and ${L}^{\mathrm{t}}({U}^{\mathrm{mv}},{\widehat{U}}^{\mathrm{mv}},{\mathbb{R}}^{n})$ computed by simulation for $n=50$, $m=100$ and different values of N. The values given by Theorem 2 are approximately $1.3814$, $2.1236$ and $3.8453$, respectively.

**Figure 2.**$L({U}^{\mathrm{mv}},{\widehat{U}}^{\mathrm{mv}},{\mathbb{R}}^{n})$ computed using Theorem 2 for $n=50$ and different values of m.

**Figure 4.**$L({U}^{\mathrm{mv}},{\widehat{U}}_{C}^{\mathrm{mv}},{\mathbb{R}}^{n})$ computed by simulation ($N=1000$) for $n=50$ and different values of m. The adjustment factors ${c}_{1},\dots ,{c}_{n}$ are selected according to Equation (7).

**Figure 7.**Adjustment factors Equation (7) computed by simulation for $N=1000$ and $m=100$, for each eigenvariance.

**Figure 8.**Adjustment factors (9) computed by simulation for $N=1000$ and $m=100$, for each eigenvariance.

**Figure 9.**Adjustment factors (11) computed by simulation for $N=1000$ and $m=100$, for each eigenvariance.

**μ**and Σ, whereas problems (ii)–(iv) require only estimation of Σ. We compare the behavior of the loss functions in (ii)–(iv) for different values of m, since they are based on estimation of the same parameters. In Figure 10, Figure 11 and Figure 12 we present these loss functions. We observe that these problems are not severely deteriorated by model risk even in the case of n close to m or non-Gaussian observations because they only require estimation of Σ. The equal contributions to risk and maximum diversification portfolios are less affected by the distribution of the observations compared to the minimum variance portfolio.

**Figure 10.**$L({U}^{\mathrm{v}},{U}^{\mathrm{v}},\mathcal{A})$ for $N=1000$ and different values of m.

**Figure 11.**$L({U}^{\mathrm{ecr}},{U}^{\mathrm{ecr}},\mathcal{A})$ for $N=1000$ and different values of m.

**Figure 12.**$L({U}^{\mathrm{md}},{U}^{\mathrm{md}},\mathcal{A})$ for $N=1000$ and different values of m.

#### 5.2. Large-Cap U.S. Equity Portfolios

**μ**and Σ. Note that the estimation of

**μ**is required in this problem, unlike in problems (ii)–(iv) of Section 5.1. As shown in Section 5.1, the estimator ${\widehat{\Sigma}}_{C}$, where ${c}_{1},\dots ,{c}_{n}$ are selected according to Equations (7) and (11), considerably reduces the negative effects of model risk in the unconstrained mean-variance problem. In this section we apply Equations (7) and (11) to the problem of long only fully invested constraints. Observe that for non-Gaussian observations the adjustment factors (7) depend on the unobservable distribution of W. To estimate these values we calibrate a specific normal variance mixture model to the data. The adjustment factors (11) are also unobservable since they depend on the true covariance matrix. In order to estimate them we rely on the following simulation procedure. For each trading day, we estimate $\widehat{\Sigma}$ and consider it as the true covariance matrix. We then simulate $N=1000$ samples of 100 observations using a specific normal variance mixture model calibrated to the data. These samples are used, on the other hand, to estimate sample covariance matrices ${\Sigma}^{\left(1\right)},\dots ,{\Sigma}^{\left(N\right)}$. Diagonalization of these matrices provide the eigenvariances

- (i)
- The classical mean-variance portfolio ${\mathit{w}}^{*}({\widehat{U}}^{\mathrm{mv}},\mathcal{A})$ based on the sample estimators $\widehat{\mathit{\mu}}$ and $\widehat{\Sigma}$.
- (ii)
- Portfolio ${\mathit{w}}^{*}({\widehat{U}}_{\overline{c}}^{\mathrm{mv}},\mathcal{A})$ based on the estimators $\widehat{\mathit{\mu}}$ and ${\widehat{\Sigma}}_{\overline{c}}$, where $\overline{c}$ is determined by Equation (7) under a multivariate Gaussian model.
- (iii)
- As in (ii), where $\overline{c}$ is determined by Equation (7) under a multivariate Student-t model.
- (iv)
- Portfolio ${\mathit{w}}^{*}({\widehat{U}}_{C}^{\mathrm{mv}},\mathcal{A})$ based on the estimators $\widehat{\mathit{\mu}}$ and ${\widehat{\Sigma}}_{C}$, where ${c}_{1},\dots ,{c}_{n}$ are determined numerically by Equation (12) under a multivariate Gaussian model.
- (v)
- As in (iv), where ${c}_{1},\dots ,{c}_{n}$ are determined numerically by Equation (12) under a multivariate Student-t model.

**Table 1.**Out-of-sample backtesting results of the strategies (i)–(iii). All numbers are expressed in percent except for Sharpe ratios and average degrees of diversification. The return and volatility values are annualized assuming 250 trading days per year.

Return Annualized | Volatility Annualized | Sharpe Ratio | Maximum Drawdown | Average Turnover | Average Diversification | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

(i) | (ii) | (iii) | (i) | (ii) | (iii) | (i) | (ii) | (iii) | (i) | (ii) | (iii) | (i) | (ii) | (iii) | (i) | (ii) | (iii) | |

All | 7.63 | 8.03 | 8.66 | 15.06 | 15.01 | 15.00 | 0.51 | 0.53 | 0.58 | 47.80 | 47.16 | 45.31 | 13.38 | 12.78 | 12.25 | 10.05 | 10.10 | 10.15 |

2011 | 10.85 | 11.85 | 12.01 | 14.09 | 13.97 | 13.88 | 0.77 | 0.85 | 0.87 | 10.04 | 9.58 | 9.75 | 13.75 | 12.95 | 12.31 | 8.87 | 8.98 | 9.05 |

2010 | 9.13 | 9.26 | 8.90 | 11.12 | 10.97 | 10.86 | 0.82 | 0.84 | 0.82 | 7.63 | 7.35 | 7.33 | 10.39 | 9.94 | 9.80 | 7.74 | 7.88 | 8.05 |

2009 | 22.10 | 21.35 | 24.27 | 17.94 | 18.05 | 18.19 | 1.23 | 1.18 | 1.33 | 19.56 | 20.06 | 18.83 | 10.84 | 10.69 | 10.28 | 9.66 | 9.57 | 9.54 |

2008 | −27.67 | −26.57 | −25.93 | 28.82 | 28.74 | 28.91 | neg | neg | neg | 39.68 | 38.59 | 38.05 | 12.95 | 12.31 | 11.60 | 8.03 | 7.97 | 7.94 |

2007 | 10.29 | 10.06 | 10.72 | 11.33 | 11.23 | 11.09 | 0.91 | 0.90 | 0.97 | 7.05 | 6.87 | 6.70 | 14.27 | 13.27 | 12.80 | 8.39 | 8.34 | 8.38 |

2006 | 9.30 | 10.04 | 10.91 | 8.05 | 7.96 | 7.89 | 1.16 | 1.26 | 1.38 | 5.93 | 5.37 | 4.78 | 15.06 | 13.62 | 12.40 | 8.43 | 8.23 | 8.03 |

2005 | 1.66 | 2.02 | 1.94 | 9.59 | 9.51 | 9.41 | 0.17 | 0.21 | 0.21 | 7.29 | 6.79 | 6.62 | 15.75 | 15.16 | 14.48 | 9.38 | 9.47 | 9.53 |

2004 | 9.17 | 9.76 | 10.38 | 9.45 | 9.35 | 9.26 | 0.97 | 1.04 | 1.12 | 7.68 | 7.18 | 6.79 | 14.28 | 13.50 | 12.59 | 8.52 | 8.68 | 8.86 |

2003 | 20.49 | 21.05 | 21.44 | 12.16 | 12.10 | 12.07 | 1.69 | 1.74 | 1.78 | 8.57 | 8.59 | 8.80 | 13.03 | 12.64 | 12.05 | 9.24 | 9.20 | 9.15 |

2002 | −13.33 | −13.29 | −12.20 | 18.12 | 18.25 | 18.26 | neg | neg | neg | 27.35 | 27.56 | 27.19 | 12.33 | 12.05 | 11.74 | 9.39 | 9.52 | 9.56 |

2001 | −7.23 | −6.52 | −5.41 | 13.97 | 13.89 | 13.94 | neg | neg | neg | 16.49 | 16.20 | 16.00 | 13.93 | 13.45 | 13.14 | 11.85 | 11.93 | 12.02 |

2000 | 6.64 | 6.29 | 6.83 | 17.99 | 17.99 | 18.01 | 0.37 | 0.35 | 0.38 | 22.37 | 22.42 | 22.34 | 14.70 | 14.49 | 13.95 | 12.80 | 12.94 | 13.00 |

1999 | −2.76 | −1.62 | −0.41 | 15.32 | 15.30 | 15.25 | neg | neg | neg | 15.12 | 14.52 | 14.01 | 14.45 | 13.77 | 13.47 | 12.00 | 12.16 | 12.35 |

1998 | 18.57 | 19.22 | 19.09 | 17.63 | 17.57 | 17.61 | 1.05 | 1.09 | 1.08 | 17.41 | 17.57 | 17.88 | 13.82 | 13.53 | 13.35 | 11.09 | 10.99 | 10.92 |

1997 | 26.78 | 26.58 | 26.09 | 15.91 | 15.76 | 15.59 | 1.68 | 1.69 | 1.67 | 8.05 | 8.07 | 8.13 | 12.54 | 11.82 | 11.12 | 11.32 | 11.31 | 11.25 |

1996 | 18.62 | 19.32 | 20.01 | 12.01 | 11.94 | 11.88 | 1.55 | 1.62 | 1.68 | 7.33 | 7.13 | 7.17 | 14.93 | 14.22 | 13.70 | 12.03 | 12.33 | 12.61 |

1995 | 17.80 | 18.60 | 19.35 | 7.76 | 7.60 | 7.47 | 2.29 | 2.45 | 2.59 | 3.77 | 3.72 | 3.78 | 9.72 | 9.26 | 8.83 | 12.92 | 13.15 | 13.40 |

**Table 2.**Out-of-sample backtesting results of the strategies (i)–(v). All numbers are expressed in percent except for Sharpe ratios and average degrees of diversification. The return and volatility values are annualized assuming 250 trading days per year.

Return annualized | Volatility annualized | Sharpe ratio | Maximum drawdown | Average turnover | Average diversification | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

(i) | (iv) | (v) | (i) | (iv) | (v) | (i) | (iv) | (v) | (i) | (iv) | (v) | (i) | (iv) | (v) | (i) | (iv) | (v) | |

All | 7.63 | 7.64 | 9.34 | 15.06 | 14.74 | 14.55 | 0.51 | 0.52 | 0.64 | 47.80 | 45.43 | 41.17 | 13.38 | 11.40 | 9.55 | 10.05 | 11.78 | 12.13 |

2011 | 10.85 | 11.46 | 13.43 | 14.09 | 13.95 | 13.35 | 0.77 | 0.82 | 1.01 | 10.04 | 10.10 | 9.65 | 13.75 | 11.63 | 8.65 | 8.87 | 10.19 | 10.64 |

2010 | 9.13 | 9.42 | 7.85 | 11.12 | 10.83 | 10.43 | 0.82 | 0.87 | 0.75 | 7.63 | 7.09 | 6.83 | 10.39 | 9.34 | 8.34 | 7.74 | 8.26 | 8.32 |

2009 | 22.10 | 17.63 | 18.27 | 17.94 | 16.48 | 16.00 | 1.23 | 1.07 | 1.14 | 19.56 | 19.49 | 19.72 | 10.84 | 8.77 | 7.83 | 9.66 | 10.94 | 10.13 |

2008 | −27.67 | −25.08 | −20.81 | 28.82 | 28.09 | 28.06 | neg | neg | neg | 39.68 | 37.17 | 33.29 | 12.95 | 11.06 | 10.74 | 8.03 | 9.05 | 8.48 |

2007 | 10.29 | 11.40 | 13.34 | 11.33 | 11.23 | 10.77 | 0.91 | 1.02 | 1.24 | 7.05 | 6.67 | 5.60 | 14.27 | 12.58 | 9.21 | 8.39 | 9.82 | 9.91 |

2006 | 9.30 | 9.41 | 12.84 | 8.05 | 7.98 | 7.80 | 1.16 | 1.18 | 1.65 | 5.93 | 6.12 | 4.26 | 15.06 | 12.41 | 9.22 | 8.43 | 11.00 | 11.12 |

2005 | 1.66 | 2.60 | 0.69 | 9.59 | 9.55 | 9.28 | 0.17 | 0.27 | 0.07 | 7.29 | 6.80 | 7.42 | 15.75 | 13.71 | 11.14 | 9.38 | 11.37 | 11.87 |

2004 | 9.17 | 9.63 | 10.66 | 9.45 | 9.39 | 9.10 | 0.97 | 1.03 | 1.17 | 7.68 | 7.27 | 6.63 | 14.28 | 12.32 | 9.32 | 8.52 | 10.48 | 11.45 |

2003 | 20.49 | 20.85 | 22.23 | 12.16 | 11.98 | 11.90 | 1.69 | 1.74 | 1.87 | 8.57 | 7.95 | 8.03 | 13.03 | 11.12 | 9.81 | 9.24 | 10.07 | 9.54 |

2002 | −13.33 | −15.84 | −15.64 | 18.12 | 17.72 | 18.16 | neg | neg | neg | 27.35 | 28.48 | 29.53 | 12.33 | 10.59 | 9.98 | 9.39 | 9.95 | 10.26 |

2001 | −7.23 | −10.55 | −5.53 | 13.97 | 13.75 | 13.67 | neg | neg | neg | 16.49 | 17.88 | 15.71 | 13.93 | 10.92 | 9.44 | 11.85 | 13.61 | 14.53 |

2000 | 6.64 | 8.22 | 11.63 | 17.99 | 17.71 | 17.74 | 0.37 | 0.46 | 0.66 | 22.37 | 21.88 | 21.99 | 14.70 | 12.09 | 10.29 | 12.80 | 16.04 | 17.02 |

1999 | −2.76 | −0.29 | 4.73 | 15.32 | 14.97 | 14.79 | neg | neg | 0.32 | 15.12 | 13.74 | 10.51 | 14.45 | 11.69 | 9.65 | 12.00 | 14.46 | 14.84 |

1998 | 18.57 | 16.90 | 15.64 | 17.63 | 17.57 | 17.45 | 1.05 | 0.96 | 0.90 | 17.41 | 18.08 | 18.41 | 13.82 | 12.30 | 11.17 | 11.09 | 12.08 | 11.89 |

1997 | 26.78 | 26.78 | 27.04 | 15.91 | 15.78 | 15.10 | 1.68 | 1.70 | 1.79 | 8.05 | 7.86 | 7.87 | 12.54 | 10.87 | 8.82 | 11.32 | 12.65 | 12.57 |

1996 | 18.62 | 18.54 | 22.43 | 12.01 | 12.03 | 11.77 | 1.55 | 1.54 | 1.91 | 7.33 | 6.94 | 7.78 | 14.93 | 13.33 | 11.13 | 12.03 | 15.06 | 16.62 |

1995 | 17.80 | 19.73 | 20.95 | 7.76 | 7.73 | 7.10 | 2.29 | 2.55 | 2.95 | 3.77 | 3.62 | 4.05 | 9.72 | 8.49 | 6.94 | 12.92 | 16.74 | 19.58 |

**Remark 14.**

## Author Contributions

## Appendix

## 1. Proofs

#### 1.1. Proof of Proposition 1

#### 1.2. Proof of Theorem 2

#### 1.3. Proof of Corollary 1

- (i)
- Multivariate Gauss: the statement follows directly setting $W=1$.
- (ii)
- 2-points mixture: set $W\sim {x}_{2}+({x}_{1}-{x}_{2})\mathrm{Bernoulli}\left(p\right)$, where ${x}_{1}\in (0,{p}^{-1})$ and ${x}_{2}=\frac{1-p{x}_{1}}{1-p}$. Then, $E\left[{W}^{-1}\right]=\frac{p}{{x}_{1}}+\frac{{(1-p)}^{2}}{1-p{x}_{1}}$ and $E\left[{W}^{-2}\right]=\frac{p}{{x}_{1}^{2}}+\frac{{(1-p)}^{3}}{{(1-p{x}_{1})}^{2}}$.
- (iii)
- Multivariate Student-t: set $W\sim \frac{\nu -2}{\nu}\mathrm{Ig}\left(\frac{\nu}{2},\frac{\nu}{2}\right)$. Then, ${W}^{-1}\sim \frac{\nu}{\nu -2}\mathrm{Gamma}\left(\frac{\nu}{2},\frac{\nu}{2}\right)$, $E\left[{W}^{-1}\right]=\frac{\nu}{\nu -2}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}\mathrm{and}\phantom{\rule{0.222222em}{0ex}}\phantom{\rule{0.222222em}{0ex}}E\left[{W}^{-2}\right]=\frac{{\nu}^{2}}{{(\nu -2)}^{2}}\left(\frac{2}{\nu}+1\right)$.

#### 1.4. Proof of Corollary 2

#### 1.5. Proof of Proposition 3

#### 1.6. Proof of Theorem 4

#### 1.7. Proof of Theorem 5

## 2. Wishart Distribution

**Proposition 6.**

**Proposition 7.**

**μ**and covariance matrix Σ. Let $\widehat{\mathit{\mu}}$ and $\widehat{\Sigma}$ be the sample mean and sample covariance matrix given by

**Proposition 8.**

- (i)
- $E\left[{M}^{-1}\right]=\frac{\alpha (m,n)}{m}{\Sigma}^{-1}$; and
- (ii)
- $E\left[{M}^{-1}A{M}^{-1}\right]=\frac{\alpha (m,n)\beta (m,n)}{{m}^{2}(m-1)}\mathrm{tr}\left({\Sigma}^{-1}A\right){\Sigma}^{-1}+\frac{\beta (m,n)}{m(m-1)}{\Sigma}^{-1}A{\Sigma}^{-1}$;

## 3. Equity Universe for Case Studies

Nr. | Company | Bloomberg Ticker | Industry Sector |
---|---|---|---|

1 | Wells Fargo & Company | WFC US Equity | financials |

2 | JP Morgan Chase & Co. | JPM US Equity | financials |

3 | Citigroup, Inc. | C US Equity | financials |

4 | Bank of America Corporation | BAC US Equity | financials |

5 | American Express Company | AXP US Equity | financials |

6 | American International Group | AIG US Equity | financials |

7 | PNC Financial Services Group | PNC US Equity | financials |

8 | General Electric Company | GE US Equity | industrials |

9 | United Technologies Corporation | UTX US Equity | industrials |

10 | 3M Company | MMM US Equity | industrials |

11 | Caterpillar, Inc. | CAT US Equity | industrials |

12 | Boeing Company | BA US Equity | industrials |

13 | Union Pacific Corporation | UNP US Equity | industrials |

14 | Honeywell International | HON US Equity | industrials |

15 | Wal-Mart Stores, Inc. | WMT US Equity | consumer staples |

16 | McDonald’s Corporation | MCD US Equity | consumer discretionary |

17 | Comcast Corporation | CMCSA US Equity | consumer discretionary |

18 | Walt Disney Company | DIS US Equity | consumer discretionary |

19 | Home Depot, Inc. | HD US Equity | consumer discretionary |

20 | CVS Caremark Corporation | CVS US Equity | consumer staples |

21 | Costco Wholesale Corporation | COST US Equity | consumer staples |

22 | Apple Inc. | AAPL US Equity | information technology |

23 | Microsoft Corporation | MSFT US Equity | information technology |

24 | International Business Machines (IBM) | IBM US Equity | information technology |

25 | Oracle Corporation | ORCL US Equity | information technology |

26 | Intel Corporation | INTC US Equity | information technology |

27 | Hewlett-Packard Company | HPQ US Equity | information technology |

28 | EMC Corporation Common | EMC US Equity | information technology |

29 | Exxon Mobil Corporation | XOM US Equity | energy |

30 | Chevron Corporation Common | CVX US Equity | energy |

31 | Schlumberger N.V. | SLB US Equity | energy |

32 | ConocoPhillips | COP US Equity | energy |

33 | Occidental Petroleum | OXY US Equity | energy |

34 | Anadarko Petroleum | APC US Equity | energy |

35 | Apache Corporation | APA US Equity | energy |

36 | Procter & Gamble Company | PG US Equity | consumer staples |

37 | Coca-Cola Company | KO US Equity | consumer staples |

38 | Pepsico, Inc. | PEP US Equity | consumer staples |

39 | Altria Group, Inc. | MO US Equity | consumer staples |

40 | Colgate-Palmolive Company | CL US Equity | consumer staples |

41 | Ford Motor Company | F US Equity | consumer discretionary |

42 | Nike, Inc. | NKE US Equity | consumer discretionary |

43 | Kimberly-Clark Corporation | KMB US Equity | consumer staples |

44 | Johnson & Johnson | JNJ US Equity | health care |

45 | Pfizer, Inc. Common Stock | PFE US Equity | health care |

46 | Merck & Company, Inc. | MRK US Equity | health care |

47 | Abbott Laboratories | ABT US Equity | health care |

48 | Bristol-Myers Squibb Company | BMY US Equity | health care |

49 | Amgen Inc. | AMGN US Equity | health care |

50 | UnitedHealth Group | UNH US Equity | health care |

## Conflicts of Interest

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Stefanovits, D.; Schubiger, U.; Wüthrich, M.V.
Model Risk in Portfolio Optimization. *Risks* **2014**, *2*, 315-348.
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**AMA Style**

Stefanovits D, Schubiger U, Wüthrich MV.
Model Risk in Portfolio Optimization. *Risks*. 2014; 2(3):315-348.
https://doi.org/10.3390/risks2030315

**Chicago/Turabian Style**

Stefanovits, David, Urs Schubiger, and Mario V. Wüthrich.
2014. "Model Risk in Portfolio Optimization" *Risks* 2, no. 3: 315-348.
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