Inference for performance measures for financial assets

Abstract

In this work the precision of point and interval estimators for some performance measures for risky financial assets is analyzed and the conditions under which the point estimators are asymptotically normally distributed are provided. The findings of this research suggest that a huge number of observations is needed to get reasonably precise point and interval estimates. Therefore, the considered performance measures may be surely employed as descriptive statistics for ex-post performance comparisons but they should be employed with caution in ex-ante evaluations for investment choices.

Appendix

Lemma 1

Let $F$ be a continuous df and let $(X_1,...,X_n)$ a sample with possibly dependent elements drawn from $F$. Let $\widehat{F}_n$ be the ecdf based on $(X_1,...,X_n)$ and assume that $\widehat{F}_{n}(x)=F(x)+O_{p}(n^{-1/2})$ for all $x\in \mathbb {R}$ as $n\rightarrow \infty$. Let $\widehat{z}_{n}$ be an estimator of $z$ such that $\widehat{z}_{n}=z+O_{p}(n^{-1/2})$ for $n\rightarrow \infty$. Consider the functional $I_{X}=\int _{-\infty }^{z}(z-x) dF(x)$ and its estimator $\widehat{I}_{n}=\int _{-\infty }^{\widehat{z}_{n}}(\widehat{z}_{n}-x) d\widehat{F}_{n}(x)$. The following asymptotic representation for $\widehat{I}_{n}$ holds:

$$\begin{aligned} \widehat{I}_{n}=\left( \widehat{z}_{n}-z\right) F(z)-\frac{1}{n}\sum _{t=1}^{n}\mathbf {1}_{(X_{t}\le z)}(X_{t}-z)+o_{p}(n^{-1/2}) \end{aligned}$$

Proof

$$\begin{aligned} \widehat{I}_{n}&= \int _{z}^{\widehat{z}_{n}}(\widehat{z}_{n}-x) d\widehat{F}_{n}(x)+\int _{-\infty }^{z}(\widehat{z}_{n}-x) d\widehat{F}_{n}(x)\\&-\int _{-\infty }^{z}(z-x) d\widehat{F}_{n}(x)+\int _{- \infty }^{z}(z-x) d\widehat{F}_{n}(x) \end{aligned}$$

and since

$$\begin{aligned} \int _{-\infty }^{z}(\widehat{z}_{n}-x) d\widehat{F}_{n}(x)-\int _{-\infty }^{z}(z-x) d\widehat{F}_{n}(x)&= (\widehat{z}_{n}-z)\widehat{F}_{n}(z)\\&= (\widehat{z}_{n}-z)F(z)\\&+(\widehat{z}_{n}-z)(\widehat{F}_{n}(z)-F(z)), \end{aligned}$$

we may write

$$\begin{aligned} \widehat{I}_{n}=(\widehat{z}_{n}-z)F(z)+\int _{-\infty }^{z}(z-x) d\widehat{F}_{n}(x)+R_{1; n}+R_{2;n} \end{aligned}$$

where

$$\begin{aligned} R_{1;n}=(\widehat{z}_{n}-z)(\widehat{F}_{n}(z)-F(z)) \end{aligned}$$

and

$$\begin{aligned} R_{2;n}=\int _{z}^{\widehat{z}_{n}}(\widehat{z}_{n}-x) d\widehat{F}_{n}(x). \end{aligned}$$

Thus, it remains to show that $R_{1;n}$ and $R_{2;n}$ are both of order $o_{P}(n^{-1/2})$. $R_{1;n}=o_{P}(n^{-1/2})$ follows immediately from the assumptions that $\widehat{F}_{n}(x)=F(x)+O_{p}(n^{-1/2})$ for all $x\in \mathbb {R}$ and $\widehat{z}_{n}=z+O_{p}(n^{-1/2})$, while proving $R_{2;n}=o_{P}(n^{-1/2})$ requires a further argument: we first estimate $|R_{2;n}|$ by

$$\begin{aligned} \left| R_{2;n}\right| \le \left| \widehat{z}_{n}-z\right| |\widehat{F}_{n}(\widehat{z}_{n})-\widehat{F}_{n}(z)| \end{aligned}$$

and since, by assumption, $\widehat{z}_{n}-z=O_{p}(n^{-1/2})$, it remains to show that

$$\begin{aligned} \widehat{F}_{n}(\widehat{z}_{n})-\widehat{F}_{n}(z)=o_{p}(1). \end{aligned}$$

To this end we observe that

$$\begin{aligned} |\widehat{F}_{n}(\widehat{z}_{n})-\widehat{F}_{n}(z)|\le |\widehat{F}_{n}(\widehat{z}_{n})-F(\widehat{z}_{n})|+|F(\widehat{z}_{n})-F(z)|+|F(z)-\widehat{F}_{n}(z)|, \end{aligned}$$

where $\widehat{F}_{n}(z)-F(z)=O_{p}(n^{-1/2})$ by assumption, $F(\widehat{z}_{n})-F(z)=o_{p}(1)$ by the continuity assumption on $F$ and by the assumption that $\widehat{z}_{n}=z+O_{P}(n^{-1/2})$, and finally

$$\begin{aligned} |\widehat{F}_{n}(\widehat{z}_{n})-F(\widehat{z}_{n})|\le \sup _{x\in \mathbb {R}}|\widehat{F}_{n}(x)-F(x)|=o_{p}(1) \end{aligned}$$

since pointwise convergence of $\widehat{F}_{n}(x)$ to $F(x)$ implies uniform convergence. $\square$