# CAPM模型中残差与市场组合收益率协方差的一个证明

\begin{eqnarray} E(\tilde{r}_{i}) & = & r_{f}+(E(\tilde{r}_{m})-r_{f})\beta_{im}\\  \tilde{r}_{i} & = & r_{f}+\beta_{im}(\tilde{r}_{m}-r_{f})+\tilde{\epsilon}_{i} \end{eqnarray}

\begin{eqnarray*} cov(\tilde{\epsilon}_{i},\tilde{r}_{m}) & = & E[(\tilde{r}_{i}-E(\tilde{r}_{i}))(\tilde{r}_{m}-E(\tilde{r}_{m}))]\\ & = & E[((\tilde{r}_{m}-E(\tilde{r}_{m}))\beta_{im}+\tilde{\epsilon}_{i})(\tilde{r}_{m}-E(\tilde{r}_{m}))]\\ & = & \beta_{im}\sigma_{m}^{2}+cov(\tilde{\epsilon}_{i},\tilde{r}_{m})\\ & = & \beta_{im}\sigma_{m}^{2} \end{eqnarray*}