Let $r$ be the risk free rate. Assume that we have a U$.$S$.$ stock portfolio with price $P(t)$ and

the portfolio excess return on the market over the risk free rate is $R_P($i$.$e$.R_p=$

$\{$:$\frac{P(T)-P(t)e^{r(T-t)}}{P(t)e^{r(T-t)}}).$ We would like to hedge the risk of the portfolio using short position of

S&P $500$ index futures. Let $R_F$ be the return of S&P $500$ index futures. Applying

regression model, we obtain $R_P=R_F+lon$ with $=1\mathrm{.}2.$ The current portfolio price $P(t)=$

$5,000,000,$ current $S$&$P500$ index price is $S(t)=2,000.$ We know the contract size of the

S&P $500$ index futures is $$250($S&P $500$ index price$).$ How many share of shorts of the

futures we need to do the perfect hedge?