Individual investors, asset management firms and institutional investors are subject to $40\%,10\%$ and $5\%$ taxation on dividends and $20\%,25\%$ and $10\%$ on capital gains respectively. They all invest in growth stocks, neutral stocks and blue chips. Growth stocks are expected to pay dividends of $2$ USD and have capital gains of $30$ every year perpetually. The same forecasts for neutral stocks are $7$ and $7$ and for blue chips $40$ and $5.$ The institutional investors are the marginal investors and can determine the price of the stocks. Every type of investor maximises their after$-$tax income.

$($i$)$ Suppose that institutional investors require a $10\%$ after$-$tax return. What are the prices of the growth, neutral, and bluechip stocks?

$($ii$)$ Calculate the after$-$tax returns of the three types of stocks for each investor group.