Here are data on three hedge funds. Each fund charges its investors an incentive fee of $10\%$ of total returns. Suppose initially that a fund of funds $($FF$)$ manager buys equal amounts of each of these funds, also charging its investors a $10\%$ incentive fee. For simplicity, assume also that management fees other than incentive fees are zero for all funds.

Hedge Fund $1$ Hedge Fund $2$ Hedge Fund $3$

Start of year value $($millions$)$ $ $160$ $ $160$ $ $160$

Gross portfolio rate of return $10\%20\%30\%$

Required:

a$.$ Compute the rate of return after incentive fees to an investor in the fund of funds.

b$.$ Suppose that instead of buying shares in each of the three hedge funds, a stand$-$alone $($SA$)$ hedge fund purchases the same portfolio as the three underlying funds. The total value and composition of the SA fund is therefore identical to the one that would result from aggregating the three hedge funds. Consider an investor in the SA fund. After paying $10\%$ incentive fees, what would be the value of the investor$$s portfolio at the end of the year?

d$.$ Now suppose that the return on the portfolio held by hedge fund $3$ were $30\%$ rather than $+30\%.$ Recalculate your answers to parts $($a$)$ and $($b$).$