Suppose we know that an investment fund invests in a combination of stocks and

bonds. We observe the funds monthly return $r_i,$ the return on stocks $s_i,$ and the return on

bonds $b_i$ for $i=1,$dots,$n$ time periods. We do not observe the investment strategy of the

fund, though. We can try to back out the relative weights on stocks and bonds employed by

the fund by trying to match the model

hat$(r{)}_i=\frac{1}{1+w}{s}_i+\frac{w}{1+w}{b}_i$

to the data as well as possible. Note that for all $w,$ the weights on $s_i$ and $b_i$ are guaranteed

to add up to one.

Find the $w$ that minimizes the sum of squared prediction errors, that is find $w$ that

minimizes

$f(w)={\displaystyle \underset{i=1}{\overset{n}{}}}(r_i-\frac{1}{1+w}{s}_i-\frac{w}{1+w}{b}_i{)}^2.$

$($This reverse engineering exercise is called a "style regression" and was introduced by William

Sharpe in $1988.)$

Please do not use ChatGPT for the answer $($it$\text{'}$s pretty obvious when it's used$)$