A competitive lender makes loans to a pool of borrowers that are identical. After borrowers have received their loans they choose one of two investment projects. Project G pays the borrower a rate of return of $r_g$ with probability $p_g.$ With probability $1-p_g,$ the project earns a zero rate of return, the borrower defaults on the loan, and the lender receives back the initial loan amount. Project B pays the borrower a rate of return of $r_b$ with probability $p_b.$ With probability $1-p_b,$ the project earns a zero rate of return, the borrower defaults on the loan, and the lender receives back the initial loan amount. We assume that $r_g{p}_b,$ and $p_g(1+r_g)>p_b(1+r_b).$

The lender can't observe in which project the borrower invests and so it charges all borrowers the same interest rate $r_L.$ The lender lends an amount $L$ and pays interest $r_D$ on funds acquired from depositors.

Suppose that when $r_L>r_L^{**}$ the lender requires the borrower to post collateral. Let $c_L$ denote the collateral$-$to$-$loan ratio. That is$,c_L=\frac{C}{L}.$ And let $c_L^{**}$ denote the minimum collateral$-$to$-$loan ratio such that the borrower is indifferent between projects $G$ and $B$ even though $r_L>r_L^{**}.$

Round all numeric answers to at least three decimal places.

Q$1\mathrm{.}1$

$2$ Points

Suppose $r_g=0\mathrm{.}06,r_b=0\mathrm{.}12,p_g=0\mathrm{.}99,p_b=0\mathrm{.}4,r_D=0\mathrm{.}01,L=100.$ Find the value for $r_L^{**}$ such that the borrower is indifferent between projects $G$ and $B.$ Round to three decimal places.

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Q$1\mathrm{.}2$

$1$ Point

Suppose that the lending rate is $r_L=0\mathrm{.}16.$ Find $c_L^{**}.$

Q$1\mathrm{.}3$

$1$ Point

Suppose that the lending rate is $r_L=0\mathrm{.}20.$ Find $c_L^{**}.$

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