You are considering making a loan for $$100,000$ at an interest rate of $5\%.$ The loan can either

$($a$)$ be prepaid immediately, in which case you receive back your $$100,000$ at zero profit

or

$($b$)$ be repaid in one period at interest rate $5\%($a repayment of $$105,000$ in one year$).$

You do not know the right rate to discount cashflows, but will learn that appropriate discount rate at the moment after loan closing. There is a $50\%$ chance your discount rate will be $\mathrm{.}03$ and a $50\%$ chance your discount rate will be $\mathrm{.}07.($thus repayment a year from now would be worth $$105,000/1\mathrm{.}03$ with $50\%$ probability and $$105,000/1\mathrm{.}07$ with the remaining $50\%$ probability$).$

Suppose that there is a $25\%$ chance that the borrower will repay the loan immediately regardless of what you learn about discounting.

Of the remaining $75\%$ of borrowers, none will prepay immediately should your discount rate be $7\%.$ But if your discount rate is $3\%,25\%$ of remaining borrowers will be "woodheads" and only repay next year. The remaining $75\%$ will prepay immediately.

In other words, $25\%$ of borrowers prepay randomly, and $25\%*75\%$ are woodheads, and $75\%*75\%$ are rational prepayers.

What is the expected value $($Net of the initial $$100,000,$ so expected profitability$)$ to you of this loan before uncerainty is revealed?In the question above, the value of the loan to you would rise if the proportion of woodheads also rose. True or false$$