Assume the probability of default for a P$2$P consumer loan is $5\%,$ while the probability of default for a consumer loan at a commercial bank is $1\mathrm{.}5\%.$

Further assume origination and servicing costs for a P$2$P loan is $0\mathrm{.}1\%$ while the loan orgination and servicing costs for a commercial bank is $1\mathrm{.}5\%.$

$($a$)$ What is the relative cost of the P$2$P loan compared to the commercial bank? $($Show your calculation$)$

$($b$)$ What is the ratio of default rate for a P$2$P loan vs commercial bank loan for the two offerings to be cost competitive to the lender?