An inventor has two prototypes for an invention, A and B, and needs the same amount of loan L for each. It is expected that either one of these prototypes will take a year to develop and will yield a reward of either RA or 0 for prototype A, and either RB or 0 for prototype B. The probabilities of a positive payoff are given by prob(RA) = pA prob(RB) = pB and these probabilities are known to the bank. The bank will only lend if the expected repayment at the end of the year generates a yield of i. If the prototype gives a 0 reward, then the bank receives no repayment. (a) Assuming risk neutrality on the part of the bank, explain why the repayments when either prototype is a success are given by (1 + i)L/pA, (1 + i)L/pB for A, B respectively. (b) If the loan were granted, show that the expected payoff for the inventor after the bank has been repaid is given by pARA − (1 + i)L, pBRB − (1 + i)L for A, B respectively.
An inventor has two prototypes for an invention, A and B, and needs the same amount of loan L for each. It is expected that either one of these prototypes will take a year to develop and will yield a reward of either RA or 0 for prototype A, and either RB
or 0 for prototype B. The probabilities of a positive payoff are given by
prob(RA) = pA prob(RB) = pB
and these probabilities are known to the bank. The bank will only lend if the expected repayment at the end of the year generates a yield of i. If the prototype gives a 0 reward, then the bank receives no repayment.
(a) Assuming risk neutrality on the part of the bank, explain why the repayments when either prototype is a success are given by
(1 + i)L/pA, (1 + i)L/pB for A, B respectively.
(b) If the loan were granted, show that the expected payoff for the inventor after the bank has been repaid is given by
pARA − (1 + i)L, pBRB − (1 + i)L for A, B respectively.
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