Consider two local banks. Bank A has 77 loans? outstanding, each for? $1.0 milli
ID: 2616134 • Letter: C
Question
Consider two local banks. Bank A has
77 loans? outstanding, each for? $1.0 million, that it expects will be repaid today. Each loan has a 6%
probability of? default, in which case the bank is not repaid anything. The chance of default is independent across all the loans. Bank B has only one loan of $77
million? outstanding, which it also expects will be repaid today. It also has a 6%
probability of not being repaid. Calculate the? following:
a. The expected overall payoff of each bank.
b. The standard deviation of the overall payoff of each bank.
Explanation / Answer
Solution:-
a) Calculation of expected overall pay off
Bank A:-
Expected pay off of each loan
= Summation of expected amount of recovery*probability
=$1.0 million*0.94+0*0.06
=0.94 million
Hence the overall expected pay off = Expected payoff of each loan*Number of loans = 0.94 million *77 = 72.38 million
Bank B:-
Expected pay off of bank
= Summation of expected amount of recovery*probability
=$77.0 million*0.94+0*0.06
=72.38 million
Hence the expected overall pay off of both bank is $73.38 million.
b) Calculation of the standard deviation of each bank
Bank A:-
Variance of each loan =( CF1-Mean CF)2*P1+( CF2-Mean CF)2*P2
Where
CF1= Possible cash flow 1 =1 million
Mean CF= 0.94 million
P1=0.94
P2= 0.06
Hence
Variance of each loan =(1-0.94)2*0.94+(0-0.94)2*0.06
=0.003384+0.05316
=0.0564
Overall variance= Variance of each loan *Number of loans (Since independent )
= 0.0564*77= 4.3428
Standard deviation overall = (Variance overall)(1/2) = (4.3428)(1/2) =2.084
Hence the standard deviation of bank A= 2.084
Bank B:-
Variance of loan =( CF1-Mean CF)2*P1+( CF2-Mean CF)2*P2
=(77-72.38)2*0.94+(0-72.38)*0.06
=20.063736+314.331864
=334.3956
Standard deviation = (Variance )(1/2) = (334.3956)(1/2) =18.286
Hence the standard deviation of overall pay off of bank A= 2.084 and bank B= 18.286
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