Math 1324 Sections 7.5-7.6 3. Mortgage Defaults: A bank finds that the relations

Question

Math 1324 Sections 7.5-7.6 3. Mortgage Defaults: A bank finds that the relationship between mortgage defaults (cannot pay off the loan, the house is repossessed) and the size of the down payment is. given by the following table. Down Payment of Mortgages w/ This ProbabilityofDefault_ Down Payment 5% 10% 20% 25% 1260 700 560 280 0.06 0.04 0.02 0.01 Using the chart above, answer the following questions. a. Find the probability of mortgage that will default if a person makes a down payment of 1 0%. b. Find the probability of a mortgage that will be paid in full if a person makes a down payment of20%. C. If a default occurs, what is the probability that it is on a mortgage with a 5% down payment? d. If a mortgage is paid to maturity (paid in full), what is the probability that the mortgage had a 25% down payment? e. Fromthe data, how many mortgages will default if the down pay ment was 10% or 20%?

Explanation / Answer

a) Given that a person makes a down payment of 10%, the probability of default is 0.04 as can be directly seen from the table

b) Given that a person makes a down payment of 20%, probability of default is 0.02, therefore the probability that the loan would be repaid is computed as: = 1 - 0.02 = 0.98

c) Total number of mortgages = 1260 + 700 + 560 + 280 = 2800

Now using law of total probability, probability that a deafult will occur is computed as:

P( default ) = P( 5% down ) P( default | 5% down ) + P( 10% down ) P( default | 10% down ) + P( 20% down ) P( default | 20% down ) + P( 25% down ) P( default | 25% down )

P( default ) = (1260/2800)*0.06 + (700/2800)*0.04 + (560/2800)*0.02 + (280/2800)*0.01

P( default ) = 0.042

Now given that a default occurs, probability that it is on a mortgage with a 5% down payment is computed as:

P( 5% down | default ) = P( 5% down ) P( default | 5% down )/P( default )

P( 5% down | default ) = (1260/2800)*0.06 / 0.042 = 0.6429

Therefore 0.6429 is the required probability here.

d) Now as we know that: P( default ) = 0.042

Therefore, P( paid in full ) = 1 - 0.024 = 0.958

P( 25% down | paid in full ) = P( paid in full | 25% down ) / P( paid in full )

P( 25% down | paid in full ) = 0.99*(280/2800) / 0.958

P( 25% down | paid in full ) = 0.99*0.1 / 0.958

P( 25% down | paid in full ) = 0.1033

Therefore 0.1033 is the required probability here.

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