Mortgage Defaults A bank finds that the relationship between...

Name: Problem 22, Chapter 7: Finite Mathematics and Calculus with Applications
Uploaded: 2020-06-09T03:05:46-08:00
Duration: 9 min 24 s
Channel: Haggai Liu
Description: Mortgage Defaults A bank finds that the relationship between mortgage defaults and the size of the down payment is given by the following table. TABLE CANNOT COPY a. If a default occurs, what is the probability that it is on a mortgage with a 5% down payment? b. What is the probability that a mortgage that is paid to maturity has a 10% down payment?

Mortgage Defaults A bank finds that the relationship between mortgage defaults and the size of the down payment is given by the following table. TABLE CANNOT COPY a. If a default occurs, what is the probability that it is on a mortgage with a 5% down payment? b. What is the probability that a mortgage that is paid to maturity has a 10% down payment?

Mortgage Defaults A bank finds that the relationship between mortgage defaults and the size of the down payment is given by the following table.
TABLE CANNOT COPY
a. If a default occurs, what is the probability that it is on a mortgage with a 5% down payment?
b. What is the probability that a mortgage that is paid to maturity has a 10% down payment?

Key Concepts

Conditional Probability

Conditional probability measures the chance of an event occurring given that another event has already occurred. In this context, it helps determine the probability of a mortgage default being associated with a specific down payment percentage when it's known that a default occurred, or identifying the likelihood of a specific down payment for mortgages that reach maturity.

Joint Probability

Joint probability involves the likelihood of two events happening at the same time, such as a mortgage having a particular down payment and also resulting in default or maturity. This concept is essential for building and interpreting the probability table that categorizes different down payment sizes against mortgage outcomes.

Marginal Probability

Marginal probability refers to the probability of a single event occurring regardless of the outcome of another event. In problems like these, it is used to calculate the overall probability of a mortgage defaulting or being paid to maturity, which in turn is necessary for applying conditional probability formulas.

Total Probability Rule

The total probability rule is used to determine the overall probability of an event by summing the probabilities across all possible conditions or categories. In this scenario, it allows one to aggregate the probabilities of all defaults or all payments to maturity across different down payment sizes, forming the basis for calculating the conditional probabilities in question.

Transcript

00:01 So the first thing here we would like to do is we need to compute the proportion of mortgages with each particular down payment.

00:12 So to do that, we need to select the total number of mortgages available in the whole system.

00:20 So let us add total down this column here.

00:23 1 -260 plus 700 plus 560 plus 280 giving us 2 ,800 mortgages.

00:38 Okay, so see.

00:41 So now we want to find proportions, portion of mortgages with down payment, with this down payment.

01:00 Okay, so the first one would be 1 ,260 divided by 2 ,800.

01:18 That's 0 .45.

01:21 Okay, so in a similar manner, we can now obtain the next one as 700 divided by 2 ,800, which should be 0 .25.

01:34 Yes, that's 0 .25.

01:37 The next one, again, in the similar manner.

01:43 Give 0 .2 the last one 0 .1 okay let's add that together real quick to make sure it's 1 7 and so that does add up to 1 so we are good okay so that now with that complete and so now we have enough information to solve part a and so what i want to do now is for convenience let's label this these columns as let's say column one column two column three and for column four i would like to compute the product of the column two time column three okay and so let's do that so 0 .06 times 0 .027.

02:58 That's our first product there.

03:02 So the second one is .04 times .25, that's 0 .01 .3 .2 times .2 times .2, which should be 0 .04.

03:21 And the last one is 0 .0 .01.

03:31 Okay, now let's add those up together...

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