What is the probability that the bank will recover less than the principal advanced on this loan; assuming the probability of the home buyer’s default is independent of the value of the house?

A bank extends a loan of $1m to a home buyer to buy a house currently worth $1.5m, with the house serving as the collateral. The volatility of returns (assumed normally distributed) on house prices in that neighborhood is assessed at 10% annually. The expected probability of default of the home buyer is 5%.

What is the probability that the bank will recover less than the principal advanced on this loan; assuming the probability of the home buyer’s default is independent of the value of the house?
A . More than 1%
B . Less than 1%
C . More than 5%
D . 0

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Answer: B

Explanation:

The bank will not be able to recover the principal advanced on this loan if both the home buyer defaults, and the house value falls to less than $1m, ie the price moves adversely by more than $500k, which is $-500k/$150k = -3.33. (Note that 150k is the 1 year volatility in dollars, ie $1.5m * 10%).

The probability of both these things happening together is just the product of the two probabilities, one of which we know to be 5%. The other is also certainly a small number, and intuitively it is clear that the probability of both the things happening together will be less than 1%.

For a more precise answer, we can calculate the probability of the house price falling by 3.33 standard deviations by calculating the area under the standard normal curve to the left of -3.33. This indeed is a very small number (actually equal to NORMSINV(-3.33)=0.00043), which when multiplied by the probability of default of the home buyer at 5% is certainly going to be less than 1%. Therefore Choice ‘b’ is the correct answer.