[ 2009 FRM Sample Exam ] Quantitative Analysis Q4 defaults, second

lalamei

 

4. Consider a portfolio of six equally weighted bonds. Each bond has a default probability of 4% per annum, and the defaults are independent of each other. What is the probability that this portfolio will see its first default by the end of the second year? Assume the same default probabilities for first and second years for the surviving bonds.

A. 43%

B. 39%

C. 78%

D. 24%

lalamei

 

Correct answer is B

A is incorrect:  this results from the wrong formulA   P(first default in year 1) + P(at least one default in year 2) =(1-0.96^6) + (1-0.96^6)= 0.217 + 0.217= 43.44%

B is correct: this is the correct formula : P (first default in year 1) + P(first default in year 2)=P(first default in year 1)+P(no default in first year) * P(at least one default in second year) =(1-0.96^6)+0.96^6*(1-0.96^6) = 38.73%

C is incorrect: this results from the wrong formula :  (1 - P(default in a year))^6 = P(no default in a year)^n°bond  = (1-0.96)^6=78.28%

D is incorrect: this results from the mere product:  n° bond * P(no default in a year)= 6*4%= 24%

Reference: John Caouette, Edward I. Altman, and Paul Narayanan, Managing Credit Risk (New York: Wiley, 1998), Chapter 7, Chapter 15.