FC 9 9. Using the normal distribution to approximate binomial probabilities Aa A

Question

FC 9

9. Using the normal distribution to approximate binomial probabilities Aa Aa Corporate bonds have higher default rates than municipal bonds with the same rating. Moody's Investors Service measured 10-year default rates for bonds between 1970 and 2006. They found that 69% of Caa-C-rated corporate bonds defaulted, whereas only 17% of Caa-C-rated municipal bonds defaulted. [Source: Moody's Investors Service, The US. municipal bond rating scale: Mapping to the global rating scale and assigning global ratings to municipal obligations (Moody's Investors Service, 2007).] A random sample of 300 Caa-C-rated municipal bonds is selected. Of the 300 bonds in the sample, the expected and the standard deviation of the number of defaults is number of defaults (during the next 10 years) is Use the Distributions tool to help you answer the questions that follow. Normal Distribution Mean 75 Standard Deviation 5.5 10 20 30 40 50 60 70 80 90 100 110 120 Use the normal distribution to approximate the probability of obtaining exactly 58 defaults in the sample. The probability is

Explanation / Answer

mean =np=300*0.17=51

and std deviation =(np(1-p))1/2 =6.506

P(X=58)=P(57.5<X<58.5)=P((57.5-51)/6.506<Z<(58.5-51)/6.506)=P(0.9991<Z<1.1528)=0.8755-0.8411=0.0344

P(X<=44) =P(Z<(44.5-51)/6.506)=P(Z<-0.9991)=0.1587

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