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DEFAULT CORRELATIONS WITHIN A PORTFOLIO Cara Herlihy DEFAULT CORRELATION EXPLAINED Default correlation measures whether risky credit assets are more likely to default together or separately. Three types: zero,positive and negative default correlation. Zero Default Correlation P(A,B) P(A) x P(B)= P(A,B) 10% x 10% = 1% P(A)= 10% P(B)= 10% Zero Default Correlation Cont’d. P(A)=10 % P(A ,B) = 1% P(A not B)= P(A)-P(A,B)= 9% P(A or B)=P(A)+P(B)-P(A,B)=19% P(neither A nor B)= 100%-P(A or B)=81% 9% Credit A 1% Credit B 9% NON ZERO DEFAULT CORRELATION Default Correlation (A,B)= Covariance(A,B) Standard Dev. (A) * Standard Dev. (B) Standard Dev. (A)= whether or not Credit A defaults Standard Dev. (A)= {P(A) * [1-p(a)]}1/2 Covariance(A,B) = How far the P(A,B) is from P(A,B) at zero correlation Covariance(A,B)= P(A,B)-P(A)*P(B) WHY IS DEFAULT CORRELATION IMPORTANT? Default correlation is one of three factors in determining the credit risk of a portfolio. It directly affects the risk-return profile of investors in credit risky assets Compares the expected returns of an investment to the amount of risk undertaken to get these returns. INVESTMENT PORTFOLIO What does a portfolio consist of? CDO ABS, MBS,RMBS etc. CDO is divided into tranches Tranches are paid out through a cash flow waterfall “Magic of Diversification” DEFAULT CORRELATION WITHIN A PORTFOLIO How does a correlation of defaults within a portfolio affect the overall volatility and return of the portfolio? If default correlation is low: volatility If default correlation is high: volatility CAUSES OF DEFAULT CORRELATION State of the economy State of a particular industry/region A direct relationship between the parties DEFAULT CORRELATION AND THE RECESSION Sub-prime mortgage crisis is at the center of financial crisis Bank Involvement A.R.M. and SIVA, NIVA and NINA loans were given by banks Housing bubble burst which in turn made people default Defaults on mortgages went from being completely random, to being highly correlated EXAMPLE OF DEFAULT CORRELATION SHOWN THROUGH A DEFAULTS ON MORTGAGES Owner 1 defaults on his mortgage because he loses his job. His house goes into foreclosure, which brings down the values of the houses in his neighborhood. Living in an area with decreasing house prices, owner 2 feels the repercussions of owner 1 foreclosing. The price of his house goes down and he cannot afford his mortgage payments, so he must foreclose as well. With both of his neighbors foreclosing, the neighborhood becomes less desirable and the price of owner 3’s house declines. After refinancing, owner 3’s house is worth significantly less than what anticipated. As a result, he can’t pay his mortgage and he defaults on his loan. R- CODE Independent Version n<-100 #number of bonds. pavg<-0.02 #average default probability of 2% numdef<-rbinom(100,n,pavg) #number of defaults is taken from the binomial distribution numdef #number of defaults with correlation. Correlated Version n<-100 pavg<-0.02 c<-1 numdef2<-double(100) #constant that is chosen #number of defaults is taken from an array of 100 numbers for (i in 1:100){ pact<-rbeta(1,c*pavg,c*(1-pavg)) #default probabilities taken from Beta distribution numdef2[i]<-rbinom(1,n,pact)} #number of defaults is taken from the binomial distribution with various default probabilities Binomial Distribution Binomial is used because we are concerned with only two options: whether the loan is being defaulted on or not. Beta Distribution Beta Distribution is used because we are interested in the events which take place within a set interval (0,1) PDF: F(x)= P(α+Β) P(α) +P(Β) x α-1 * (1-x) Β-1 Let P1~ Beta(α=cp, Β=c(1-p)) Mean= α (α+ Β) =P O<x<1 Formula: Let P1~ Beta(α=cp, Β=c(1-p)) My code: pact<-rbeta(1, c*pavg, c*(1-pavg)) (α) (Β) numdef2[i]<-rbinom(1,n,pact)} VARYING VALUES OF CONSTANT C Left two graphs, when C=1 Right two graphs, when C= 10 VARYING VALUES OF CONSTANT C Left two, when c=1 Right two, when c=10 Changing the Average Rate of Default Keeping C constant Average Default Probability 2% Average Default Probability10 % Changing the Average Rate of Default Keeping C constant Average Default Probability 2% Average Default Probability10 % IN THE FUTURE.. For the next part of my presentation, I would like to focus on Credit Default swaps and simulating the price of a default swap. In order to do this I would have to find a way to simulate the pricing of a CDS. After I have done this, I then want to compare the pricing of one bond to the pricing of more than 1 bond.