### Bond Portfolios and Default Correlations

```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.
```