Multi-Variate Bernoulli Model

Abstract

Bernoulli distribution, a random variable distribution known for centuries, serves a fundamental knowledge in the actuary science and financial credit derivatives. In this article, we will present its multi-dimension version.

Background

A Bernoulli distribution is a distribution that the random variables only take two values: 1 and 0 with probability p and 1-p. p is also the mean of the random variable. The standard deviation is (p.(1-p))^0.5

In many applications, a realization of 1 might mean an occurrence of a single event, say, bankruptcy or death. In case there is a group/basket of the event, say, the first-to-default credit derivative swap or the joint-life insurance, we come to the multi-dimension version.

Notation

We have a group of size n, each member in the group is defined an event. Xi=1 means the event occurs for i-th member, Xi=0 means the event does not occur for the i-th member.

We start from an independent group which means each event for each member is independent. In this specialized case, we have:

P[X1=x1, X2=x2, ..., Xn=xn]=Product[pi^xi.(1-pi)^(1-xi) , {i=1..n}]

where

pi is the probability of event occurrence for i-th member
xi=1 means occurrence, xi=0 means not occurrence

Sol

The multi-dimension is defined this way:

P[X1=x1, X2=x2, ..., Xn=xn]=Product[pi^xi.(1-pi)^(1-xi) , {i=1..n}] + Sum[Dij.(-1)^(xi+xj), {i,j=1..n, i<j}]

where Dij is the deviation from the independent group case. Also note that the summation of Dij over all event is zero.

Therefore, we can calculate any event of the group, for example:

P[X1=x1, X2=x2]=Sum[Product[pi^xi.(1-pi)^(1-xi) , {i=1..n}] + Sum[Dij.(-1)^(xi+xj), {i,j=1..n, i<j}] , {j=3..n, xj=0..1}]
=Sum[Product[pi^xi.(1-pi)^(1-xi) , {i=1..n}], {j=3..n, xj=0..1}] + Sum[Dij.(-1)^(xi+xj), {i,j=1..n, i<j}] , {j=3..n, xj=0..1}]
=p1^x1.(1-p1)^(1-x1).p2^x2.(1-p2)^(1-p2) + D12.(-1)^(x1+x2).2^(n-2)

covariance matrix

The covariance is

E[X1.X2]-p1.p2
=p1^1.p2^1.1.1+D12.(-1)^(1+1).2^(n-2)-p1.p2
=D12.2^(n-2)

And the correlation of X1 and X2 is

(E[X1.X2]-p1.p2)/Sqrt[p1.(1-p1).p2.(1-p2)]
=D12.2^(n-2)/Sqrt[p1.(1-p1).p2.(1-p2)]

We may impose an assumption that the correlation of each pair of members is the same, just like financial credit default swap practice. Suppose the correlation is r then

r=D12.2^(n-2)/Sqrt[p1.(1-p1).p2.(1-p2)]

and therefore,

D12=r.Sqrt[p1.(1-p1).p2.(1-p2)]/2^(n-2)

and therefore

P[X1=x1, X2=x2]=p1^x1.(1-p1)^(1-x1).p2^x2.(1-p2)^(1-p2) + r.Sqrt[p1.(1-p1).p2.(1-p2)].(-1)^(x1+x2).

and the covariance matrix is

1	r.(p1.(1-p1).p2.(1-p2))^0.5	r.(p1.(1-p1).p3.(1-p3))^0.5	...	r.(p1.(1-p1).pn.(1-pn))^0.5
r.(p2.(1-p2).p1.(1-p1))^0.5	1	r.(p2.(1-p2).p3.(1-p3))^0.5	...	r.(p2.(1-p2).pn.(1-pn))^0.5
...
r.(pn.(1-pn).p1.(1-p1))^0.5	r.(pn.(1-pn).p2.(1-p2))^0.5	...	1

Of course, r can not be too extreme, it must make all the probability positive. When r is 0, the case is specialized as independent group case.

An application

Suppose we have a first-to-default credit default swap for a basket of 100 names. Event probability for first-to-default is

P[X1=1 or X2=1 or ... X100=1]
=1 - P[X1=0, X2=0, ..., X100=0]
=1 - (Product[1-pi , {i=1..n}] + Sum[r.Sqrt[pi.(1-pi).pj.(1-pj)/2^(n-2)] , {i,j=1..n, i<j}])
=1 - (Product[1-pi , {i=1..n}] + r/2^(n-2).Sum[Sqrt[pi.(1-pi).pj.(1-pj)] , {i,j=1..n, i<j}])

Once the correlation r and each member's probability of default pi are estimated, we have the event probability of first-to-default with which we can then calculate out the actuary present value by typical single-life actuary method.