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= 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.

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