Description of Equity Buffer Approach
Let
, and
denote the standardized returns of each risk factor�s asset class against which the counterparty is exposed (s): Interest Rates, Commodities, Foreign Exchange and Equities respectively.
As usual the asset�s return is expressed as the log of price changes of the asset.
The Return is then
unitized or standardized/centralized to fit the standardized Normal distribution of mean zero and variance one.
Hence, for every Return
R computed
we compute the unitized or centralized return:
. i.e.
Now, lets denote C
as the counterparty and the return on the �portfolio� or synthetic asset of Counterparty C
. The model assumes the decomposition of counterparty�s portfolio
in terms of each individual risk factor returns:
We seek to map the weight(s) of the counterparty�s individual exposure to risk factor(s), including his own idiosyncratic /specific risk OSR so that the weight sum�s up to 1. (i.e. 100%).
This ensures that the sum of returns for each individual risk factor, which are unitized to fit the standard Normal distribution with mean 0 and variance 1, will also be unitized or standardized to fit the normal distribution with mean zero and variance one
where
are the weights associated with the systematic (or factor) risk, and
is the weight of the counterparty�s / obligor specific risk (OSR). The variable
is a standard normal random variable with mean zero and variance one.
For each counterparty, the weights of each risk factor, including the counterparty / obligor specific risk is re-based so that the overall return fits the normal distribution with expected mean zero and variance one.
For example, if the obligor is exposed to two interest rate factors,
, one Commodity factor
, two foreign exchange factors,
, and one equity risk factors,
then his decomposition will become:
The standard deviation of the above sum of correlated returns is obtained either from the dot product of individual weighted returns or a correlation matrix
or decomposed via cholesky technology accordingly.
In general, suppose the vector is represented by decomposition factors
and
. It is required that
. To solve this we re-base the weights by applying.
where
are standard normal variables and <
is the correlation between
<
.
The specific risk factor,
, is a standard normal random variable related to the OSR .