35 Understanding Key Terms and Concepts

The following are the key terms and concepts used in the OFS Loan Loss Forecasting and Provisioning application.

• Poisson Process and Exponential Distribution

  The Poisson process is a counting process for the number of events that occurred up to a particular time. It is at times called a jump process, as it jumps up to a higher state each time an event occurs. It is also a special case of a continuous Markov process.

  It has potential applications in the Financial Industry. For example, Total Credit default amounts consist usually of a sum of individual default amounts. The number of defaults is usually assumed to occur according to a Poisson process.

  The exponential distribution plays a very important role in the Poisson process partly because the time between events or jumps follows an exponential distribution.

  Random variable X is said to be an exponential distribution if density has the form:

  fX(x) = e?x, for x0.

• Splitting of Poisson Processes

  For example, Times between births (in a family) follows an exponential distribution. The births are categorized by gender.

  For example, Times between back pains follow an exponential distribution. However, the degree of pain may be categorized as per the required medication, which depends on the degree of pain.

  Consider a Poisson Process fN(t); where in addition to observing an event, the event can be classified as belonging to one of r possible categories.

  Define Ni(t) = no. of events of type i during (0; t] for i = 1; 2; : : : ; r)

  N(t) = N1(t) + N2(t) + … + Nr(t)

  This process is referred to as splitting the process.

  The LLFP Application makes use of the Poisson process as one of the techniques to interpolate the PD term structures, PD curves, from a lower frequency to a higher frequency, for example, from an annual frequency to a monthly frequency.

  Note:

  For PD interpolation, the input frequency must be lower, the period must be higher, than the output frequency. For example, the PD term structure cannot be monthly while the bucket frequency is annual).
• Marginal Transition Matrix versus Cumulative Transition Matrix

  Cumulative Transition Matrix refers to a matrix, which includes transitions from previous years as well. Marginal Matrix refers to transitions that are incremental or for only one unit of time.

  In the LLFP Application, for computing Point in Time Probability of Default through the inbuilt PD Model, it is required to provide Marginal Transition Matrix on an annual basis for multiple periods as required by you.

  For example, Marginal Transition Matrix - M1 can be used for Years 1, 2, and 3 while Marginal Transition Matrix M2 can be used for years 4 and 5.

  Examples of Marginal Transition Matrices:

  Table 35-1 Fields in the Marginal Transition Matrix (Year 1) table

  Year 1 - Transition Matrix
  From or To	AAA	AA	A	BBB	BB	B	D
  AAA	88.53%	7.75%	0.47%	0.00%	0.00%	0.00%	3.25%
  AA	0.60%	87.50%	7.33%	0.54%	0.06%	0.50%	3.47%
  A	0.40%	2.07%	87.21%	5.36%	0.39%	0.16%	4.41%
  BBB	0.01%	0.17%	3.96%	84.13%	4.03%	0.72%	6.98%
  BB	0.02%	0.05%	0.21%	5.32%	75.62%	7.15%	11.63%
  B	0.00%	0.05%	0.16%	0.28%	5.92%	73.00%	20.59%
  CCC or C	0.00%	0.00%	0.24%	0.36%	1.02%	11.74%	86.64%