Systems and methods for modeling credit risks of publicly traded companies

• US 7,236,951 B2
• Filed: 03/08/2004
• Issued: 06/26/2007
• Est. Priority Date: 07/24/2003
• Status: Expired due to Fees

• Current Assignee: Credit Suisse Securities USA LLC (Credit Suisse)
• Original Assignee: Credit Suisse First Boston Gilts (Credit Suisse)
• Inventors: Lipton, Alexander, Song, Jonathan Z., Lee, Shinghoi
• Primary Examiner(s): Borlinghaus; Jason M.
• Application Number:
  US10/795,541
• Publication Number:
  US 20050021452A1
• Time in Patent Office: 1,205 Days
• Field of Search: 705 35- 39
• US Class Current: 705/35
• CPC Class Codes: G06Q 10/10 Office automation; Time management
• CPC Class Codes: G06Q 40/00 Finance; Insurance; Tax strategies; Processing of corporate or income taxes
• CPC Class Codes: G06Q 40/03 Credit; Loans; Processing thereof

Abstract

There are provided new structural default models for modeling the likely default of publicly traded companies. In a first embodiment, the invention is straight-forward to implement and allows the capture of some important ingredients of the actual default, including positive short-term CDSs. In a second embodiment the model is somewhat more versatile and complex. Provided is a very efficient method for dealing with the timing of a default boundary, that is, jumps in the company'"'"'s value, etc. Further provided is a process using Fast Fourier Transform matrix processing for processing the structural default models in a computationally efficient manner.

Claims

1. A method, using a computer, of calculating the financial status of a company, comprising the steps of:

determining using a computer, the value of the company in accordance with a formula

$\frac{d<br><br><br><br>V}{V}=\left(r-\mathrm{\lambda <br><br>}<br><br><br><br>\mathrm{\kappa <br><br>}\right)<br><br><br><br>d<br><br><br><br>t+\mathrm{\sigma <br><br>}<br><br><br><br>d<br><br><br><br>W+\left({\mathrm{e<br><br>}}^J-1\right)<br><br><br><br>d<br><br><br><br>N$

wherein

N is a standard Poisson process with intensity λ

,

J is a normal variable with mean j and standard deviation k;

W is a standard Wiener process,

V is the value of the company,

r is an interest rate,

σ

is a company volatility,

λ

is an intensity of jump arrival,

t is a calendar time between today and maturity T;

determining, using a computer, that the company defaults if at a sequence of discrete observational times t₀=0(today),t₁, t₂, . . . ,t_N=T(maturity) the value of the company V_n=V(t_n) falls below a corresponding barrier level B₁, B₂, . . . ,B_N=D , the barrier levels selected to represent different debt amounts which come due at corresponding times t₀=0(today),t₁,t₂, . . . ,t_N=T(maturity);

calculating a transitional probability density function (TPDF) for the value of the company conditional on no default occurring between time t=0 and an observational time t_X; and

determining, using the TPDF, a transitional probability that the company will have a value of V_m at time t_n.

2. The method of claim 1 further comprising calculating the transitional probability density function (TPDF) for the value of the company conditional on no default occurring between time t=0 and an observational time t_x comprising the steps of:

selecting a range for a natural logarithm of the value of the company U=ln(V);

U_min<

U<

U_max

where

U_min=−

nstdiv√

{square root over ((σ

²+k²)T)}+min(r−

λ

κ

−

σ

²/2,0)T

and

U_max=nstdiv√

{square root over ((σ

²+k²)T)}+max(r−

λ

κ

−

σ

²/2,0)T

and where nsd is a number of standard deviations used to characterize the extreme values of the natural logarithm of the value of the company;

dividing the range into an equidistant grid u₀=U_min,u₁, . . . ,u_m, . . . ,u_M=U_max of M steps, with each grid step, denoted by h, equal to h=(U_max−

U_min)/M, with a total number of points in the grid, which is equal to M+1, being a power of 2;

defining an integer μ

=floor(−

U_min/h), where floor(.) is a mathematical function, which for every number defines an integer less than or equal to this number;

constructing a modified grid as;

ũ

_m=(m−

μ

)h, m=0,1, . . . ,M.;

defining a grid on the line representing the value of the company using a following formula

v_m=V₀exp(u_m)m=0,1, . . . ,M

where an initial value of the company V₀ is equal to V_μ

; and

calculating on the grid a probability that the company will have a value of v_m at time t_n by a vector P_n=(P_0,n, . . . , P_m,n, . . . , P_M,n) where initial vector P₀=(0, . . . ,1/_μ

h, . . . ,0).

3. The method of claim 2 wherein the step of determining the transitional probability that the company will have the value of v_m at time t_n comprises:

determining an unconditional probability vector P₁ in accordance with a formula

P₁={circumflex over (T)}_0,1P₀

wherein {circumflex over (T)}_0,1 is a transition operator between the times to and t₀ and t₁ for the equation

$\frac{d<br><br><br><br>V}{V}=\left(r-\mathrm{\lambda <br><br>\kappa <br><br>}\right)<br><br>d<br><br><br><br>t+\mathrm{\sigma <br><br>}<br><br><br><br>d<br><br><br><br>W+\left(e^J-1\right)<br><br>d<br><br><br><br>N$

wherein like variables to those defined above define the same values and wherein; and

applying a projection operator {circumflex over (Π

)}₁ to vector P₁ in order to obtain a vector P₁;

P₁={circumflex over (Π

)}_{1 P1}.

4. The method of claim 3 further comprising a step of determining selected financial characteristics of the company as a function of an at least one probability vector P.

5. The method of claim 4 further comprising a step of calculating the survival probability of the company in accordance with an equation

$Q_n=\sum _{}^{}$

m = 0 M ⁢

P m , n

where

m is an index changing from 0 to M,

M is the total number of grid points,

n is an index changing from 0 to N, and

N is the total number of times when company value is observed.

6. The method of claim 4 further comprising a step of calculating an equity value of the company in accordance with an equation

$S=e^{}$

- rT ⁢

∑

m = 0 M ⁢

P m , N ⁢

max ⁡

( v m - B N , 0 ) .

where

r is the interest rate,

T is the date of maturity of debt,

m is an index changing from 0 to M,

V_m represents the value of the company,

N is the total number of times when company value is observed,

M is the total number of grid points, and

B_N is a terminal debt level.

7. The method of claim 4 further comprising a step of calculating a present value of company debt δ

in accordance with an equation

$\mathrm{\delta <br><br>}={\mathrm{e<br><br>}}^{-\mathrm{rT}}<br><br>\underset{m=0}{\overset{M}{\sum <br><br>}}<br><br>P_{m,N}$

where

r is the interest rate,

T is the date of maturity of debt,

N is the total number of times when company value is observed,

m is an index changing from 0 to M, and

M is the total number of grid points.

8. The method of claim 5 further comprising a step of calculating a credit default spread CDS in accordance with an equation

$\begin{array}{c}{\mathrm{CDS}}_n=\end{array}$

( 1 - R ) ⁢

( 1 - ⅇ

- rt n ⁢

Q n 1 2 ⁢

ⅇ

- rt 0 ⁡

( t 1 - t 0 ) + 1 2 ⁢

∑

n ′

= 1 n - 1 ⁢

ⅇ

- rt n ′

⁢

( t n ′

+ 1 - t n ′

- 1 ) + 1 2 ⁢

ⅇ

- rt n ⁡

( t n - t n - 1 ) - r )

R is a recovery level for a selected debt seniority,

n is an index changing from 0 to N, and

n′

is an index changing from 1 to n−

1.

9. The method of claim 4 further comprising a step of determining the value of the vector P_n is performed with the transition operator {circumflex over (T)}_0,1 being a Toeplitz matrix.

10. The method of claim 9 further comprising a step of determining the value of the vector P_n is performed using a Fast Fourier Transformation.

11. A system for calculating the financial status of a company, comprising:

a processor;

a memory connected to the processor and storing instructions for controlling the operation of the processor;

the processor operative with the instructions in the memory to perform the steps of determining the company'"'"'s value in accordance with a formula

$\frac{d<br><br>V}{V}=\left(r-\mathrm{\lambda <br><br>}<br><br><br><br>\mathrm{\kappa <br><br>}\right)<br><br>d<br><br>t+\mathrm{\sigma <br><br>}<br><br><br><br>d<br><br>W+\left({\mathrm{e<br><br>}}^J-1\right)<br><br>d<br><br>N$

wherein

N is a standard Poisson process with intensity λ

,

J is a normal variable with mean j and standard deviation k;

W is a standard Wiener process,

V is the value of the company,

r is an interest rate,

σ

is a company volatility,

λ

is an intensity of a jump arrival,

t is a calendar time between today and maturity T;

determining that the company defaults if at a sequence of discrete observational times t₀=0(today), t₁, t₂, . . . , t_N=T(maturity) the value of the company V_n=V(t_n) falls below corresponding barrier level B₁, B₂, . . . , B_N=D, the barrier levels selected to represent different debt amounts which come due at corresponding times t₀=0(today), t₁, t₂, . . . , t_N=T(maturity);

calculating a transitional probability density function (TPDF) for the value of the company conditional on no default occurring between time t=0 and an observational time t_X; and

determining, using the TPDF, a transitional probability that the company will have a value of v_m at time t_n.

12. The system of claim 11 further comprising calculating the transitional probability density function (TPDF) for the value of the company conditional on no default occurring between time t=0 and an observational time t_x comprising a steps of:

selecting a range for a natural logarithm of the value of the company U=ln(V);

U_min<

U<

U_max

where

U_min=−

nstdiv√

{square root over ((σ

²+k²)T)}+min(r−

λ

κ

−

σ

²/2,0)T

and

U_max=nstdiv√

{square root over ((σ

²+k²)T)}+max(r−

λ

κ

−

σ

²/2,0)T

and where nsd is a number of standard deviations used to characterize extreme values of a natural logarithm of the value of the company;

dividing the range into an equidistant grid u₀=U_min, u₁, . . . , u_m, . . . u_M=U_max of M steps with each grid step, denoted by h, equal to h=(U_max−

U_min)/M, with a total number of points in the grid, which is equal to M+1, being a power of 2;

defining an integer μ

=floor(−

U_min/h), where floor(.) is a mathematical function, which for every number defines an integer less than or equal to this number;

constructing a modified grid as;

ũ

_m=(m−

μ

)h, m=0, 1, . . . , M.;

defining a grid on the line representing the value of the company using a following formula

v_m=V₀exp(u_m)m=0, 1, . . . , M

where the initial value of the company V₀ is equal to V_μ

; and

calculating on the grid a probability that the company will have a value of v_m at time t_n by a vector P_n=(p_0,n, . . . , p_m,n, . . . , p_M,n) where initial vector P₀=(0, . . . , 1_μ

/h, . . . ,

     0).

13. The system in accordance with claim 12 wherein the step of determining the transitional probability that the company will have the value of v_m at time t_n comprises a steps of:

determining an unconditional probability vector P₁ in accordance with a formula

P₁={circumflex over (T)}_0,1P₀

wherein {circumflex over (T)}_0,1 is a transition operator between the times t₀ and t₁ for the equation

$\frac{d<br><br><br><br>V}{V}=\left(r-\mathrm{\lambda <br><br>\kappa <br><br>}\right)<br><br>d<br><br><br><br>t+\mathrm{\sigma <br><br>}<br><br><br><br>d<br><br><br><br>W+\left(e^J-1\right)<br><br>d<br><br><br><br>N$

wherein like variables to those defined above define the same values and wherein; and

applying a projection operator {circumflex over (Π

)}₁ to vector P₁ in order to obtain a vector P₁;

P₁={circumflex over (Π

)}₁P₁.

14. The system in accordance with claim 13 further comprising a step of determining selected financial characteristics of the company as a function of an at least one probability vector P.

15. The system in accordance with claim 14 further comprising a step of calculating a survival probability of the company in accordance with an equation

$Q_n=\sum _{}^{}$

m = 0 M ⁢

P m , n

where

m is an index changing from 0 to M,

M is the total number of grid points,

n is an index changing from 0 to N, and

N is the total number of times when company value is observed.

16. The system in accordance with claim 14 further comprising a step of calculating an equity value of the company in accordance with an equation

$S=e^{}$

- rT ⁢

∑

m = 0 M ⁢

P m , N ⁢

max ⁡

( v m - B N , 0 ) .

where

r is the interest rate,

T is the date of maturity of debt,

m is an index changing from 0 to M,

V_m represents the value of the company,

N is the total number of times when company value is observed,

M is the total number of grid points, and

B_N is a terminal debt level.

17. The system in accordance with claim 14 further comprising a step of calculating a present value of company debt δ

in accordance with an equation

$\mathrm{\delta <br><br>}={\mathrm{e<br><br>}}^{-\mathrm{rT}}<br><br>\underset{m=0}{\overset{M}{\sum <br><br>}}<br><br>P_{m,N}$

where

r is the interest rate,

T is the date of maturity of debt,

N is the total number of times when company value is observed,

m is an index changing from 0 to M, and

M is the total number of grid points.

18. The system in accordance with claim 15 further comprising a step of calculating a credit default spread CDS in accordance with an equation

${\mathrm{CDS}}_n=\left(1-R\right)$

⁢

( 1 - ⅇ

- rt n ⁢

Q n 1 2 ⁢

ⅇ

- rt 0 ⁡

( t 1 - t 0 ) + 1 2 ⁢

∑

n ′

= 1 n - 1 ⁢

ⅇ

- rt n ′

⁢

( t n ′

+ 1 - t n ′

- 1 ) + 1 2 ⁢

ⅇ

- rt n ⁡

( t n - t n - 1 ) ⁢

- ⁢

r )

where

r is the interest rate,

t_n is the observational time between today and maturity T,

R is a recovery level for a selected debt seniority,

n is an index changing from 0 to N, and

n′

is an index changing from 1 to n−

1.

19. The system in accordance with claim 14 further comprising a step of determining the value of the vector P_n is performed with the transition operator {circumflex over (T)}_0,1 being a Toeplitz matrix.

20. The system in accordance with claim 19 wherein a step of determining the value of the vector P_n is performed using a Fast Fourier Transformation.

21. A system for calculating the financial status of a company, comprising:

means for determining the company'"'"'s value in accordance with the formula

$\frac{d<br><br><br><br>V}{V}=\left(r-\mathrm{\lambda <br><br>\kappa <br><br>}\right)<br><br>d<br><br><br><br>t+\mathrm{\sigma <br><br>}<br><br><br><br>d<br><br><br><br>W+\left(e^J-1\right)<br><br>d<br><br><br><br>N$

wherein

N is a standard Poisson process with intensity λ

,

J is a normal variable with mean j and standard deviation k;

W is a standard Wiener process,

V is the value of the company,

r is an interest rate,

σ

is the company volatility,

λ

is a intensity of a jump arrival,

t is calendar time between today and maturity T;

means for determining, using the computer, that the company defaults if at a sequence of discrete observational times t₀=0(today), t₁, t₂, . . . , t_N=T(maturity) the value of the company V_n=V(t_n) falls below a corresponding barrier level B₁, B₂, . . . , B_N=D, the barrier levels selected to represent different debt amounts which come due at corresponding times t₀=0(today), t₁, t₂, . . . , t_N=T(maturity);

means for calculating a transitional probability density function (TPDF) for the value of the company conditional on no default occurring between time t=0 and an observational time t_{X; and}

means for determining, using the TPDF, a probability that the company will have a value of V_m at time t_n.

22. The method operable on a computer for calculating the financial status of a company, comprising a steps of:

calculating on a computer the value over time of a company in accordance with a model wherein

$\frac{\mathrm{dV}}{V}=\left(r-\mathrm{\lambda <br><br>}<br><br><br><br>\mathrm{\kappa <br><br>}\right)<br><br>\mathrm{dt}+\mathrm{\sigma <br><br>}<br><br><br><br>\mathrm{dW}+\left(e^J-1\right)<br><br>\mathrm{dN}$

V is the value of the company,

r is a risk free interest rate,

σ

is a standard deviation of the underlying asset,

t is an intermediate time between today (t=0) and maturity (t=T),

N is a standard Poisson process with intensity λ

,

J is a lognormal jump size, which is assumed to be a random variable with a known probability density function φ

(J), and

κ

is an expected value of the jump size,

$\mathrm{\kappa <br><br>}={\int <br><br>}_{-\mathrm{\infty <br><br>}}^{\mathrm{\infty <br><br>}}<br><br>\left(e^J-1\right)<br><br>\mathrm{\varphi <br><br>}<br><br>\left(J\right)<br><br>d<br><br>J;<br><br>$

determining that the company defaults if at a sequence of discrete observational times the value of the company falls below a corresponding barrier level;

the barrier levels selected to represent different debt amounts which come due at corresponding times; and

calculating on a computer a transitional probability density function for the value of the company conditional on no default occurring between an initial time and an observational time using a probability vector P.

23. The method in accordance with claim 22 wherein the probability vector P is determined using a projection operator nullifying the components of the probability vector below the barrier further comprising a step of determining selected financial characteristics of the company as a function of the probability vector P.

24. The method in accordance with claim 23 wherein the financial characteristic is selected from a group comprising an equity value of a company, a debt value of a company, a survival probability of a company and a value of a credit default spread.

25. The method in accordance with claim 23 wherein a step of determining the value of the vector P_n is performed with a transition operator {circumflex over (T)}_0,1 being a Toeplitz matrix.

26. The method in accordance with claim 25 wherein the step of determining the value of the vector P_n is performed using a Fast Fourier Transformation.

27. A system for calculating the financial status of a company, comprising:

a processor;

a memory connected to the processor and storing instructions for controlling the operation of the processor to perform a steps of

calculating the value over time of the company in accordance with a model;

$\frac{\mathrm{dV}}{V}=\left(r-\mathrm{\lambda <br><br>}<br><br><br><br>\mathrm{\kappa <br><br>}\right)<br><br>\mathrm{dt}+\mathrm{\sigma <br><br>}<br><br><br><br>\mathrm{dW}+\left(e^J-1\right)<br><br>\mathrm{dN}$

V is the value of the company,

r is a risk free interest rate,

σ

is a standard deviation of the underlying asset,

t is an intermediate time between today (t=0) and maturity (t=T),

N is a standard Poisson process with intensity λ

,

J is a lognormal jump size, which is assumed to be a random variable with a known probability density function φ

(J), and

κ

is an expected value of the jump size,

$\mathrm{\kappa <br><br>}={\int <br><br>}_{-\mathrm{\infty <br><br>}}^{\mathrm{\infty <br><br>}}<br><br>\left(e^J-1\right)<br><br>\mathrm{\varphi <br><br>}<br><br>\left(J\right)<br><br>d<br><br>J;<br><br>$

determining that the company defaults if at a sequence of discrete observational times the value of the company falls below a corresponding barrier level;

the barrier levels selected to represent different debt amounts which come due at corresponding times; and

calculating on a computer a transitional probability density function for the value of the company conditional on no default occurring between an initial time and an observational time using a probability vector P.

28. The system in accordance with claim 27 wherein the probability vector P is determined using a projection operator nullifying the components of the probability vector below the barrier further comprising a step of determining selected financial characteristics of the company as a function of the probability vector P.

29. The system in accordance with claim 28 wherein the financial characteristic is selected from a group comprising an equity value of a company, a debt value of a company, a survival probability of a company and a value of a credit default spread.

30. The system in accordance with claim 29 wherein a step of determining the value of the vector P_n is performed with a transition operator {circumflex over (T)}_0,1 being a Toeplitz matrix.

31. The system in accordance with claim 30 wherein the step of determining the value of the vector P_n is performed using a Fast Fourier Transformation.

32. A system for calculating the financial status of a company, comprising:

means for calculating the value over time of the company in accordance with a model wherein

$\frac{\mathrm{dV}}{V}=\left(r-\mathrm{\lambda <br><br>}<br><br><br><br>\mathrm{\kappa <br><br>}\right)<br><br>\mathrm{dt}+\mathrm{\sigma <br><br>}<br><br><br><br>\mathrm{dW}+\left(e^J-1\right)<br><br>\mathrm{dN}$

V is the value of the company,

r is a risk free interest rate,

σ

is a standard deviation of the underlying asset,

t is an intermediate time between today (t=0) and maturity (t=T),

N is a standard Poisson process with intensity λ

,

J is a lognormal jump size, which is assumed to be a random variable with a known probability density function φ

(J), and

κ

is an expected value of the jump size,

$\mathrm{\kappa <br><br>}={\int <br><br>}_{-\mathrm{\infty <br><br>}}^{\mathrm{\infty <br><br>}}<br><br>\left(e^J-1\right)<br><br>\mathrm{\varphi <br><br>}<br><br>\left(J\right)<br><br>d<br><br>J;<br><br>$

means for determining that the company defaults if at a sequence of discrete observational times the value of the company falls below a corresponding barrier level;

the barrier levels selected to represent different debt amounts which come due at corresponding times; and

means for calculating on a computer a transitional probability density function for the value of the company conditional on no default occurring between an initial time and an observational time using a probability vector P.

33. A program product operable on a computer to control the operation of the computer to calculate the financial status of a company, the program product comprising a computer-readable medium storing instructions to perform the steps of:

calculating the value over time of the company in accordance with a model wherein

$\frac{\mathrm{dV}}{V}=\left(r-\mathrm{\lambda <br><br>}<br><br><br><br>\mathrm{\kappa <br><br>}\right)<br><br>\mathrm{dt}+\mathrm{\sigma <br><br>}<br><br><br><br>\mathrm{dW}+\left(e^J-1\right)<br><br>\mathrm{dN}$

V is the value of the company,

r is a risk free interest rate,

σ

is standard deviation of the underlying asset,

t is an intermediate time between today (t=0) and maturity (t=T),

N is a standard Poisson process with intensity λ

,

J is a lognormal jump size, which is assumed to be a random variable with a known probability density function φ

(J), and

κ

is an expected value of the jump size,

$\mathrm{\kappa <br><br>}={\int <br><br>}_{-\mathrm{\infty <br><br>}}^{\mathrm{\infty <br><br>}}<br><br>\left(e^J-1\right)<br><br>\mathrm{\varphi <br><br>}<br><br>\left(J\right)<br><br>d<br><br>J;<br><br>$

means for determining that the company defaults if at a sequence of discrete observational times the value of the company falls below a corresponding barrier level;

the barrier levels selected to represent different debt amounts which come due at corresponding times; and

means for calculating on a computer a transitional probability density function for the value of the company conditional on no default occurring between an initial time and an observational time using a probability vector P.