Here, we model credit risk and present Insurance function in order to obtain the expected present value of the payments of the Insurance. This will be done by applying the binomial model (discrete model). Consider a borrower with a loan term of N months given by a financial institution. Let a stochastic process Y represent the number of defaultable periods (delinquency index) as of time t. Then,
Y = ( Y t ) t ∈ T , for T = ( 0 , 1 , 2 , ⋯ , N − 1 )
Under certain circumstances, the value of Yt can be negative. In such scenarios we treat Y as absolute value. Let pi represent the probability that the borrower fails to make his credit payment for time i. Denote a function f ( Y t ) which is a function that depends on the delinquency index. Consider cash bond Bt to be another stochastic process to represent the time-value of money. Assume a constant risk-free rate r. Thus B t = B 0 e r t with condition that B0 = 1. We thus analyze the simple case of the process Yt by applying binomial trees. That is, when N = 1 we can represent the compensation by the insurer f ( Y t ) of the delinquency index at time 1 as follows:
From the above figure, we can obtain the expected present value of the compensation by the insurance using the following formula:
E = E [ f ( Y t ) ] = e − r [ p 0 ∗ f ( u ) + ( 1 − p 0 ) ∗ f ( d ) ] (2.1)
where;
f ( u ) represents compensation paid by an Insurance company in which the borrower defaults payment; and
f ( d ) represents compensation paid by an Insurance company in which the borrower pay the corresponding payment of his credit.
Thus by Kolmogorov’s strong law of large numbers, if the insurance company has a large number of portfolio, then that company can expect a loss given by formula (2.1).
We use the binomial tree of two steps to generalize the binomial model. In this case we will do our analyzation by considering the case when N = 2, (t = 0, 1, 2) under the probability measures ℙ and ℚ .
Where N is the loan term.
Based on the diagram above we can name the outcomes of the delinquency index and its proportion of upward and downward movement based on the path followed by the process, for example, ud shows the outcome when the process had an upward movement in the first time and then a downward movement in the second step.
In our generalization and considering the methods in the previous section, we can calculate the expected present value of the possible losses of the insurance company which issues f(.) as follows. At time 0, we can estimate the expected present value of the losses of the company at time as follows;
E ℙ { 1 } = e − r E ℙ [ f ( Y 1 ) | F 0 = { 1 } ] = e − r ( p 0 E ℙ { 1,2 } + ( 1 − p 0 ) E ℙ { 1,3 } ) (2.2)
We now apply the method of replicating asset portfolio in order to make change of probability measure. This enables us find another way of expressing the expected present value of the losses to the insurance company.
Suppose an insurance company wishes to match its losses by using the following portfolio:
C t = a t A ( Y t + 1 ) + b t B t
where A(Yt) is an asset whose value depends on the delinquency index Yt and Bt is a cash bond that has the risk-free rate r.
Then the above portfolio Ct should replicate the losses of the insurance company. Thus we have;
a t A ( Y t + 1 | F t ) + b t B 0 e r = f ( Y t + 1 | F t ) (2.3)
From Equation (2.2) we have;
a t A ( Y t + 1 | F t + ) + b t B 0 e r = f ( Y t + 1 | F t + ) (i)
and
a t A ( Y t + 1 | F t − ) + b t B 0 e r = f ( Y t + 1 | F t − ) (ii)
solving Equations (i) and (ii) simultaneously, we obtain;
a t = f ( Y t + 1 | F t + ) − f ( Y t + 1 | F t − ) A ( Y t + 1 | F t + ) − A ( Y t + 1 | F t − )
b t = 1 e r B t [ f ( Y t + 1 | F t + ) − ( f ( Y t + 1 | F t + ) − f ( Y t + 1 | F t − ) A ( Y t + 1 | F t + ) − A ( Y t + 1 | F t − ) ) A ( Y t + 1 | F t + ) ]
where; F t + is the filtration at time t when in the last path of that filtration there was a rise in the value of the delinquency index, and F t − is the filtration at time t when in the last step of that filtration there was a decrease in the value of the delinquency index. If we substitute at and bt in Ct we obtain:
C t = e − r [ q t f ( Y t + 1 | F t + ) + ( 1 − q t ) f ( Y t + 1 | F t − ) ]
We thus can be able to change the probability measure from ℙ to ℚ and the following is now the generalization of the results.
q t = A ( Y t ) e r − A ( Y t + 1 | F t − ) A ( Y t + 1 | F t + ) − A ( Y t + 1 | F t − )
A ( Y t ) = q t e − r A ( Y t + 1 | F t − ) + ( 1 − q t ) e − r A ( Y t + 1 | F t − ) = E ℚ [ e − r A ( Y t + 1 ) | F t ] = e 0 A ( Y t )
This is a very significant result that will be used in the following section.
Let’s modify the binomial tree shown in the previous sections. If we take changes
in time to correspond to 1 N where N is the term of the loan (asset-based) and suppose that the borrower can increase or decrease its “delinquency index” y = 1 N with probability p = 1 2 . Let xi be a random variable:
X i = { 1 withprobability p = 1 2 − 1 withprobability 1 − p = 1 2
Therefore the delinquency index at time t can be written down as:
Y t = x 1 y + x 2 y + x 3 y + ⋯ + x t y = x 1 1 N + x 3 1 N + ⋯ + x t 1 N
Based on the central limit theorem: Y t ~ N ( 0 , t ) .
The unconditional probability density function which follows Y t ~ N ( 0 , t ) at a fixed time t for Wiener process (Brownian Motion process) is given by;
f ( W t ( x ) ) = 1 2 π t e − x 2 2 t
The expectation is zero,
E ( W t ) = 0
E 2 [ W t ] = E [ W t 2 ] − 0 = E [ W t 2 ] = t
Thus
Y t ~ N ( 0, t )
It’s a continuous time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian Motion with drift. Thus a stochastic process is said to follow a Geometric Brownian Motion if it satisfies the following stochastic differential equation;
d Y t = μ Y t d t + σ Y t d W t
where W t is a Wiener process (Brownian Motion) and ( μ the drift) and ( σ the volatility) are constants. The expression of this diffusion is
d Y t Y t = σ d W t + μ d t (2.4)
We can solve this stochastic differential Equation (2.4) by applying Itô’s lemma (shown below); This model is very important in the formulas developed further.
Assume that Y has a stochastic differential given by
d Y t = μ t d t + σ t d W t (2.5)
where μ and σ are adapted processes. Define the process Z by Z t = f ( t , Y t ) . Then Z has a stochastic differential given by
d f ( t , Y t ) = { ∂ f ∂ t + μ ∂ f ∂ y + 1 2 σ 2 ∂ 2 f ∂ y 2 } d t + σ ∂ f ∂ y d W t (2.6)
Proof. Taking Taylor expansion including second order terms, we obtain;
d f = ∂ f ∂ t d t + ∂ f ∂ y d Y + 1 2 ∂ 2 f ∂ y 2 ( d Y ) 2 + 1 2 ∂ 2 f ∂ t 2 ( d t ) 2 + ∂ 2 f ∂ t ∂ y d t d Y
Squaring Equation (2.5) we obtain;
( d Y ) 2 = μ 2 ( d t ) 2 + 2 μ σ d t d W + σ 2 ( d W ) 2 (2.7)
Substituting Equation (2.5) and (2.7) into the Taylor expansion taking into account the following conditions,
( d t ) 2 = 0
d t × d W = 0
( d W ) 2 = d t
We obtain the result (2.6). □
We can obtain the solution to the Geometric Brownian Motion SDE shown above by applying Itô’s lemma as follows;
d ( ln Y t ) = ( ln Y t ) ′ d Y t + 1 2 ( ln Y t ) ′ ′ ( d Y t ) 2 (2.8)
= d Y t Y t − 1 2 1 Y t 2 ( d Y t ) 2 (2.9)
where ( d Y t ) 2 is the quadratic variation of the SDE
( d Y t ) 2 = μ 2 Y t 2 ( d t ) 2 + 2 σ Y t 2 μ d W t d t + σ 2 ( Y t ) 2 ( d W t ) 2
Assuming;
( d t ) 2 = 0
d t ⋅ d W = 0
( d W ) 2 = d t
Thus ( d Y t ) 2 = σ 2 ( Y t ) 2 d t .
Substituting the value of d Y t and ( d Y t ) 2 in Equation (2.9) we obtain;
d ( ln Y t ) = ( μ − 1 2 σ 2 ) d t + σ d W t
Written in the integral form, this leads to ln Y t = Y 0 exp [ ( μ − 1 2 σ 2 ) t + σ W t ]
Thus this equation can also be written as;
Y t = Y t 0 e σ ( W t − W t 0 ) + ( μ − 1 2 σ 2 ) ( t − t 0 ) (2.10)
In this study we are looking at credit insurance in the case of asset-based lending where a borrower borrows money to buy a car. Thus the definition for credit insurance becomes; a financial tool for transferring credit risk of a credit from a financial institution to an insurance company. The financial institution has to pay a premium and the insurance company will pay a percentage of the outstanding balance of the loan plus interests if there is a default. In this case Credit insurance will pay the benefit only when the borrower defaults in their payments and the financial institution takes over or recover the underlying car (collateral) for that loan.
Therefore if we denote the loss reserve at time t as OCRt, X the random variable representing the losses, and F t the filtration of the delinquency index we can write
O C R t = g ( F t ) = E ( X | F t )
We can say that this OCRt exhibits markovian property. We can build it further in the following sections.
Assume that the delinquency index can be modeled as a geometric Brownian motion, we will have;
d Y t Y t = σ d W t + μ d t (2.11)
⇔ Y t = Y t 0 e σ ( W t − W t 0 ) + ( μ − 1 2 σ 2 ) ( t − t 0 ) (2.12)
E ℙ [ Y t | Y t 0 ] = Y t 0 e μ ( t − t 0 )
Consider cash bond Bt as the stochastic process that give the risk-free rate, that is:
d B t B t = r d t ⇔ B t = B t 0 e r ( t − t 0 ) (2.13)
Rememeber we need an asset that depends on the delinquency index whose present value is a martingale. Denote that asset as A(Yt) and we suppose that its linearly proportional to the delinquency index as follows;
A ( Y t ) = A t ∗ Y t
Let’s also denote D = (Dt) t ≥ 0 to be the stochastic process representing the present value of At:
D t = B t − 1 A ( Y t ) = B t − 1 A t Y t
when we apply ltô’s lemma to Dt we get:
d D t D t = σ d W t + ( μ − r ) d t (2.14)
Proof. Recall that d Y t = μ Y t d t + σ Y t d W t
d D t = d ( B t − 1 A t Y t )
B t = exp ( r t ) B t − 1 = exp ( − r t )
d B t = r exp ( r t ) d t
d B t − 1 = − r exp ( − r t ) d t
d D t = B t − 1 A t d Y t + B t − 1 Y t d A t + d B t − 1 Y t A t = B t − 1 A t ( μ Y t d t + σ Y t d W t ) + Y t A t ( − r B t − 1 ) = B t − 1 A t μ Y t d t + B t − 1 A t σ Y t d W t − Y t A t r B t − 1 d t (2.15)
But D t = B t − 1 A t Y t .
Thus Equation (2.15) becomes;
d D t D t = ( μ − r ) d t + σ d W t as above □
Based on the construction of asset portfolio strategy we were able to make change of probability measure. This implied the present value of the asset which depends on the delinquency index was a martingale. Therefore, we will make a change in probability measure from ℙ to ℚ on the process Dt in order to make Dt to be a martingale. Based on the Cameron-Martin-Girsanov theorem, for each probability measure ℚ equivalent to ℙ and if we have a previsible process γ = ( γ t ) then: W ˜ t = W t + ∫ 0 t γ s is also ℚ Brownian motion with d W ˜ t = d W t + γ t d t .
Substituting in (2.14) we have:
d D t D t = σ d W ˜ t + ( μ − r − γ t σ ) d t
for Dt be a martingale we need ( μ − r − γ t σ ) = 0 since drift equals 0. The solution of this equation is the Market price of risk;
γ t = μ − r σ (2.16)
But Dt is a martingale;
d D t D t = σ d W ˜ t (2.17)
Considering the concepts shown so far, the delinquency index under the probability measure ℚ is:
d Y t Y t = σ d W ˜ t + r d t
Proof. Recall
d Y t Y t = σ d W t + μ d t (2.18)
But d W t = d W ˜ t − γ t d t
and from γ t = μ − r σ
r = μ − σ γ t
Thus Equation (2.17) becomes
= σ ( d W ˜ t − γ t d t ) + μ d t = σ d W ˜ t − σ γ t d t + μ d t = σ d W ˜ t + ( μ − σ γ t ) d t = σ d W ˜ t + r d t
as above. □
Thus under the probability measure ℚ the drift of the delinquency index is the risk-free rate. Solving this stochastic differential equation, we can have estimate of the delinquency index;
Y t = Y t 0 e σ ( W ˜ t − W ˜ t 0 ) + ( r − 1 2 σ 2 ) ( t − t 0 ) (2.19)
This is an important result which will enable us forecast future losses of a credit insurance which pays f(T):
O C R t = E ℚ [ B u − 1 f ( T ) | F t 0 ] (2.20)
where:
OCRt is the reserve at time t given information of the filtration F t 0 .
Bt = Is cash bond which give the risk-free rate r.
T = t + u and u = T − t .
f(T) = Is the compensation by the insurance company. T is the random variable that represents the time to paying the sum insured. Remember, the credit insurance pays claims when the car is taken over. In the case the outstanding loan balance upon default is more than the value of the collateral, the insured lender will exercise the right to sell the collateral at the outstanding loan balance. This is achieved by the lender receiving from the insurer the difference of the outstanding loan and the market value of the car at the time of default. The contract is therefore settled by the difference.
In this case, the total cost of the policy holder will be the premium paid and the value of the collateral upon default. In addition to the time of valuation, t we must consider another random variable u representing the time to repossession of the car and thus T = t + u as denoted above.
The time to repossess a car is not certain, it can take more than a year. That’s why we use the random variable u. Moreover, the financial institutions which insure itself with the insurance company have delays in giving out the information of the delinquency index and this gives the reason for use of t0 which is about one month or two. Based on these reasons, we are going to forecast the future losses of credit insurance at time t, with the information given by the financial institution at time t0 with the filtration F t 0 . We can see that t 0 < t < T . Consider t0 to be independent of t, we can develop Equation (2.19). Consider f(T) to be the function that represents compensation by the insurance company (assuming that the car is recovered after R months defaultable period). It will be defined as a percentage of the outstanding balance at time of recovering the car if the delinquency index is greater than R. Where R is the delinquency index threshold. That is:
f ( T ) = { % C o v ∗ ( O B ) I T for Y t ≥ R 0 for Y t < 0 = % C o v ∗ ( O B ) I T ∗ I Y t ≥ R
%Cov = Is the percentage of the covered outstanding balance by the insurance company.
( O B ) I T = Is the outstanding balance of the credit at time T.R is the delinquency index threshold.
I Y t = Is an indicative random variable:
I Y t = { 1 if event Y t occurs 0 if event Y t does not occur
Then, we have;
O C R t = E ℚ [ B u − 1 f ( T ) | F t 0 ] = E ℚ [ B u − 1 ∗ % C o v ∗ ( O B ) I T ∗ I Y t ≥ R | F t 0 ]
Since Y t is modelled as Geometric Brownian motion, it conforms to a diffusion process, and it thus possess the Markovian property. Consequently we can reorganize the terms in the above equation as:
O C R t = E ℚ [ B u − 1 ∗ % C o v ∗ ( O B ) I T ∗ I Y t ≥ R | F t 0 ] = E ℚ [ B u − 1 ∗ % C o v ∗ ( O B ) I T ∗ I Y t ≥ R | Y t 0 ] (2.21)
O C R t = % C o v ∗ E ℚ [ B u − 1 ∗ ( O B ) I T | Y t 0 ] ∗ E ℚ [ I Y t ≥ R | Y t 0 ] = % C o v ∗ E ℚ [ B u − 1 ∗ ( O B ) I T ] ∗ ℚ [ Y t ≥ R | Y t 0 ] (2.22)
Since Y t is a Geometric Brownian Motion, we have;
Y t = Y t 0 e σ ( W ˜ t − W ˜ t 0 ) + ( r − 1 2 σ 2 ) ( t − t 0 ) = Y t 0 e σ t − t 0 ∗ ε + ( r − 1 2 σ 2 ) ( t − t 0 ) , with ε ~ N ( 0 , 1 )
Y t ≥ R ⇔ ln Y t = ln Y t 0 + σ t − t 0 ∗ ε + ( r − 1 2 σ 2 ) ( t − t 0 ) ≥ ln R
η = − ε ≤ ln Y t 0 R + ( r − 1 2 σ 2 ) ( t − t 0 ) σ t − t 0 , η ~ N ( 0 , 1 )
Taking the notation of the Black-Scholes we have;
d 2 = ln Y t 0 R + ( r − 1 2 σ 2 ) ( t − t 0 ) σ t − t 0
d 2 captures the idea of credit risk in the Merton Model. It denotes the probability of default. Formula (2.22) turns into:
O C R t = % C o v ∗ E ℚ [ B u − 1 ∗ ( O B ) I T ] ∗ ℚ [ η ≤ d 2 | Y t 0 ] = % C o v ∗ E ℚ [ B u − 1 ∗ ( O B ) I t + u ] ∗ Φ ( d 2 ) (2.23)
We thus have managed to obtain the formula to forecast loss reserves in credit insurance if we know just the percentage of outstanding balance covered by the insurer, the value of the delinquency index, a risk-free rate, and the distribution of the random variable u. We can write Equation (2.23) as:
O C R t = % C o v ∗ ( ∫ 0 n − t e − r u ( O B ) I t + u p ( u ) d u ) ∗ Φ ( d 2 ) (2.24)
where:
p(u) is the probability density function of u.
n is the loan term.
( O B ) I t + u is the outstanding balance of the loan at time t plus interest at the rate c:
( O B ) I t + u = ( O B ) I t ∗ ( 1 + t c ) u
We can simplify this calculation by taking u as a constant:
O C R t = % C o v ∗ e − r u ∗ ( O B ) I t + u ∗ Φ ( d 2 ) (2.25)
Since this formula was obtained under the probability measure ℚ we have a replicating portfolio P t , which matches the future losses of the insurance company. Thus:
P t = ϕ t A ( Y t ) + ψ t B t = ϕ t A t Y t + ψ t B t (2.26)
d P t = ϕ t A t d Y t + ψ t d B t (2.27)
where; ϕ t denotes number of units of A ( Y t ) .
ψ t denotes number of units of the cash bond. Let G t be the process representing the present value of the portfolio:
G t = B t − 1 P t = ϕ t D t + ψ t (2.28)
Considering the following equation and applying Itô’s lemma we have;
O C R t = h ( t , x ) = E ℚ [ B u − 1 ∗ % C o v ∗ ( O B ) I t + u ∗ I Y t ≥ R | Y t 0 = x ]
ϕ t A t = d d Y t ( % C o v ∗ e − r u ∗ ( O B ) I t + u ∗ Φ ( d 2 ) ) = % C o v ∗ e − r u ∗ ( O B ) I t + u ∗ d d Y t ( Φ ( d 2 ) ) = % C o v ∗ e − r u ∗ ( O B ) I t + u ∗ 1 2 π e d 2 2 2
ψ t can be obtained from (2.28). We’ve therefore found an alternative method for projecting losses in credit insurance products and a replicating portfolio to match them.