2. THE MODEL

We describe U_n, the company's discounted surplus at time n, in the form

where u = U₀ is the initial surplus and G_j is the gain of the company in the jth year discounted to the beginning of year j and v = (1 + i)⁻¹ is the discount factor, where i is the 1-year interest rate.

We suppose that

where X_n is the annual premium received in the nth year, Y_n is the annual claim paid in the nth year, and ρ₁ and ρ₂ are factors for discounting the quantities to the beginning of the year.

In Gerber [2,3] it is assumed that i = 0. It is immaterial when the payment is made within a year, so ρ₁ = ρ₂ = 1. In Yang and Zhang [6], it is assumed that the premium is received at the beginning of the year and the claim is paid at the end of the year, so ρ₁ = 1 and ρ₂ = v. Since we are considering the aggregate income and expense of an insurance company, it might be better to assume that the premium and the claim arrive at more than one time point throughout a year. If we assume that the premium and the claim are received uniformly and continuously within a year, ρ₁ = ρ₂ = ρ and

Let

denote the time of ruin. If U_t > 0 for all t, we write T = ∞. We are interested in finding the probability of ruin (i.e.,

).

3. MODELS WITH BOTH CLAIM AND PREMIUM ARE MOVING AVERAGE PROCESSES

In this section we consider the case where both premium and claim processes are moving average processes.

We assume that W₁, W₂, … are independent and identically distributed random variables having generic random variable

. Z₁, Z₂, … are independent and identically distributed random variables having generic random variable

. Also, {W_t} and {Z_t} are independent for all t. We have to specify the initial values w₀,w₋₁,…,w_−p+1,z₀,z₋₁,…, z_−q+1. Let

We assume the existence of an adjustment coefficient that is the positive solution, R, of

We need two assumptions on the α_i's and β_i's.

Condition (10) is assumed so that the derivative of the left-hand side of (9) with respect to R evaluated at R = 0 is negative; this guarantees that the positive solution R exists. Condition (11) is essentially equivalent to the net profit condition

Inequality (12) is a sufficient condition for the ruin probability being less than one. The converse is not true for i > 0 because the insurance company also receives interest as a source of income. For i = 0, conditions (10) and (11) are equivalent. In that case, U_t tends to infinity almost surely when t tends to infinity. For i > 0, we conjecture that U_t will converge to a finite random variable like the case considered in Boogaert, Haezendonck, and Delbaen [1].

Let

be the σ-field generated by {W_i; i = 1,2,…, n} and {Z_i; i = 1,2,…, n}; that is,

We have the following results.

Lemma 1:

where

is a

-supermartingale.

Proof:

The inequality in the second to last line follows from Jensen's inequality, as vⁿ⁻¹ ≤ 1 and −x^vn−1 is a convex function for x > 0. █

We need one more lemma before giving the main result.

Lemma 2: Let

Then

, is a

-supermartingale.

Proof:

Similarly,

From (1), (2), (17), and (18), we have

rearranging terms, we have

Therefore

which is a constant. Multiplying both sides of (20) by −R and taking the exponential, we have

Since exp{−Rs₀} is a constant and

, is a

-supermartingale by Lemma 1, then

, is a

-supermartingale. █

Now we state one of the main results in this article.

Theorem 1:

Proof: By Lemma 2,

, is a

-supermartingale. Let n₀ be a positive integer; then T ∧ n₀ is a bounded

-stopping time. By Doob's optional stopping theorem,

which implies

Let n₀ → ∞; by the monotone convergence theorem,

The ruin probability is a function of p + q + 1 variables: u,w₀,w₋₁,…, w_−p+1,z₀,z₋₁,…, z_−q+1. Note that equality holds in (15) when i = 0, so

, are martingales. Therefore, it is plausible that equality holds in (23), but we need a stronger condition to facilitate the proof.

Suppose W and Z have finite support; that is,

for some

.

Corollary 1: If (26) and (27) hold in addition to other conditions stated earlier and i = 0,

Proof: Now

, is a

-martingale. Let n₀ be a positive integer. Then for all

, similar to (24) we have

Let n₀ → ∞; the first term on the right-hand side of (29) converges as earlier by the monotone convergence theorem and the second term converges to zero by the dominated convergence theorem because it is bounded by the constant

We can find conditions other than conditions (26) and (27) such that the second term on the right-hand side of (29) is bounded. For example, if α₁,…, α_p ≥ 0 and β₁,…, β_q ≤ 0, X and Z are positive random variables with infinite support. The second term on the right-hand side of (29) is bounded by one. In such cases, Corollary 1 still holds because the dominated convergence theorem can be applied in the proof.

So far we have assumed that p and q are finite. To prove the result in the next section, we need to extend our result to the infinite number of lag terms case. In this case, the number of initial values is still finite because we cannot collect information for infinite periods in reality. We set W_t = Z_t = 0 for t less than the earliest time of the available information. Previous results still hold for this case if some further conditions are satisfied. First, we need the following conditions:

We also need (10) and (11) to be satisfied with p and q being replaced by infinity, and we need S_t to be convergent. Sufficient conditions for this are

and

Conditions (32) and (33) essentially imply (31).

4. GRANGER'S SIMPLE CAUSAL MODEL

Granger [4] proposed a notion of causality in time series models. The use of the causal model attempts to lower the variance of predicted values compared to the models without causality. This model has useful applications because in many real-world economic situations, some economic variables tend to affect the value of other variables. So using past data of several variables to predict the future value of a series might be more accurate than using past data of the variable that we want to predict. In this article we will adopt the bivariate simple causal model to describe the dynamics of the premium process and the claim process.

The premium process {X_t} and the claim process {Y_t} can be expressed as

respectively, where W and Z are defined as earlier.

We have specified past data of premium and claim for m years, (i.e., x₀,x₋₁,…, x_−m+1,y₀,y₋₁,…, y_−m+1).

Let ∪ be the time shift operator (i.e., ∪X_t = X_t−1) and let

Note that all constant terms of polynomials a(x),b(x),c(x), and d(x) are zero. Then (34) and (35) can be expressed in matrix form:

We give some definitions of causality for this model as stated in Granger [4].

Definition: For a bivariate time series model described by (37), we have the following:

1. Y_t causes X_t provided that b(∪) is not the zero polynomial.
2. X_t causes Y_t provided that c(∪) is not the zero polynomial.
3. There is a feedback relationship between X_t and Y_t if both X_t causes Y_t and Y_t causes X_t.

For the model in Yang and Zhang [6], we have a(∪) = b∪, b(∪) = 0, c(∪) = 0, and d(U) = aU, where 0 < a < 1 and 0 < b < 1. So X_t and Y_t do not have causal relationship in Granger's sense. Let

Since the constant term of μ(∪) is always one, A⁻¹ exists and

From (37) and (40) we have

We make some assumptions about the coefficients of a(x), b(x), c(x), and d(x):

Since μ(0) = 1 and, by assumption 1, μ(v) > 0 and μ(1) > 0, (44) and (45) are essentially the same as (10) and (11). Let a*(x) and b*(x) have coefficients a_k* and b_k*, defined as the expansion of partial fractions:

a_k* and b_k* converge because of assumption 1. Equation (43) becomes

We set W_t = Z_t = 0 for t ≤ −m; using (41) and (42), we obtain X_t = Y_t = 0 for t ≤ −m. We can uniquely determine w₀,w₋₁,…,w_−m+1,z₀,z₋₁,…, z_−m+1 from x₀,x₋₁,…, x_−m+1,y₀,y₋₁,…, y_−m+1 by (34) and (35).

From the expansion of partial fractions in (45) and (46), we can see that a_k* and b_k* tend to zero exponentially. So (31)–(33) are satisfied.

We can obtain results for ruin probabilities using the result in Section 3 by replacing α_k and β_k by a_k* and b_k*, respectively. We can also formulate the result in terms of a_k,b_k,c_k, and d_k.

We obtain the adjustment coefficient by (9) with

We also need to express S_t in terms of a_k,b_k,c_k, and d_k. Let X_t^d and Y_t^d be deterministic components of X_t and Y_t, respectively. From (34) and (35), we set a_k = b_k = c_k = d_k = 0 for k > m, then we have

Observe that

Let

From (51) and (52), we have

From (49), (50), (53), (57), and (58), we have

We obtain the desired expression for s₀ in terms of a_k, b_k, c_k, and d_k. Let

From the results of Section 3, we have

, is a

-supermartingale and

If (26) and (27) are satisfied and i = 0,

Note that the ruin probability is a function of initial surplus and past values of the premium and claim.

5. EXAMPLES

5.1. The Yang–Zhang Model

We consider the model in Yang and Zhang [6]. The model can be written as

where 0 < a < 1 and 0 < b < 1. Furthermore, ρ₁ = 1 and ρ₂ = v. We can easily check that assumption 1 in Section 4 holds because 0 < a < 1 and 0 < b < 1. Suppose that

; then both (44) and (45) hold.

We have α = 1/(1 − bv) and β = v/(1 − av); so from (9), (49), and (50) the adjustment coefficient R satisfies

We can obtain the bound for ruin probability, which is a function of u, x₀, and y₀. Since the definition of U_n in Yang and Zhang [6] is the accumulated value of surplus after time n, so our result gives the same expression as Yang and Zhang [6] if we have taken this into account.

The Yang–Zhang model can be thought of as a model for changing exposure of a company, where a proportion of customers in a year will stay to the next year and a random number of new customers will arrive.

5.2. Credibility Model

Next we consider a credibility model. For simplicity, let the annual claim Y_i be nonnegative, independent, and identically distributed with generic random variable Y. We assume that the company charges a credibility premium that is a combination of a manual rate P_d and the average of past n claims, for 0 < Z < 1 being the credibility factor:

We have a(∪) = c(∪) = d(∪) = 0 and b(∪) = (Z/n)(∪ + ··· + ∪ⁿ). Clearly, assumption 1 in Section 4 is satisfied. Let i = 0 such that ρ₁ = ρ₂ = 1 for simplicity. If

, then (45) and (46) hold.

From (9), (49), and (50), the adjustment coefficient R satisfies

Suppose R′ is the adjustment coefficient for the case in which a constant premium P_d is received each year and an annual claim amount follows Y_i. Then the adjustment coefficient R for the credibility model is

We also have

Since

which satisfies β₁,…, β_n ≤ 0, Corollary 1 holds and the ruin probability is a function of u,y₀,…, y_−n+1.