How the Premium Rate Affects Ruin

Why the Premium Rate Matters

An insurer receives premium income over time and pays claims when they occur. If the claim process and initial capital are held fixed, increasing the premium rate adds more income to the surplus path. The key question is not whether this makes each possible path safer—it does—but what that implies for the probability of ruin, and whether the probability must fall strictly.

The comparison is meaningful only when the assumptions are controlled. In this tutorial, the claim arrivals and claim amounts are unchanged when the premium rate changes. Premiums are received continuously at a constant rate, there is no interest on surplus, and the insurer starts with the same nonnegative initial capital. These assumptions isolate the effect of the premium rate.

Definition: Let $u\geq 0$ be initial surplus, $c\geq 0$ the premium rate per unit time, and $S(t)$ the aggregate claims paid by time $t$. The surplus at time $t$ is $$ U_c(t)=u+ct-S(t). $$ Ruin occurs if the surplus is negative at some time. The ultimate ruin probability is $\psi_c(u)=P(\text{there is a }t\geq 0\text{ such that }U_c(t)<0)$. For a fixed horizon $T$, the finite-horizon ruin probability is $\psi_c(u,T)=P(\text{there is a }t\in[0,T]\text{ such that }U_c(t)<0)$.

Here $S(t)$ is the same process under every premium rate being compared. In particular, changing $c$ does not change the number, timing, or sizes of claims. A surplus of exactly zero is not negative, so it is not ruin under this convention. Whether a future claim then causes ruin depends on the surplus available when that claim occurs.

The Pathwise Comparison

Take two premium rates $c_2>c_1$, and construct both surplus processes using exactly the same initial capital and claim path. At every time $t$,

U_{c_2}(t)-U_{c_1}(t)=(c_2-c_1)t\geq 0.

Thus the higher-rate surplus is never below the lower-rate surplus. If the higher-rate process is negative at a time $t$, then the lower-rate process must also be negative at that time. Equivalently, for each fixed claim path, ruin under $c_2$ implies ruin under $c_1$. Taking probabilities preserves this event inclusion.

Theorem: With the same initial capital and the same claim process, if $c_2\geq c_1$, then $$ \psi_{c_2}(u,T)\leq\psi_{c_1}(u,T) \quad\text{and}\quad \psi_{c_2}(u)\leq\psi_{c_1}(u). $$ The result applies to any fixed claim process for which the surplus is defined; it does not require a particular claim-size distribution. It does not say that the decrease must be strict.

Fix a claim path.
Use the same arrivals and claim amounts for both premium rates. This is a pathwise comparison, not a comparison of two unrelated claim samples.

Compare surplus at time $t$.
The higher-rate process exceeds the lower-rate process by $(c_2-c_1)t$, which is nonnegative for all $t\geq 0$.

Compare ruin events.
If the higher-rate surplus is negative at any time, the lower-rate surplus is negative at that same time. The higher-rate ruin event is therefore a subset of the lower-rate ruin event.

Take probabilities.
Subset inclusion gives a weak inequality for both finite-horizon and ultimate ruin probabilities.

The comparison is weak because the added premium may not change whether a path ruins. For example, a path may avoid ruin under both rates, or a claim may be so large that it causes ruin under both. A higher rate can lower ruin probability without eliminating ruin on every path.

Following the Same Claims Under Two Rates

Worked Example: A Higher Rate Prevents Ruin on a Specified Path

A hypothetical insurer starts with $u=\$10$, receives premiums continuously, and earns no interest. Its fixed claim path has a claim of $\$8$ at time $1.5$ years and a claim of $\$12$ at time $3$ years, with no other claims. Compare $c_1=\$2$ per year and $c_2=\$4$ per year. Determine whether ruin occurs under each rate.

Method. Calculate surplus immediately after each claim; between claims, surplus increases at the premium rate. The claim path is deterministic, so there is no probability calculation in this example.

At $t=1.5$, the surplus just before the first claim is $10+1.5c$. Under $c_1=2$, this is $10+1.5(2)=13$, so surplus just after the $\$8$ claim is $13-8=\$5$. Under $c_2=4$, it is $10+1.5(4)=16$ before the claim and $16-8=\$8$ after it. Neither path has yet ruined.

At $t=3$, the lower-rate path has received another $1.5(2)=\$3$ in premiums. Its surplus just before the second claim is $5+3=\$8$, and just after the claim it is $8-12=-\$4$. Ruin occurs at the second claim.

The higher-rate path receives another $1.5(4)=\$6$. Its surplus just before the second claim is $8+6=\$14$, and just after the claim it is $14-12=\$2$. It does not ruin on this path.

Check. At $t=3$, the two surpluses differ by $(4-2)3=\$6$: $\$2-(-\$4)=\$6$. The calculation agrees with the pathwise comparison. The common mistake is to compare only the claim amounts; the surplus available when each claim arrives includes premiums earned since the previous claim.

Calculating a Finite-Horizon Probability

The pathwise result orders probabilities, but it does not always give their numerical values. A simple model with at most one claim illustrates how the arrival time affects the calculation. Suppose there is no claim with probability $1-q$; with probability $q$, exactly one claim occurs, and its time is uniform on $[0,T]$. Let its fixed amount be $x$. The claim process, including $q$, $T$, and $x$, is held unchanged when $c$ changes.

On a path with a claim at time $t$, the surplus immediately after the claim is $u+ct-x$. Thus the claim causes ruin exactly when $t<(x-u)/c$, for $c>0$. Intersect this interval with $[0,T]$ to obtain the probability. The single time $t=(x-u)/c$, if it lies in the interval, has probability zero under a continuous uniform distribution, so the strict inequality does not change the probability calculation.

Worked Example: One Possible Claim With a Uniform Arrival Time

A hypothetical insurer has initial surplus $u=\$5$, no interest, and a two-year horizon. With probability $q=0.6$, exactly one claim of $\$9$ occurs at a time uniform on $[0,2]$; otherwise there is no claim. Compare rates $c_1=\$1$ and $c_2=\$3$ per year. Find the finite-horizon ruin probability under each rate.

Method. Condition on whether the claim occurs, then find the fraction of possible claim times that cause negative surplus. If there is no claim, surplus only increases and ruin is impossible.

Under $c_1=1$, a claim at time $t$ leaves surplus $5+t-9=t-4$. For every $t\in[0,2]$, this is negative. Conditional on a claim, the ruin probability is $1$, so the unconditional probability is $0.6(1)=0.6000$.

Under $c_2=3$, the surplus after a claim at time $t$ is $5+3t-9=3t-4$. It is negative when $t<4/3$. The fraction of the two-year interval satisfying this is $(4/3)/2=2/3$. Hence the unconditional ruin probability is $0.6(2/3)=0.4000$.

Check. The higher premium rate reduces the probability from $0.6000$ to $0.4000$, consistent with the theorem. The common mistake is to use surplus at the end of year two, $5+3(2)-9=\$2$, and conclude that no ruin can occur. The claim may arrive before the insurer has earned enough premium; the relevant surplus is the surplus at the claim time.

A Discrete-Time Comparison

The same logic applies when a model records surplus at discrete claim or accounting times. For the next example, premiums are received at a constant rate throughout each year, and claims, if any, are paid at each year-end. Ruin is checked immediately after those payments. This timing convention is part of the model; a different payment schedule would require different surplus calculations.

Worked Example: Two Annual Claim Opportunities

A hypothetical insurer begins with $u=\$3$, earns no interest, and has two year-end claim opportunities. In each year, independently, a claim of $\$8$ occurs with probability $0.25$, and no claim occurs with probability $0.75$. Compare annual premium rates $c_1=\$2$ and $c_2=\$5$. Find the probability of ruin by the end of year two.

Method. Enumerate the four claim patterns. Each year’s premium income is received before that year-end claim. The probabilities of no claim and claim in a year are $0.75$ and $0.25$.

Under $c_1=2$, if a claim occurs in year one, surplus after that claim is $3+2-8=-\$3$, so ruin occurs. If there is no first-year claim, surplus is $3+2=\$5$. A claim in year two then leaves $5+2-8=-\$1$, also causing ruin. Thus ruin occurs whenever at least one of the two claims occurs. The probability is $1-(0.75)^2=1-0.5625=0.4375$.

Under $c_2=5$, a year-one claim leaves $3+5-8=\$0$, which is not ruin. If year one has no claim, surplus at the end of that year is $3+5=\$8$. A year-two claim then leaves $8+5-8=\$5$, also not ruin. If both years have claims, the surplus after the second is $3+10-16=-\$3$, so that pattern does cause ruin. Its probability is $0.25(0.25)=0.0625$.

Check. The ruin probabilities are $0.4375$ at $c_1$ and $0.0625$ at $c_2$. Under the higher rate, only the pattern with claims in both years ruins; under the lower rate, every pattern with at least one claim ruins. The common mistake is to treat a surplus of exactly zero after the first claim as ruin. Under the stated definition, ruin requires negative surplus.

What the Comparison Does—and Does Not—Establish

The pathwise argument is stronger than a comparison of average surpluses. It says that, for each fixed claim path, the higher-rate surplus is at least as large at every time. Consequently, raising the rate cannot increase ruin probability under the stated assumptions, whether ruin is assessed through a fixed horizon or over an unlimited horizon.

It does not give the size of the probability change by itself. That requires information about the claim process and the way its paths interact with the surplus. Nor does the result say the higher rate will strictly reduce ruin probability. If a claim path avoids ruin at both rates, or ruins at both rates, its ruin outcome is unchanged. A strict decrease in the overall probability requires some probability of paths that ruin at the lower rate but not at the higher rate.

The fixed-process assumption matters. If a higher premium changes policyholder behavior, the number or size of claims, or the timing of claims, then the claim process is no longer held fixed. The pathwise comparison alone does not cover that situation. Likewise, if premium income is not constant over time, the added income must be compared at each time rather than represented by the simple term $ct$.

A frequent exam trap is to confuse the premium rate with the total premium earned by the time horizon. Ruin is checked along the surplus path, not only at the horizon. A positive terminal surplus does not rule out an earlier negative surplus after a claim. A useful calculation sequence is to state the claim path, calculate premiums accumulated by each claim time, and then check surplus immediately after each claim.

Key takeaway: With initial capital and the claim process fixed, increasing the constant premium rate raises surplus by $(c_2-c_1)t$ at time $t$. This gives a pathwise inclusion of ruin events and therefore a nonincreasing finite-horizon and ultimate ruin probability. The decrease may be zero, and a numerical probability still depends on the claim process.

Check Your Understanding

Use the stated surplus convention: ruin occurs only when surplus is negative, and premiums accumulate continuously unless a question specifies year-end accounting.

A hypothetical insurer has $u=\$6$, no interest, and claims of $\$5$ at $t=0.5$ and $\$10$ at $t=2$, with no other claims. What is the minimum constant premium rate that avoids ruin on this path? (Answer: $\$4.50$ per year.)
With probability $0.8$, a hypothetical insurer has exactly one claim of $\$10$ at a time uniform on $[0,3]$; otherwise it has no claim. Initial surplus is $\$4$, there is no interest, and the claim process is unchanged. Find the ruin probability for $c=\$2$ and $c=\$4$ per year. (Answer: $0.8000$ at $c=\$2$, and $0.4000$ at $c=\$4$.)
A two-year hypothetical model starts at $u=\$2$. At each year-end, independently, a $\$6$ claim occurs with probability $0.2$. Premiums are received during each year, with no interest. Find the probability of ruin by year two for $c=\$3$ and $c=\$5$ per year. (Answer: $0.2000$ at $c=\$3$, and $0.0000$ at $c=\$5$.)
For the same claim path and initial surplus, rates are $c_1=\$1$ and $c_2=\$3$ per year. By how much does the higher-rate surplus exceed the lower-rate surplus at $t=2.5$ years? (Answer: $\$5.00$.)
Explain why a path with a positive surplus at the horizon might still have ruined earlier. (Answer: A claim before the horizon can reduce surplus below zero; later premium income cannot undo an earlier ruin event.)

What You'll Learn