Updating Predictive Claims After New Experience

Why New Experience Changes a Forecast

A claim forecast is conditional on the information available when it is made. If an insurer first observes a risk for several periods and then receives another period of claim experience, that new evidence can change the estimated claim rate. The appropriate comparison is between the forecast based on the earlier information and a revised forecast based on all observed information, using the same future exposure in both calculations.

This tutorial uses a Poisson model for claim counts and a Gamma prior for the unknown claim rate. The focus is the update after new count and exposure data arrive, and the resulting change in the expected future count. A forecast is an expectation, not an observed count: it is not added to the data when the posterior is updated.

Definition: Let \(\lambda\) be an unknown claim frequency per unit of exposure. Under the Gamma shape-rate convention, a prior \(\lambda\sim\operatorname{Gamma}(\alpha,\beta)\) has density proportional to \(\lambda^{\alpha-1}e^{-\beta\lambda}\), with shape \(\alpha>0\), rate \(\beta>0\), and mean \(\alpha/\beta\). If a count \(N=n\) is observed over exposure \(e\), and \(N\mid\lambda\sim\operatorname{Poisson}(e\lambda)\), the posterior is \(\operatorname{Gamma}(\alpha+n,\beta+e)\).

Here exposure might be policy-years, vehicle-years, or another consistently defined measure. The claim rate \(\lambda\) is measured per that same unit. The Gamma parameter \(\beta\) is a rate parameter, not a scale parameter: the prior mean is \(\alpha/\beta\), not \(\alpha\beta\). The prior mean and the exposure units must therefore be interpreted together.

Updating Once More Experience Arrives

Suppose initial experience consists of \(n_0\) claims over exposure \(e_0\), and additional experience later consists of \(n_1\) claims over exposure \(e_1\). Starting with \(\operatorname{Gamma}(\alpha_0,\beta_0)\), the posterior after the initial experience is

\lambda\mid\text{initial data}\sim \operatorname{Gamma}(\alpha_0+n_0,\ \beta_0+e_0).

After the additional experience, update that posterior again:

\lambda\mid\text{all observed data}\sim \operatorname{Gamma}(\alpha_0+n_0+n_1,\ \beta_0+e_0+e_1).

The same result follows by combining all the observed counts and exposures at once. This makes the update order-independent, provided that the data follow the same claim-rate model and use compatible exposure units.

Forecast rule: For future exposure \(H\), let \(N_H\mid\lambda\sim\operatorname{Poisson}(H\lambda)\). Given observed data with posterior \(\operatorname{Gamma}(\alpha^*,\beta^*)\), the expected future count is \(E[N_H\mid\text{data}]=H\alpha^*/\beta^*\). To compare forecasts before and after new experience, use the same \(H\) in both calculations.

This forecast rule concerns the expected count. It does not assert that the next observed count will equal that expectation, nor does it incorporate a change in underlying rate over time. It assumes that the future exposure is known and that the same risk-level rate continues to apply.

Why the Update Formula Holds

The conjugate update follows by multiplying the prior density by the likelihood of the observed count. For a Poisson count \(n\) over exposure \(e\), the likelihood as a function of \(\lambda\) is

P(N=n\mid\lambda) =\frac{(e\lambda)^n e^{-e\lambda}}{n!} \ \propto\ \lambda^n e^{-e\lambda}.

The Gamma prior density is proportional to \(\lambda^{\alpha-1}e^{-\beta\lambda}\). Multiplying the two kernels gives

\lambda^{\alpha-1}e^{-\beta\lambda} \lambda^n e^{-e\lambda} = \lambda^{\alpha+n-1}e^{-(\beta+e)\lambda}.

This is the kernel of a Gamma density with shape \(\alpha+n\) and rate \(\beta+e\). The omitted factors do not depend on \(\lambda\), so they are absorbed into the posterior normalizing constant. In particular, the count adds to the shape and exposure adds to the rate.

For the forecast, condition first on the unknown rate. The conditional Poisson mean is \(E[N_H\mid\lambda]=H\lambda\). Averaging over the posterior gives

E[N_H\mid\text{data}] =E\!\left[E[N_H\mid\lambda,\text{data}]\mid\text{data}\right] =E[H\lambda\mid\text{data}] =H\frac{\alpha^*}{\beta^*}.

This argument establishes the expected count without requiring the unknown rate to be replaced by a fixed value. It applies when the conditional count model is Poisson, exposure is known, and the future rate is represented by the same posterior rate parameter.

Worked Comparisons

Worked Example: A Higher Revised Forecast

A hypothetical portfolio has a prior claim rate \(\lambda\sim\operatorname{Gamma}(3,2)\), with rate measured per vehicle-year. Initially, 8 claims are observed over 4 vehicle-years. Later, another 5 claims are observed over 2 vehicle-years. Compare the expected count for the next 6 vehicle-years using the information available before and after the later experience.

Method. Update the Gamma shape by adding observed claims and the rate parameter by adding exposure. Then multiply each posterior mean rate by the same future exposure, \(H=6\).

Before the additional experience: the initial posterior parameters are \(\alpha^*=3+8=11\) and \(\beta^*=2+4=6\). Its mean rate is \(11/6=1.8333\) claims per vehicle-year. The forecast for 6 vehicle-years is \(6(11/6)=11\) claims.

After the additional experience: the updated shape is \(11+5=16\), and the updated rate parameter is \(6+2=8\). The posterior mean rate is \(16/8=2\) claims per vehicle-year. The revised forecast is \(6(2)=12\) claims.

Check and common mistake avoided. The later experience raises the expected rate from \(1.8333\) to \(2\), so the forecast rises from 11 to 12. The 11-claim forecast is not an observation and is not added to the posterior shape; only the 5 newly observed claims are added.

Worked Example: Keeping Fractional Exposure in Years

A hypothetical property segment has prior \(\lambda\sim\operatorname{Gamma}(4,5)\), where \(\lambda\) is claims per policy-year. Initial experience is 9 claims over 10 policy-years. Subsequent experience is 2 claims over 18 policy-months. Compare expected counts for a future 3 policy-years.

Method. Convert all exposure to policy-years before updating. The new exposure is \(18/12=1.5\) policy-years; it is not 18 policy-years.

Before the subsequent experience: the posterior is \(\operatorname{Gamma}(4+9,5+10)=\operatorname{Gamma}(13,15)\). Its mean rate is \(13/15=0.8667\) claims per policy-year. The 3-year forecast is \(3(13/15)=39/15=2.6\) claims.

After the subsequent experience: add 2 claims and 1.5 policy-years. The posterior is \(\operatorname{Gamma}(15,16.5)\), with mean rate \(15/16.5=10/11=0.9091\). The revised 3-year forecast is \(3(10/11)=30/11=2.7273\) claims.

Check and common mistake avoided. Because the new count of 2 over 1.5 policy-years is above the current posterior mean rate, the forecast increases. Treating 18 policy-months as 18 policy-years would incorrectly add too much exposure and depress the revised rate.

Worked Example: Updating After a Zero-Claim Period

A hypothetical commercial account has prior \(\lambda\sim\operatorname{Gamma}(2,1)\), with rate per account-year. Its first observed period has 3 claims over 2 account-years. The next period has zero claims over 1 account-year. Compare expected counts for the next 8 account-years before and after the zero-claim period.

Method. Apply the same conjugate update to each period. A zero count adds nothing to the shape, but its positive exposure still adds to the rate parameter.

Before the zero-claim period: after the initial data, the posterior is \(\operatorname{Gamma}(2+3,1+2)=\operatorname{Gamma}(5,3)\). Its mean rate is \(5/3\) claims per account-year. The forecast for 8 account-years is \(8(5/3)=40/3=13.3333\) claims.

After the zero-claim period: the posterior is \(\operatorname{Gamma}(5+0,3+1)=\operatorname{Gamma}(5,4)\). Its mean rate is \(5/4=1.25\). The revised forecast is \(8(5/4)=10\) claims.

Check and common mistake avoided. The zero-claim observation reduces the posterior mean because it contributes exposure without adding claims. Leaving the posterior unchanged would ignore evidence in the observed claim-free account-year.

Conditions and Practical Checks

These updates are appropriate only when the model reasonably represents the experience being used. In particular, the count for each period is conditionally Poisson given the rate, and periods contribute conditionally independent counts given that rate. The claim-rate parameter is treated as stable across the periods being combined, and exposure is measured consistently. If rates have changed because of a material coverage, mix, or operational change, mechanically pooling the periods can misstate the current rate.

Before calculating, check the following:

Confirm the likelihood. The update uses a claim count with Poisson mean equal to rate times exposure. It is not the update for a binary claim/no-claim observation per policy.
Align exposure units. A rate per policy-year requires exposure in policy-years. Convert months, days, or other units before adding exposures.
Separate observed from forecast counts. Only realized counts update the posterior. A forecast for a future period remains a forecast unless that period's actual experience is subsequently observed.
Compare like with like. Use the same future exposure when measuring how the new data changed the forecast. If the future exposure itself changes, the difference combines a rate update with a change in volume.
Keep the Gamma convention straight. Under the shape-rate convention, exposure is added to \(\beta\), and the mean is \(\alpha/\beta\). Using a shape-scale prior formula without converting conventions gives the wrong result.

Key takeaway: With a Gamma shape-rate prior and a Poisson count likelihood, observed claims add to the shape and observed exposure adds to the rate. Recalculate the posterior mean rate after each new period, then multiply by the specified future exposure to obtain the revised expected count.

Check Your Understanding

For each problem, update the posterior using observed data only, then calculate the expected count for the stated future exposure.

A hypothetical risk has prior \(\operatorname{Gamma}(2,3)\) per exposure unit. It first has 4 claims over 2 units, then 1 claim over 1 unit. Compare the expected counts for 5 future units before and after the second period. (Answer: before, 6 claims; after, \(35/6=5.8333\) claims.)
A hypothetical class has prior \(\operatorname{Gamma}(5,10)\) per policy-year. It has 7 claims over 5 policy-years, followed by 0 claims over 2 policy-years. Compare forecasts for 4 future policy-years. (Answer: before, 3.2 claims; after, \(48/17=2.8235\) claims.)
A hypothetical book has prior \(\operatorname{Gamma}(3,2)\) per exposure unit. Initial experience is 6 claims over 3 units, and new experience is 4 claims over 1 unit. Compare forecasts for 2.4 future units. (Answer: before, 4.32 claims; after, 5.2 claims.)
A hypothetical portfolio has prior \(\operatorname{Gamma}(2,4)\) per policy-year. It observes 12 claims over 24 policy-months, then 3 claims over 6 policy-months. Compare forecasts for 1.5 future policy-years. (Answer: before, 3.5 claims; after, \(51/13=3.9231\) claims.)
A hypothetical risk has prior \(\operatorname{Gamma}(1,2)\) per exposure unit. It observes 2 claims over 1 unit, then 3 claims over 2 units, then 0 claims over 0.5 unit. Compare forecasts for 2 future units immediately after the first period and after all three periods. (Answer: after the first period, 2 claims; after all three periods, \(24/11=2.1818\) claims.)

What You'll Learn