Worked Calculation of Finite Risk Economics

Why Finite Risk Economics Requires a Cash-Flow Schedule

A finite risk contract may combine a premium, claim reimbursements, experience-account credits, and repayments by the insurer. Each component has a different economic direction, and the components may occur at different times. A reliable comparison therefore begins with a dated cash-flow schedule, not with a single net figure.

The calculations in this tutorial value the insurer’s expected cash outflows under specified scenarios. They distinguish the present value of the insurer’s retained claims from its total outflow after contract premiums, credits, and repayments. The examples use hypothetical contracts and deterministic annual effective discount rates. They do not include expenses, taxes, counterparty default, or other cash flows unless stated.

Definition: Let $X_t$ be the insurer’s gross claims at time $t$, and let $C_t$ be claim reimbursements paid to the insurer by the reinsurer at that time. The insurer’s retained claims are $X_t-C_t$. Let $P_t$ be premiums paid by the insurer, $K_t$ be experience-account credits paid to the insurer, and $R_t$ be repayments made by the insurer. Premiums and repayments are outflows; claim reimbursements and account credits are inflows.

All cash flows must be measured consistently. A credit $K_t$ is an additional payment to the insurer, separate from claim reimbursement $C_t$. A repayment $R_t$ is an additional payment by the insurer. If a contract’s account balance is already included in a claim settlement or premium adjustment, it must not also be counted as a separate credit or repayment.

Present-Value Framework

With effective annual rate $i$, the discount factor is $v=1/(1+i)$. A cash flow at time $t$, measured in years, has present value equal to its amount multiplied by $v^t$. For a deterministic discount rate, expected present value is found by discounting each dated cash flow and taking its expectation. Because discounting is linear, this is equivalent to discounting the expected amount at each date.

\operatorname{EPV}(\text{net insurer outflow}) = \sum_t v^t E[P_t] +\sum_t v^t E[X_t-C_t] -\sum_t v^t E[K_t] +\sum_t v^t E[R_t].

The insurer’s expected present value of retained claims alone is

\operatorname{EPV}(\text{retained claims}) = \sum_t v^t E[X_t-C_t].

These are different comparisons. Retained claims measure the amount of gross claims left with the insurer after claim reimbursements. Net insurer outflow also includes premiums, account credits, and repayments. For context, the present value of gross claims without the contract is $\sum_t v^t E[X_t]$. It is useful to report these quantities separately rather than describe the total contract outflow as “retained loss.”

Derivation: In a scenario, the insurer’s net outflow at time $t$ is $P_t+X_t-C_t-K_t+R_t$. Its present value is the sum of each dated net outflow multiplied by $v^t$. Taking expectations and using linearity, $E[A+B]=E[A]+E[B]$, gives the EPV formula above. This works even when cash flows depend on the same loss scenario; independence among them is not required. The formula assumes the listed cash flows are complete and that the discount rate and payment dates are specified.

Build the Calculation in a Fixed Order

Write the contract cash flows.
For each scenario, record gross claims, claim reimbursements, account credits, and repayments separately. Record premiums at their payment dates.

Calculate expectations by date.
Multiply each scenario amount by its probability, then add the probability-weighted amounts for each cash-flow category.

Discount each date.
Use $v^t$ for a payment at time $t$, including premiums paid after time zero.

Report separate totals.
Show expected PV of retained claims and expected PV of total net outflow, with gross claims PV as a useful baseline when relevant.

Worked Calculations

Worked Example: One-Year Claims, a Credit, and a Later Repayment

A hypothetical insurer pays a premium of $70 thousand at time 0. At the end of year 1, there is a mild-loss outcome with probability 0.6000 and a severe-loss outcome with probability 0.4000. In the mild outcome, gross claims are $60 thousand, the reinsurer pays $50 thousand of claim reimbursement, and the insurer receives an experience-account credit of $20 thousand. In the severe outcome, gross claims are $180 thousand, the reinsurer pays $120 thousand, and there is no credit. The insurer repays $30 thousand at the end of year 2 only in the severe outcome. Assume an effective annual discount rate $i=5\%$. Calculate retained-claim EPV and total net-outflow EPV. All amounts are in thousands of dollars.

Method. First find each expected cash flow by date. Then discount the retained claims, credits, and repayments at their respective times. The premium is paid at time 0 and is not discounted.

Expected gross claims at time 1 are $0.6000(60)+0.4000(180)=36+72=108$. Expected claim reimbursements are $0.6000(50)+0.4000(120)=30+48=78$. Thus expected retained claims at time 1 are $108-78=30$. Expected account credits are $0.6000(20)+0.4000(0)=12$. Expected repayments at time 2 are $0.6000(0)+0.4000(30)=12$.

The discount factors are $v=1/1.05=0.952381$ and $v^2=1/(1.05)^2=0.907029$. Therefore, retained-claim EPV is $30(0.952381)=28.5714$, or $28.57 thousand.

The total net-outflow EPV is $70+28.5714-12(0.952381)+12(0.907029)$. The credit has present value $11.4286$, and the repayment has present value $10.8844$. The total is $70+28.5714-11.4286+10.8844=98.0272$, or $98.03 thousand.

As a baseline, gross-claim EPV without the contract is $108(0.952381)=102.8571$, or $102.86 thousand. Check: the difference between total net-outflow EPV and gross-claim EPV is $98.0272-102.8571=-4.8299$, consistent with the combined net effect of premiums, reimbursements, credits, and repayments. A common mistake is to add the credit as an outflow or discount the time-2 repayment using $v$. The signed, date-specific calculation avoids both errors.

Worked Example: A Two-Year Contract with Scenario-Dependent Account Terms

A hypothetical two-year contract has an upfront premium of $100 thousand. At the end of year 1, the reinsurer reimburses some claims. At the end of year 2, the contract may pay an account credit; a repayment, if any, is due at the end of year 3. There are two equally likely outcomes. In outcome A, year-1 gross claims are $40 thousand and reimbursement is $30 thousand; year-2 claims are $60 thousand and reimbursement is $40 thousand; the year-2 credit is $20 thousand and there is no repayment. In outcome B, the corresponding year-1 amounts are $100 thousand and $70 thousand; the year-2 amounts are $120 thousand and $80 thousand; the credit is $10 thousand and the year-3 repayment is $16 thousand. Assume $i=4\%$. Calculate retained-claim EPV and total net-outflow EPV. All amounts are in thousands of dollars.

Method. Calculate the expected retained claims separately for each year. Because the account credit and repayment have different payment dates, discount them separately even though they arise from the same contract.

The expected year-1 gross claims are $0.5000(40)+0.5000(100)=70$, and expected reimbursement is $0.5000(30)+0.5000(70)=50$. Expected retained claims at year 1 are $70-50=20$. At year 2, expected gross claims are $0.5000(60)+0.5000(120)=90$, and expected reimbursement is $0.5000(40)+0.5000(80)=60$. Expected retained claims at year 2 are $90-60=30$.

Expected account credit at year 2 is $0.5000(20)+0.5000(10)=15$. Expected repayment at year 3 is $0.5000(0)+0.5000(16)=8$. The discount factors are $v=1/1.04=0.961538$, $v^2=1/(1.04)^2=0.924556$, and $v^3=1/(1.04)^3=0.888996$.

Retained-claim EPV is $20(0.961538)+30(0.924556)=19.2308+27.7367=46.9675$, or $46.97 thousand. The total net-outflow EPV is $100+46.9675-15(0.924556)+8(0.888996)$. The credit EPV is $13.8683$, and the repayment EPV is $7.1120$. Thus total net outflow is $100+46.9675-13.8683+7.1120=140.2112$, or $140.21 thousand.

Check: the time-2 credit reduces total outflow, while the time-3 repayment increases it. Discounting the repayment at year 2 instead would overstate its present value. The expected amounts are valid even though claims, credits, and repayments are linked by scenario; linearity of expectation does not require independence.

Worked Example: Installment Premiums and Three Loss Outcomes

A hypothetical insurer pays premiums of $40 thousand at time 0 and $40 thousand at the end of year 1. At the end of year 1, gross claims and reimbursements are respectively $30 thousand and $25 thousand with probability 0.5000, $80 thousand and $60 thousand with probability 0.3000, and $150 thousand and $100 thousand with probability 0.2000. At the end of year 2, the respective gross claims and reimbursements in those same outcomes are $50 thousand and $40 thousand, $90 thousand and $70 thousand, and $170 thousand and $110 thousand. The year-2 account credit is respectively $18 thousand, $8 thousand, and zero; the year-2 repayment is respectively zero, zero, and $25 thousand. Assume $i=6\%$. Calculate the EPV of retained claims and total net outflow. All amounts are in thousands of dollars.

Method. Preserve the three outcome probabilities when finding each expectation, then discount the year-1 and year-2 flows at different factors. Include the second premium at year 1.

Expected year-1 gross claims are $0.5000(30)+0.3000(80)+0.2000(150)=15+24+30=69$. Expected reimbursement is $0.5000(25)+0.3000(60)+0.2000(100)=12.5+18+20=50.5$. Expected retained claims at year 1 are $69-50.5=18.5$.

Expected year-2 gross claims are $0.5000(50)+0.3000(90)+0.2000(170)=25+27+34=86$. Expected reimbursement is $0.5000(40)+0.3000(70)+0.2000(110)=20+21+22=63$, so expected retained claims at year 2 are $86-63=23$. Expected account credit is $0.5000(18)+0.3000(8)+0.2000(0)=9+2.4=11.4$. Expected repayment is $0.5000(0)+0.3000(0)+0.2000(25)=5$.

The discount factors are $v=1/1.06=0.943396$ and $v^2=1/(1.06)^2=0.889996$. Retained-claim EPV is $18.5(0.943396)+23(0.889996)=17.4528+20.4699=37.9227$, or $37.92 thousand.

Premium EPV is $40+40(0.943396)=77.7358$. Credit EPV is $11.4(0.889996)=10.1459$, and repayment EPV is $5(0.889996)=4.4500$. Total net-outflow EPV is $77.7358+37.9227-10.1459+4.4500=109.9626$, or $109.96 thousand.

Gross-claim EPV without the contract is $69(0.943396)+86(0.889996)=65.0943+76.5397=141.6340$, or $141.63 thousand. Check: the net-outflow total includes both premiums, and the second premium is discounted because it is paid at year 1. Omitting or treating that installment as a time-0 payment would change the comparison.

Common Errors and Interpretation

A frequent error is to combine all expected amounts before assigning payment dates. In the first worked example, the expected retained claims are $30 thousand at year 1, the expected credit is $12 thousand at year 1, and the expected repayment is $12 thousand at year 2. The correct net outflow is $70+v(30-12)+v^2(12)=98.03$ thousand. If all three post-premium amounts are incorrectly treated as outflows and discounted at year 1, the calculation becomes $70+v(30+12+12)=121.43$ thousand. That result both reverses the credit’s sign and uses the wrong date for the repayment.

Other checks help catch errors before interpreting the result:

Check the sign: premiums and repayments increase insurer outflow; claim reimbursements and account credits reduce it.
Check the date: a time-0 premium is not discounted, while a time-2 payment is multiplied by $v^2$.
Check the category: retained claims are $X_t-C_t$, not $X_t-C_t-K_t+R_t$. Credits and repayments belong in total net outflow, not in the claim measure.
Check the units: if amounts are stated in thousands, keep all intermediate amounts in thousands and label the final result accordingly.
Check what the comparison means: a lower total outflow than gross-claim EPV is a cash-flow comparison under the stated assumptions. It does not, by itself, establish any broader conclusion about the contract.

Key takeaway: A finite risk calculation is a dated, signed cash-flow valuation. Compute retained claims from gross claims less claim reimbursements; then add premiums and repayments and subtract experience-account credits, discounting each item at its own payment date.

Check Your Understanding

For each problem, calculate the requested expected present value. Unless noted otherwise, amounts are in thousands of dollars, the annual effective discount rate is deterministic, and there are no cash flows beyond those stated.

A contract has a $25 premium at time 0. At year 1, expected gross claims are $40, expected reimbursement is $25, and an account credit of $4 is paid with certainty. A repayment of $3 is due at year 2 with certainty. If $i=5\%$, find the retained-claim EPV and total net-outflow EPV. (Answer: retained-claim EPV $14.29 thousand; total net-outflow EPV $38.20 thousand.)
A contract has a $60 premium at time 0. At year 1, two equally likely outcomes have gross claims of $40 and $100, with reimbursements of $30 and $70, respectively. The account credit at year 1 is $10 in the first outcome and $2 in the second. A repayment of $8 is due at year 2 only in the second outcome. Find total net-outflow EPV at $i=5\%$. (Answer: $76.96 thousand.)
A contract charges premiums of $30 at time 0 and $30 at year 1. Expected retained claims are $12 at year 1 and $20 at year 2. An expected account credit of $5 is paid at year 2, and an expected repayment of $6 is paid at year 3. Find total net-outflow EPV at $i=4\%$. (Answer: $89.59 thousand.)
A contract has a $50 premium at time 0. At year 1, a severe outcome has probability 0.2500 and produces gross claims of $80 and reimbursement of $50; otherwise, gross claims are $20 and reimbursement is $10. An account credit of $5 is paid at year 1 only in the non-severe outcome. A $12 repayment is due at year 2 only in the severe outcome. At $i=10\%$, calculate retained-claim EPV and total net-outflow EPV. (Answer: retained-claim EPV $13.64 thousand; total net-outflow EPV $62.71 thousand.)

What You'll Learn