Editing Openai/695d4253-5060-8005-95db-586e777927f4 (section)

=== Assistant:  ===
==== Prospective experience rating ====
* Premium is set in advance for the rating period using expected claims (often credibility-weighted experience, trend, benefit changes, etc.).
* Carrier retains favorable results (profit) and bears unfavorable results (loss) during the period.
* Simpler to administer; aligns premium with expected cost, but the customer may feel they “overpaid” if experience is good.

Retrospective experience rating
* Premium is adjusted after the fact (or a refund/dividend is paid) based on actual experience (often with corridors, minimum/maximum, and pooling).
* Shares results between carrier and customer; reduces customer’s cost when experience is favorable but also can introduce settlement timing/complexity.
* If not symmetric (e.g., refunds on gains but no surcharge on losses), it is effectively a one-sided option granted to the customer.

==== Note: The screenshots do not include the actual net premium and developed claims by year, so I can’t produce numeric totals. Below is the exact calculation method (what a grader expects), which you can apply once the 20X1–20X3 PyP_yPy and CyC_yCy are provided. ====

Let PyP_yPy = net premium in year yyy, CyC_yCy = fully developed claims in year yyy. Target net loss ratio = 96% ⇒ target profit margin = 4%.

===== Profit in year yyy: =====

Πycurrent=Py−Cy\Pi^{\text{current}}_y = P_y - C_yΠycurrent=Py−Cy
Total 20X1–20X3:

Πcurrent=∑y=20X120X3(Py−Cy)\Pi^{\text{current}} = \sum_{y=20X1}^{20X3} (P_y - C_y)Πcurrent=y=20X1∑20X3(Py−Cy)
===== - “Excess profit” = profit above 4% of premium. =====
* Refund in year yyy:

Ry=max⁡(0, (Py−Cy)−0.04Py)=max⁡(0, 0.96Py−Cy)R_y=\max\big(0,\ (P_y-C_y)-0.04P_y\big)=\max\big(0,\ 0.96P_y-C_y\big)Ry=max(0, (Py−Cy)−0.04Py)=max(0, 0.96Py−Cy)
* Green’s profit after refund:

Πyproposal=(Py−Cy)−Ry=min⁡(Py−Cy, 0.04Py)\Pi^{\text{proposal}}_y = (P_y-C_y)-R_y = \min\big(P_y-C_y,\ 0.04P_y\big)Πyproposal=(Py−Cy)−Ry=min(Py−Cy, 0.04Py)
Total 20X1–20X3:

Πproposal=∑ymin⁡(Py−Cy, 0.04Py)\Pi^{\text{proposal}}=\sum_{y}\min\big(P_y-C_y,\ 0.04P_y\big)Πproposal=y∑min(Py−Cy, 0.04Py)
(Operationally: compute LR each year =Cy/Py=C_y/P_y=Cy/Py. If LR < 96%, cap profit at 4% of premium; if LR ≥ 96%, no refund.)

==== Key concerns: ====
* Asymmetric risk sharing: Green gives away upside when experience is good, but still absorbs full downside when experience is bad. That is equivalent to giving the customer a free call option on results, which reduces expected profit and increases required capital.
* Volatility / capital strain: With large groups, random fluctuation is smaller than small groups, but severity risk (shock claim years) remains material; removing upside reduces Green’s ability to offset bad years.
* Lapse/anti-selection risk: Groups may stay after good years (collect refunds) and shop/terminate after bad years, worsening Green’s long-run results.
* Settlement lag & disputes: Requires fully developed claims and clear definitions (runout, pooling, eligibility changes), increasing administrative burden and the risk of disagreement.
* Pricing implication: If offered, the expected cost of the refund feature should be explicitly priced (either via higher premium or explicit risk charge), otherwise target LR will not be achieved.

==== Again, the numeric inputs (annual Py,CyP_y, C_yPy,Cy and the list of claimants ≥$1M by year) are not shown, so below is the required calculation structure. ====

===== For each year yyy: =====
* Target claims threshold: Ty=0.96PyT_y = 0.96P_yTy=0.96Py
* Refund: Ry=max⁡(0, Ty−Cy)R_y=\max(0,\ T_y-C_y)Ry=max(0, Ty−Cy)
* Green profit: Πy=min⁡(Py−Cy, 0.04Py)\Pi_y=\min(P_y-C_y,\ 0.04P_y)Πy=min(Py−Cy, 0.04Py)

===== Let high-claimant annual costs be Hy,iH_{y,i}Hy,i for claimants with ≥$1M in claims. =====
* Stop-loss recoveries:

Recovy=∑imax⁡(0, Hy,i−1,000,000)\text{Recov}_y=\sum_i \max(0,\ H_{y,i}-1{,}000{,}000)Recovy=i∑max(0, Hy,i−1,000,000)
* Net claims after stop loss:

CySL=Cy−RecovyC^{\text{SL}}_y = C_y - \text{Recov}_yCySL=Cy−Recovy
* Add pool charge cost:

Pooly=0.01Py\text{Pool}_y = 0.01P_yPooly=0.01Py
* “All-in” cost used for profit sharing:

C~y=CySL+Pooly\tilde{C}_y = C^{\text{SL}}_y + \text{Pool}_yC~y=CySL+Pooly
Then apply the same profit-sharing rule (96% target unless changed):

ΠySL=min⁡(Py−C~y, 0.04Py)\Pi^{\text{SL}}_y = \min\big(P_y-\tilde{C}_y,\ 0.04P_y\big)ΠySL=min(Py−C~y, 0.04Py)
Interpretation: lowers volatility by capping severity, but reduces expected profit by the pool charge; net benefit depends on how large Recovy\text{Recov}_yRecovy is relative to 0.01Py0.01P_y0.01Py.

===== This changes the target net LR from 96% to 95%. =====
* Target claims threshold: Ty=0.95PyT_y = 0.95P_yTy=0.95Py
* Green profit under proposal becomes:

Πy5%=min⁡(Py−Cy, 0.05Py)\Pi^{5\%}_y = \min\big(P_y-C_y,\ 0.05P_y\big)Πy5%=min(Py−Cy, 0.05Py)
(If combined with stop loss, replace CyC_yCy with C~y\tilde{C}_yC~y above.)

Interpretation: improves Green’s retained upside (higher cap), reduces refund frequency/amount, better compensates for volatility and the “optionality” granted to the group.

===== Define (using target LR 96% unless changed): =====
* Excess profit available for sharing in year yyy:

EPy=max⁡(0, 0.96Py−C~y)EP_y=\max(0,\ 0.96P_y-\tilde{C}_y)EPy=max(0, 0.96Py−C~y)
* Excess loss above target in year yyy:

ELy=max⁡(0, C~y−0.96Py)EL_y=\max(0,\ \tilde{C}_y-0.96P_y)ELy=max(0, C~y−0.96Py)
Let fund balance at start of year yyy be Fy−1F_{y-1}Fy−1, with F0=0F_0=0F0=0, cap Fmax⁡=500,000F_{\max}=500{,}000Fmax=500,000.

Use fund to offset bad years first:

Uy=min⁡(Fy−1, ELy)U_y=\min(F_{y-1},\ EL_y)Uy=min(Fy−1, ELy)
Remaining excess loss borne by Green:

ELynet=ELy−UyEL^{\text{net}}_y = EL_y - U_yELynet=ELy−Uy
Then credit favorable-year share into the fund up to the cap:

Deposity=min⁡(EPy, Fmax⁡−(Fy−1−Uy))\text{Deposit}_y = \min(EP_y,\ F_{\max}- (F_{y-1}-U_y))Deposity=min(EPy, Fmax−(Fy−1−Uy))
Any excess beyond filling the fund is distributed:

Distributiony=EPy−Deposity\text{Distribution}_y = EP_y - \text{Deposit}_yDistributiony=EPy−Deposity
Fund recursion:

Fy=min⁡(Fmax⁡, Fy−1−Uy+Deposity)F_y = \min\big(F_{\max},\ F_{y-1}-U_y + \text{Deposit}_y\big)Fy=min(Fmax, Fy−1−Uy+Deposity)
Green profit each year (profit-sharing with fund):
* Green retains the target margin, less any net excess loss not covered by the fund:

Πyfund=0.04Py−ELynet\Pi^{\text{fund}}_y = 0.04P_y - EL^{\text{net}}_yΠyfund=0.04Py−ELynet
* If the policy terminates after 20X3, add the retained fund balance as additional profit:

Πfund,total=∑yΠyfund+F20X3\Pi^{\text{fund,total}} = \sum_y \Pi^{\text{fund}}_y + F_{20X3}Πfund,total=y∑Πyfund+F20X3
Interpretation: reduces refund payouts in early good years, creates a buffer against future bad years, and (because Green keeps the balance at termination) materially improves Green’s expected value—though it may be viewed as less favorable/less transparent to the customer.

===== 1. Net premium PyP_yPy and developed claims CyC_yCy for 20X1–20X3 =====
# For the stop-loss option: the list of Hy,iH_{y,i}Hy,i (annual claim totals by claimant for those ≥$1M)

If you paste those figures, I’ll compute the exact totals and show the full arithmetic.