Calculation of Premium Development to Loss Development Ratios for Retrospectively Rated Insurance Policies: Practice Questions and Solutions

Formula 57.1: Retrospective Premium Calculation
P_n = (BP+(CL_n *LCF))*TM

Definitions of Variables
P_n = Premium at the nth retro adjustment
BP = Basic premium
CL_n = Capped loss at the nth retro adjustment
LCF = Loss conversion factor
TM = Tax multiplier

Formula 57.2: Premium Development to Loss Development (PDLD) Ratio at First Retro Adjustment
PDLD₁ = ((BP/L₁)*TM)+((CL₁/L₁)*LCF*TM)

Formula 57.3: Approximation of Premium Development to Loss Development (PDLD) Ratio at First Retro Adjustment
PDLD₁ ≈ BP*TM/(SP*ELR*%Loss₁)

Definitions of Variables
SP = Standard premium
ELR = Expected loss ratio
%Loss₁ = Percent of loss emerged for the first retro adjustment

Formula 57.4: Premium Development to Loss Development (PDLD) Ratio for Second and Subsequent Retro Adjustments
PDLD_n = ((CL_n - CL_n-1)/(L_n - L_n-1))*LCF*TM

Definitions of Variables
L_n = Uncapped loss at the nth retro adjustment

Sources:
Teng, M.T.S.; and Perkins, M.E., "Estimating the Premium Asset on Retrospectively Rated Policies," PCAS LXXXIII, 1996, pp. 611-647, excluding Section 5.

Original Problems and Solutions from The Actuary's Free Study Guide

Problem S6-57-1. Consider a retrospectively rated book of business. The premium at the 3^rd retro adjustment is $142,140, the basic premium is $70,600, and the capped loss at the 3^rd retro adjustment is $58,250. The tax multiplier is 1.04. What is the loss conversion factor?

Solution S6-57-1. We use Formula 57.1: P_n = (BP+(CL_n *LCF))*TM, rearranging it in terms of LCF: P₃/TM = BP+(CL₃ *LCF) → P₃/TM - BP = CL₃ *LCF →
LCF = (P₃/TM - BP)/CL₃ = (142140/1.04 - 70600)/58250 = LCF = 1.13430175.

Problem S6-57-2. Consider a retrospectively rated book of business. At the first retro adjustment, the uncapped loss is $130,000, and the capped loss is $110,000. The tax multiplier is 1.08, and the loss conversion factor is 1.2. The basic premium is $45,000. What is the PDLD ratio at the first retro adjustment?

Solution S6-57-2. We use Formula 57.2: PDLD₁ = ((BP/L₁)*TM)+((CL₁/L₁)*LCF*TM) =
((45000/130000)*1.08)+((110000/130000)*1.2*1.08) = PDLD₁ = 1.470461538.

Problem S6-57-3. Consider a retrospectively rated book of business. At the first retro adjustment, the basic premium is $55,000, and the standard premium is $65,000. The expected loss ratio is 80%, and the expected percentage of loss emerged for the first retro adjustment is 56%. The tax multiplier is 1.05. Estimate the PDLD ratio at the first retro adjustment.

Solution S6-57-3. We use Formula 57.3: PDLD₁ ≈ BP*TM/(SP*ELR*%Loss₁) = 55000*1.05/(65000*0.8*0.56) = PDLD₁ =1.983173077.

Problem S6-57-4. Consider a retrospectively rated book of business. At the first retro adjustment, the uncapped loss is $130,000, and the capped loss is $110,000. The tax multiplier is 1.08, and the loss conversion factor is 1.2. At the second retro adjustment, the uncapped loss is $150,000, and the capped loss is $125,000. The basic premium is $45,000. What is the PDLD ratio at the second retro adjustment?

Solution S6-57-4. We use Formula 57.4: PDLD_n = ((CL_n - CL_n-1)/(L_n - L_n-1))*LCF*TM.
Here, PDLD₂ = ((CL₂ - CL₁)/(L₂ - L₁))*LCF*TM = ((125000 - 110000)/(150000 - 130000))*1.2*1.08 = PDLD₂= 0.972.

Problem S6-57-5. The PDLD ratio at the third retro adjustment is 0.555, while the PDLD ratio at the second retro adjustment is 1.222. There are no taxes, and the loss conversion factor is 1. It is known that 20000 in uncapped losses emerged between the first and second retro adjustments, and 60000 in uncapped losses emerged between the second and third retro adjustments. Capped losses at the first retro adjustment were 12200. What was the magnitude of capped losses at the third retro adjustment?

Solution S6-57-5. We use Formula 57.4: PDLD_n = ((CL_n - CL_n-1)/(L_n - L_n-1))*LCF*TM. Here, LCF*TM = 1, so PDLD_n = ((CL_n - CL_n-1)/(L_n - L_n-1)). We are given the following:
L₂ - L₁ = 20000
L₃ - L₂ = 60000
PDLD₂ = 1.222
PDLD₃ = 0.5555
CL₁ = 12200
We want to find CL₃.
1.222 = (CL₂ - CL₁)/(20000), so CL₂ - CL₁= 24440.
0.5555 = (CL₃ - CL₂)/(60000), so CL₃ - CL₂ = 33330.
(CL₃ - CL₂) + (CL₂ - CL₁) = CL₃ - CL₁ = 33330 + 24440 = 57770, so CL₃ = 57770 + CL₁ = 57770 + 12200 = CL₃ = 69970.

See other sections of The Actuary's Free Study Guide for Exam 6.