**This section of sample problems and solutions is a part of** **The Actuary’s Free Study Guide for Exam 6, authored by Mr. Stolyarov. This is Section 57 of the Study Guide. See an index of all sections by following the link in this paragraph.**

**Formula 57.1: Retrospective Premium Calculation**

Pn = (BP+(CLn *LCF))*TM

*Definitions of Variables*

Pn = Premium at the nth retro adjustment

BP = Basic premium

CLn = 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**

PDLDn = ((CLn – CLn-1)/(Ln – Ln-1))*LCF*TM

*Definitions of Variables*

Ln**=** 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 3rd retro adjustment is $142,140, the basic premium is $70,600, and the capped loss at the 3rd 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: Pn = (BP+(CLn *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: PDLDn = ((CLn – CLn-1)/(Ln – Ln-1))*LCF*TM.

Here, PDLD2 = ((CL2 – CL1)/(L2 – L1))*LCF*TM = ((125000 – 110000)/(150000 – 130000))*1.2*1.08 = **PDLD****2****=** **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: PDLDn = ((CLn – CLn-1)/(Ln – Ln-1))*LCF*TM. Here, LCF*TM = 1, so PDLDn = ((CLn – CLn-1)/(Ln – Ln-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****.**