# Free Tools for Rational Education

# Free Study Materials for the Casualty Actuarial Society (CAS) Exam 7

# (Old Exam 6)

# Calculation of Premium-Development-to Loss-Development Ratios for Retrospectively Rated Insurance Policies

**This section is part of Mr. Stolyarov's Free Study Materials for the CAS Exam 7.**

**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**

PDLD_{1} = ((BP/L_{1})*TM)+((CL_{1}/L_{1})*LCF*TM)

**Formula 57.3: Approximation of Premium Development to Loss Development (PDLD) Ratio at First Retro Adjustment**

PDLD_{1} ≈ BP*TM/(SP*ELR*%Loss_{1})

*Definitions of Variables*

SP = Standard premium

ELR = Expected loss ratio

%Loss_{1} = 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: P_{3}/TM = BP+(CL_{3} *LCF) → P_{3}/TM - BP = CL_{3} *LCF →

LCF = (P_{3}/TM - BP)/CL_{3} = (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: PDLD_{1} = ((BP/L_{1})*TM)+((CL_{1}/L_{1})*LCF*TM) =

((45000/130000)*1.08)+((110000/130000)*1.2*1.08) = **PDLD _{1} =**

**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: PDLD_{1} ≈ BP*TM/(SP*ELR*%Loss_{1}) = 55000*1.05/(65000*0.8*0.56) = **PDLD _{1} =**

**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, PDLD_{2} = ((CL_{2} - CL_{1})/(L_{2} - L_{1}))*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: 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:

L_{2} - L_{1} = 20000

L_{3} - L_{2} = 60000

PDLD_{2} = 1.222

PDLD_{3} = 0.5555

CL_{1} = 12200

We want to find CL_{3}.

1.222 = (CL_{2} - CL_{1})/(20000), so CL_{2} - CL_{1}= 24440.

0.5555 = (CL_{3} - CL_{2})/(60000), so CL_{3} - CL_{2} = 33330.

(CL_{3} - CL_{2}) + (CL_{2} - CL_{1}) = CL_{3} - CL_{1} = 33330 + 24440 = 57770, so CL_{3} = 57770 + CL_{1} = 57770 + 12200 = **CL _{3} = 69970.**

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