Section 57

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

G. Stolyarov II
October 4, 2010 - Republished July 11, 2014

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

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

Definitions of Variables
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
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 = 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: 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.

Gennady Stolyarov II (G. Stolyarov II) is an actuary, science-fiction novelist, independent philosophical essayist, poet, amateur mathematician, composer, and Editor-in-Chief of The Rational Argumentator, a magazine championing the principles of reason, rights, and progress.

In December 2013, Mr. Stolyarov published Death is Wrong, an ambitious children’s book on life extension illustrated by his wife Wendy. Death is Wrong can be found on Amazon in paperback and Kindle formats.

Mr. Stolyarov has contributed articles to the Institute for Ethics and Emerging Technologies (IEET), The Wave Chronicle, Le Quebecois Libre, Brighter Brains Institute, Immortal Life, Enter Stage RightRebirth of Reason, The Liberal Institute, and the Ludwig von Mises Institute. Mr. Stolyarov also published his articles on Associated Content (subsequently the Yahoo! Contributor Network) from 2007 until its closure in 2014, in an effort to assist the spread of rational ideas. He held the highest Clout Level (10) possible on the Yahoo! Contributor Network and was one of its Page View Millionaires, with over 3.1 million views.

Mr. Stolyarov holds the professional insurance designations of Associate of the Society of Actuaries (ASA), Associate of the Casualty Actuarial Society (ACAS), Member of the American Academy of Actuaries (MAAA), Chartered Property Casualty Underwriter (CPCU), Associate in Reinsurance (ARe), Associate in Regulation and Compliance (ARC), Associate in Personal Insurance (API), Associate in Insurance Services (AIS), Accredited Insurance Examiner (AIE), and Associate in Insurance Accounting and Finance (AIAF).

Mr. Stolyarov has written a science fiction novel, Eden against the Colossus, a philosophical treatise, A Rational Cosmology,  a play, Implied Consent, and a free self-help treatise, The Best Self-Help is Free. You can watch his YouTube Videos. Mr. Stolyarov can be contacted at gennadystolyarovii@yahoo.com.

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