The Sad Man
09-21-2005, 12:09 AM
This article initially gave me fits since I tried learning it solely from All10's summary - a big mistake. Here's my take, focusing mostly on computation type questions.
The article revolves around the quantity called the "Premium Asset" which can be thought of as a "negative reserve" or how much premium you expect to gain in the future from the retro policy. Since retro policies give premium as a function of losses, most of the computation questions revolve around the term PDLD which stands for Premium Development to Loss Developent. One very important fact to remember is that the PDLD ratios are incremental
I see computation questions broken into three categories:
1. Computer PDLD ratio's based on formulas (without table M).
Define P_n and L_n to be the total premium and loss, respectively at the nth retro adjustment.
These formulas are critical to memorize:
P_n = [BP + (CL_n * LCF)]*TM
BP = Basic Premium, a predetermined value that remains constant
LCF = Loss Conversion factor, which converts a loss amount to a premium
TM = Tax Multiplier
CL_n = Capped loss at nth adj. (losses are capped due to a total loss max/min or per occurence limitation all deifned in the retro policy)
The PDLD at n=1 is simply P_n/L_n. Using the above formula we can rewrite PDLD at n=1 as
[(BP/L_1)]*TM + [(CL_1/L_1)*LCF*TM]
Note that you can estimate L_1 as:
L_1 = SP*ELR*%Loss_1
SP = Standard Premium, based off an adj. manual premium, given.
ELR = Expected loss ratio
%Loss_1 = Expected % of loss you'd see in _1 adjustment. Moving some terms around gives:
P_1/L_1 = (BP/SP)*TM/(ELR*%Loss_1) + [(CL_1/L_1)*LCF*TM]
This is important to know because (BP/SP) = Basic premium factor and CL_1/L_1 = loss caping factor which could be given in the problem. Losses are usually emerged as of 18 months to compute the first retro adj.
The PDLD_2 (for the 2nd retro adj) refers to both incremental premium and losses developed between the 1st and 2nd adj. It is simply:
(P_2-P_1)/(L_2-L_1) which simplifies to
[(CL_2-CL_1)/(L_2-L_1)]*LCF*TM
2. Empirical calculation of PDLD ratios
To create empirical PDLD ratios you use booked premium/reported loss. The most important thing to remember is that the development of premium is 9 months ahead of losses for each retro adjustment. Since you usually start with losses at 18 months, you will then use premium at 27 months for the 1st retro adj. Just remember that for subsequent adjustments, you use incremental premium and losses, e.g. 2nd adj would be premium between 27 and 39 months and losses between 18 and 30 months.
Cumulative PDLD (CPDLD) tie into the ultimate goal which is to estimate the premium asset (negative reserve). CPDLD's are the weighted average of the current (you can start at any nth adj) and all subsequent PDLDs, weighted by the % of losses to emerge in each of those periods. The CPDLD ratio states how much premium one can expect for every $1 of loss that emerges.
3. Compute PDLD ratios based on Table M formulas.
This formula is extremely important and should be memorized:
CL = L*(1-x-LER) -> CL/L = (1-x-LER)
CL = capped losses
L = losses
x = Tab. M net ins charge = Tab. M charge at max - Table M saving at min
LER = % of losses due to per accident limit.
I'm hoping we won't have to go deeper than this in calculating x but there's a good chance we will, so...
To calculate the charge at max and saving at min, you need an "entry ratio at max" and "entry ratio at min." These two terms are defined as:
entry ratio at max = (loss ratio at max)/(actual loss ratio)
entry ratio at min = (loss ratio at min)/(actual loss ratio)
I'm hoping the loss ratio at min/max will be given. The actual loss ratio is actual loss at each retro adj divided by the standard premium OR it can be estimated as expected loss ratio * expected % of loss to emerge by given retro adj.
This formula CL/L = (1-x-LER) allows us to directly computer the loss capping ratios (CL/L). We can then compute the incremental PDLD from the 2nd retro adj on through the following formula (memorize it)
[(CL_n/L_n)*%Loss_n - (CL_n-1/L_n-1)*%Loss_n-1]/[%Loss_n - %Loss_n-1]*LCF*TM
If n=1, then use the above formula [(BP/L_1)]*TM + [(CL_1/L_1)*LCF*TM]
The article revolves around the quantity called the "Premium Asset" which can be thought of as a "negative reserve" or how much premium you expect to gain in the future from the retro policy. Since retro policies give premium as a function of losses, most of the computation questions revolve around the term PDLD which stands for Premium Development to Loss Developent. One very important fact to remember is that the PDLD ratios are incremental
I see computation questions broken into three categories:
1. Computer PDLD ratio's based on formulas (without table M).
Define P_n and L_n to be the total premium and loss, respectively at the nth retro adjustment.
These formulas are critical to memorize:
P_n = [BP + (CL_n * LCF)]*TM
BP = Basic Premium, a predetermined value that remains constant
LCF = Loss Conversion factor, which converts a loss amount to a premium
TM = Tax Multiplier
CL_n = Capped loss at nth adj. (losses are capped due to a total loss max/min or per occurence limitation all deifned in the retro policy)
The PDLD at n=1 is simply P_n/L_n. Using the above formula we can rewrite PDLD at n=1 as
[(BP/L_1)]*TM + [(CL_1/L_1)*LCF*TM]
Note that you can estimate L_1 as:
L_1 = SP*ELR*%Loss_1
SP = Standard Premium, based off an adj. manual premium, given.
ELR = Expected loss ratio
%Loss_1 = Expected % of loss you'd see in _1 adjustment. Moving some terms around gives:
P_1/L_1 = (BP/SP)*TM/(ELR*%Loss_1) + [(CL_1/L_1)*LCF*TM]
This is important to know because (BP/SP) = Basic premium factor and CL_1/L_1 = loss caping factor which could be given in the problem. Losses are usually emerged as of 18 months to compute the first retro adj.
The PDLD_2 (for the 2nd retro adj) refers to both incremental premium and losses developed between the 1st and 2nd adj. It is simply:
(P_2-P_1)/(L_2-L_1) which simplifies to
[(CL_2-CL_1)/(L_2-L_1)]*LCF*TM
2. Empirical calculation of PDLD ratios
To create empirical PDLD ratios you use booked premium/reported loss. The most important thing to remember is that the development of premium is 9 months ahead of losses for each retro adjustment. Since you usually start with losses at 18 months, you will then use premium at 27 months for the 1st retro adj. Just remember that for subsequent adjustments, you use incremental premium and losses, e.g. 2nd adj would be premium between 27 and 39 months and losses between 18 and 30 months.
Cumulative PDLD (CPDLD) tie into the ultimate goal which is to estimate the premium asset (negative reserve). CPDLD's are the weighted average of the current (you can start at any nth adj) and all subsequent PDLDs, weighted by the % of losses to emerge in each of those periods. The CPDLD ratio states how much premium one can expect for every $1 of loss that emerges.
3. Compute PDLD ratios based on Table M formulas.
This formula is extremely important and should be memorized:
CL = L*(1-x-LER) -> CL/L = (1-x-LER)
CL = capped losses
L = losses
x = Tab. M net ins charge = Tab. M charge at max - Table M saving at min
LER = % of losses due to per accident limit.
I'm hoping we won't have to go deeper than this in calculating x but there's a good chance we will, so...
To calculate the charge at max and saving at min, you need an "entry ratio at max" and "entry ratio at min." These two terms are defined as:
entry ratio at max = (loss ratio at max)/(actual loss ratio)
entry ratio at min = (loss ratio at min)/(actual loss ratio)
I'm hoping the loss ratio at min/max will be given. The actual loss ratio is actual loss at each retro adj divided by the standard premium OR it can be estimated as expected loss ratio * expected % of loss to emerge by given retro adj.
This formula CL/L = (1-x-LER) allows us to directly computer the loss capping ratios (CL/L). We can then compute the incremental PDLD from the 2nd retro adj on through the following formula (memorize it)
[(CL_n/L_n)*%Loss_n - (CL_n-1/L_n-1)*%Loss_n-1]/[%Loss_n - %Loss_n-1]*LCF*TM
If n=1, then use the above formula [(BP/L_1)]*TM + [(CL_1/L_1)*LCF*TM]