February 24, 2019

## Purpose

There are a few reasons for the calculation of premiums liabilities. The main purpose is to calculate an offset for the Unearned Premiums (20.20 Line 12). This offset we’ll cover later is called: Deferred Policy Acquisition Expenses (20.10 Line 43) and if negative we would use the liability called Premium Deficiency (20.20 Line 15)

Secondly, this is used twice in the MCT calculation as there is margin added for insurance risk, which is based on the net premium liabilities (less any PfADs). We also use premium liabilities when calculating interest rate risk (including the PfADs).

Finally, the only time you would need duration is when calculating interest rate risk

Net unpaid claims and adjustment expenses, which include PfADs, are net of reinsurance, salvage and subrogation, and self-insured retentions. Net premium liabilities, which also include PfADs, are after reinsurance recoverables.

OSFI – MCT Guide (2018)

1. Expected Claims APV(L+LAE)
2. Unpaid Reinsurance
3. Policy Maintenance

## Expected Claims

This is the ONLY component used for the MCT calculation: we exclude re-insurance costs and maintenance costs there.

$E(L+LAE) = \mbox{PV}_i=\left[\mbox{LR}\times\left(P-R\right)+U\right]$

• LR = Loss Ratio (net premiums)
• U = ULAE (sometimes included in LR)
• R = Unpaid re-insurance premiums

\begin{aligned}APV(L+LAE) =&\left[\mbox{LR}\times\left(P-R\right)+U\right]\times\mbox{Unearn.Factor}_{i}\\&+\mbox{PfAD}_{re}+\mbox{PfAD}_{i}+\mbox{PfAD}_{claims}\end{aligned}

$\mbox{Unearn.Factor}_{i}$ will be calculated below after including a adjustment for differences in timing for the premium liabilities.

## Unpaid Reinsurance

As seen above, we deducted “unpaid re-insurance”. This only relates to re-insurance we have to pay to cover the unearned premiums.

This amount was also deducted from the unearned premiums to ensure we were working with NET Unearned Premiums

Note: this is deducted out of claims and included into total premium liabilities

## Maintenance Expense

Maintenance expenses also need to be included to reflect the future cost of servicing the policies in force. These expenses include expenses associated with endorsements, mid-term cancellations, changes in reinsurance contracts, etc. Maintenance expenses are generally expressed as a percentage of gross UPR and are evaluated as a portion of general expenses with typical ratios ranging from 25% to 50%.

Appendix B provides a sample calculation for maintenance expenses:

$M = E\times X$

Assumptions

• X = Portion of Policy Servicing Costs : Assumption that X% of general expenses are used for servicing in-force policies
• E = General Expense Ratio = Trended General Expenses / Total On-level Premiums

## Present Value Factor

However, an adjustment would be required to reflect differences in the average accident date (AAD) of a future accident period versus the average accident date underlying the UPR (which is a partial accident period).

$\frac{1}{3} = \frac{\int_0^1 x\cdot\left(1-x\right)dx}{\int_0^1 \left(1-x\right)dx}$

$\frac{1}{2} = \frac{\int_0^1 x\cdot\left(1\right)dx}{\int_0^1 \left(1\right)dx}$

The difference being:

$-\frac{1}{6}=\frac{1}{3}-\frac{1}{2}$

## Unearned Present Value Factor

A few important points:

1. At year-end the claim liabilities for the current year are at age = 12 months (even though some may have just occurred)
2. At year-end the premium liabilities haven’t started yet and are at age = 0 months.
 Age (mn) % Paid % Paid (incremental) 12 20% 20% 24 45% 25% 36 80% 35% 48 100% 20%

## Normal Present Value Factor

For unpaid claims starting at age = 12 months (20%):

$\mbox{PV.Factor}_i=\frac{0.45-0.2}{0.8\cdot(1+i)^{0.5}}+\frac{0.8-0.45}{0.8\cdot(1+i)^{1.5}}+\frac{1.0-0.8}{0.8\cdot(1+i)^{2.5}}$

## Unearned Premium Present Value Factor

For unearned claims starting at age = 0 months (0%):

$\mbox{Unearn.Factor}_i=\left[ \frac{0.2}{(1+i)^{0.333}}+\frac{0.45-0.2}{(1+i)^{1.333}}+\frac{0.8-0.45}{(1+i)^{2.333}}+\frac{1.0-0.8}{(1+i)^{3.333}}\right]$

$\mbox{Unearn.Factor}_i=\left[ \frac{0.2}{(1+i)^{0.5}}+\frac{0.45-0.2}{(1+i)^{1.5}}+\frac{0.8-0.45}{(1+i)^{2.5}}+\frac{1.0-0.8}{(1+i)^{3.5}}\right]\cdot\frac{1}{(1+i)^{-1/6}}$