Role of Generalised Linear Model in non-life pricing Phase3

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Before reading this article, make sure that you read phase1 and phase2. Here are the link:
Phase1: http://www.actuarysense.com/2018/10/role-of-generalised-linear-model-in-non.html
Phase2: http://www.actuarysense.com/2018/11/role-of-generalised-linear-model-in-non.html So we know that the purpose of GLM is to find the relationship between mean of the response variable and covariates.

In this Article we are going to talk about Linear Predictors.
Linear Predictor: Let’s denote it with, “η” (eta). So, linear predictor is actually a function of covariates. For example, in the normal linear model where function is Y = B0 + B1x. So linear predictor will be η = B0 + B1x. Always note that linear predictor has to be linear in its parameter. In this case parameters are B0 and B1. But still the question is how I came up with B0 + B1x as a function? First of all, note that broadly there are two types of Covariates. 1. Variables: It takes the numerical value. For example: age of policyholder, years of ex…

How an Actuary calculates Expense for Pension Fund

Let’s see some Actuarial Terminologies from Accounting Valuations point of view under US GAAP/IAS 19R:
1.   Accumulated Benefit Obligations (ABO): ABO is an approximate amount of a company's pension plan liability at a single point in time. ABO is estimated based on the assumption that the pension plan is to be terminated immediately; it does not consider any future salary increases. Changes in annual ABO are mainly determined by changes in service costs, interest costs, contributions by plan participants.
2.   Projected Benefit Obligation (PBO): A pension's projected benefit obligation (PBO) is an actuarial liability equal to the present value of liabilities earned and the present value of liability from future compensation increases. 
Note: ABO differs from PBO as ABO does not includes any assumption about future compensation levels. For plans with flat benefit or non-pay related pension benefit formulas, the ABO and PBO will be same.

Today we will see Accounting Valuation through US GAAP and IAS 19R.
US Accounting Standard: US GAAP
International Accounting Standard: IAS 19R


Now note that there are two accounting elements basically: Expense and Disclosure.

Expense: Cost of a plan for income statement purpose. Pension expense is the amount that a business charges to expense in relation to its liabilities for pensions payable to employees.
Disclosure: Display of plan’s assets and liabilities for company’s balance sheet. Assets = Liabilities + Shareholder equity.


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Today we will cover Expense
By expense we mean NPPC i.e. Net Periodic Pension Cost. If NPPC is +ve it indicates a cost otherwise an income. Simple CT2 rule right !!.

Let’s look at the components of NPPC:
1.  Service Cost(SC): It is expected increase in the plan’s liabilities due to additional year of benefit accrual.
Formula: SC = NC(Beginning of year)*(1+i)+Expense*(1+0.5i).
So here NC(Normal Cost) is cost for plan for one more year of benefit accrual and expenses represents administrative expenses and we use 0.5 because we assume expenses to be evenly distributed throughout the year.

Let’s see the example
:
XYZ company has NC = 24L on 1/1/2018
Expected expenses for 2018 = 1 lac and discount rate is 6%
Service cost = 24,00,000*(1+.06)+1,00,000(1+0.5*.06) = $2,647,000

2.  Interest Cost(IC): This is the interest on the projected benefit obligation. It is a financial item, rather than a cost related to employee compensation.
Formula: PBO(Beginning of year)*i – [Expected Benefit payments*i*13/24]
Here PBO is the liability at the beginning of year and now when we multiplied it with interest rate we get interest(simple maths) but during the year we paid benefit payments to pensioners due to which liability reduced and therefore interest cost will also reduce. So now question is why we multiplied it with 13/24 and not 0.5. Well! The answer is that benefits are paid on the first day of each month.

Let’s see the example:
XYZ company has PBO = 90 lacs on 1/1/2018
Expected Benefit Payments for 2018 = 7 lacs
Discount rate = 6%
Interest Cost(IC): 90,00,000*(1+.06)-[7,00,000*.06*13/24] = $517,250


3.    Expected Return on Assets(EROA): This is the difference between the fair values of beginning and ending plan assets, adjusted for contributions and benefit payments.
Formula: EROA = (Assets*r) – (Expected Benefit payments*r*13/24) – (Expected Expenses*0.5*r) + (Expected contributions*r*contribution weighting)
Here “r” represents return on asset; it is different from “i” which is the discount rate for liabilities used above. Expected Contributions are the amount paid into the plan done by employer. Contribution weighting is the weights assigned to each contribution made by employer.

Let’s see the example:
XYZ company has Assets = 75,00,000
Expected Employer Contribution = 7,50,000 he pays on 1 July 2018 and plan follows calendar year
Return on plan assets = 7%
Expected Benefit payments = 7,00,000
Expected Expenses = 1,00,000
EROA: 75,00,000*.07-7,00,000*.07*13/24-1,00,000*0.5*.07+7,50,000*.07*6/12 = $518,208.33

Note that under US GAAP, NPPC calculation includes SC, IC, EROA and Amortizations. But under IAS 19R amortizations are not present because any gain/loss is recognized immediately.

So for US GAAP Amortizations are:
Concept behind amortization is to spread the impact of certain events over a period of time
.

1. Amortization of Transition Asset Obligation/(Asset): Any gain/loss arises due to transition from one accounting standard to another. If assets exceeds the liabilities at implementation, leads to reduction in NPPC.

2. Amortization of Prior service cost: When an employer issues a plan amendment, it may contain increases in benefits that are based on services rendered by employees in prior periods. If so, the cost of these additional benefits are amortized over the future periods in which those employees active on the amendment date are expected to receive benefits.

3. Amortization of Gain/Loss: This is the gain or loss resulting from a change in the value of a projected benefit obligation from changes in assumptions, or changes in the value of plan assets.



SO NPPC = SC+IC+EROA+Amortizations


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