### Role of Generalised Linear Model in non-life pricing Phase3

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.

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 to Calculate Minimum Required Contribution that has to be paid by Employer to fund the Plan

As we know that Under DB pension plans, Employer has to contribute towards the plan, so today we are going to see how much an employer has to contribute and up to how much employer can avail the Tax benefit.
1.Stabilized rates: Pension regulators has set the Segmented interest rates that we can use to calculate our Liability.

2.Non-Stabilized rates: Non-stabilized rates are lower than stabilized rates, so technically Liability will be higher under non-stabilized rates.

3.FT= Funding Target, it means the liability of a plan calculated using 3 segment stabilized interest rates.

4.FT(max) = Funding Target, it means the liability of a plan calculated using non-stabilized interest rates.

5.TNC = Target Normal Cost = Present value of benefits accruing during the plan year.

6.EIR = Effective interest rate; it is the single rate of interest which is used to determine the PV of plan’s accrued benefit, would result in an amount equal to the Funding target of the plan for such plan year.

7.AVA = Actuarial Value of Assets

8.COB = Carryover Balance: Continuation of Plan’s Credit balance as of the end of 2007 plan year.

9.PFB = Prefunding Balance: Balance due to excess of contributions made in 2008 and subsequent plan years

10.MRC= Minimum Required Contribution

11.MDC = Maximum Deductible Contribution

12.SAC= Shortfall amortization charge

13.WAC = Waiver Amortization Charge

14.Segment Yield Curve = these are the 24 month rates in to 3 segments: 0-5 years, 5-20 years and then thereafter

Calculation of MRC (Minimum Required Contribution):

·       It is the amount that employer has to pay in quarterly instalments. Before start with its calculation let’s understand the meaning of Funding Shortfall. It means where the plan’s assets are not enough to cover liabilities.
·       FS = FT-NA; NA means Net Assets. And FS cannot be negative, it is either zero or positive.
·       Net Assets= AVA-COB-PFB
·       But COB, PFB balance cannot be used to reduce MRC if there is a binding agreement with PBGC not to use it.
·       So, if funding shortfall is greater than Zero, then calculation of MRC is MRC= TNC+SAC+WAC
·       If Funding Shortfall is Zero, then calculation of MRC is MRC = TNC-(Net Assets-FT)
3       points to remember:
1.If NA>FT+TNC, then funding shortfall =0 and MRC = 0
2.If FT<NA<FT+TNC, then FS = 0 and MRC = TNC+FT-NA
3.If FT>NA, then FS>0 and MRC = TNC+SAC+WAC

Let’s understand the concept of SAC:
·       Here we set up a SAI (Shortfall amortization instalments) to amortize each SAB (shortfall amortization base) in level instalments over a period of 7 years and must use yield curve segment rates for the next 7 years (it will be 9 or 15 years if Pension Relief Act 2010 is used)

·       Note to test if SAB base is exempted or not, check the calculation of AVA-PFB, if it is greater than FT , then it will not apply; which further means SAC will not be included in the calculation of MRC.

·       Let’s see the Calculation of SAC:
1. Calculate the SAB, which is equal to the Funding Shortfall
2.Subsequesnt years SAB = FT-PV of remaining amortization payments
3. For calculating PV, current year segment rates will be used.
4. In any year , if FS = 0, then SAB are considered Fully Amortized.
5. Now SAI = SAB/SAF; where SAF is Shortfall amortization factor
6. SAC = sum of SAI
7. SAI = now suppose if amortize for 7 years, then it is the sum of discounting back of segment rates of first 7 years from now onwards.
8. MRC = TNC+∑SAI
·       But we cannot use PFB to satisfy MRC, if it was not used in the shortfall amortization base exemption test. We can only use COB or PFB if the plan was at least 80% funded in the prior year.
·       (AVA-PFB)/FT >=80%, then MRC-COB-PFB

Maximum Deductible Contribution:
It is the maximum amount upto which an employer gets Tax deduction.
The MDC is greater of

1. MRC
2. At risk FT+At risk TNC – AVA
3. FT(max)+Cushion amount (which is 50% of FT(max)) + TNC – AVA

Let’s see an Example:
Given: FT = 18,957,466    FT(max)= 23,961,983 TNC = 160,000   AVA = 18,347,261 COB = PFB = 0.
PPA segment rates are 4.16% for 5 years, 5.72% for other 15 years and 6.48% thereafter.
Solution:
·       Let’s calculate FS= FT-AVA = 18957466-(18347261-0-0) = 610205
·       So, our funding shortfall is \$610,205 which is our SAB (shortfall amortization base) too.
·       SAC = ∑SAI
·       SAI= SAB/SAF
·       So let’s calculate SAF =
·
 Year Segment rates Discounting back 1 1.0416 1 2 1.0416 0.960061444 3 1.0416 0.921717976 4 1.0416 0.884905891 5 1.0416 0.849564028 6 1.0572 0.757206316 7 1.0572 0.716237529 Total 6.089693183
·       Now SAI = 610205/6.0896 = 100,203
·       So, SAC = \$100,203
·       Now let’s calculate MRC = TNC+SAC+WAC = 160,000+100,203+0 = \$260,203

Let’s Calculate MDC:
FT(max)+TNC+ Cushion amount-AVA = 23,961,983+160,000+50%(23,961,983)-18,347,261 = 17,755,713.50