Valuation of Life Insurance Liabilities - Chapter 3

Valuation of Life Insurance Liabilities - Chapter 3 - TYPES OF RESERVE FACTORS

Mean and Mid-Terminal Reserves and Their Relationshiop to Deferred and Unearned Premiums

Mean Reserves
- Interpolated Mean Reserve - (1-h)([t-1]V(x) + [t]P(x)) + h*[t]V(x)
- Mean Reserve h = 1/2 -> [t]MV(x) = 0.5*([t-1]V(x) + P(x) + [t]V(x))
Deferred Premiums
- modal premiums due after valn date & befroe next policy anniv
- an Asset in US (neg liab in Canada)
  - b/c mean reserves based on annual net prem payment @ boy overstate reserves on policies w/ more freq modes of payment
  - use net prems since reserves based on net prem
  - usually calced using an inventory
  - can use avg def premium (m-)/2m * P(x)
  - using avg def prem, net liab = 0.5*([t-1]V(x) + [t]V(x)) + 1/2m*P(x) (mean reserve - net def prem)
Mid-Terminal Reseres and Unearned Premiums
- interpolated terminal reserve (1-h)[t-1]V(x) + h*[t]V(x)
- mid-terminal reserve 0.5([t-1]V(x) + [t]V(x))
- used by some companies instead of mean reserves
- understates reserves unles sprems paid very frequently
- Unearned Premium Liablility - used as an adjustmetn
- usual practice - 1/2 modal prm for each policy
- can also calc exact for each policy
- or estimate as 1/2m * P(x)
- net liability is saem as formula shown under def prems

Graphic Representatin of Reserves

show start-step of modal premium reserves

Other Reserve Adjustments and Alternative Types of Factors

Curtate and SemiContinuous Factors
- reserve formulas assume prems payable at BOY and claims paid at EOY
- reality: prems paid modally and claims paid at death
- Immediate Payment of CLaims Reserve (IPC)
  - AG32 - need to establish IPC reserve if basic reserves calculated using curtate functin
    - IPC reserve = i/2*death portion of basic reserve (term portion if an endowment) if policy credits interest from DOD to date of payment
      - i/3 if payment made immediately upon receipt of due proof of death
      - 0 if payments made @ EOY or Vx calculated using continuous or semi-cont or disc cont
    - Semi-Continuous Reserves - prems payable @ BOY, claims paid at moment of death
      - P(Abar(x)) = Abar(x)/adue(x) [t]V(Abar(x)) = Abar(x+t) - P(Abar(x))*adue(x+t)
Fully Continuous and Discounted Continuous Factors
- Fully Continuous - continuous prems adn IPC
- Discounted Continuous - prems paid BOY, IPC and refund of unearned portion of Px @ death
  - two net prems - one for terminal reserves and one for mean reserves
- discounted continuous is what most cos using "continuous" actually use
- difference between fully continuous and dicounted continuous
  - fully cont based on NP payable continuously throughout the year
  - reserve factors are mid-terminals adn unearned prem reserve must be set up for prems paid beyond valn date
  - disc cont terminal reserves = full continuous terminals
  - mean reserves calculated using annual NP = continuous NP discounted w/ int only
  - def prem reserve calculated if mode <> annual
Relationship of the Expense Allowance to type of Reserve Factor
- NP under semi-cont and cont methods > curtate
- SOA published tables using abar(1|)*Pbar(Abar(x)) + (P(x+1) - c(x))/adue(x) <- cont renewal prem for WL
- theoretically correct s/b abar(1|)*Pbar(Abar(x+1)) <- s/b used per AG18
Nondeduction Reserve
- under curate or semi-cont - assumes full annual premium collected each year
  - not true since std practice is to not collect modals due after death
- reserve for this -> term insurance for avg # remaining def prems @ date of death
  - for endowment, this woudl be[t]V'(x:n|)*((m-1)/2m * P_modal(x:n|))
Refund Reserve
- many cos refund pro-rata share of prems for periods beyond DOD
- reserve for both this and non-ded reserve -> 0.5*P_modal(x:n|)*[t]V'(x:n|)
  - where P_modal(x:n|) approx = P(x:n|) / (1 = (m-1)/2m*d - 0.5P(x:n|))
  - co's often omit (m-1)*d/2m term so 1 factor applies regardless of mode
- these reserves not necessary for fully and discounted coint reserves - already built in
- nice grid goes here showing various reserve components for curtate, fully, disc and semi cont

Copyright © 2004 Steve Welander.
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