### 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))
• 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
• 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
• 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