Life Contingencies and Survival Models
A survival model describes the random future lifetime of a life aged x through its survival function and its force of mortality. The life table tabulates survival on an integer-age grid. The expected present value combines the time value of money with survival probability, giving the price of insurances and annuities, and the equivalence principle sets the level premium so that the EPV of premiums equals the EPV of benefits.
Survival Function And Force Of Mortality
Let T_x be the future lifetime of a life now aged x. The survival probability that the life reaches age x plus t is denoted by the actuarial symbol for t-year survival. The force of mortality is the instantaneous rate of death at age x plus t, the continuous analogue of a hazard rate. Dickson, Hardy and Waters, Actuarial Mathematics for Life Contingent Risks, 3rd ed., Ch. 2 to 3 is the canonical reference and the SOA FAM and ALTAM source text.
The survival function recovers from the force of mortality by integrating the force over the elapsed age interval and exponentiating the negative. A constant force mu reduces this to an exponential survival model, which is the tractable special case used in many exam questions.
The Life Table
The life table starts from a radix of lives at an initial age and applies survival probabilities to tabulate the expected number alive at each integer age. The expected deaths between consecutive ages follow by differencing. Quantities such as the curtate expectation of life are then sums over the table.
Fractional-age assumptions, uniform distribution of deaths or constant force within each year of age, bridge the integer table to continuous quantities. ALTAM expects fluent movement between the table and the continuous model.
Expected Present Value Of Insurances And Annuities
An insurance pays a benefit on death; an annuity pays while the life survives. The expected present value discounts each possible payment by both interest and the probability that the payment is made. For a whole-life insurance with unit benefit at the moment of death and constant force of interest delta, the EPV is the expectation of the discount factor evaluated at the random time of death.
The equivalence principle sets the level annual premium so that the EPV of future premiums equals the EPV of future benefits at issue. This is the pricing identity underneath the profit-testing page.
Worked Example: Whole-Life EPV Under Constant Force
Assume a constant force of mortality mu equal to 0.02 and a constant force of interest delta equal to 0.05. Under the constant-force model the whole-life insurance EPV has the closed form mu divided by the sum of mu and delta.
Substituting gives EPV equal to 0.02 divided by 0.07, which equals about 0.2857 per unit of benefit. The matching whole-life annuity EPV is 1 divided by 0.07, about 14.2857, and the continuous equivalence-principle premium rate is the ratio, mu equal to 0.02 per unit per year. The premium equals the force of mortality exactly because, under constant force, insurance and annuity share the same denominator.