Principles of Actuarial Mathematics: Cashflow Models, Survival and Present-Value Random Variables
Cashflow and survival models
This domain tests how actuaries model uncertain cashflows and value them. A survival model uses a future lifetime random variable T_x for a life aged x, with survival probability t p_x=P(T_x>t) and density f_T(t)=t p_x μ_{x+t}. The curtate future lifetime K_x is the integer number of complete future years. A present-value random variable discounts each possible payment using v^t or v^{K_x+1}; its expected value is the EPV. The equivalence principle sets expected present value of premiums equal to expected present value of benefits plus expenses.
Valuation logic
Benefits are valued using the timing and probability of payment. A death benefit paid at the end of the year of death uses v^{k+1} k p_x q_{x+k}. An annuity-due paid while alive uses sum v^k k p_x. Premiums are often level and payable annually in advance. Reserves are prospective EPV of future benefits and expenses minus future premiums, or retrospectively accumulated past premiums minus benefits.
Worked example
A 2-year term assurance pays 1,000 at the end of the year of death. Let q_x=0.10 and q_{x+1}=0.20, with i=5%. The EPV is 1,000[v q_x+v^2 p_x q_{x+1}]=1,000[0.95238(0.10)+0.90703(0.90)(0.20)]=1,000[0.095238+0.163265]=258.50. If a single premium is charged, the equivalence principle gives 258.50. If premiums are payable annually in advance, divide by the appropriate annuity-due EPV.
Traps
Valuing a death benefit at the start of the year of death when it is paid at the end; using q_x instead of p_x q_{x+1} for second-year death; omitting expenses; and comparing a single premium with a level annual premium without an annuity factor.