Recursion Formulas for Insurance and Annuity Values
Apply recursion formulas to efficiently compute insurance and annuity values for Exam LTAM.
Why Recursions Matter
Recursion formulas express the value of a life insurance or annuity contract in terms of the value at a later age or duration. They are fundamental in actuarial science because they provide an efficient computational method for building tables of values, reveal the economic structure of insurance contracts, and serve as the basis for reserve development. On Exam LTAM, recursions appear in both computational and conceptual questions.
The core idea behind every recursion is the same: condition on what happens in the first year. Either the insured survives the year (with probability p_x) or dies during the year (with probability q_x). Each outcome produces a different cash flow and a different continuation value, and the current value is the expected present value across both outcomes.
Insurance Recursions
The whole life insurance recursion is A_x = v * q_x + v * p_x * A_{x+1}. The first term, v * q_x, is the expected present value of the death benefit if death occurs in the first year. The second term, v * p_x * A_{x+1}, is the expected present value of the remaining insurance if the insured survives. This recursion enables backward computation: starting from the oldest age in the life table (where A_omega = v * 1 = v, since death is certain), you can compute A_x for every younger age.
For n-year term insurance, the recursion is A_{x:n}^1 = v * q_x + v * p_x * A_{x+1:n-1}^1, with the boundary condition A_{x:0}^1 = 0 (a zero-year term insurance has no coverage). For endowment insurance, A_{x:n} = v * q_x + v * p_x * A_{x+1:n-1}, with boundary A_{x:0} = 1 (the pure endowment pays 1 at maturity). The endowment boundary differs from term insurance because the maturity benefit is payable upon survival to the end of the term.
Annuity Recursions
The whole life annuity-due recursion is a-ddot_x = 1 + v * p_x * a-ddot_{x+1}. The "1" represents the current payment (due immediately), and v * p_x * a-ddot_{x+1} is the discounted, survival-weighted value of future payments. This recursion starts from the oldest age where a-ddot_omega = 1 (only the immediate payment, since death is certain within the year) and works backward.
For a temporary annuity-due, a-ddot_{x:n} = 1 + v * p_x * a-ddot_{x+1:n-1}, with boundary a-ddot_{x:0} = 0 (no payments in a zero-duration annuity). For a deferred annuity, the recursion follows from the identity: n|a-ddot_x = v * p_x * (n-1)|a-ddot_{x+1}, reflecting that survival through the first year brings the deferral one year closer.
Reserve Recursions
The reserve recursion is one of the most important formulas in life contingencies. For a policy with net premium P and death benefit b, the recursion is: (tV + P) * (1 + i) = q_{x+t} * b + p_{x+t} * (t+1)V. This equation has a clear financial interpretation. The left side is the fund available at the end of the year: the reserve at the start of the year plus the premium, accumulated with interest. The right side shows how this fund is allocated: part covers the expected death claims, and the rest provides the reserve for survivors.
Rearranging gives (t+1)V = [(tV + P)(1+i) - q_{x+t} * b] / p_{x+t}, which allows forward computation of reserves. Starting with 0V = 0 (for a net level premium policy), you can compute the reserve at every subsequent duration. This forward recursion is especially useful for building reserve tables and for profit testing.
Connecting Insurance and Annuity Recursions
The fundamental relationship A_x = 1 - d * a-ddot_x connects insurance and annuity values. Substituting this into the insurance recursion and simplifying produces the annuity recursion, confirming that these are not independent relationships. This connection also means that any recursion for insurance values implies a corresponding recursion for annuity values, and vice versa.
The net premium P_x = A_x / a-ddot_x can also be expressed recursively. Using the reserve recursion with 0V = 0, the first-year equation gives P * (1+i) = q_x * b + p_x * 1V. This determines the reserve at duration 1 in terms of the premium and first-year mortality, which is often a useful starting point for reserve calculations.
Common Pitfalls
A frequent mistake is applying the wrong boundary condition. Term insurance has boundary value 0 (coverage ends), while endowment insurance has boundary value 1 (maturity benefit is paid). Confusing these changes the entire table of values. Another common error is forgetting the discount factor v in each term of the insurance recursion. Every future cash flow must be discounted by one year.
In reserve recursions, students sometimes place the premium on the wrong side of the equation or forget to accumulate with interest. Remember that the reserve plus premium is the available fund at the start of the year, and it must be accumulated to the end of the year before comparing to outflows.
Study Tips
Exam LTAM uses recursions extensively. Practice computing tables of insurance values, annuity values, and reserves using recursions starting from known boundary values. Be comfortable working both forward (from duration 0) and backward (from the oldest age or the end of the term). Make sure you can write down the recursion for any standard product (whole life, term, endowment) from first principles by conditioning on first-year outcomes. Understanding the recursion structure deeply will help you tackle unfamiliar problems by applying the same conditioning logic.