Whole Life Insurance: Continuous and Discrete Models
Compare continuous and discrete whole life insurance models and their APV calculations for Exam LTAM.
Continuous Whole Life Insurance
A continuous whole life insurance pays a benefit of 1 at the exact moment of death, regardless of when death occurs. Since the payment is certain to be made eventually (everyone dies), the only uncertainty is the timing. The present value random variable is Z = v^{T_x} = exp(-delta * T_x), where T_x is the continuous future lifetime of the insured aged x and delta = ln(1+i) is the force of interest.
The APV (actuarial present value) is the expected value of this random variable: A-bar_x = E[v^{T_x}] = integral from 0 to infinity of v^t * t_p_x * mu_{x+t} dt. This integral does not generally have a closed form for tabular mortality, so it is evaluated numerically or approximated from discrete values. However, under certain parametric mortality assumptions, closed-form solutions exist.
Under constant force of mortality mu, A-bar_x = mu/(mu + delta). This elegant formula has clear intuition: the APV is the ratio of the mortality force to the total force (mortality plus interest). When mu is large relative to delta, the APV is close to 1 because death is expected soon and there is little discounting. When delta is large relative to mu, the APV is small because the distant expected death date means heavy discounting.
Variance and Higher Moments
The variance of the present value random variable Z = v^{T_x} measures the insurer's risk on a single policy. Using the standard formula Var(Z) = E[Z^2] - (E[Z])^2, we need the second moment E[Z^2] = E[v^{2T_x}]. Since v^{2T_x} = (v^2)^{T_x} = exp(-2*delta*T_x), this is the same as computing A-bar_x but with the force of interest doubled.
We denote this second moment as ^2A-bar_x, computed at force 2*delta. Under constant force: ^2A-bar_x = mu/(mu + 2*delta). The variance is then Var(Z) = mu/(mu + 2*delta) - (mu/(mu + delta))^2. For example, with mu = 0.04 and delta = 0.05: A-bar_x = 0.04/0.09 = 0.4444, ^2A-bar_x = 0.04/0.14 = 0.2857, and Var(Z) = 0.2857 - 0.1975 = 0.0882.
The standard deviation sqrt(0.0882) = 0.297 is substantial relative to the mean 0.4444, indicating significant risk on a single policy. This variability is why insurers write large numbers of policies: the variance of the average present value decreases as 1/n, making the portfolio outcome more predictable.
Discrete Whole Life Insurance
A discrete whole life insurance pays 1 at the end of the year of death. The present value random variable is Z = v^{K_x + 1}, where K_x is the curtate future lifetime (the number of complete years survived). The APV is A_x = sum from k=0 to infinity of v^{k+1} * k_p_x * q_{x+k}. In practice, this sum terminates at k = omega - x - 1, where omega is the limiting age of the mortality table.
The discrete APV can be computed directly from a life table by multiplying v^{k+1} * d_{x+k} / l_x for each year and summing. This is straightforward but tedious for a long remaining lifetime. The recursion A_x = v*q_x + v*p_x*A_{x+1} provides a more efficient approach. Starting from the end of the table (where A_{omega-1} = v, since death is certain within one year at the penultimate age), work backward to compute A_x at any desired age.
The recursion has a clear interpretation: A_x equals the expected cost of one year of coverage (v*q_x, the discounted probability of paying the death benefit this year) plus the expected present value of continuing the insurance (v*p_x*A_{x+1}, the discounted probability of surviving and then having a remaining life insurance obligation worth A_{x+1}).
The 1/m-thly Insurance Model
The 1/m-thly whole life insurance pays the benefit at the end of the 1/m-th of a year of death. For m = 12 (monthly), the benefit is paid at the end of the month of death. The present value random variable is Z = v^{(K_x^{(m)} + 1/m)}, where K_x^{(m)} is the 1/m-thly curtate future lifetime. The APV is denoted A_x^{(m)}.
As m increases, the 1/m-thly model approaches the continuous model. The relationship A-bar_x = lim as m goes to infinity of A_x^{(m)} connects the discrete and continuous models. In practice, the 1/m-thly APV is approximated from the annual APV using the relationship A_x^{(m)} is approximately (i/i^{(m)}) * A_x, where i^{(m)} = m*((1+i)^{1/m} - 1) is the nominal interest rate compounded m times per year.
Converting Between Models
Three key conversion relationships are essential for Exam LTAM. First, under the Uniform Distribution of Deaths (UDD) assumption within each year of age: A-bar_x = (i/delta) * A_x and A_x^{(m)} = (i/i^{(m)}) * A_x. The ratio i/delta is approximately 1 + delta/2 for small delta, reflecting that the continuous insurance pays on average half a year earlier than the discrete insurance.
Second, the exact relationship A_x = (1+i) * A-bar_x * (delta/i) under UDD can be rearranged to A-bar_x = (i/delta) * A_x. For i = 5%: delta = ln(1.05) = 0.04879, i/delta = 0.05/0.04879 = 1.0248. So the continuous APV is about 2.5% larger than the discrete APV, reflecting the earlier expected payment.
Third, the claims acceleration approach approximates A_x^{(m)} by shifting the payment time earlier by (m-1)/(2m) years: A_x^{(m)} is approximately (1+i)^{(m-1)/(2m)} * A_x. For monthly insurance (m=12), this shifts payment earlier by 11/24 of a year. Both the UDD-based and claims acceleration approximations give similar results for practical interest rates.
Common Pitfalls
A common error is confusing A_x (discrete, end of year) with A-bar_x (continuous, moment of death). The continuous APV is always larger because the benefit is paid sooner on average. Another frequent mistake is doubling the interest rate i instead of the force delta when computing the second moment. The correct approach is to compute ^2A at force 2*delta, which corresponds to an interest rate of (1+i)^2 - 1, not 2i.
When using the recursion A_x = v*q_x + v*p_x*A_{x+1}, students sometimes forget to discount both terms by v. Both the death benefit term and the continuation term are discounted one year because we are computing the APV at time 0, not at time 1.
Study Tips
Exam LTAM tests whole life insurance from multiple angles: computing APVs from life tables, deriving them under constant force assumptions, converting between continuous and discrete models, and computing variances. Practice the backward recursion until it becomes automatic. Memorize the UDD conversion formulas and the constant force formulas, as these provide quick solutions and useful verification checks. Work through problems that require computing the variance of Z for a portfolio of independent policies, which combines the single-policy variance with the law of large numbers.