Why Reconcile the Reserve and Profit Projections?
A universal life projection contains two related but distinct calculations. The policy account tracks the amount credited to the policyholder, after premiums, charges, and credited interest. The insurer’s projection tracks assets, expenses, death claims, and the liability held for policies still in force. Projected profit is the amount left after the insurer’s closing assets support that liability.
These calculations must use the same timing and policy population. In particular, a charge deducted from an account reduces the closing liability; it is not automatically a second cash receipt for the insurer. The method below makes that distinction explicit. It is a simplified annual model for a hypothetical contract, not a complete model of every universal life provision.
The Two Rollforwards
For a policy that survives the year, the account charge is deducted before credited interest. The per-survivor closing account value, used here as the closing reserve, is therefore
For the insurer’s asset projection, assume assets at the start equal the opening reserve. Premiums are received and expenses are paid at the start of the year, and those assets earn the gross investment return. The account charge is an internal transfer: it reduces the policyholder’s account liability but does not add a second amount to the insurer’s assets. Expected end-of-year claims per policy in force at the start are \(q_tB_t\). Thus, closing assets per starting policy are
Only the fraction \(p_t\) of policies survives and carries a closing reserve. The expected closing reserve per starting policy is \(p_tV_t\). Projected profit for the year, per policy in force at the start, is consequently
If there are \(N_{t-1}\) policies in force at the start, multiply this per-policy result by \(N_{t-1}\) to obtain total expected profit. The projection assumes that profit is released at the end of each year; if profit is retained in assets, those assets must be tracked separately in later years.
Why the Reconciliation Formula Holds
The formula follows from a simple balance-sheet identity. Start with assets equal to the opening reserve. Add the premium and subtract the expense; the resulting amount earns investment income. Subtract expected death claims to obtain closing assets. At year-end, the insurer must still hold the closing reserve for surviving policies. The difference between those closing assets and that liability is the year’s projected profit.
This derivation also explains the role of the account charge. It lowers \(V_t\), so it lowers the amount of the closing liability. But, in this model, it does not independently increase the assets in the profit equation. If a projection instead records a charge as an actual cash receipt, its reserve and asset definitions must be adjusted consistently; counting the same deduction in both places overstates profit.
Worked Example: Reconciling One Year of Charges and Claims
A hypothetical policy has an opening reserve of $1,000. At the start of the year it receives a premium of $200, incurs an expense of $30, and deducts a $20 account charge. The insurer earns 5% effective annual interest, while the account is credited at 3%. The annual death probability is \(q=0.01\), and a death benefit of $1,500 is paid at year-end. Assume no lapses. Calculate the closing account reserve and projected profit per policy in force at the start.
Method. First project the policy account using the credited rate. Separately project insurer assets using the gross investment return, then subtract expected claims and the reserve for survivors.
1. Closing account value. The amount after the start-of-year premium and charge is \(1{,}000+200-20=1{,}180\). Credited interest is \(1{,}180(0.03)=35.40\). Therefore, \(V_1=1{,}180+35.40=\$1{,}215.40\).
2. Closing assets before claims. The amount invested after the premium and expense is \(1{,}000+200-30=1{,}170\). Investment income is \(1{,}170(0.05)=58.50\), so assets before claims are \(1{,}170+58.50=\$1{,}228.50\).
3. Expected claims and closing reserve. Expected claims per starting policy are \(0.01(1{,}500)=\$15.00\). The survival probability is \(0.99\), so the expected closing reserve is \(0.99(1{,}215.40)=\$1{,}203.246\).
4. Profit and check. Closing assets after expected claims are \(1{,}228.50-15.00=\$1{,}213.50\). Thus profit is \(1{,}213.50-1{,}203.246=\$10.254\), or $10.25 to the nearest cent. The check is direct: closing assets equal the closing reserve plus profit, \(1{,}203.246+10.254=1{,}213.50\).
The $20 charge is reflected in the account value of $1,215.40. Adding it again as cash income would count its effect twice under the stated reserve convention.
Extending the Reconciliation Across Years
For a portfolio projection, both the number of surviving policies and the per-survivor reserve change from year to year. If \(N_{t-1}\) policies are in force at the start of year \(t\), then, under the model’s mortality assumption,
The expected closing liability is \(N_tV_t\). If profits are released at year-end, the next year’s opening assets equal that liability. The year’s total expected profit is \(N_{t-1}\Pi_t\). This approach keeps the changing policy count visible and prevents a per-survivor reserve from being confused with the total reserve for the cohort.
Worked Example: A Two-Year Projection With Survival-Weighted Reserves
A hypothetical cohort begins with one policy in force and a reserve of $400 per policy. In each year, the premium is $100, the account charge is $12, and the expense is $10. The insurer earns 4% effective annual interest, and the account is credited at 2%. In each year, the death probability is \(q=0.02\), and the end-of-year death benefit is $600. There are no lapses, and annual profits are released at year-end. Calculate the expected profit in each year and the two-year total per initial policy.
Method. Calculate each per-survivor account value, update the expected number in force, and reconcile assets to the survival-weighted closing reserve in each year.
Year 1 account and assets. The account amount before crediting is \(400+100-12=488\). Credited interest is \(488(0.02)=9.76\), giving \(V_1=488+9.76=\$497.76\). Assets before claims are \((400+100-10)(1.04)=490(1.04)=\$509.60\). Expected claims are \(0.02(600)=\$12.00\), so closing assets are \(509.60-12.00=\$497.60\).
The survival probability is \(0.98\), so the expected closing reserve is \(0.98(497.76)=\$487.8048\). Year 1 profit per initial policy is \(497.60-487.8048=\$9.7952\), or $9.80 to the nearest cent. The expected number in force at the start of year 2 is \(0.98\).
Year 2 account and assets. For each year-2 survivor, the account amount before crediting is \(497.76+100-12=585.76\). Credited interest is \(585.76(0.02)=11.7152\), giving \(V_2=585.76+11.7152=\$597.4752\).
Per year-2 policy in force, assets before claims are \((497.76+100-10)(1.04)=587.76(1.04)=\$611.2704\). Expected claims per year-2 policy in force are again \(0.02(600)=\$12.00\), leaving \(611.2704-12.00=\$599.2704\). Multiplying by the \(0.98\) expected year-2 policies in force gives closing assets per initial policy of \(0.98(599.2704)=\$587.284992\).
The expected number of policies at the end of year 2 is \(0.98(0.98)=0.9604\). The expected closing reserve per initial policy is \(0.9604(597.4752)=\$573.81518208\). Year 2 profit per initial policy is \(587.284992-573.81518208=\$13.46980992\), or $13.47 to the nearest cent.
Two-year check. Total expected profit per initial policy is \(9.7952+13.46980992=\$23.26500992\), or $23.27 to the nearest cent. In each year the reconciliation is closing assets minus closing reserve; the second-year values are weighted by the policies surviving to that year.
Solving for a Charge That Reconciles to a Profit Target
The same equations can be rearranged to find a charge consistent with a specified profit target. This is a reconciliation calculation, not a claim that a charge should be set by this simplified model alone. The result is meaningful only under the stated benefit, expense, interest, and mortality assumptions.
Worked Example: Finding the Charge for Zero Projected Profit
A hypothetical policy has an opening reserve of $500 and a start-of-year premium of $100. The expense is $10. The insurer earns 4% effective annual interest, and the account is credited at 2%. Death probability is \(q=0.05\), the death benefit is $1,500, and there are no lapses. Find the start-of-year account charge \(C\) that produces zero expected profit for the year.
Method. Calculate closing assets, which do not depend on this internal account charge under the stated asset convention. Set those assets equal to the expected closing reserve for survivors, which does depend on the charge.
Assets before claims are \((500+100-10)(1.04)=590(1.04)=\$613.60\). Expected claims are \(0.05(1{,}500)=\$75.00\). Closing assets are therefore \(613.60-75.00=\$538.60\).
The account value per survivor is \(V_1=(500+100-C)(1.02)=1.02(600-C)\). Since the survival probability is \(0.95\), the expected closing reserve per starting policy is \(0.95(1.02)(600-C)=0.969(600-C)\).
Zero profit requires closing assets to equal the expected closing reserve: \(0.969(600-C)=538.60\). Dividing gives \(600-C=538.60/0.969=555.830752\) to six decimal places. Hence \(C=600-555.830752=44.169248\), so the charge is $44.17 to the nearest cent.
Check. At the unrounded charge, the expected closing reserve is $538.60, equal to closing assets. Thus the implied profit is zero before rounding.
Common Mistake: Counting the Charge Twice
A frequent error is to include a deducted account charge as an extra cash receipt in the asset projection and also use an account value that has already been reduced by that charge. The charge is then counted once through the lower closing reserve and again as added assets.
For the first worked example, the incorrect calculation would add the $20 charge to closing assets: \(1{,}228.50+20-15=1{,}233.50\). Subtracting the correct closing reserve of $1,203.246 would give incorrect profit of $30.254. The correction is to recognize that the charge has already reduced the account value: closing assets are \(1{,}228.50-15=\$1{,}213.50\), and profit is \(1{,}213.50-1{,}203.246=\$10.254\), or $10.25 to the nearest cent.
Check Your Understanding
For each problem, use the annual timing assumptions stated in the question and reconcile closing assets to the expected reserve for survivors.
- A hypothetical policy has \(V_0=\$600\), premium $80, charge $8, expense $12, insurer return 3%, credited rate 2%, death probability \(q=0.01\), and end-of-year death benefit $700. There are no lapses. Find the closing reserve per survivor and profit per policy in force at the start. (Answer: closing reserve $685.44; profit $2.45.)
- A hypothetical policy has \(V_0=\$1{,}000\), no premium, charge $20, expense $15, insurer return 5%, credited rate 3%, death probability \(q=0.02\), and death benefit $1,200. Find the closing account reserve and projected profit per starting policy. (Answer: closing reserve $1,009.40; profit $21.04.)
- A hypothetical policy has \(V_0=\$500\), premium $100, expense $10, insurer return 4%, credited rate 2%, death probability \(q=0.05\), and death benefit $1,500. Find the account charge that gives zero expected profit. (Answer: $44.17.)
- A hypothetical policy has \(V_0=\$900\), premium $150, charge $15, expense $20, insurer return 4%, credited rate 3%, death probability \(q=0.02\), and death benefit $1,000. Find the closing account reserve and profit per policy in force at the start. (Answer: closing reserve $1,066.05; profit $6.47.)