Why Compare Surrender Payments With Asset Shares?
When a policy ends by surrender, the insurer makes a payment under the policy’s surrender terms. The accumulated asset share at that date provides a retrospective measure of the assets attributed to the policy under the chosen cash-flow and allocation conventions. Comparing the two shows whether the attributed assets exceed the payment, or whether the payment is larger.
The comparison depends on using the same policy, date, and accounting basis for both amounts. In particular, the asset share must be measured immediately before the surrender payment, and cash flows must be accumulated for the correct number of investment periods. A positive difference is not automatically a distributable profit: it is the amount remaining under the stated asset-share convention after paying the surrender value.
Accumulating Cash Flows to the Surrender Date
Asset shares are built from cash flows and investment returns according to a stated projection convention. For the examples here, cash flows occur at the beginning of each policy year, annual investment returns are credited at year-end, and the surrender payment occurs at the termination date after all relevant returns have been credited. Net cash flow means amounts entering the asset share minus amounts leaving it, such as expenses included in the projection.
If \(C_k\) is the net cash flow at the beginning of year \(k\), and \(j_k\) is the effective return for the year from time \(k\) to \(k+1\), the asset share just before a surrender payment at time \(n\) is
Each cash flow earns returns only for the periods after it enters the asset share. A cash flow at time 0 earns \(n\) years of return by time \(n\); a cash flow at time 1 earns \(n-1\) years. If a projection uses a different timing convention, the accumulation factors must be adjusted to match it.
Worked Example: Counting Investment Periods Correctly
A hypothetical policy has net cash flows of $275 at time 0 and $325 at time 1, each at the beginning of a policy year. The effective annual return is 4%. Surrender occurs at time 2, after the return for the second year has been credited and before the surrender payment. The stipulated surrender payment is $620. Find the pre-payment asset share and the residual.
Method. Accumulate each cash flow from its own entry date to time 2, then subtract the surrender payment. This avoids the common mistake of giving both cash flows the same accumulation period.
The time-0 cash flow earns two years of return:
The time-1 cash flow earns one year:
Therefore, the asset share immediately before payment is
The residual is
Check. The time-0 amount has earned two returns and the time-1 amount has earned one; the resulting asset share exceeds the payment by $15.44. The amount remaining under these assumptions is therefore $15.44.
Why the Residual Formula Holds
At the termination date, \(A_t^-\) is the amount attributed to the policy immediately before the surrender transaction. Paying \(W_t\) removes that amount from the assets attributed to the policy. The balance immediately after the payment is consequently the pre-payment amount less the payment:
This is an accounting identity, not a claim that the asset share is itself a contractual benefit. The surrender payment is determined by the policy terms and any applicable surrender-value basis. The asset share is a retrospective measure determined by the specified cash flows, returns, and allocation rules. Their difference quantifies the gap on those bases.
Identify when the final investment return is credited, when any final cash flow is included, and when the surrender payment is made.
Accumulate each net cash flow for the periods it remains in the asset share, using the specified returns and allocation convention.
Apply the stated guarantee, additions, and deductions to obtain the amount actually paid.
Subtract the payment from the pre-payment asset share. State whether the result is an amount remaining or a shortfall.
Worked Example: Including Earlier Net Expenses
A hypothetical policy has net cash flows at the beginning of each year: $900 at time 0, $400 at time 1, and $350 at time 2. These figures already reflect the expenses included in the asset-share projection. The annual effective return is 3%, credited at each year-end. The policy surrenders at time 3. Its stipulated payment is a guaranteed amount of $1,700 plus a vested addition of $45. Calculate the asset share and residual.
Method. Accumulate each net cash flow separately to time 3. The surrender payment is calculated from its stated components and is not used as an asset-share cash flow before termination.
The time-0 amount earns three years of return:
The time-1 amount earns two years:
The time-2 amount earns one year:
Thus the asset share immediately before payment is
The payment is
So
Answer and check. Rounding only the final monetary amounts to the nearest cent gives an asset share of $1,768.31 and a residual of $23.31. The payment components sum to $1,745, which is less than the unrounded asset share by $23.3143.
The mistake to avoid is subtracting the earlier expenses again: they are already included in the given net cash flows. Subtracting them a second time would understate the asset share.
Surrender Terms and the Basis of the Comparison
A surrender payment may be specified as a guaranteed amount, as a fund value less a charge, or as a combination of guaranteed and non-guaranteed components. Use the payment actually due under the terms given in the problem. If a surrender charge is deducted from a gross amount to determine the payment, compare the asset share with that net payment. Do not deduct the same charge from the asset share as well unless the projection explicitly records it as a separate cash flow into or out of the asset share.
Worked Example: Applying a Surrender Charge Once
A hypothetical policy has net cash flows of $1,000 at time 0 and $150 at time 1, both at the beginning of the year. The annual effective return is 5%, and surrender occurs at time 2. The contract specifies a gross surrender amount of $1,300, reduced by a charge equal to 5% of that gross amount. Find the payment and compare it with the asset share.
Method. First accumulate the asset-share cash flows. Separately calculate the contract payment after its stated charge.
The time-0 cash flow earns two years of return:
The time-1 cash flow earns one year:
Therefore,
The charge is \(0.05(1300)=65\), so the payment is
The residual is
Check and trap. The correct comparison is $1,260 against the net payment of $1,235, leaving $25. Subtracting the $65 charge from the asset share as well and then comparing with $1,235 would apply the charge twice. The charge has already reduced the payment in this problem.
Allocation Conventions Can Change Policy-Level Results
A shared amount, such as an expense included in the asset-share projection, may be allocated among policies according to a stated convention. Different allocations can change individual asset shares and therefore individual surrender residuals. They need not change the aggregate asset share if the same total expense is allocated across the whole group and all policies are included.
The following example shows why an aggregate comparison cannot always be used to infer that every individual policy has a positive residual.
Worked Example: Comparing Two Policies Under an Allocation Rule
Two hypothetical policies have asset-share inflows of $600 and $300 at time 0. A shared expense of $30 is charged at time 1, and the annual effective return is 5%. Both policies surrender at time 2. The expense is allocated in proportion to the time-0 inflows. The surrender payments are $630 for Policy A and $330 for Policy B. Find each pre-payment asset share and residual, and check the aggregate.
Method. Allocate the shared expense first, accumulate each policy’s inflow for two years, and accumulate its allocated expense for the one year from time 1 to time 2.
The allocation proportions are \(600/(600+300)=2/3\) for Policy A and \(300/(600+300)=1/3\) for Policy B. Thus the expense allocations are
For Policy A, the time-0 inflow accumulates to \(600(1.05)^2=661.50\), and its allocated expense accumulates to \(20(1.05)=21.00\). Therefore,
For Policy B, the time-0 inflow accumulates to \(300(1.05)^2=330.75\), and its allocated expense accumulates to \(10(1.05)=10.50\). Therefore,
The aggregate asset share and aggregate payment are
The aggregate residual is \(960.75-960=0.75\), which also equals \(10.50-9.75=0.75\). Policy A has $10.50 remaining, while Policy B’s payment exceeds its allocated asset share by $9.75.
Check and trap. The two allocated expenses total $30, and their accumulated value is \(30(1.05)=31.50\). The total accumulated inflow is \(900(1.05)^2=992.25\); subtracting $31.50 gives the aggregate asset share of $960.75. A positive aggregate residual does not eliminate the individual shortfall for Policy B.
Common Exam Traps
- Using the wrong number of return periods. A cash flow earns returns only after it enters the asset share. Trace each cash flow from its date to the surrender date.
- Comparing amounts at different dates. Accumulate the asset share to the payment date, or otherwise put both quantities on the same date and basis.
- Using the gross surrender amount instead of the payment due. Apply stipulated charges and additions to identify the actual amount paid.
- Double-counting a charge or expense. Establish whether it has already been reflected in the asset-share cash flows, the surrender payment, or both under an explicitly stated convention.
- Reading an aggregate result as an individual result. Shared allocations can produce different policy-level residuals even when the group total is positive.
Check Your Understanding
For each problem, compare the asset share immediately before surrender with the payment due. Unless stated otherwise, cash flows occur at the beginning of each year and returns are credited at year-end.
- A hypothetical policy has a net inflow of $450 at time 0 and $200 at time 1. The annual effective return is 4%, and surrender occurs at time 2 with a payment of $700. Find the pre-payment asset share and residual. (Answer: Asset share $694.72; residual −$5.28, a shortfall.)
- A policy has net inflows of $500 at time 0, $250 at time 1, and $100 at time 2. The annual effective return is 3%, and surrender occurs at time 3 for $880. Find the asset share and residual, rounding to cents. (Answer: Asset share $914.59; residual $34.59.)
- Two policies have time-0 inflows of $400 and $200. At time 1, a shared expense of $18 is allocated in proportion to those inflows. The annual effective return is 5%, and both surrender at time 2. Their payments are $430 and $220, respectively. Find the two residuals and the aggregate residual. (Answer: First residual −$1.60; second residual −$5.80; aggregate residual −$7.40.)
- A policy has a net inflow of $1,200 at time 0, earns an annual effective return of 4%, and surrenders at time 2. Its gross surrender amount is $1,500, reduced by a charge of 6% of the gross amount. Find the asset share, actual payment, and residual. (Answer: Asset share $1,297.92; payment $1,410.00; residual −$112.08, a shortfall.)