Emerging Surplus as a Reconciliation
An asset share records the accumulated value of premiums and other inflows, less outgo, with investment income allowed for under the chosen method. Comparing actual and benchmark asset shares can help explain how experience emerged over a period. The comparison is useful only if the cash-flow timing, policy groups and basis of comparison are clear.
This tutorial develops two linked reconciliations. The first rolls aggregate assets forward from the opening value to the closing value. The second explains a difference between actual and benchmark aggregate closing asset shares by separating per-policy share differences from policy-count differences. These are two views of the same overall comparison; they must not be added together as if they were independent sources of surplus.
The benchmark must be specified. It might use expected cash flows and investment returns for the same group, or a prior-period basis carried forward under stated assumptions. A meaningful comparison requires consistent valuation dates, group definitions and treatment of expenses, benefits, taxes and other cash flows. The earlier material on per-policy versus aggregate asset shares and asset shares versus prospective reserves provides useful context: an asset share is not interchangeable with a prospective reserve.
Roll Forward Aggregate Asset Shares
For a group of policies, let \(A_0\) be opening aggregate assets, \(I\) be inflows during the period, \(O\) be outflows, \(J\) be investment earnings, and \(A_1\) be closing aggregate assets. With inflows and outflows recorded at their stated amounts and investment earnings recorded separately, the accounting identity is:
This identity is a bookkeeping reconciliation, not a prescription for calculating earnings. The method for \(J\) depends on when cash flows occur and how investment income is allocated. If cash flows happen at different times, each must be accumulated or discounted using the appropriate timing convention. The examples below state their timing and earnings bases explicitly.
State the period, policy groups, benchmark basis and whether amounts are aggregate or per policy.
Start with opening assets, add inflows and investment earnings, and subtract outflows, keeping the timing convention consistent.
Subtract benchmark closing assets from actual closing assets; then, if appropriate, explain that difference by policy group.
Worked Example: Explaining a Zero Net Difference
A hypothetical block has opening aggregate assets of $100,000. A premium of $20,000 is received at the start of the year. Actual investment earnings are 4% of the $120,000 invested immediately after that premium is received. At year-end, expenses of $3,000 and claims of $5,000 are paid. The benchmark uses the same opening assets and premium, a 3% investment return on $120,000, expenses of $2,800 and claims of $4,000. Find the actual and benchmark closing assets, and reconcile the surplus by source.
Method. Apply the roll-forward identity separately to actual and benchmark experience. Since the premium is invested at the start of the year, the stated return applies to opening assets plus that premium. Expenses and claims are paid at year-end, so they are not included in the earnings base.
Actual investment earnings: \(0.04(120{,}000)=4{,}800\). Therefore actual closing assets are:
Benchmark investment earnings: \(0.03(120{,}000)=3{,}600\). Therefore benchmark closing assets are:
The emerging surplus is \(116{,}800-116{,}800=0\). Its components relative to the benchmark are investment earnings \(4{,}800-3{,}600=+1{,}200\), expenses \(-(3{,}000-2{,}800)=-200\), and claims \(-(5{,}000-4{,}000)=-1{,}000\). Their sum is \(1{,}200-200-1{,}000=0\), matching the direct closing comparison. The common mistake avoided here is to treat a favorable investment difference as the whole surplus while overlooking unfavorable expense and claim differences.
Separating Per-Policy Shares from Policy Counts
An aggregate difference can arise because the average asset share per policy differs from the benchmark, because the number of policies differs, or because both change. For a single group, let \(N_a\) and \(s_a\) be the actual count and actual closing share per policy, and let \(N_b\) and \(s_b\) be their benchmark counterparts. The aggregate difference is \(N_as_a-N_bs_b\).
For multiple groups \(g\), apply the identity within each group and sum:
This allocation makes the two effects add exactly to the total. It does not assert that count changes are independent of experience in an economic or causal sense. It is a transparent arithmetic bridge, and its interpretation depends on the group definitions and chosen benchmark.
Why the Decomposition Holds
The identity follows by adding and subtracting the intermediate aggregate \(N_as_b\). This intermediate amount uses the actual count but the benchmark share. It creates a two-step bridge: first change the share at the actual count, then change the count at the benchmark share.
The proof also shows why the convention matters. If instead the count effect were valued at the actual share, the experience effect would need to be valued at the benchmark count to preserve the same total. Either can be a valid bridge when stated clearly; mixing terms from the two conventions can leave an unexplained interaction.
Worked Example: Two Groups with Offsetting Share Experience
A hypothetical portfolio has two groups. For Group A, the benchmark count is 5 and benchmark share is $2,000; actual count is 6 and actual share is $2,100. For Group B, the benchmark count is 4 and benchmark share is $3,000; actual count is 3 and actual share is $2,800. Calculate the aggregate difference directly and decompose it into actual-count experience and benchmark-share count effects.
Method. Calculate each group’s actual and benchmark aggregate, then apply the stated decomposition within each group. This makes signs and group contributions visible.
Group A actual aggregate is \(6(2{,}100)=\$12{,}600\); its benchmark aggregate is \(5(2{,}000)=\$10{,}000\). The direct difference is \(+\$2{,}600\). Its experience effect is \(6(2{,}100-2{,}000)=+\$600\), and its count effect is \((6-5)(2{,}000)=+\$2{,}000\). These sum to \(+\$2{,}600\).
Group B actual aggregate is \(3(2{,}800)=\$8{,}400\); its benchmark aggregate is \(4(3{,}000)=\$12{,}000\). The direct difference is \(-\$3{,}600\). Its experience effect is \(3(2{,}800-3{,}000)=-\$600\), and its count effect is \((3-4)(3{,}000)=-\$3{,}000\). These sum to \(-\$3{,}600\).
Across both groups, actual aggregate assets are \(12{,}600+8{,}400=\$21{,}000\), and benchmark aggregate assets are \(10{,}000+12{,}000=\$22{,}000\). The total difference is \(-\$1{,}000\). Total experience is \(600-600=\$0\), and total count effect is \(2{,}000-3{,}000=-\$1{,}000\); together they give \(-\$1{,}000\). The common mistake avoided is reporting Group A’s favorable share experience without also including Group B’s offsetting unfavorable share experience.
Worked Example: Reconciling Cash Flows and Closing Shares
A hypothetical benchmark block starts with $14,000 of aggregate assets and receives $2,000 of premiums at the start of the year. Benchmark investment earnings are $450; year-end outgo is $1,450. At closing, the benchmark has 10 policies with a share of $1,500 each. Actual experience starts with the same $14,000, receives $2,400 of start-of-year premiums, earns $600, and has year-end outgo of $1,050. Actual closing count is 11 and the actual share is $1,450 per policy. Reconcile the emerging surplus in both ways.
Method. First compare the roll-forward totals by source. Then independently compare the closing aggregates using the count-and-share identity. The two calculations should agree, but they are not separate amounts to add.
The benchmark roll-forward gives \(14{,}000+2{,}000+450-1{,}450=\$15{,}000\), which also equals \(10(1{,}500)=\$15{,}000\). The actual roll-forward gives \(14{,}000+2{,}400+600-1{,}050=\$15{,}950\), which also equals \(11(1{,}450)=\$15{,}950\). The direct emerging surplus is therefore \(15{,}950-15{,}000=+\$950\).
By cash-flow source, the premium difference is \(2{,}400-2{,}000=+\$400\), the investment-earnings difference is \(600-450=+\$150\), and the outgo effect is \(-(1{,}050-1{,}450)=+\$400\). Total: \(400+150+400=+\$950\).
By closing share and count, the experience effect is \(11(1{,}450-1{,}500)=-\$550\); the count effect is \((11-10)(1{,}500)=+\$1{,}500\). Their sum is \(-550+1{,}500=+\$950\), matching the roll-forward. The common mistake avoided is to add the $950 cash-flow reconciliation to the $950 closing-share reconciliation; both explain the same difference.
Interpretation and Common Traps
A surplus bridge is only as informative as its basis. In particular, a difference in closing counts can reflect entries, exits, coverage changes or grouping choices. The count effect is not automatically a mortality, lapse or new-business gain: interpreting its cause requires a suitable exposure and cash-flow analysis. If a group is heterogeneous, broad grouping can also disguise offsetting per-policy differences, so a finer grouping may be needed for a useful explanation.
When actual and benchmark roll-forwards use different cash-flow timing, the investment difference may combine changes in return with changes in the amount or timing of invested funds. Separate these effects only after specifying an earnings allocation method. A simple rate times a single asset base is valid only when the stated timing supports that calculation.
- Reversing the comparison: define surplus as actual less benchmark; a negative result means actual is lower.
- Using the wrong count in a component: under this convention, experience uses actual count and count effect uses benchmark share.
- Dropping a group’s negative contribution: calculate and sum group effects with their signs intact.
- Double counting: cash-flow and endpoint bridges are alternative reconciliations of one total difference.
- Calling every difference profit: emerging surplus is relative to a stated asset-share benchmark and does not by itself establish distributability or solvency status.
This reconciliation approach is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. In an exam setting, state the benchmark, define the sign convention, show the aggregate totals, and verify the component sum.
Check Your Understanding
For each problem, use the stated sign convention and check the components against a direct calculation.
- A block begins with $50,000. A premium of $8,000 is received at the start of the year and earns 5% on the resulting $58,000 investment base. Year-end expenses are $1,800 and claims are $6,000. Find closing aggregate assets. (Answer: $53,100)
- Benchmark count and share are 8 and $1,250; actual count and share are 9 and $1,300. Using actual-count experience and benchmark-share count, find the aggregate difference and both components. (Answer: Difference +$1,700; experience +$450; count +$1,250)
- Group A has benchmark count 4 and share $1,000, and actual count 5 and share $1,100. Group B has benchmark count 3 and share $2,000, and actual count 2 and share $1,900. Find total experience effect, count effect and aggregate difference. (Answer: Experience +$300; count -$1,000; difference -$700)
- A benchmark starts with $80,000, receives a $10,000 premium at the start of the year, earns $2,700, and pays $1,800 of expenses and $4,500 of claims at year-end. Actual experience has the same opening assets and premium, earns $3,600, and pays $2,000 of expenses and $5,000 of claims. Find the actual and benchmark closing assets and emerging surplus. (Answer: Actual $86,600; benchmark $86,400; surplus +$200)
- Benchmark count and share are 12 and $900; actual count and share are 10 and $980. Find the direct aggregate difference and its decomposition using actual-count experience and benchmark-share count. (Answer: Difference -$1,000; experience +$800; count -$1,800)