New-Business Capital Strain

Why New Business Can Require Capital

A new policy may receive a premium at issue, but that premium is not automatically available as free capital. Acquisition expenses and other immediate outflows reduce the assets generated by the contract, while a technical provision must be held for the policy’s future obligations. The insurer may also need to hold own funds against a separate capital requirement. The difference between the capital requirement and the own funds generated by the new business is its initial capital strain, if that difference is positive.

The calculation is an issue-date balance-sheet comparison. It is not a projection of later profits, and it does not mean that the technical provision and capital requirement are the same thing. The provision values policy obligations; the capital requirement specifies the own funds needed in addition to the provision under the stated capital basis.

Definition: For an incremental new-business calculation, let $P_0$ be premiums received at issue, $E_0$ be issue expenses, and $B_0$ be any other immediate outflows, such as an immediate benefit. With no opening assets assigned to the policy, issue-date assets are $A_0=P_0-E_0-B_0$. Let $V_0$ be the issue-date technical provision and $C_0$ the capital requirement allocated to the policy. Issue-date own funds are $OF_0=A_0-V_0$. The initial capital strain is $S_0=\max(0,C_0-OF_0)$.

The calculation uses amounts measured on a consistent basis and at the same valuation date. In this tutorial, the technical provision and capital requirement are inputs supplied by the problem or by an explicitly stated valuation basis. They are not inferred from the premium. A policy-level capital allocation is also assumed to have been provided where necessary: in practice, a capital requirement for an insurer may be calculated for the business as a whole and may not be additive by policy.

Deriving the Capital-Strain Formula

The formula follows from the balance sheet and the amount of additional capital needed to meet the requirement. With assets $A_0$ and technical provisions $V_0$, own funds are the excess of assets over provisions, $OF_0=A_0-V_0$. Suppose the insurer contributes an amount $K$ to the policy’s assets. The contribution does not change the provision, so post-contribution own funds are $OF_0+K$. To meet the requirement, they must satisfy $OF_0+K\geq C_0$, and a capital contribution cannot be negative. Therefore the smallest adequate contribution is the greater of zero and $C_0-OF_0$.

OF_0+K\geq C_0,\qquad K\geq 0 \quad\Longrightarrow\quad S_0=\min\{K:K\geq0,\ OF_0+K\geq C_0\} =\max(0,C_0-OF_0).

This is a funding interpretation of strain: it is the minimum additional capital needed to bring own funds up to the stated requirement. If own funds already meet or exceed the requirement, strain is zero. Any amount above the requirement is a surplus relative to that requirement, not negative strain.

Calculate issue-date assets.
Add the issue-date cash flows with their correct signs: premiums are inflows; expenses and immediate benefits are outflows.

Deduct the technical provision.
Subtract $V_0$ from the assets to obtain issue-date own funds $OF_0$.

Compare own funds with required capital.
If $OF_0<C_0$, strain is $C_0-OF_0$. If $OF_0\geq C_0$, strain is zero and the excess is $OF_0-C_0$.

Worked Example: Strain on a Single-Premium Policy

A hypothetical insurer issues a policy for a single premium of $12,000. It incurs $850 of acquisition expense and pays a $250 immediate benefit. The issue-date technical provision is $10,200, and the allocated capital requirement is $1,300. Calculate the issue-date own funds and capital strain. The method is to build the balance sheet from cash flows, then compare own funds with the requirement.

Issue-date assets. The premium is an inflow and both the expense and benefit are outflows:

A_0=12{,}000-850-250=10{,}900.

Own funds. Deduct the technical provision from assets:

OF_0=10{,}900-10{,}200=700.

Capital strain. Own funds are $600 below the $1,300 requirement:

S_0=\max(0,1{,}300-700)=600.

The check is $700+600=1{,}300$: after a $600 capital contribution, own funds exactly meet the requirement. The common mistake this method avoids is treating the $10,200 provision itself as the capital requirement; they are distinct balance-sheet quantities.

Negative Own Funds and Capital Surplus

Issue-date own funds can be negative. This occurs when issue-date assets are less than the technical provision. It does not make the capital strain negative: the capital required to reach $C_0$ must cover both the own-funds deficit and the requirement. Conversely, own funds can exceed the requirement, in which case the policy has no initial strain under the stated measure.

Relationship at issue	Capital strain	Amount to check
$OF_0<C_0$	$C_0-OF_0$	$OF_0+S_0=C_0$
$OF_0=C_0$	0	Own funds equal the requirement
$OF_0>C_0$	0	Excess is $OF_0-C_0$, so $OF_0=C_0+\text{excess}$

Worked Example: Strain When Own Funds Are Negative

A hypothetical long-term contract produces a $7,000 premium at issue. Acquisition expenses are $900 and an immediate benefit of $300 is paid. The technical provision is $6,200, and the allocated capital requirement is $800. Calculate the strain, including the effect of negative own funds. The method is to retain the sign of own funds before applying the strain formula.

Assets. After the two outflows, the issue-date assets are:

A_0=7{,}000-900-300=5{,}800.

Own funds. Assets are $400 below the provision:

OF_0=5{,}800-6{,}200=-400.

Strain. The gap from negative $400 to the required positive $800 is $1,200:

S_0=\max(0,800-(-400))=1{,}200.

After a $1,200 contribution, own funds are $-400+1{,}200=800$, exactly the requirement. A common error is to discard the negative sign and calculate $800-400=400$. That understates the necessary capital: the contribution must first cover the $400 shortfall in assets relative to the provision, as well as provide $800 of own funds.

Worked Example: Finding the Minimum Premium

For a hypothetical policy with premium $P$, acquisition expenses are $0.10P+\$100$, a further immediate outflow is $50, and the technical provision is $0.75P$. The capital requirement is $200. Premiums must be quoted in whole cents. Find the minimum premium that produces no capital strain. The method is to express own funds as a function of $P$, solve the inequality exactly, and then round the premium upward to an available cent amount.

Own funds as a function of premium. Issue-date assets are the premium less expenses and the immediate outflow. Subtracting the provision gives:

\begin{aligned} A_0&=P-(0.10P+100)-50=0.90P-150,\\ OF_0&=A_0-0.75P=0.15P-150. \end{aligned}

Set the no-strain condition. Own funds must be at least the $200 requirement:

0.15P-150\geq200 \quad\Longrightarrow\quad 0.15P\geq350 \quad\Longrightarrow\quad P\geq\frac{350}{0.15}=2{,}333.333\ldots.

The exact threshold is $2,333.333…; it is not itself an available whole-cent premium. The least whole-cent premium meeting the inequality is therefore $2,333.34. Check using unrounded intermediate values: $OF_0=0.15(2{,}333.34)-150=200.001$, which is at least $200. At $2,333.33, $OF_0=0.15(2{,}333.33)-150=199.9995$, which is below $200. Thus rounding the threshold to the nearest cent would not produce a sufficient premium; the minimum must be rounded upward.

Checks, Assumptions, and Common Traps

The strain formula is an issue-date measure under the stated cash-flow, provision and capital assumptions. Use it only when those amounts are measured consistently and the capital requirement represents the relevant requirement for the business being assessed. It does not, by itself, describe the policy’s future profits or future capital needs. Those would require additional projection assumptions.

A reliable check starts with the balance-sheet identity $A_0=V_0+OF_0$. Then check the relationship between own funds, strain and the requirement: when own funds are below the requirement, $OF_0+S_0=C_0$. When own funds exceed the requirement, strain remains zero; instead, verify $OF_0=C_0+\text{excess}$. Do not assert that own funds plus strain always equals the requirement.

Subtracting the provision twice. First calculate assets from cash flows. Then subtract the provision once to obtain own funds.
Using the provision as the capital requirement. A technical provision and a capital requirement are separate inputs with different roles.
Ignoring negative own funds. If $OF_0<0$, retain its sign when calculating $C_0-OF_0$.
Reporting negative strain. If own funds exceed required capital, report zero strain and identify the excess separately.
Rounding a threshold in the wrong direction. When a minimum amount must satisfy an inequality and is restricted to cents, round upward if ordinary nearest-cent rounding falls below the threshold.

Key takeaway: Initial capital strain is the shortfall, if any, between issue-date own funds and the stated capital requirement. Calculate assets from initial cash flows, subtract the technical provision to find own funds, and only then compare those own funds with required capital.

The method is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. In an exam calculation, state the basis and signs, show the asset and own-funds steps, and interpret the result as either a funding shortfall or no strain with an excess.

Check Your Understanding

For each hypothetical case, calculate issue-date own funds and capital strain or excess as requested. Keep unrounded values until the final reported amount.

A policy receives a $5,000 premium at issue, has $400 of issue expenses and no other immediate outflow, and has a $4,300 technical provision and a $500 capital requirement. Find own funds and strain. (Answer: own funds $300; strain $200.)
A contract generates $12,000 of premium, incurs $700 of expenses and a $300 immediate benefit, and has a $9,500 provision and a $1,000 requirement. Find own funds, strain and any excess. (Answer: own funds $1,500; strain $0; excess $500.)
At issue, a hypothetical policy has a $3,000 premium, $600 of expenses, and a $2,200 technical provision. Its capital requirement is $450. Calculate assets, own funds and strain, and verify the post-contribution own funds. (Answer: assets $2,400; own funds $200; strain $250; post-contribution own funds $450.)
A product has premium $P$, issue expenses $0.20P+\$100$, no other immediate outflow, a provision of $0.70P$, and a $100 capital requirement. Find the minimum premium that eliminates strain, assuming premiums may be any positive amount. (Answer: $P=\$2,000$; own funds are $P-(0.20P+100)-0.70P=0.10P-100$, which equal $100 at that premium.)
A policy has $8,000 of premium, $500 of expenses and $300 of immediate outflows. Its provision is $6,500 and its capital requirement is 10% of that provision. Find own funds, the capital requirement, strain and excess. (Answer: own funds $700; requirement $650; strain $0; excess $50.)

Relationship at issue	Capital strain	Amount to check
\(OF_0<C_0\)	\(C_0-OF_0\)	\(OF_0+S_0=C_0\)
\(OF_0=C_0\)	0	Own funds equal the requirement
\(OF_0>C_0\)	0	Excess is \(OF_0-C_0\), so \(OF_0=C_0+\text{excess}\)

What You'll Learn