Deriving Capital From A Prescribed Stress

From a Prescribed Stress to a Capital Charge

A prescribed stress becomes a capital calculation only after its effects have been translated into values on the solvency balance sheet. The central task is to compare the balance sheet before and after the stress: an asset-value decline reduces surplus, while a liability-value increase also reduces surplus. A liability decrease or asset increase can offset that loss.

This tutorial derives a hypothetical charge for one specified stress. It assumes the stress and the valuation basis have already been prescribed. It does not assign a probability to the stress or determine its calibration. The calculation is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Definition: Let $A_0$ and $L_0$ be the values of assets and liabilities before a prescribed stress, and let $A_1$ and $L_1$ be their values after revaluation under that stress. Initial surplus is $S_0=A_0-L_0$; stressed surplus is $S_1=A_1-L_1$. The signed stress loss is $S_0-S_1$. Under the positive-part convention used here, the hypothetical capital charge is $K=\max(0,S_0-S_1)$.

The positive-part convention means that a stress producing a surplus gain does not create a negative charge. A particular regulatory rule may specify a different treatment, such as allowing gains to offset charges elsewhere. Use the stated rule if one is provided. The charge $K$ is also distinct from stressed surplus $S_1$: the first measures how much surplus changed, while the second is the balance-sheet position after the stress.

Why the Change in Surplus Gives the Charge

The result follows directly from the definition of surplus. It is useful to show the steps explicitly because the liability term is a frequent source of sign errors.

Write surplus before the stress.
$S_0=A_0-L_0$.

Write surplus after revaluation.
$S_1=A_1-L_1$.

Subtract stressed surplus from initial surplus.
$S_0-S_1=(A_0-L_0)-(A_1-L_1)=(A_0-A_1)+(L_1-L_0)$.

Apply the stated charge convention.
$K=\max(0,(A_0-A_1)+(L_1-L_0))$. Asset decreases and liability increases contribute positively; asset increases and liability decreases contribute negatively.

\text{signed stress loss} = S_0-S_1 = (A_0-A_1)+(L_1-L_0)

This identity is exact when all four values use the same balance-sheet scope and valuation basis, and the stressed values include all changes required by the scenario. It does not require every asset to fall or every liability to rise. The signed terms preserve the direction of each revaluation. The positive-part charge is then applied to the total signed loss, not separately to each line unless the prescribed method explicitly requires that.

Revalue Each Balance-Sheet Component

For each item, determine its starting value and how the prescribed stress changes that value. A percentage shock is applied to the affected item’s stated base value. An item not affected by the stress remains at its base value if the problem says so. If the problem instead provides stressed values directly, use those values without applying the shock a second time.

The calculation also depends on scope. For example, a liability component that decreases in value is a reduction in liabilities and therefore improves surplus, all else equal. Do not treat a decrease in liabilities as an asset loss. Unless specified otherwise, the examples below assume no taxes, management actions, new business, or other changes during the stress. They treat every listed amount as part of the same hypothetical solvency balance sheet.

Worked Example: Revalue Assets and Liabilities Under One Stress

A hypothetical insurer reports balance-sheet amounts in millions of dollars. Before a prescribed stress, it has equity investments of $60 million, bonds of $100 million, and cash of $90 million. Under the stress, equities decline by 15%, bonds decline by 3%, and cash is unchanged. Its liabilities are a claims provision of $150 million and an expense provision of $80 million. Under the same stress, the claims provision increases by 6% and the expense provision decreases by 4%. Calculate initial surplus, stressed surplus, and the positive-part charge.

Method. Revalue each asset and liability from its own base value, sum the stressed components, and compare the two surplus amounts.

Assets. The equity decline is $60(0.15)=9$, so stressed equities are $60-9=51$. The bond decline is $100(0.03)=3$, so stressed bonds are $100-3=97$. Cash remains $90$. Thus $A_0=60+100+90=250$, and $A_1=51+97+90=238$. The total asset decline is $250-238=12$.

Liabilities. The claims provision increases by $150(0.06)=9$, giving $150+9=159$. The expense provision decreases by $80(0.04)=3.2$, giving $80-3.2=76.8$. Thus $L_0=150+80=230$, and $L_1=159+76.8=235.8$. The net liability increase is $235.8-230=5.8$.

Surplus and charge. Initial surplus is $S_0=250-230=20$. Stressed surplus is $S_1=238-235.8=2.2$. The signed stress loss is $20-2.2=17.8$, which is positive, so the charge is $K=\max(0,17.8)=17.8$ million dollars.

Check. The asset decline of $12$ plus the net liability increase of $5.8$ is $17.8$, matching $S_0-S_1$. The expense provision’s decrease is already reflected in the net liability increase; it offsets part of the claims-provision increase.

Common mistake to avoid. Subtracting the liability increase from the asset decline gives $12-5.8=6.2$, which is wrong: both an asset decline and a liability increase reduce surplus. The correct signed loss is $12+5.8=17.8$.

Worked Calculations with Different Revaluation Patterns

The same balance-sheet identity applies when some assets increase or some liabilities decrease. These changes reduce the signed loss; they do not change the underlying formula.

Worked Example: Include an Asset Gain and a Liability Reduction

A hypothetical insurer has assets of $180 million: loans of $100 million, government securities of $50 million, and cash of $30 million. Under a prescribed stress, loans fall by 12%, government securities rise by 2%, and cash is unchanged. Liabilities total $160 million: claims provisions of $100 million rise by 5%, expense provisions of $40 million fall by 2%, and other liabilities of $20 million are unchanged. Find the stressed surplus and charge. All amounts are in millions of dollars.

Method. Apply each shock to its corresponding component, including increases and decreases, then compare total surplus before and after.

Assets. Loans fall by $100(0.12)=12$, so their stressed value is $88$. Government securities rise by $50(0.02)=1$, so their stressed value is $51$. Cash remains $30$. Hence $A_0=100+50+30=180$, and $A_1=88+51+30=169$. The net asset decline is $11$.

Liabilities. Claims provisions rise by $100(0.05)=5$, giving $105$. Expense provisions fall by $40(0.02)=0.8$, giving $39.2$. Other liabilities remain $20$. Hence $L_0=100+40+20=160$, and $L_1=105+39.2+20=164.2$. The net liability increase is $4.2$.

Surplus and charge. Initial surplus is $180-160=20$. Stressed surplus is $169-164.2=4.8$. Therefore, the signed loss is $20-4.8=15.2$, and the positive-part charge is $15.2$ million dollars.

Check. The component changes give a net asset decline of $12-1=11$, and a net liability increase of $5-0.8=4.2$. Their sum, $11+4.2=15.2$, agrees with the surplus comparison.

Common mistake to avoid. Counting the $1 million securities gain as an additional loss would overstate the charge. It offsets part of the loan decline: the net asset decline is $12-1$, not $12+1$.

Worked Example: A Stress That Leaves Surplus Negative

A hypothetical insurer has assets of $120 million: property of $40 million, a structured asset of $30 million, and cash of $50 million. The prescribed stress reduces property by 10% and the structured asset by 20%; cash is unchanged. Liabilities are a benefit obligation of $70 million, a claims liability of $25 million, and an expense liability of $10 million. The stress increases the benefit obligation by 8% and the claims liability by 12%, and decreases the expense liability by 5%. Calculate the stressed surplus and charge, in millions of dollars.

Method. Revalue all affected components, calculate both surpluses, and do not confuse a negative stressed surplus with the amount of the stress loss.

Assets. Property becomes $40-40(0.10)=36$. The structured asset becomes $30-30(0.20)=24$. Cash remains $50$. Thus $A_0=40+30+50=120$, and $A_1=36+24+50=110$.

Liabilities. The benefit obligation becomes $70+70(0.08)=75.6$. The claims liability becomes $25+25(0.12)=28$. The expense liability becomes $10-10(0.05)=9.5$. Thus $L_0=70+25+10=105$, and $L_1=75.6+28+9.5=113.1$.

Surplus and charge. Initial surplus is $120-105=15$. Stressed surplus is $110-113.1=-3.1$. The signed stress loss is $15-(-3.1)=18.1$. Since the loss is positive, the charge is $18.1$ million dollars.

Check. Assets decline by $120-110=10$, and liabilities increase by $113.1-105=8.1$. The combined reduction in surplus is $10+8.1=18.1$, as required.

Common mistake to avoid. Reporting the charge as $3.1 million confuses the deficit after stress with the change in surplus. The stressed deficit is $3.1 million, but surplus fell from $15 million to negative $3.1 million, a loss of $18.1 million.

Interpretation, Scope, and Exam Traps

The revaluation method matters because the charge is sensitive to both sides of the balance sheet. A stress affecting only one asset class may still change liabilities, and a liability response can materially add to or offset the asset-side result. A calculation that applies the shock to assets but leaves a stated stressed liability change out is incomplete.

A prescribed percentage shock is ordinarily applied to the specified starting value unless the problem states otherwise. Check whether the stated percentage is a decline or increase, what component it applies to, and whether it is a direct change to value or a parameter that must first be translated into value. If stressed values are given, use them as the post-stress values; do not apply the percentage again.

The calculation should keep the following distinctions clear:

Surplus versus charge. Stressed surplus is $A_1-L_1$; the signed stress loss is $S_0-S_1$.
Liability direction. A rise in liability value reduces surplus; a fall in liability value improves surplus.
Signed loss versus positive-part charge. A signed loss can be negative. Under the convention here, the charge cannot be negative.
Balance-sheet scope. All asset and liability components must use consistent valuation bases and units.
Scenario assumptions. Include taxes, reinsurance, management actions, or other effects only when the problem specifies how to treat them.

A final arithmetic check is to calculate the signed loss in two independent ways: first as $S_0-S_1$, and second as $(A_0-A_1)+(L_1-L_0)$. If the results differ, likely causes include a sign error, an omitted component, a percentage applied to the wrong base, or inconsistent units. This check verifies the balance-sheet arithmetic; it does not establish that the prescribed stress or valuation assumptions are appropriate.

Key takeaway: Revalue the full specified balance sheet, calculate stressed surplus, and compare it with initial surplus. A loss of assets and an increase in liabilities both reduce surplus; the charge is the resulting loss subject to the stated convention.

Check Your Understanding

For each hypothetical scenario, calculate stressed surplus and the positive-part charge. Amounts are in millions of dollars unless stated otherwise; assume no unlisted changes.

An insurer has assets of $130 million, including an asset of $90 million that falls by 10% and cash of $40 million that is unchanged. Its $100 million liability increases by 4%. Calculate the stressed surplus and charge. (Answer: stressed surplus is $17 million; charge is $13 million.)
An insurer has equities of $80 million, bonds of $70 million, and cash of $50 million. Equities fall by 20%, bonds fall by 4%, and cash is unchanged. Its liabilities are claims provisions of $120 million, expense provisions of $30 million, and other liabilities of $20 million. Claims provisions rise by 5%, expense provisions fall by 10%, and other liabilities are unchanged. Calculate stressed surplus and charge. (Answer: stressed surplus is $8.2 million; charge is $21.8 million.)
An insurer has real estate of $100 million, equities of $150 million, bonds of $200 million, and cash of $50 million. Under stress, these become $92 million, $132 million, $194 million, and $50 million, respectively. Liabilities comprise provisions of $300 million that rise by 4%, an annuity obligation of $120 million that falls by 2%, and expenses of $50 million that are unchanged. Calculate stressed surplus and charge. (Answer: stressed surplus is negative $11.6 million; charge is $41.6 million.)
An insurer has $300 million of assets: equities of $100 million rise by 5%, bonds of $100 million fall by 2%, and cash of $100 million is unchanged. Its $280 million of liabilities comprise claims of $180 million that fall by 3% and other liabilities of $100 million that rise by 1%. Calculate stressed surplus and the signed loss, then apply the positive-part convention. (Answer: stressed surplus is $27.4 million; signed loss is negative $7.4 million; charge is $0.)
An insurer has a mortgage portfolio of $120 million, equities of $60 million, and cash of $80 million. Under stress, mortgages fall by 7%, equities fall by 15%, and cash is unchanged. Liabilities are claims provisions of $150 million that rise by 6%, contract liabilities of $50 million that fall by 4%, and expenses of $40 million that are unchanged. Calculate stressed surplus and charge. (Answer: stressed surplus is negative $4.4 million; charge is $24.4 million.)

What You'll Learn