Why Market Risk Starts With the Factor
An insurer can be exposed to several market risks at once, and one portfolio item can be sensitive to more than one market factor. A bond issued by a foreign company, for example, can respond to changes in interest rates, foreign-exchange rates, and credit spreads. To understand the exposure, first identify which factor changes, then determine how that change affects asset and liability values.
The distinction matters because a factor’s effect on the insurer is not determined by the asset alone. A fall in an asset’s value reduces surplus, but a fall in the value assigned to a liability can increase surplus. The same market shock may therefore have offsetting effects on the two sides of the balance sheet. This tutorial is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.
From Balance-Sheet Values to Surplus
Let \(A\) denote the market value of assets, \(L\) the value assigned to liabilities, and \(S\) surplus, with \(S=A-L\). For any market scenario, define \(\Delta A\), \(\Delta L\), and \(\Delta S\) as the changes from the initial values. Then the change in surplus is
This relation is exact when the changes are measured consistently. It is also the essential sign check in a market-risk analysis: a liability-value decrease means \(\Delta L<0\), so subtracting that decrease contributes positively to surplus.
When an insurer evaluates several factors separately, it may report a contribution from each factor. Adding those contributions gives a useful total only under the stated aggregation assumption—for example, that the effects are being measured as separate scenario contributions and interaction effects are ignored. In a nonlinear valuation, the result of applying several shocks together need not equal the sum of the separately measured effects.
Specify what changes, such as a parallel increase in a yield curve, a fall in an equity index, a currency appreciation, or a widening of credit spreads.
Separate assets from liabilities and note which items are assumed unaffected by that scenario.
A decrease in an asset’s value is negative \(\Delta A\); a decrease in a liability’s value is negative \(\Delta L\).
Apply \(\Delta S=\Delta A-\Delta L\), and check that the sign agrees with the balance-sheet interpretation.
Four Market Risk Factors and Their Balance-Sheet Channels
| Factor | Typical asset channel | Possible liability channel | Surplus question |
|---|---|---|---|
| Interest rates | Values of fixed-income investments and the income earned when assets mature or are reinvested | Present values of obligations whose valuation depends on interest rates; product guarantees may also respond to rates | Do asset and liability values move by different amounts or in different directions? |
| Equity prices | Direct holdings of shares and other equity-sensitive investments | Liabilities or guarantees linked to equity performance may change in value | Does an equity-price change affect only assets, or does it also change obligations? |
| Foreign-exchange rates | Foreign-currency investments and cash flows, after translation into the reporting currency | Foreign-currency obligations and cash flows, also translated into the reporting currency | Do foreign-currency assets and liabilities offset, or is there a net currency exposure? |
| Market spreads | Corporate and other spread-sensitive securities whose values respond to changes in market spreads | Some liability valuations may respond to spread changes depending on the valuation basis | Is the spread shock affecting asset values, liability values, or both? |
Interest-rate risk concerns changes in interest rates used in markets or valuation. It can affect fixed-income asset values and the valuation of interest-sensitive liabilities. It also affects future investment income when cash flows are reinvested. The impact on surplus depends on the difference between asset and liability responses; an increase in rates does not automatically increase or decrease surplus.
Equity risk is exposure to changes in equity prices. It is direct when the insurer owns equities, and can also arise through liabilities or guarantees linked to an equity measure. Whether a particular liability is affected depends on its terms and on how it is valued.
Foreign-exchange risk arises when assets, liabilities, or cash flows are denominated in a currency other than the reporting currency. A foreign-currency asset and liability can partially offset if they are in the same currency, but the size and timing of their cash flows may differ. The exposure should therefore be assessed on both sides of the balance sheet, not from the foreign assets alone.
Spread risk is the risk that market spreads change and alter the value of spread-sensitive investments or liabilities. It is distinct from a borrower actually defaulting or migrating to a different credit rating. A spread widening can reduce a bond’s market value even if no default occurs. Earlier material on credit risk addresses default and rating-migration losses; here, the focus is the market-value channel of a spread change.
Worked Example: Separating Factor Contributions to Surplus
Worked Example: Separating Factor Contributions to Surplus
A hypothetical insurer reports values in millions of dollars. A deterministic one-period scenario uses a flat effective annual valuation rate of 4.0% initially and 4.5% after a parallel increase of 50 basis points. The insurer’s valuation team supplies the following changes under that rate shock: bond assets decrease by $4.2 million and interest-sensitive liabilities decrease by $3.1 million. In the same scenario, equity assets decrease by $2.4 million, spread-sensitive assets decrease by $0.7 million, and foreign-currency assets increase by $0.3 million after translation. Liabilities are unaffected by the equity, spread, and foreign-exchange shocks. Find the total surplus change under an additive-contribution assumption.
Assumptions and method. These are deterministic scenario changes, not probability estimates. No mortality assumption or loss distribution is used. The rate-shock valuation changes are supplied, so no bond-pricing or liability-revaluation formula is needed. Treat the reported factor contributions as additive and assume no interaction effects.
Step 1: Calculate the interest-rate contribution. The asset change is \(\Delta A_{\text{rate}}=-4.2\), and the liability change is \(\Delta L_{\text{rate}}=-3.1\). Therefore:
Step 2: Calculate the other factor contributions. These liability values are unchanged, so each contribution equals its asset change:
Step 3: Add the contributions.
The result is a $3.9 million decrease in surplus. The method avoids treating the $3.1 million liability decrease as a further loss: because liabilities are subtracted in surplus, that decrease offsets part of the asset loss. The check is that the rate shock alone reduces surplus by $4.2 million less the $3.1 million liability reduction, or $1.1 million.
Worked Example: Translating a Foreign-Currency Position
Worked Example: Translating a Foreign-Currency Position
A hypothetical insurer reports in U.S. dollars and holds euro assets worth €18 million. It also has euro-denominated liabilities worth €7 million. The initial exchange rate is $1.10 per euro. In a deterministic scenario, the euro appreciates by 6% against the dollar. Find the changes in the translated asset value, liability value, and surplus. Assume no hedges, no other market movements, and no cash flows during the scenario.
Assumptions and method. Values are translated at the stated spot exchange rates. No interest rate, mortality assumption, or probability distribution is required; this is a one-period deterministic currency scenario.
Step 1: Find the new exchange rate. The appreciation is applied to the initial rate:
Step 2: Translate the assets before and after the shock.
Thus, \(\Delta A=20.988-19.80=1.188\) million dollars.
Step 3: Translate the liabilities.
Thus, \(\Delta L=8.162-7.70=0.462\) million dollars.
Step 4: Calculate the surplus change.
The euro appreciation increases both translated assets and translated liabilities, but the asset increase is larger. Surplus therefore increases by $726,000. As a check, the net euro position is €18 million minus €7 million, or €11 million; 6% of its initial dollar value, €11 million times $1.10 per euro, is $726,000.
Worked Example: Identifying Spread Risk Without Assuming Default
Worked Example: Identifying Spread Risk Without Assuming Default
A hypothetical insurer holds two spread-sensitive bond portfolios: domestic corporate bonds valued at $25 million and foreign-currency corporate bonds valued at $10 million after translation to the reporting currency. The initial effective annual risk-free valuation rate is 3.0%, and the initial market spread is 1.2%. A deterministic scenario widens the market spread by 40 basis points, to 1.6%, while holding the risk-free rate and exchange rates fixed. The insurer’s valuation team supplies spread-only asset-value decreases of $0.80 million and $0.36 million, respectively. Liability values are unchanged under this scenario. Calculate the spread contribution to surplus and identify what is not being measured.
Assumptions and method. The supplied valuation changes isolate spread effects; no bond-pricing approximation or probability distribution is needed. There is no assumed default, rating migration, mortality change, or exchange-rate movement.
Step 1: Classify the shock. The spread changes from 1.2% to 1.6%, a widening of \(1.6\%-1.2\%=0.4\%\), or 40 basis points. Because the risk-free rate is held fixed, the scenario isolates spread risk rather than interest-rate risk.
Step 2: Add the asset changes. The total asset-value change is:
Step 3: Apply the surplus relation. Since liabilities are unchanged, \(\Delta L=0\). Thus:
The spread widening reduces surplus by $1.16 million under the stated scenario. This result measures a market-value response to wider spreads. It does not estimate losses from actual borrower defaults or rating migrations, which are different credit-risk outcomes.
Common Exam Trap: Reversing the Liability Sign
Suppose a market shock changes assets by \(-\$2.0\) million and liabilities by \(-\$1.4\) million. A common error is to report \(-2.0-1.4=-\$3.4\) million, treating the liability decrease as another loss. That calculation uses the sign of the liability change incorrectly. The correct calculation is:
The decrease in liabilities offsets part of the decrease in assets. Another frequent error is to label every loss on a corporate bond as a spread loss. A market spread widening is a valuation shock; a default or rating migration is a credit event or credit outcome. State the factor being shocked before classifying the resulting loss.
Check Your Understanding
For each deterministic scenario, classify the market factor and calculate the requested change. Unless a question says otherwise, assume no other factors change and no interaction effects.
- A parallel rate shock changes an insurer’s bond assets by \(-\$1.6\) million and its interest-sensitive liabilities by \(-\$2.1\) million. What is the rate contribution to surplus? (Answer: \(+\$0.5\) million.)
- A hypothetical insurer has CAD assets of C$9 million and CAD liabilities of C$4 million. The initial exchange rate is $0.75 per CAD, and the Canadian dollar appreciates by 4%. There are no hedges. What is the change in translated surplus? (Answer: \(+\$150{,}000\).)
- An insurer holds $8.5 million of equities. An equity-price shock reduces their value by 12%; assume liabilities are unaffected. What is the surplus change? (Answer: \(-\$1.02\) million.)
- A spread-widening scenario reduces spread-sensitive assets by $0.45 million and $0.28 million in two portfolios. A spread-linked liability valuation also falls by $0.10 million. What is the net surplus change? (Answer: \(-\$0.63\) million.)
- In an additive scenario, a rate shock changes assets by \(-\$2.0\) million and liabilities by \(-\$1.4\) million. Separate equity, FX, and spread shocks change assets by \(+\$0.5\) million, \(-\$0.2\) million, and \(-\$0.6\) million, respectively; those three shocks do not change liabilities. What is the total surplus change? (Answer: \(-\$0.9\) million.)