Gross and Net Loss Bases for Capital

Why the Loss Basis Matters

An economic capital result depends not only on the risk measure and confidence level but also on which losses the measure is applied to. A gross basis measures losses before recoveries from risk transfer. A net basis measures the portion retained after those recoveries. For the same hypothetical portfolio and horizon, the two bases can produce materially different capital amounts.

A comparison is meaningful only when the other choices are held consistent: the portfolio, measurement horizon, valuation assumptions, confidence level, and capital convention. If those also change, the difference cannot be attributed to the gross-versus-net basis alone. The earlier material on VaR-based economic capital provides the general quantile framework; here the key task is to define the loss variable to which that framework is applied.

Definition: Let $X$ be the aggregate loss over the stated horizon before risk-transfer recoveries. Let $R$ be the recovery actually recognized for that same loss basis and horizon. The gross loss is $X$; the net retained loss is $N=X-R$. In the basic examples below, $0\le R\le X$, and premiums, expenses, taxes, and counterparty default effects are excluded unless stated. In practice, $R$ should reflect the recovery assumptions appropriate to the chosen model and may need adjustment for limits, exclusions, collection risk, and timing.

A gross basis does not give credit for the risk transfer in the loss amount being measured. A net basis does give credit for recoveries, but only to the extent the model recognizes them. For example, a contractual recovery that is limited by an attachment point or exhaustion limit is not the same as full reimbursement. If the intended basis includes reinsurance premium, reinstatement premium, expenses, or counterparty nonpayment, those items need to be specified and incorporated consistently rather than silently mixed into $X$ or $R$.

Applying a Capital Measure to Each Basis

For a loss variable $L$, define its Value-at-Risk at confidence level $\alpha$ by the quantile convention

\operatorname{VaR}_{\alpha}(L)=\inf\{l:F_L(l)\ge\alpha\}.

The gross VaR capital amount is $\operatorname{VaR}_{\alpha}(X)$, while the net VaR capital amount is $\operatorname{VaR}_{\alpha}(N)$, using the same $\alpha$. Some capital presentations instead report unexpected-loss capital, defined here as VaR less expected loss. On that convention, the amounts are $\operatorname{VaR}_{\alpha}(X)-E[X]$ and $\operatorname{VaR}_{\alpha}(N)-E[N]$. State which convention is being used: VaR itself and VaR less the mean are not interchangeable quantities.

When $X$ and $R$ have finite means, linearity of expectation gives

$$ E[N]=E[X-R]=E[X]-E[R]. $$

This identity does not require independence between gross losses and recoveries. In contrast, there is generally no corresponding subtraction rule for VaR. VaR is a quantile of a distribution, not an expectation, and the distribution of $X-R$ depends on the joint outcomes of $X$ and $R$. A recovery that is large in one scenario may occur precisely when gross losses are large, or it may be limited, delayed, or unavailable in that scenario.

Key result: Calculate the net loss scenario by scenario as $N=X-R$, then find the quantile of the resulting net-loss distribution. In general, $\operatorname{VaR}_{\alpha}(X-R)\ne \operatorname{VaR}_{\alpha}(X)-\operatorname{VaR}_{\alpha}(R)$. A fixed, certain recovery amount $c$ is a special case: if $R=c$ in every scenario, then $\operatorname{VaR}_{\alpha}(X-c)=\operatorname{VaR}_{\alpha}(X)-c$.

Worked Comparisons

Worked Example: Comparing Gross and Net VaR on the Same Scenarios

A hypothetical insurer models one-year losses using four mutually exclusive scenarios. The probabilities are 0.70, 0.15, 0.10, and 0.05. The corresponding gross losses are $0, $100,000, $300,000, and $600,000. Recoveries are $0, $0, $150,000, and $300,000. Calculate gross and net VaR at 90%, and calculate unexpected-loss capital on both bases.

Method. First subtract the recovery in each scenario, then sort each loss distribution and locate the smallest loss whose cumulative probability is at least 0.90.

The net losses are, in order, $0, $100,000, $150,000, and $300,000. The cumulative probabilities for the gross losses are 0.70 at $0, 0.85 at $100,000, 0.95 at $300,000, and 1.00 at $600,000. Therefore, $\operatorname{VaR}_{0.90}(X)=\$300{,}000$.

For net losses, the cumulative probabilities are 0.70 at $0, 0.85 at $100,000, 0.95 at $150,000, and 1.00 at $300,000. Thus, $\operatorname{VaR}_{0.90}(N)=\$150{,}000$.

The expected gross loss is $0.70(0)+0.15(\$100{,}000)+0.10(\$300{,}000)+0.05(\$600{,}000)=\$75{,}000$. The expected net loss is $0.70(0)+0.15(\$100{,}000)+0.10(\$150{,}000)+0.05(\$300{,}000)=\$45{,}000$. The unexpected-loss amounts are therefore $\$300{,}000-\$75{,}000=\$225{,}000$ gross and $\$150{,}000-\$45{,}000=\$105{,}000$ net.

Check. Expected recovery is $\$75{,}000-\$45{,}000=\$30{,}000$, which matches $0.10(\$150{,}000)+0.05(\$300{,}000)=\$30{,}000$. The calculation avoids treating the recovery as a flat reduction to every scenario: the first two scenarios have no recovery.

Worked Example: Why Recovery VaR Cannot Usually Be Subtracted

A hypothetical risk-transfer arrangement produces the following one-year outcomes: with probability 0.60, $X=\$0$ and $R=\$0$; with probability 0.20, $X=\$100{,}000$ and $R=\$90{,}000$; with probability 0.10, $X=\$200{,}000$ and $R=\$0$; and with probability 0.10, $X=\$500{,}000$ and $R=\$450{,}000$. Find the 90% VaR of gross losses, recoveries, and net losses.

Method. Calculate each distribution separately. The net scenario losses are $0, $10,000, $200,000, and $50,000. Ordering net losses gives $0 with probability 0.60, $10,000 with probability 0.20, $50,000 with probability 0.10, and $200,000 with probability 0.10.

For gross losses, the cumulative probability reaches 0.90 at $200,000, so $\operatorname{VaR}_{0.90}(X)=\$200{,}000$. Recoveries in increasing order are $0 with probability 0.70, $90,000 with probability 0.20, and $450,000 with probability 0.10. Hence $\operatorname{VaR}_{0.90}(R)=\$90{,}000$. For net losses, the cumulative probability is 0.80 at $10,000 and 0.90 at $50,000, so $\operatorname{VaR}_{0.90}(N)=\$50{,}000$.

Subtracting the two separate VaRs would give $\$200{,}000-\$90{,}000=\$110{,}000$, not the correct net VaR of $50,000. The recovery VaR does not identify the recovery paired with the gross loss at the gross-loss quantile. The joint scenario outcomes, not separate marginal quantiles, determine the net distribution.

Check. The net losses listed above retain their original scenario probabilities, which total $0.60+0.20+0.10+0.10=1.00$. Their cumulative probability reaches 0.90 at $50,000, confirming the net VaR.

Applying Treaty Terms to the Net Basis

A recovery under an excess-of-loss treaty is commonly determined by applying the contract terms to the covered loss. For a simple per-occurrence layer with attachment $a$ and maximum recovery $u$, the recovery on a covered loss $X$ is

R=\min\bigl((X-a)^+,u\bigr), \qquad (z)^+=\max(z,0).

This expression assumes the stated loss is the treaty’s covered subject loss and that no other contractual adjustments apply. The attachment is applied before the recovery is calculated, and the limit caps the recovery. Net retained loss is then $X-R$. For an aggregate model, the contract may apply to occurrence losses, annual aggregate losses, or another defined unit; the model must match the treaty wording.

Worked Example: An Attachment and Limit Change the Capital Basis

A hypothetical annual gross loss has four possible outcomes: $0 with probability 0.50, $80,000 with probability 0.25, $200,000 with probability 0.15, and $500,000 with probability 0.10. A covered excess layer has a $100,000 attachment and a $250,000 maximum recovery. Calculate gross and net 90% VaR and unexpected-loss capital.

Method. Apply $R=\min((X-\$100{,}000)^+,\$250{,}000)$ to each scenario. At $X=\$0$ and $X=\$80{,}000$, the loss is below attachment, so recovery is $0 and net loss equals gross loss. At $X=\$200{,}000$, recovery is $\min(\$100{,}000,\$250{,}000)=\$100{,}000$, leaving $100,000 net. At $X=\$500{,}000$, recovery is $\min(\$400{,}000,\$250{,}000)=\$250{,}000$, leaving $250,000 net.

The gross cumulative probabilities are 0.50 at $0, 0.75 at $80,000, 0.90 at $200,000, and 1.00 at $500,000. Thus gross 90% VaR is $200,000. Net losses are $0, $80,000, $100,000, and $250,000 with the same respective probabilities. The cumulative probability reaches 0.90 at $100,000, so net 90% VaR is $100,000.

Expected gross loss is $0.25(\$80{,}000)+0.15(\$200{,}000)+0.10(\$500{,}000)=\$100{,}000$. Expected net loss is $0.25(\$80{,}000)+0.15(\$100{,}000)+0.10(\$250{,}000)=\$60{,}000$. Therefore unexpected-loss capital is $\$200{,}000-\$100{,}000=\$100{,}000$ gross and $\$100{,}000-\$60{,}000=\$40{,}000$ net.

Check. Expected recovery is $\$100{,}000-\$60{,}000=\$40{,}000$, also equal to $0.15(\$100{,}000)+0.10(\$250{,}000)=\$40{,}000$. The calculation avoids applying the $250,000 limit to the gross loss itself; it caps the recovery, not the retained loss.

Derivation and Interpretation

The reason net capital must be calculated from the net distribution follows directly from the definition of VaR. For each scenario $j$, let gross loss be $x_j$, recovery be $r_j$, and probability be $p_j$. The net loss in that scenario is $n_j=x_j-r_j$. Its cumulative distribution at any amount $t$ is

F_N(t)=P(N\le t)=\sum_{j:n_j\le t}p_j =\sum_{j:x_j-r_j\le t}p_j.

The net VaR is the smallest $t$ for which this cumulative probability is at least $\alpha$. The set of scenarios satisfying $x_j-r_j\le t$ depends on paired gross losses and recoveries. Sorting gross losses alone, or sorting recoveries alone, does not determine that set. This is why scenario-level netting is required before finding a quantile. If recovery is the same certain amount $c$ in every scenario, the condition becomes $x_j-c\le t$, equivalently $x_j\le t+c$; shifting every loss by $c$ shifts every quantile by $c$. A variable recovery does not generally permit that shift.

Define the basis.
Specify the horizon, what counts as gross loss, and whether recoveries, premiums, expenses, and collection risk are included.

Build consistent outcomes.
For every scenario, calculate the recovery under the assumed terms and set net loss equal to gross loss minus recognized recovery.

Apply the same capital convention.
Use the same confidence level and loss measure on both distributions; if reporting VaR less expected loss, calculate each mean from its own basis.

Check the comparison.
Confirm that the expected gross-minus-net difference equals expected recovery under the stated model, and verify that scenario probabilities and contract limits have been applied correctly.

Common Traps and Practical Checks

The most common error is to calculate gross VaR and subtract the VaR of recoveries as if quantiles were additive. They are not generally additive, as the second worked example demonstrates. A related error is to subtract expected recovery from gross VaR and label the result net VaR. Expected recovery does support the identity $E[N]=E[X]-E[R]$, but it does not determine a tail quantile.

Other checks help keep the basis interpretable:

Do not mix capital conventions. Compare VaR with VaR, or unexpected-loss capital with unexpected-loss capital. Subtracting the mean on only one basis does not produce a like-for-like comparison.
Apply treaty terms in the correct order. Respect attachment, limit, exclusions, and the unit to which the contract applies. A per-occurrence limit cannot automatically be treated as an annual aggregate limit.
Use recoveries that belong to the modeled horizon. If timing or collection risk matters, the model must reflect the assumed treatment rather than counting an uncertain contractual amount as certain cash.
Do not assume net capital must fall by the expected recovery. The expected loss falls by expected recovery, but the change in VaR depends on the tail scenarios and their probabilities.

Gross and net bases answer different questions. A gross view describes loss before risk transfer; a net view describes the modeled retained loss after risk transfer. Neither number is self-explanatory without its basis, horizon, capital convention, and recovery assumptions. Stating those choices makes the result more useful for comparing risk and for explaining what the capital estimate does and does not recognize.

Check Your Understanding

For each problem, use the stated scenario probabilities and the quantile definition given above. Report dollar amounts to the cent where needed.

A hypothetical portfolio has gross losses of $0, $120,000, and $400,000 with probabilities 0.70, 0.20, and 0.10. Recoveries are $0, $90,000, and $250,000, respectively. Find gross and net 90% VaR and unexpected-loss capital on both bases. (Answer: Gross VaR $120,000; net VaR $30,000; gross unexpected-loss capital $56,000; net unexpected-loss capital $9,000.)
For a hypothetical annual loss, the outcomes are $X=\$0,R=\$0$ with probability 0.50; $X=\$150{,}000,R=\$100{,}000$ with probability 0.30; and $X=\$350{,}000,R=\$200{,}000$ with probability 0.20. Calculate gross and net 80% VaR and their unexpected-loss amounts. (Answer: Gross VaR $150,000 and net VaR $50,000; gross expected loss $115,000 and net expected loss $45,000; unexpected-loss amounts $35,000 gross and $5,000 net.)
A hypothetical treaty has a $50,000 attachment and a $120,000 recovery limit. Gross loss is $0, $100,000, $250,000, or $500,000 with respective probabilities 0.60, 0.20, 0.15, and 0.05. Calculate recoveries by scenario, then gross and net 90% VaR and unexpected-loss capital. (Answer: Recoveries $0, $50,000, $120,000, and $120,000; net losses $0, $50,000, $130,000, and $380,000; gross VaR $250,000 and net VaR $130,000; gross expected loss $82,500 and net expected loss $48,500; unexpected-loss capital $167,500 gross and $81,500 net.)
In a hypothetical model, gross loss and recovery are $20,000 and $0 with probability 0.50; $100,000 and $80,000 with probability 0.25; $300,000 and $100,000 with probability 0.15; and $600,000 and $500,000 with probability 0.10. Calculate the 90% VaR of gross loss, recovery, and net loss. Does subtracting recovery VaR from gross VaR give net VaR? (Answer: Gross VaR $300,000; recovery VaR $100,000; net losses $20,000, $20,000, $200,000, and $100,000, so net VaR $200,000. Subtracting the separate VaRs gives $200,000 here, equal by coincidence; this does not establish a general subtraction rule.)

What You'll Learn