Why Reimbursement Creates Counterparty Credit Risk
Under a large-deductible arrangement, an insurer may pay covered claims and seek reimbursement from the insured. Until reimbursement is received, the insurer has a credit exposure to that counterparty. The exposure can include amounts already due and unpaid, as well as eligible obligations expected to be outstanding if the counterparty defaults. Its size is not the same as the amount ultimately lost: collateral and recoveries after default can reduce the loss.
A useful analysis keeps four quantities distinct: exposure at default (EAD), probability of default (PD), the value of collateral that is actually available to satisfy the obligation, and loss given default (LGD). This tutorial develops the exposure and conditional loss measures; it does not combine them into a prospective expected credit loss or pricing load.
The word “eligible” matters. A claim reserve is not automatically a reimbursement receivable: the amount must be allocable to the counterparty under the arrangement, and the relevant reimbursement obligation must be included under the stated measurement basis. Where the analysis includes estimated unpaid obligations, state the estimation basis and avoid counting the same amount both as a case reserve and as a separate development provision. The earlier material on reimbursable loss and expense allocation and claim development is useful for establishing those amounts.
EAD is measured before collateral recovery. Collateral is analyzed separately, because its face amount may not equal the amount available at default. A haircut may reflect valuation uncertainty or realization risk, and legal or contractual limitations may make some collateral unavailable for a particular obligation. Only collateral that is enforceable, accessible, and applicable to the reimbursement obligation should be credited.
From Gross Exposure to Unsecured Exposure
Let \(E\) be gross EAD at the chosen default date. Let \(C\) be collateral value eligible to satisfy that exposure after any applicable haircut, capped at the exposure. The portion not covered by eligible collateral is the unsecured exposure \(U\):
If collateral consists of several instruments, calculate each instrument’s eligible value under the stated assumptions, then add the eligible values that can actually be applied to the obligation. For example, if cash is fully available and a letter of credit with face amount \(L\) has haircut \(h\), its eligible value is \(L(1-h)\). Do not apply a haircut twice, and do not count collateral in excess of EAD as protection for this exposure.
Unsecured exposure is not necessarily the final loss. The counterparty may make a recovery after default from assets outside the collateral. For a simple severity model, let \(r\) be the recovery rate on unsecured exposure, so \(0\le r\le1\). Assume eligible collateral is realized in full at its eligible value, is applied to the same exposure, and does not compete with another claim. Under these assumptions, the dollar loss given default is \(L_{\text{def}}=(1-r)U\). The LGD rate is \(L_{\text{def}}/E\), using gross EAD as the denominator.
Why the LGD Formula Holds
The formula follows by separating the exposure into the part satisfied by collateral and the part left unsecured. With \(C\le E\), the remaining amount is \(U=E-C\). If the counterparty recovers a fraction \(r\) of \(U\), the recovery on the unsecured amount is \(rU\). Total recovery is therefore \(C+rU\), and the loss is gross EAD less that recovery:
Dividing the dollar loss by gross EAD gives the LGD rate:
This expression assumes full realization of eligible collateral and a single recovery rate on the unsecured balance. If collateral is disputed, delayed, subject to additional costs, or not fully realizable, the eligible value or recovery assumptions must reflect that limitation. If EAD is zero, the ratio is undefined; report no exposure rather than calculating an LGD percentage.
Worked Example: Reconciling Claims into Gross EAD
At a hypothetical default date, an insurer has paid covered losses and allocated expenses under a large-deductible arrangement. Claim A has an eligible reimbursement obligation of $118,000, of which $76,000 has already been reimbursed. Claim B has an eligible obligation of $64,000, of which $24,000 has already been reimbursed. A further $6,000 of allocated expense is eligible and unpaid, and is not included in either claim amount. Determine gross EAD before collateral.
Method. For each obligation, subtract reimbursements received from the eligible amount. Add the separate unpaid expense once.
For Claim A, the outstanding balance is \( \$118{,}000-\$76{,}000=\$42{,}000\). For Claim B, it is \( \$64{,}000-\$24{,}000=\$40{,}000\). The separately identified allocated expense is $6,000 and has not been included in either claim balance.
Thus gross EAD is $88,000. This is before considering collateral or any recovery after default. A common mistake is to use the gross eligible obligations, \(118{,}000+64{,}000+6{,}000=\$188{,}000\), as EAD. That ignores $100,000 already reimbursed and overstates the amount still exposed.
Measuring Default Probability over a Stated Horizon
PD has meaning only with a stated horizon and starting condition. An annual conditional PD of \(p\) is the probability of default during a year, given that the counterparty has survived without default to the start of that year. If the same conditional PD applies in each year, the probability of surviving \(n\) years is \((1-p)^n\), provided the conditional annual probability is constant. The cumulative probability of default by the end of year \(n\) is its complement:
This is not the same as \(np\), except as a rough approximation when the probabilities are small. If conditional PD varies by year, with annual probabilities \(p_1,\ldots,p_n\), cumulative PD is \(1-\prod_{j=1}^{n}(1-p_j)\). A PD estimate should not be applied to an exposure measured at a different date or over an unstated horizon without explaining the alignment.
A default probability describes how likely default is; it does not describe how much is lost if default occurs. Likewise, a large EAD does not imply a high PD. Keeping these measures separate helps identify whether the main concern is the size of the obligation, the likelihood of default, or the protection and recoveries available after default.
Worked Example: Collateral, Unsecured Exposure, and LGD
A hypothetical insured has gross reimbursement EAD of $240,000. The insurer holds $54,000 of cash collateral and a letter of credit with a face amount of $100,000. Apply a 10% haircut to the letter of credit. Assume all eligible collateral is fully realized and applied to this exposure. After default, assume recovery of 30% of the remaining unsecured balance. Calculate eligible collateral, unsecured exposure, dollar LGD, and LGD as a percentage of gross EAD.
Method. Apply the haircut to the letter of credit, add cash collateral, cap the collateral credit at EAD, and then apply the recovery assumption to the unsecured balance.
The letter of credit’s eligible value is \( \$100{,}000(1-0.10)=\$90{,}000\). Adding the $54,000 cash gives total eligible collateral of \( \$90{,}000+\$54{,}000=\$144{,}000\). This is below the $240,000 EAD, so the collateral amount credited is $144,000.
The assumed recovery on the unsecured balance is \(0.30(\$96{,}000)=\$28{,}800\). Therefore dollar LGD is \( \$96{,}000-\$28{,}800=\$67{,}200\), equivalently \(0.70(\$96{,}000)\). Relative to gross EAD:
The results are eligible collateral of $144,000, unsecured exposure of $96,000, dollar LGD of $67,200, and LGD of 28% of gross EAD. The calculation checks because total assumed recovery is \( \$144{,}000+\$28{,}800=\$172{,}800\), and \( \$240{,}000-\$172{,}800=\$67{,}200\). A common mistake is to report \(70\%\) as the LGD rate against gross EAD. It is 70% of the unsecured balance; gross-EAD LGD is 28%.
Worked Example: Combining a Multi-Year PD with Conditional Loss Severity
A hypothetical portfolio has gross EAD of $180,000 at the relevant default date. Eligible collateral is $60,000. Assume the counterparty’s annual conditional PD is 2% in each of the next three years, and assume a 40% recovery rate on unsecured exposure if default occurs. Calculate the three-year cumulative PD and conditional dollar LGD and LGD rate. Do not calculate an expected credit loss.
Method. First measure the exposure and loss conditional on default. Separately calculate the probability of default over the three-year horizon.
Unsecured exposure is \( \$180{,}000-\$60{,}000=\$120{,}000\). The assumed unsecured recovery is \(0.40(\$120{,}000)=\$48{,}000\), leaving dollar LGD of \( \$120{,}000-\$48{,}000=\$72{,}000\). As a proportion of gross EAD:
For three years, survival probability is \(0.98^3=0.941192\). Thus the cumulative three-year PD is \(1-0.941192=0.058808\), or 0.0588 to four decimal places (5.8808% before rounding).
The conditional loss severity is $72,000 and the cumulative three-year PD is 0.0588. These describe different dimensions of risk; neither result alone is an expected loss. The calculation avoids the common mistake of adding annual PDs to obtain \(3(2\%)=6\%\), rather than using the probability of surviving each year.
Practical Checks and Common Traps
A defensible reimbursement credit-risk calculation should make its measurement basis explicit. Identify the default date or horizon, the eligible unpaid obligations included in EAD, the reimbursements credited, and the collateral that can be applied. Then state any recovery assumption used to calculate LGD. The following sequence helps keep the components separate.
Determine the unpaid eligible reimbursement amounts at the stated measurement date. Credit amounts already reimbursed and avoid counting a reserve or expense twice.
Add the obligations included under the chosen basis before post-default recoveries. State whether estimates of unpaid obligations are included.
Apply the stated haircut and enforceability assumptions. Credit no more than the collateral that can satisfy this obligation, and cap the amount applied at EAD.
Subtract eligible collateral from gross EAD, then apply the stated recovery assumption to the unsecured balance. Report dollar LGD and, when EAD is positive, LGD as a proportion of gross EAD.
A frequent conceptual error is to call the balance after collateral deduction “EAD.” Under the definitions used here, that balance is unsecured exposure; EAD remains the gross unpaid obligation before post-default recoveries. Another error is to treat collateral’s stated face value as automatically available. A haircut, a limit on the obligations secured, or a condition that prevents timely access can reduce the amount that should be credited.
The methods are aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus. In practice, calculations also depend on contract terms, claim allocation, collateral documentation, and the measurement purpose. A numerical output should not be interpreted without those assumptions.
Check Your Understanding
For each problem, keep gross EAD, collateral, PD, and conditional loss severity distinct. Dollar answers are rounded to the cent where applicable; probabilities are rounded to four decimal places.
- A hypothetical insurer is owed $73,000 on one eligible reimbursement obligation and $28,000 on another. It has already received a $19,000 reimbursement that has not yet been credited in the displayed balances. What is gross EAD? (Answer: $82,000.)
- Gross EAD is $400,000. Eligible collateral consists of $35,000 cash and a $120,000 letter of credit subject to a 15% haircut. What is total eligible collateral and the resulting unsecured exposure? (Answer: eligible collateral $137,000; unsecured exposure $263,000.)
- A counterparty has a 3% conditional PD in each year, constant for three years. Assuming survival through one year is the condition for applying the next year’s PD, calculate cumulative three-year PD. (Answer: \(1-0.97^3=0.0873\).)
- Gross EAD is $250,000 and eligible collateral is $70,000. If recovery after default is 40% of unsecured exposure, find dollar LGD and LGD as a percentage of gross EAD. (Answer: $108,000; 43.2%.)
- Gross EAD is $160,000. A pledged asset has market value $100,000, but $20,000 is not eligible to secure this obligation. Assume the remaining eligible collateral is fully realized and unsecured recovery is 25%. Calculate unsecured exposure, dollar LGD, and LGD as a percentage of gross EAD. (Answer: unsecured exposure $80,000; dollar LGD $60,000; LGD rate 37.5%.)