Why Exposure at Default Depends on Timing
A credit position’s exposure is not always equal to the amount originally lent or the face amount printed on a bond. Principal can be repaid, interest can accrue, and a receivable can be partly paid or adjusted by a credit. The amount owed therefore depends on when default occurs and on which contractual amounts have become due by that time.
Exposure at default (EAD) gives a balance-sheet measure of the amount exposed to a borrower’s default at a specified time. The central task is to reconstruct that amount from the position’s terms and its payment history. This tutorial focuses on drawn loans, bonds, and trade receivables. It does not calculate expected credit loss: that requires additional inputs and is treated separately.
The exposure basis must be stated. The examples use gross contractual exposure: amounts owed before recoveries and without deductions for collateral or other credit protection. Depending on the purpose and applicable framework, an institution may define exposure differently. A fair value or accounting carrying amount can also differ from the contractual balance. Do not substitute one measure for another without being told to do so.
The default time matters even when the original balance is known. A loan default immediately before a scheduled payment includes the interest accrued since the previous payment; a default just after the payment reflects the reduced principal and a newly started accrual period. For a bond, accrued coupon interest may be part of the claim even though the next coupon has not yet been paid. For a receivable, a payment or credit reduces the outstanding amount only if it has been posted by the default date.
Rolling Forward a Loan Balance
For a loan with a fixed periodic rate, let \(B_{k-1}\) be the balance immediately after payment \(k-1\), \(j\) the effective rate per payment period, and \(C\) the payment made at the end of each period. Assuming interest accrues on the opening balance and the full payment is applied at period-end, the balance immediately before payment \(k\) and immediately after it are:
The superscript minus marks the balance immediately before the payment. If default occurs at that instant, the amount includes the interest accrued during the period. If the payment has already been made, use \(B_k\) instead. The contractual terms may require a different payment allocation, such as fees being paid before principal; in that case, follow those terms rather than automatically subtracting the entire payment from principal.
The balance formula follows by repeatedly applying the period-by-period rule. The initial balance earns interest for \(n\) periods, giving \(B_0(1+j)^n\). Payment 1 is made at the end of period 1, so it earns no interest in that period but is followed by \(n-1\) periods of accrual; its contribution to the balance at time \(n\) is \(C(1+j)^{n-1}\). Payment 2 contributes \(C(1+j)^{n-2}\), continuing to payment \(n\), which contributes \(C\). Subtracting all these accumulated payments gives the stated sum. This also shows why subtracting total payments from the starting balance alone is wrong: it omits interest on the remaining debt.
Worked Example: A Monthly-Payment Loan Before Its Fourth Payment
A hypothetical loan begins with a balance of $80,000 and charges nominal annual interest convertible monthly at \(i^{(12)}=6\%\). The borrower makes $2,000 payments at each month-end. There are no fees, extra payments, or rate changes. Find gross contractual EAD immediately before the fourth payment, after the first three payments have been made. Treat default at that instant as deterministic; no probability distribution or mortality assumption is used.
Method. Convert the nominal annual rate to the monthly rate, roll the balance forward through each payment, then accrue one more month of interest to reach the instant immediately before payment 4.
The monthly rate is \(j=0.06/12=0.005\), or \(0.5\%\). After month 1, interest is \(80{,}000(0.005)=400\), so the balance before payment 1 is $80,400 and after payment 1 is \(80{,}400-2{,}000=\)$78,400.
During month 2, interest is \(78{,}400(0.005)=392\). The balance before payment 2 is \(78{,}400+392=\)$78,792, and after payment 2 it is \(78{,}792-2{,}000=\)$76,792.
During month 3, interest is \(76{,}792(0.005)=383.96\). The balance before payment 3 is \(76{,}792+383.96=\)$77,175.96, and after payment 3 it is \(77{,}175.96-2{,}000=\)$75,175.96.
Default is immediately before payment 4, so month 4 interest has accrued but payment 4 has not. The exposure is $$ 75{,}175.96(1.005)=75{,}551.8398. $$ Rounded to the nearest cent, \(\operatorname{EAD}=\)$75,551.84. The result is consistent with the timing: it is greater than the post-payment-3 balance because interest accrued, and the fourth payment has not yet reduced it.
Common mistake avoided: Subtracting three payments directly from $80,000 gives $74,000, but ignores interest. The correct calculation rolls the balance through each interest period and applies each payment at the stated month-end.
Bond Exposure: Principal Plus Accrued Coupon
For a coupon-paying bond, start with principal still outstanding at the default date. Add coupon interest accrued since the last coupon payment if it is part of the contractual claim and remains unpaid. Subtract principal repayments that have already been made. Do not add a full future coupon when only part of its accrual period has elapsed.
Worked Example: An Amortizing Bond Between Coupon Dates
A hypothetical bond has an original face amount of $600,000 and a 6% annual coupon paid semiannually. At the most recent coupon date, the scheduled coupon was paid and a scheduled $100,000 principal installment was also paid. Four months later, default occurs, before the next coupon date. Assume a 30/360 day-count convention, no premium, discount, or fees, and no other payments. Find gross contractual EAD. Default at the stated date is deterministic; no probability distribution or mortality assumption is used.
Method. First update principal for the installment already paid. Then accrue coupon interest on the remaining principal for the four months since the last coupon payment.
Immediately after the principal installment, outstanding principal is \(600{,}000-100{,}000=\)$500,000. The 6% annual coupon accrues on that balance. Under 30/360, four months is \(4/12\) of a year, so accrued coupon interest is $$ 500{,}000(0.06)\frac{4}{12}=10{,}000. $$
The next semiannual coupon has not yet been paid; only its accrued portion is included. Thus \(\operatorname{EAD}=500{,}000+10{,}000=\)$510,000. A check on the timing: the $100,000 installment is absent because it was already paid, while the full next coupon is not included because two months of its accrual period remain.
Common mistake avoided: Using the original $600,000 principal overstates this exposure because it ignores the installment already paid. Applying the accrual to $500,000, the principal still outstanding after that payment, matches the stated timeline.
Receivables: Receipts, Credits, and Contractual Charges
A trade receivable is generally built from an amount invoiced and still unpaid at the default date. Cash receipts and credit notes posted before default reduce the outstanding invoice balance. If contract terms permit interest or another charge to accrue, include only the amount accrued by default and only if it belongs in the specified exposure basis. A charge that is merely possible, disputed, or not yet earned should not be added as though it were already owed.
Worked Example: A Partly Settled Overdue Receivable
A hypothetical supplier issues a $35,000 invoice. Before the buyer defaults, the supplier posts a $8,000 cash receipt and a $1,500 credit note. The remaining invoice balance is overdue by 24 days and earns simple late interest at 10% per year on a 30/360 basis. Assume both adjustments are valid, no other amounts are due, and the late interest is contractually owed. Find gross contractual EAD. The default date is fixed, so no probability distribution or mortality assumption is used.
Method. Adjust the invoice for the receipt and credit note first, then accrue simple interest on the resulting unpaid principal.
The unpaid invoice amount is $$ 35{,}000-8{,}000-1{,}500=25{,}500. $$ The year fraction under 30/360 is \(24/360\). Accrued late interest is therefore $$ 25{,}500(0.10)\frac{24}{360}=170. $$
Adding the contractual interest to the adjusted invoice gives \(\operatorname{EAD}=25{,}500+170=\)$25,670. The result is checked against the components: it is below the original invoice because the receipt and credit have been applied, and above the adjusted principal because late interest has accrued.
Common mistake avoided: Accruing interest on the original $35,000 invoice would ignore the posted receipt and credit. The stated terms apply interest to the unpaid balance, so the base is $25,500.
Timing and Exposure Basis: A Practical Sequence
A consistent calculation can be organized as a short sequence. The key is to use the balance at the default instant, not an earlier statement balance or a later balance after a scheduled payment.
Record whether default is before or after any payment due at that time, and whether the requested exposure is gross contractual, net of specified protection, or another defined measure.
Start from the appropriate balance and apply only repayments, receipts, and credits posted before default. Include scheduled principal reductions only if they have already taken effect.
Apply the stated periodic rate, day-count basis, and accrual convention to the correct outstanding balance. Do not accrue beyond the default date.
Add contractual amounts owed by default, exclude future unaccrued amounts, and verify that the result reflects the payment timing and the selected basis.
A common timing trap is to subtract a payment that is due on the default date even though default occurs immediately before it. Another is to omit accrued interest because it is not yet payable on the next scheduled coupon date. The contract and the precise default instant resolve both questions. If default occurs immediately after a payment has settled, use the reduced balance and start accruing from that point.
EAD is also distinct from loss given default (LGD). EAD measures the amount exposed at default; recovery assumptions determine how much of that amount is ultimately recovered. Combining the two prematurely can lead to double-counting or to subtracting recoveries from an exposure that is meant to be gross. In a particular framework, collateral or other protections may alter the defined exposure, but that adjustment should be explicit.
Check Your Understanding
For each problem, use the stated timing and exposure basis. Amounts are hypothetical, and no probability distribution or mortality assumption is needed.
- A loan has a balance of $120,000 immediately after its previous payment. It charges nominal annual interest convertible monthly at 4.8% and requires $5,000 at each month-end. There are no fees. Find EAD immediately before the third payment, after two payments. (Answer: $111,385.69)
- A hypothetical bond has $250,000 principal outstanding, pays a 5.4% annual coupon, and has its coupon paid semiannually. Default occurs three months after the most recent coupon payment. Under a 30/360 basis, no principal has been repaid and no fees apply. Find gross contractual EAD. (Answer: $253,375.00)
- A hypothetical receivable is $18,600. Before default, a $4,000 receipt and a $600 credit note have been posted. The adjusted unpaid balance earns simple late interest at 9% annually for 40 days on a 30/360 basis. Find gross contractual EAD. (Answer: $14,140.00)
- A loan starts at $50,000 and earns 1% per month. A $3,000 payment is made at each month-end, with no fees or extra payments. Find the balance immediately after payment 4. (Answer: $39,849.00)