Cash-Flow Mismatch and Basis Risk in Swap Hedges

Why a Swap Hedge Can Leave Cash-Flow Risk

A swap can be designed to offset a floating-rate asset or liability, but a hedge is not necessarily exact just because both contracts refer to floating interest rates. The liability and swap may use different indices, reset dates, payment dates, accrual factors, or notional amounts. In those cases, a change in rates can leave a cash amount that the swap does not offset. That remaining exposure is often called basis risk; the unmatched payment timing is a cash-flow mismatch.

The calculations in this tutorial treat interest payments as simple interest over each stated accrual period. Rates are annual rates, and each accrual factor is stated explicitly. A positive amount denotes a cash outflow for the hedged position; a negative amount denotes a receipt. This sign convention makes it possible to add liability and swap cash flows consistently.

When Floating Cash Flows Cancel

Suppose a borrower pays interest on a liability with notional \(N_L\), accrual factor \(\alpha_L\), reference rate \(R_L\), and spread \(s_L\). A swap pays fixed rate \(K\) and receives floating rate \(R_S\), with notional \(N_S\) and accrual factor \(\alpha_S\). Assume for now that the liability and swap cash flows settle on the same date. The net outflow at that date is

The first term is the liability payment. The second is the swap’s net payment: paying fixed contributes \(N_S\alpha_S K\), while receiving floating reduces the outflow by \(N_S\alpha_S R_S\). This formula assumes that the stated rates apply over the stated accrual periods, and that all amounts are paid on the common date. It should not be used to combine cash flows on different dates as if they occurred together.

If notional amounts and accrual factors match, and the liability and swap use the same floating rate for the same period, then \(N_L=N_S=N\), \(\alpha_L=\alpha_S=\alpha\), and \(R_L=R_S=R\). Substitution gives

The floating rate cancels because it appears once as a liability cost and once as a swap receipt, with equal dollar exposure. The spread and the fixed swap payment remain. If any cancellation condition fails, the floating components need not offset. For example, if the rates differ but notional and accrual factor match, the residual is \(N\alpha(s_L+K+R_L-R_S)\), which still depends on the difference between the two indices.

Separate the Sources of Mismatch

Before calculating a residual, identify how the contracts differ. A difference in any of the following can prevent a complete offset:

• Reference index: the liability may use one index while the swap receives another. Their realized rates can differ even over the same period.
• Reset convention: the two rates may be observed on different dates, or use different fixing lags. Even if the index names match, the rates fixed for a period may not.
• Notional schedule: the liability balance may amortize or change while the swap notional remains fixed, or the schedules may change by different amounts.
• Accrual factor: different day-count or accrual conventions can make the interest amounts differ even when the annual rates and notionals match.
• Payment date: cash flows due on different dates cannot be netted as one contemporaneous amount. They should be retained on their contractual dates.

A practical calculation keeps the contracts’ actual terms visible. Write the liability amount and swap amount separately for each payment, record the sign of each, and combine them only when their payment dates agree. If a value measure is needed, discount each dated net cash flow from its own date.

Worked Cash Flows and Present Values

Payment Dates, Resets, and Notional Schedules

If liability and swap payments occur on different dates, first build a dated cash-flow list. A liability payment due at the end of a quarter and a swap payment due at the end of a half-year are separate entries, even if both relate to some of the same underlying months. Do not combine them merely because their accrual periods overlap. If the required measure is present value, let \(c_k\) be the signed net outflow on date \(t_k\), and let \(D(0,t_k)\) be the discount factor for that date. Then

This follows from the time value of money: each dated amount is first converted to an equivalent value at time 0 using its own discount factor, and equivalent present values can then be added. The formula assumes the cash-flow amounts and the applicable discount factors are specified. A single flat effective annual rate \(i\) is one possible assumption, in which case \(D(0,t_k)=(1+i)^{-t_k}\). If the two streams have different payment dates, summing their undiscounted amounts does not give their present-value difference.

Reset mismatch is different from payment-date mismatch. If a loan rate is fixed using one observation date and the swap rate using another, use the actual fixed rates applicable to that payment period. Do not assume that equal index names imply equal coupon amounts. Likewise, when notionals change, calculate each interest amount using the notional that applies to that contract and period. An amortizing liability hedged by a constant-notional swap can leave exposure even when its index, reset, and payment dates match the swap.

Common Exam Traps

The most frequent error is to say that a swap “cancels the floating rate” without checking the amount, rate, and timing of each floating leg. The cancellation shown earlier is an algebraic result that depends on equal dollar exposures and matching rates. It is not a general property of swaps.

• Netting across different dates: add or subtract payments directly only when they are due on the same date. Otherwise preserve the dates and discount separately if present value is required.
• Ignoring the accrual factor: a quarterly coupon uses its stated quarterly accrual factor, not an annual interest amount. A mismatch in accrual conventions can itself create a residual.
• Using one rate for both legs: use the rate actually specified for each contract’s index and reset. A basis between two indices can remain even when notionals match.
• Using the wrong notional: apply each contract’s own notional for each period. Do not assume the swap follows an amortizing liability unless the terms say so.
• Dropping a sign: a swap receipt reduces the outflow, while a swap payment increases it. Evaluate rate differences and contract direction before adding amounts.

A disciplined hedge calculation is therefore a cash-flow calculation before it is a cancellation calculation. Once the dates, signs, rates, accrual factors, and notionals are explicit, the residual is visible and can be measured without assuming the hedge is exact. This approach is aligned with the SOA Exam FM (Financial Mathematics) syllabus.