Basis Swaps and the Value of Floating-Rate Spreads

Why Two Floating Rates Can Differ

A floating-rate swap need not exchange a fixed rate for a floating rate. It can exchange payments based on two different floating-rate indices. Those indices may respond differently to market conditions, or may use different reset and payment conventions. Consequently, their rates need not be equal even when they apply to the same period.

A basis swap exchanges payments tied to two floating indices, usually with a quoted spread added to one of the legs. The spread compensates for the difference between the indices under the market’s current pricing assumptions. It does not eliminate the risk that the difference will change: the future difference between the indices is the swap’s basis exposure.

Definition: For a period, define the basis as the rate on one floating index minus the rate on the other, using a stated order. In this tutorial, the basis is $L_B-L_A$. A position that receives index B plus a spread $c$ and pays index A has a positive net payment when $L_B+c>L_A$.

Unless stated otherwise, assume a notional principal $N$, no exchange of principal, and payments in arrears. For period $j$, let $\alpha_j$ be the accrual factor for the payment, $L_{A,j}$ and $L_{B,j}$ the applicable annualized index rates, and $c$ the contractual spread added to index B. Rates are applied on a simple-interest basis over their accrual periods. The payer of index A and receiver of index B has net cash flow

C_j=N\left[(L_{B,j}+c)\alpha_j-L_{A,j}\alpha_j\right] =N\alpha_j(L_{B,j}-L_{A,j}+c),

when both legs use the same accrual factor and payment date. A positive $C_j$ is a receipt for this position; a negative $C_j$ is a payment. If the legs have different accrual factors or payment dates, calculate each leg separately and discount its cash flow on its own date rather than forcing the expression into a single common factor.

From Index Fixings to Net Payments

Once a period’s index rates have fixed, the coupon amounts for that period are determined by the contract. Before the rates fix, valuation uses the relevant projected forward rates, not rates known with certainty. A forward rate is a market-implied rate for a future accrual period under the specified curve framework; it is not a guarantee of the rate that will ultimately be observed.

Worked Example: Netting Two Fixed Index Coupons

A hypothetical basis swap has notional $N=\$20{,}000{,}000$. For one quarter, the accrual factor is $\alpha=0.25$. The payer pays index A at $4.20\%$ and receives index B at $4.60\%$, plus a contractual spread of $0.15\%$ on index B. Calculate the net payment at the period end.

Method. Calculate each leg using the notional, rate, and accrual factor. Then subtract the payment from the receipt. The common mistake to avoid is applying the spread as a dollar amount rather than as an annualized rate for the accrual period.

The rate on the receiving leg, including the spread, is $4.60\%+0.15\%=4.75\%=0.0475$. The receiving coupon is

\$20{,}000{,}000(0.0475)(0.25)=\$237{,}500.

The coupon paid on index A is

\$20{,}000{,}000(0.0420)(0.25)=\$210{,}000.

Therefore, the net cash flow to the payer of A and receiver of B is $\$237{,}500-\$210{,}000=\$27{,}500$, a receipt. As a check, the net rate is $4.75\%-4.20\%=0.55\%$, so the net cash flow is $\$20{,}000{,}000(0.0055)(0.25)=\$27{,}500$, agreeing with the separate-leg calculation.

The rate difference for a coupon is not itself the coupon amount. It must be multiplied by the notional and the applicable accrual factor. For example, a difference of $0.10\%$ over a half-year period corresponds to a cash flow of $N(0.001)(0.5)$, not $N(0.001)$.

Valuing the Spread and the Basis

For valuation before rates fix, suppose the projected rates for period $j$ are $F_{A,j}$ and $F_{B,j}$, and the discount factor to that period’s payment date is $D_j$. Assume both legs have the same accrual factor $\alpha_j$, payment date, notional, and discounting treatment. Under this deterministic projection-and-discount calculation, the value to the party receiving B plus $c$ and paying A is

V=N\sum_{j=1}^{m}D_j\alpha_j(F_{B,j}-F_{A,j}+c).

The formula sums the discounted expected net payments over the remaining periods. It assumes the stated projection rates and discount factors are the appropriate inputs for the valuation framework, and that the contractual spread applies to every period as shown. If the two legs have different payment dates or accrual factors, value each projected leg separately. A spread paid on only one leg also must use that leg’s own accrual factor.

A fair spread is the spread that makes the swap’s value zero at its start, given the specified projected rates and discount factors. The spread is a single constant added to each applicable coupon; it is not generally equal to the basis in any one period.

Write each period’s projected net rate.
For a receiver of B plus $c$ who pays A, this is $F_{B,j}-F_{A,j}+c$.

Convert projected rates to discounted cash flows.
Multiply each net rate by $N\alpha_jD_j$, then sum across payment dates.

Set the initial value to zero.
Solve the resulting linear equation for $c$; that is the fair spread for the stated position.

To justify the fair-spread formula, use the value expression and collect the terms that contain $c$. Since the same spread applies to every period, its contribution is $Nc\sum_jD_j\alpha_j$. Setting the value to zero gives

0=N\sum_{j=1}^{m}D_j\alpha_j(F_{B,j}-F_{A,j}) +Nc\sum_{j=1}^{m}D_j\alpha_j.

For positive notional, positive accrual factors, and positive discount factors, the denominator below is positive. Dividing by $N\sum_jD_j\alpha_j$ therefore yields

c^*=-\frac{\sum_{j=1}^{m}D_j\alpha_j(F_{B,j}-F_{A,j})} {\sum_{j=1}^{m}D_j\alpha_j}.

Thus, the fair spread is the negative of the discounted, accrual-weighted average projected basis, when the spread is added to the received B leg in a position that pays A. Reversing which index is received changes the sign convention. Always write the cash flows from the chosen party’s perspective before interpreting the sign.

Worked Example: Finding the Fair Basis Spread

A hypothetical two-period swap has notional $N=\$10{,}000{,}000$. Both periods have accrual factor $0.25$. The discount factors are $D_1=0.98$ and $D_2=0.95$. The projected index rates are $F_{A,1}=4.00\%$, $F_{B,1}=4.20\%$, $F_{A,2}=4.40\%$, and $F_{B,2}=4.10\%$. Find the spread $c$ added to the received B leg that makes the value zero for a position receiving B and paying A.

Method. Calculate the projected basis for each period, weight it by the discount factor and accrual factor, and solve the zero-value equation. Do not average the two basis rates without weights: their payment dates have different discount factors.

The projected bases are $F_{B,1}-F_{A,1}=4.20\%-4.00\%=0.20\%=0.002$, and $F_{B,2}-F_{A,2}=4.10\%-4.40\%=-0.30\%=-0.003$. The weighted basis sum is

$$ (0.98)(0.25)(0.002)+(0.95)(0.25)(-0.003) =0.000490-0.0007125=-0.0002225. $$

The spread-weight denominator is

$$ (0.98)(0.25)+(0.95)(0.25)=0.245+0.2375=0.4825. $$

Therefore,

c^*=-\frac{-0.0002225}{0.4825} =0.0004611399\ldots.

The fair spread is $0.0004611399\ldots$ as a decimal rate, or $0.0461\%$ when rounded to four decimal places in percentage units (about $4.61$ basis points). To check, the present value of the projected basis payments before adding the spread is $\$10{,}000{,}000(-0.0002225)=-\$2{,}225$. The present value contributed by the spread is $\$10{,}000{,}000(0.0004611399\ldots)(0.4825)=\$2{,}225$. The two amounts offset.

Value of an Existing Basis Swap

After the contract begins, its spread is fixed, but projected index differences and discount factors can change. Revalue the remaining net payments using the current projections and discount factors. For the position receiving B plus $c$ and paying A, a positive value means the remaining projected receipts exceed the remaining projected payments in present-value terms. The other party’s value is the negative of this value, assuming the contract has no additional cash flows or adjustments.

Worked Example: Revaluing a Contractual Spread

At a reset date, a hypothetical swap has three remaining half-year periods, each with $\alpha_j=0.50$, and notional $N=\$8{,}000{,}000$. The receiver of B pays A and receives a contractual spread $c=0.10\%=0.001$ on B. Current discount factors are $0.97$, $0.93$, and $0.89$. Current projected bases $F_{B,j}-F_{A,j}$ are $0.15\%$, $0.25\%$, and $0.35\%$. Find the value to this party.

Method. Add the contractual spread to each projected basis, multiply by the period’s accrual factor and discount factor, and then multiply the sum by notional. The common mistake to avoid is treating the spread as though it replaced the index difference.

The net projected rates for the three periods are $0.15\%+0.10\%=0.25\%=0.0025$, $0.25\%+0.10\%=0.35\%=0.0035$, and $0.35\%+0.10\%=0.45\%=0.0045$. The discounted accrual weights are $(0.97)(0.50)=0.485$, $(0.93)(0.50)=0.465$, and $(0.89)(0.50)=0.445$. Their weighted net-rate contributions are

(0.485)(0.0025)=0.0012125,\quad (0.465)(0.0035)=0.0016275,\quad (0.445)(0.0045)=0.0020025.

The sum is $0.0012125+0.0016275+0.0020025=0.0048425$. Thus the value is $\$8{,}000{,}000(0.0048425)=\$38{,}740$, a positive value to the receiver of B plus the spread. The payer of that position has value $-\$38{,}740$. As a sign check, the projected net rate is positive in every period, so a positive value for the receiver is consistent with the cash-flow direction.

Basis Exposure and Common Traps

A basis swap’s value changes when the projected difference between its indices changes, even if each leg remains floating. A widening basis in favor of the received index generally increases the value of a position receiving that index, while a narrowing or reversal can reduce its value. This sensitivity is distinct from the effect of changes in discount factors.

For example, an executed coupon depends on the realized rates for its accrual period. A remaining, unfixed coupon is valued using the rates projected for that period under the valuation framework. Substituting today’s spot index rate for every future period ignores the term structure of each index. Likewise, using a fair spread established at inception as though it were the current projected basis confuses a contractual term with a market input.

Check the direction. Write “receive B, pay A” or “receive A, pay B” before forming the rate difference. Reversing the order reverses the sign.
Apply the spread to the specified leg. A spread added to B is not automatically added to both legs.
Use the correct accrual factors. A rate difference is annualized; multiply it by the period’s accrual factor to obtain the period coupon rate.
Discount by payment date. If payment dates or accrual factors differ, value each leg’s cash flows separately rather than netting them using an unjustified common factor.
Do not infer a realized future basis from a projected basis. Projected rates support valuation; the realized future fixings determine the eventual coupons.

Key takeaway: A basis swap is valued by comparing the two floating legs period by period, including any contractual spread, and discounting the resulting cash flows. The contractual spread is fixed; the future index difference is variable and creates basis exposure.

Check Your Understanding

For each question, use the stated position and rate conventions carefully.

A hypothetical swap has notional $\$12{,}000{,}000$ and a half-year accrual factor of $0.50$. A party pays index X at $3.80\%$ and receives index Y at $3.65\%$, plus $0.20\%$ on Y. Calculate the net cash flow and state its direction. (Answer: $\$3{,}000$ received by the party paying X and receiving Y.)
For a single period with equal accrual factors and payment dates, the projected rate on index A is $4.10\%$, and the projected rate on index B is $4.34\%$. What spread added to the received B leg makes a position that receives B and pays A have zero value? (Answer: $-0.24\%$, or $-24$ basis points.)
A two-period swap has equal accrual factors of $0.50$, discount factors $0.99$ and $0.96$, and projected bases $F_{B,1}-F_{A,1}=0.30\%$ and $F_{B,2}-F_{A,2}=-0.10\%$. Find the fair spread added to B for a receiver of B who pays A. (Answer: approximately $-0.1031\%$, or $-10.31$ basis points.)
A one-period position receives index B plus $0.30\%$ and pays index A. The notional is $\$5{,}000{,}000$, the accrual factor is $0.25$, and the discount factor is $0.97$. The projected rates are $F_B=4.80\%$ and $F_A=5.00\%$. Calculate the current value to the receiver of B. (Answer: $\$1{,}212.50$.)

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