Replicating a Swap with Forward-Rate Exposures

Why Replicate a Swap with Forward Rates?

A plain-vanilla interest rate swap exchanges a floating-rate payment for a fixed-rate payment, or vice versa, on a schedule of dates. Replication makes the connection between that exchange and a sequence of forward-rate agreements explicit: each agreement isolates the floating-versus-fixed difference for one accrual period.

This connection is useful for two reasons. It shows how period-by-period swap cash flows can be built from forward-rate exposures, and it explains the no-arbitrage condition that determines the fair fixed rate. The central discipline is to keep the rate’s accrual period, the payment date, and the direction of the cash flow consistent.

Definition: Consider payment dates $t_0<t_1<\cdots<t_n$, notional principal $N$, and accrual fractions $\alpha_i$ for periods $[t_{i-1},t_i]$. Let $L_i$ be the floating rate set for period $i$, and let $K$ be the fixed swap rate. A position that receives floating and pays fixed has net payment at $t_i$ $$ N\alpha_i(L_i-K). $$ A position that pays floating and receives fixed has the opposite payment, $N\alpha_i(K-L_i)$. The notional is not exchanged.

Unless stated otherwise, assume a single default-free discount curve, no transaction costs or counterparty risk, matching accrual periods and floating-rate index for the swap and the forward-rate agreements, and no basis spread. Write $P(0,t)$ for the time-0 price of a zero-coupon bond paying one unit at $t$. These assumptions let us compare the same future cash flows using the same discount factors. If different curves, indices, or conventions apply, the simple replication may not hold without adjustments.

Forward Rates Implied by Discount Factors

The forward rate for period $[t_{i-1},t_i]$ is the annualized rate that is consistent with the two discount factors over that accrual period. With simple interest over the period, it is

F_i=\frac{1}{\alpha_i}\left(\frac{P(0,t_{i-1})}{P(0,t_i)}-1\right).

Here $\alpha_i$ expresses the period length in years under the specified day-count convention. For example, a six-month accrual period typically has $\alpha_i=0.5$ under a simple half-year convention. Do not divide by the number of periods unless that number is the appropriate accrual fraction.

Why the Forward-Rate Formula Holds

At time $t_{i-1}$, a zero-coupon bond maturing at $t_i$ costs $P(0,t_i)/P(0,t_{i-1})$ per unit of value at $t_{i-1}$. Investing one unit for the period therefore produces the reciprocal growth factor. Under simple interest, that growth factor is $1+\alpha_iF_i$. Equating the two descriptions gives the forward-rate formula.

1+\alpha_iF_i=\frac{P(0,t_{i-1})}{P(0,t_i)} \quad\Longrightarrow\quad F_i=\frac{1}{\alpha_i}\left(\frac{P(0,t_{i-1})}{P(0,t_i)}-1\right).

This is a no-arbitrage rate implied by discount factors. It is not generally a forecast of the future floating rate under the real-world probability distribution. A forward contract struck at $F_i$ has zero value at inception under the stated assumptions.

Matching a Swap Payment with a Forward-Rate Agreement

For replication, first use an end-of-period forward-rate contract with payment at $t_i$. A receive-floating/pay-fixed contract struck at $K$ pays $N\alpha_i(L_i-K)$ at $t_i$, exactly the net cash flow of that swap period. Thus one such contract for each period reproduces the swap’s net payment schedule.

Market-standard forward-rate agreements (FRAs) are commonly cash-settled at the period’s reset date $t_{i-1}$, rather than at $t_i$. Under simple accrual, the receive-floating/pay-fixed settlement at the reset date is

\frac{N\alpha_i(L_i-K)}{1+\alpha_i L_i}.

If that settlement is invested (or financed) at the period’s floating rate $L_i$ until $t_i$, its value at $t_i$ becomes $N\alpha_i(L_i-K)$. This gives the same end-of-period economic exposure. The investment or financing is essential to the timing equivalence; the reset-date settlement by itself is not the swap’s payment at $t_i$.

Set the direction.
For receive floating/pay fixed, use $L_i-K$. For pay floating/receive fixed, reverse the sign.

Use the matching accrual fraction.
Multiply the rate difference by $N\alpha_i$; the rate alone is not a cash payment.

Align settlement timing.
Compare cash flows at the same date. Convert a reset-date FRA settlement to its end-of-period equivalent only with the stated accrual-rate financing.

Discount with the matching date’s factor.
A payment at $t_i$ is discounted with $P(0,t_i)$, not $P(0,t_{i-1})$.

Worked Example: Replicate Three Swap Payments

A hypothetical receive-floating/pay-fixed swap has notional $N=\$1{,}000{,}000$, three consecutive six-month periods, and fixed rate $K=6.00\%$ per year. The discount factors are $P(0,0)=1$, $P(0,0.5)=0.98$, $P(0,1)=0.95$, and $P(0,1.5)=0.91$. Assume simple accrual with $\alpha_i=0.5$ for each period. Find the implied forward rates and the swap’s net cash flows if the floating rates equal those forwards.

Method. Calculate each forward rate from the ratio of consecutive discount factors. Then multiply the period rate difference by $N\alpha_i=\$500{,}000$. This method avoids confusing a discount-factor ratio with its reciprocal.

Period 1.

F_1=\frac{1/0.98-1}{0.5}=0.0408163\ldots=4.0816\%.

The net payment is $ \$500{,}000(0.0408163\ldots-0.0600)=-\$9{,}591.84$, rounded to cents. The negative sign means this position pays a net amount.

Period 2.

F_2=\frac{0.98/0.95-1}{0.5}=0.0631579\ldots=6.3158\%.

The net payment is $ \$500{,}000(0.0631579\ldots-0.0600)=\$1{,}578.95$, rounded to cents.

Period 3.

F_3=\frac{0.95/0.91-1}{0.5}=0.0879121\ldots=8.7912\%.

The net payment is $ \$500{,}000(0.0879121\ldots-0.0600)=\$13{,}956.04$, rounded to cents.

Check. The rates rise across the periods because each later discount-factor ratio implies a higher period growth rate. The three rounded payments sum to $-\$9{,}591.84+\$1{,}578.95+\$13{,}956.04=\$5{,}943.15$. Each payment has the sign required by $L_i-K$, and each uses the half-year accrual fraction.

From the Replication to the Fair Fixed Rate

For a fair swap established at time 0, the present value of the receive-floating/pay-fixed cash flows must be zero. If each floating rate is represented by its corresponding forward rate, the present value of period $i$’s floating amount is $P(0,t_i)N\alpha_iF_i$. The fixed-leg present value is $P(0,t_i)N\alpha_iK$. Equating the total values and cancelling $N$ gives

\sum_{i=1}^{n}P(0,t_i)\alpha_iF_i = K\sum_{i=1}^{n}P(0,t_i)\alpha_i.

Substitute the forward-rate identity. Each discounted forward amount simplifies to the decrease in the discount factor over that period:

P(0,t_i)\alpha_iF_i = P(0,t_{i-1})-P(0,t_i).

Adding these differences makes the intermediate discount factors cancel. The resulting fair fixed rate is

K_{\mathrm{fair}} = \frac{P(0,t_0)-P(0,t_n)} {\displaystyle\sum_{i=1}^{n}\alpha_iP(0,t_i)}.

This formula assumes the swap starts at $t_0=0$, has no upfront payment, exchanges no principal, and uses the same discount curve and accrual periods throughout. It does not directly cover a swap already in progress or one with different floating and discount curves.

Worked Example: Find the Fair Rate from Forward Exposures

A hypothetical swap has three six-month periods, discount factors $P(0,0)=1$, $P(0,0.5)=0.97$, $P(0,1)=0.93$, and $P(0,1.5)=0.88$, and no exchange of notional. Assume $\alpha_i=0.5$ and the single-curve assumptions above. Find the fair fixed rate.

Method. Calculate the period forward rates, then equate the present values of the floating and fixed legs. This also checks the telescoping discount-factor result.

The forward rates are

F_1=\frac{1/0.97-1}{0.5}=0.0618557\ldots, \qquad F_2=\frac{0.97/0.93-1}{0.5}=0.0860215\ldots, \qquad F_3=\frac{0.93/0.88-1}{0.5}=0.1136364\ldots.

The discounted floating-leg rate amount, before multiplying by notional, is

\sum_i P(0,t_i)\alpha_iF_i =(1-0.97)+(0.97-0.93)+(0.93-0.88) =0.03+0.04+0.05=0.12.

The discounted fixed-leg annuity factor is $0.5(0.97+0.93+0.88)=0.5(2.78)=1.39$. Therefore

K_{\mathrm{fair}}=\frac{0.12}{1.39}=0.0863309\ldots=8.6331\%.

Check. The fixed-leg rate amount is $1.39(0.0863309\ldots)=0.12$, matching the floating-leg amount before multiplying either by the notional. The result is rounded to four decimal places as a percentage.

How No-Arbitrage Appears in the Replication

A useful way to interpret the fair-rate calculation is to compare each swap period with a forward-rate agreement struck at that period’s market forward rate $F_i$. The receive-floating/pay-fixed swap payment can be written as

N\alpha_i(L_i-K) = N\alpha_i(L_i-F_i)+N\alpha_i(F_i-K).

The first term is a forward-rate exposure struck at $F_i$, so it has zero value at inception under the assumptions. The second term is a deterministic cash flow at $t_i$. Across periods, the present value of those deterministic adjustments is zero exactly when $K=K_{\mathrm{fair}}$. The swap is therefore equivalent to zero-value forward-rate exposures plus deterministic adjustments whose total present value is zero.

If the fixed rate differs from the fair rate, the adjustment cash flows no longer have zero total present value. A contract offered at no upfront cost with that off-market rate would not match the value of its replicated cash flows. In a frictionless no-arbitrage model, the contract’s price must account for that difference; the fair rate is the rate that makes the inception value zero.

Worked Example: Verify the Replication at the Fair Rate

A hypothetical two-period swap has $\alpha_1=\alpha_2=0.5$, discount factors $P(0,0)=1$, $P(0,0.5)=0.96$, and $P(0,1)=0.90$, with notional $N=\$930{,}000$. Assume the stated single-curve conditions. Show that the fair-rate swap decomposes into zero-value forward exposures and deterministic adjustments with zero total present value.

Step 1: Find the forwards and fair fixed rate.

F_1=\frac{1/0.96-1}{0.5}=\frac{1}{12}, \qquad F_2=\frac{0.96/0.90-1}{0.5}=\frac{2}{15}.

The fixed-leg factor is $0.5(0.96+0.90)=0.93$, and the floating-leg rate amount is $(1-0.96)+(0.96-0.90)=0.10$. Thus

K_{\mathrm{fair}}=\frac{0.10}{0.93}=\frac{10}{93}=0.1075269\ldots.

Step 2: Calculate the deterministic adjustment for each period. The adjustment at $t_i$ is $N\alpha_i(F_i-K_{\mathrm{fair}})$. For period 1, $F_1-K_{\mathrm{fair}}=1/12-10/93=-3/124$, so the adjustment is $\$930{,}000(0.5)(-3/124)=-\$11{,}250$. For period 2, $F_2-K_{\mathrm{fair}}=2/15-10/93=4/155$, so the adjustment is $\$930{,}000(0.5)(4/155)=\$12{,}000$.

Step 3: Discount and check. The present value of the adjustments is $0.96(-\$11{,}250)+0.90(\$12{,}000)=-\$10{,}800+\$10{,}800=\$0$. Each forward-rate exposure struck at $F_i$ has zero value at inception, and the deterministic adjustments also have zero total present value. Therefore their combination—the fair-rate swap—has zero value at inception.

Common mistake avoided. Adding the undiscounted adjustments gives $\$750$, not zero. The no-arbitrage check is a present-value comparison, so each adjustment must be discounted to time 0 before the amounts are combined.

Common Sign and Timing Traps

Swap replication is compact, but small convention errors can reverse a result. Use the following checks before accepting a calculation:

Reversing the discount-factor ratio. The forward rate uses $P(0,t_{i-1})/P(0,t_i)$. With discount factors that fall over time, this ratio is usually greater than one and implies a positive forward rate. Using the reciprocal can produce the wrong sign.
Using $K-L_i$ for a receive-floating/pay-fixed position. That sign belongs to the opposite position. State which side receives each rate before computing cash flows.
Omitting $\alpha_i$. A quoted annual rate difference must be multiplied by the accrual fraction as well as the notional to produce a payment.
Mixing settlement dates. The standard FRA settlement at the reset date is not numerically the same as the swap payment at the period end. Convert it using the stated financing assumption or compare both cash flows at a common date.
Discounting at the reset date. A payment at $t_i$ uses $P(0,t_i)$. Discounting it with $P(0,t_{i-1})$ changes its present value.
Treating a forward rate as a guaranteed future fixing. The forward rate is the no-arbitrage contract rate implied by discount factors. A future floating fixing may differ from it.

These checks matter because the replication is a statement about matching dated cash flows and their present values, not merely about comparing quoted annual rates. Once signs, accruals, and dates agree, the forward-rate portfolio makes the swap’s no-arbitrage structure transparent.

Check Your Understanding

Assume a single default-free discount curve, simple accrual, and no transaction costs unless a question states otherwise. Round money to cents and rates to four decimal places as percentages when needed.

For a period with $\alpha=0.5$, $P(0,t_{i-1})=0.99$, and $P(0,t_i)=0.96$, find the implied forward rate. (Answer: $6.2500\%$)
A receive-floating/pay-fixed swap has $N=\$800{,}000$, $\alpha=0.25$, $L=5.2\%$, and $K=4.8\%$ for one period. Find its end-of-period net payment. (Answer: receive $\$800.00$)
For the rates and notional in Question 2, find the standard reset-date FRA settlement for the same receive-floating/pay-fixed exposure. (Answer: receive $\$789.73$)
A two-period swap has $\alpha_1=\alpha_2=0.5$, $P(0,0)=1$, $P(0,t_1)=0.96$, and $P(0,t_2)=0.90$. Find its fair fixed rate. (Answer: $10.7527\%$)
A two-period swap has accrual fractions $0.5$ and $0.5$, with discount factors $1$, $0.98$, and $0.95$ at the initial date and the two payment dates. Find the fair fixed rate using the discount-factor formula. (Answer: $5.1813\%$)

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