Carry and Roll-down of Interest Rate Swaps

In interest rate swap (IRS) trading, holding‐period returns are typically decomposed into three components:

1. Carry
2. Roll-down
3. Curve-shift impact

This article focuses on the first two and then discusses the roll-down calculation for forward-starting swaps.

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I. Core Concepts

1. Par-Swap Rate
   Given a start date a and an end date b, the par-swap rate is the fixed rate that makes the present value of the fixed leg equal to the floating leg:

   $$S(a \to b) = \frac{DF(a) - DF(b)}{\displaystyle\sum_i \tau_i\,DF(t_i)}$$

   • DF(t) is the discount factor to date t.
   • $\{t_i\}$ are the fixed-leg payment dates.
   • $\tau_i$ is the year-fraction between $t_{i-1}$ and $t_i$ under the chosen day-count convention.

2. Carry
   Measures the annualized net return of holding a spot swap until a future horizon date—ignoring any parallel shift of the curve:

   $$\mathrm{Carry} = S(\mathrm{Horizon}\to\mathrm{Maturity}) - S(\mathrm{Start}\to\mathrm{Maturity})$$

   • To express in basis points (bps), multiply by 10 000.

3. Roll-down
   Measures the annualized return due solely to the shortening of the remaining maturity, again ignoring curve-shift effects:

   $$\mathrm{Roll\!-\!down} = S(\mathrm{Start}\to\mathrm{Maturity}) - S\bigl(\mathrm{Start}\to(\mathrm{Maturity}-\mathrm{Horizon})\bigr)$$

   • Multiply by 10 000 for bps.

4. Forward-Starting Swap

   • Forward Tenor $\tau$: the delay from today before the swap begins (e.g. 1M, 3M, 6M, 1Y).
   • Underlying Tenor $T$: the nominal term of the swap once it starts (e.g. 1Y, 2Y, 5Y).
   • Forward Rate:

     $$F(\tau, T) = S\bigl(\text{Today}+\tau \;\to\;\text{Today}+\tau+T\bigr).$$

   • Forward Roll-down
     Choose a Period $\Delta$ (e.g. 1M or 3M) which defines how far you “roll” the forward start date back, then:

     $$\mathrm{Roll}_{fwd} = F(\tau, T) \;-\; F(\tau - \Delta,\,T).$$

     • Multiply by 10 000 for bps.

   Note: A forward-starting swap has no cash flows before its effective date, so it has no carry—only roll-down.

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II. Formulas at a Glance

1. Spot Swap

   • Carry:
     $\displaystyle C = S(H \to M) \;-\; S(0 \to M)$
   • Roll-down:
     $\displaystyle R = S(0 \to M) \;-\; S\bigl(0 \to M - H\bigr)$

2. Forward-Starting Swap

   • Forward Rate:
     $\displaystyle F(\tau,T) = S(\tau \to \tau+T)$
   • Forward Roll-down:
     $\displaystyle R_{fwd} = F(\tau,T) \;-\; F(\tau-\Delta,\,T)$

All rate differences can be converted to bps by multiplying by 10 000.

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III. Business-Level Calculation Steps

1. Prepare the Discount Curve
   Obtain an OIS or LIBOR discount curve that provides $DF(t)$ for any date t.

2. Define Key Dates and Parameters

   • Spot Swap:
     • Start = Today
     • Horizon = Today + H
     • Maturity = Today + M
   • Forward-Starting Swap:
     • Forward Tenor $\tau$
     • Underlying Tenor $T$
     • Roll Period $\Delta$

3. Compute Par-Swap Rates

   • Spot:
     • $S_0 = S(0\to M)$
     • $S_H = S(H\to M)$
     • $S_r = S(0\to M-H)$
   • Forward:
     • $F_\tau = F(\tau,T) = S(\tau\to\tau+T)$
     • $F_{\tau-\Delta} = S(\tau-\Delta \to \tau-\Delta + T)$

4. Take Differences and Convert to bps

   • Spot Carry = $(S_H - S_0)\times10\,000$
   • Spot Roll-down = $(S_0 - S_r)\times10\,000$
   • Forward Roll-down = $(F_\tau - F_{\tau-\Delta})\times10\,000$

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IV. Numerical Examples

1. Spot Swap (5-year swap held 1 year)

   • $S(0\to5)=3.00\%$
   • $S(1\to5)=3.75\%$
   • $S(0\to4)=2.50\%$
   • Carry = $(3.75\%-3.00\%)\times10\,000 = 75$ bps/yr
   • Roll-down = $(3.00\%-2.50\%)\times10\,000 = 50$ bps/yr

2. Forward-Starting Swap

   • Underlying = 1Y, Forward Tenor = 3M, Period $\Delta$=1M
   • $F(3M,1Y)=S(3M\to15M)=1.516\%$
   • $F(2M,1Y)=S(2M\to14M)=1.532\%$
   • Forward Roll-down = $(1.516\%-1.532\%)\times10\,000 = -1.6$ bps

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V. Reference: Bloomberg Forward-Swap Matrix

Bloomberg Forward IRS Roll Down

• Period (top-right dropdown): the roll-back interval $\Delta$ (e.g. 1M, 3M)
• Forward Tenor (column headers): the deferral τ before the swap starts (e.g. 1M, 3M, 6M, 1Y…)
• Underlying (row headers): the swap’s nominal tenor $T$ (e.g. 1Y, 2Y, 3Y…)

Each cell shows:

• Forward = $F(\tau,T)$
• Roll-down = $\bigl[F(\tau,T)-F(\tau-\Delta,T)\bigr]\times10\,000$ bps