# Curve Trades - Forward Swap vs Swap(Payer and Receiver)

let's say I want to do a steepening trade. What would be the difference between

1. entering a swap starting in 5 years and lasting for 5 years (5y5y)
2. entering a payer swap with a tenor of 10 years and receiver swap with a tenor of 5 years

Help is much appreciated:)

Option 1 is not a steepener trade. It is an outright bearish trade that the 5y5y forward rate will move upwards. Option 2 is a steepener trade, if the dv01 is equal on the 5yr and 10yr legs. Ignoring discounting , you would need to pay fixed on 50mm 10yr versus receiving on 100mm 5yr to make it duration neutral, and thus a curve trade.

We assume a single-curve environment. Let us recall that a floating LIBOR payment fixed at time $$T$$ and paid at time $$T^\prime$$ can be written in terms of zero-coupon bonds: $$L(t,T,T^\prime):=\frac{1}{T^\prime-T}\left(\frac{P(t,T)}{P(t,T^\prime)}-1\right)$$ Let $$\mathcal{T}:=\{T_0,\dots,T_n,\dots,T_m\}$$ be a schedule such that a spot 5y swap starts fixing at $$T_0$$ and makes the last payment at $$T_n$$, whereas the spot 10y swap stops paying at $$T_m$$. Let $$\delta_i:=T_i-T_{i-1}$$ be the accrual fractions. We denote by $$S_{5y}$$, $$S_{10y}$$ and $$S_{5y5y}$$ the 3 swap rates respectively $$-$$ with self-evident notation. We assume same date conventions both between the float and fix legs, as well as the 5y and 10y swaps, to avoid unnecessary complications. The swap rate for $$k\in\{1,\dots,m\}$$ is defined as:

\begin{align} S_\bullet := \frac{\sum_{i=1}^k\delta_iP(t,T_{i-1},T_i)L(t,T_{i-1},T_i)}{\sum_{i=1}^k\delta_iP(t,T_{i-1},T_i)} \end{align}

First off, note that the value of the floating leg from the 10y minus 5y trade is equal to: \begin{align} &V_{10y}^{\text{Flt}}-V_{5y}^{\text{Flt}} \\ &\qquad=\sum_{i=1}^m\delta_iP(t,T_i)L(t,T_{i-1},T_i)- \sum_{j=1}^n\delta_iP(t,T_i)L(t,T_{i-1},T_i) \\ &\qquad=(P(t,T_0)-P(t,T_m))-(P(t,T_0)-P(t,T_n)) \\[8pt] &\qquad=P(t,T_n)-P(t,T_m) \\[3pt] &\qquad=\sum_{i=n+1}^m\delta_iP(t,T_i)L(t,T_{i-1},T_i) \\[3pt]\tag{1} &\qquad=V_{5y5y}^{\text{Flt}} \end{align} Hence the value of the floating leg from the 10y minus 5y is equal to that from the 5y5y swap. Now, the value of the fixed leg from the 10y minus 5y position is: \begin{align} &V_{10y}^{\text{Fix}}-V_{5y}^{\text{Fix}} \\ &\qquad=\sum_{i=1}^m\delta_iP(t,T_i)S_{10y}- \sum_{j=1}^n\delta_iP(t,T_i)S_{5y} \\ &\qquad=\sum_{i=n+1}^m\delta_iP(t,T_i)S_{10y} +\sum_{i=1}^n\delta_iP(t,T_i)S_{10y} -\sum_{j=1}^n\delta_iP(t,T_i)S_{5y} \\ &\qquad=\frac{S_{10y}}{S_{5y5y}}V_{5y5y}^{\text{Fix}} +\left(\frac{S_{10y}}{S_{5y}}-1\right)V_{5y}^{\text{Fix}} \\ &\qquad=V_{5y5y}^{\text{Fix}}\left(\frac{S_{10y}}{S_{5y5y}} +\left(\frac{S_{10y}}{S_{5y}}-1\right) \frac{V_{5y}^{\text{Fix}}}{V_{5y5y}^{\text{Fix}}}\right) \end{align} Note that if the curve if flat, that is $$S_{5y}=S_{10y}=S_{5y5y}$$, you recover @emot's statement that both strategies are the same. Another way of thinking about the 5y5y steepener is by introducing the fixed leg annuity, which is defined as: \begin{align} A_\bullet&:=\sum_{i=\bullet}^\bullet\delta_iP(t,T_{i-1},T_i) \\[4pt]\tag{2} &=\frac{V_\bullet^{\text{Flt}}}{S_\bullet} \end{align} Then using $$(2)$$, $$(1)$$ and noting that $$A_{10y}=A_{5y5y}+A_{5y}$$, you can rewrite the 5y5y rate as follows: \begin{align} S_{5y5y} &=S_{10y}+(S_{10y}-S_{5y})\frac{A_{5y}}{A_{5y5y}} \end{align}

I would expect the ratio of annuities to be close to but above 1 in a rising curve, hence you see you seem more exposed to an increase in the 10 year rate when you’re long a 5y5y swap: \begin{align} S_{5y5y}&\approx S_{10y}+(S_{10y}-S_{5y}) \end{align} On the other hand: \begin{align} S_{10y}-S_{5y}&=(S_{5y5y}-S_{10y})\frac{A_{5y5y}}{A_{5y}} \\ &\approx S_{5y5y}-S_{10y} \end{align} From the definition of the swap rate, you observe it can be interpreted as an average of the LIBOR rates during the swap’s period. Hence the 10y-5y trade profits if the average over the second 5y period is greater than the average over the whole period, that is rates over the second period are greater than over the first one. Hence a 10y-5y seems a better strategy as steepener.

• What’s missing here (or I’m missing it in the answer) is that 5v10yr is traded duration weighted. So you would buy/sell roughly 2x the 5yr nominal as you would sell/buy the 10yr swap. Jul 10, 2021 at 9:02

I am assuming that in option 1 you are entering into payer swap. If the curve is flat then option a) and b) are the same because you will get the same cashflows in both cases. Why? In option b) both the floating legs and fixed legs on 10Y swap and 5Y will cancel for the first 5 years i.e. the cashflows will be opposite sign, effectively making it 5y5y swap. If the curve is not flat, then the resulting cashflows in 10Y and 5Y swap in option b) will be different and won't offset each other. The forward swap rate in option a) will also be different than the swap rates of 10Y and 5Y spot deals.