# How would this 10s/20s steepener work

Say I'm interested in a trade that wants to execute a 10s/20s steepener

This is done via a receiver leg on the 10s and a payer leg on 20s

Look at the following example (the figures are all indicative) I understand the basic logic of the trade, that we are long the short end and short the long end, because the fixed cashflows we receive on the short-end become more valuable if demand spikes and yields fall. On the long-end, paying fixed cashflows is advantageous when prices fall and subsequently yields rise.

But how is my carry on 6months, a year or 5 years calculated?

Would it simply be $$500,000\times0,04\times\frac{6}{12}-1,000,000\times0,02\times\frac{6}{12}$$ for the $$6m$$ carry, for example?

Furthermore, why would the PV01 of the receiver leg be positive? The way I understand PV01: If market yields move up a point, then what is the change in PVs of the cashflows. If this is the definition, then my PV01 on the receiver leg should be negative because my present value would decrease?

This may be a trivial trade but it is my first time seeing this and I'd be interested to hear how carry etc. is calculated, and how the notionals are changed to ensure that such swaps with different tenors are indeed initially "fair", i.e. have an NPV of $$0$$

Carry is typically only associated with known cashflows - its closely related cousin, roll, is typically associated with unknown cashflows, assuming the state of the world is unchanged.

Given this, carry is typically only analyzed for the current period of the swap or bond. If we assume your swaps are fixed Semi vs a 6m Ibor index, then the natural period to analyze carry for is 6m. If we just focus on the 10y swap where you're receiving, its carry will be the fixed swap rate minus the 6m Ibor fixing for the first period only - if, for example, the fixing were 1.75%, then the swap would have a carry of (2% - 1.75%) = 25 basis points (annualized) for the first 6m.

Since your 20y payer is presumably against the same 6m Ibor index, its 6m carry would be (1.75% - 4%) = -225 bps. Once you've expressed the carry for each swap, you can multiply by notionals / accrual factors to figure out your total cash carry if that's required:

$$(+25 bps * 1mm * \frac{6}{12}) + (-225 bps * 0.5mm * \frac{6}{12}) = -4,375$$

As for the risk on each trade, PV01 and DV01 are similar but different in a crucial way - PV01 is the Present Value (PV) change from moving the fixed rate on the swap up by one basis point, whereas DV01 is the PV change from shifting the floating rates across the term structure of the yield curve down by one basis point. Therefore, in general, they are not the same numerical value.

Typically, at trade inception the ratio of the PV01s defines the notional amount to do in each swap - for your 10s20s steepener, if you want 200 of risk to the spread then you would need to pay on $$1mm * (200 / 120) = 1.66mm$$ of 20y

• Can you elaborate on your last point wrt 200 of risk to the spread? What does the $1.66mm$ of $20$y actually mean? – MinaThuma Sep 5 '19 at 5:07
• Let's assume you're talking about USD. If you want to have 200 USD DV01 to the 10s20s spread, and the DV01 per leg of the trade is as specified in your problem statement, then receiving the 10y swap rate on a notional of 1 million USD is 200 DV01 - to make the trade DV01 neutral, you need to pay 20y fixed rate on a notional of 1.66 million USD for a risk of -200 USD on 20y, so that your +200 on 10y cancels out with the -200 on 20y. This way, to first order, you have +200 USD exposure to the difference in the 20y swap rate and the 10y swap rate. – thetableed Sep 6 '19 at 0:43