Mini Chapter One
Interest Rate Derivatives
 Interest Rate Swap (IRS) – is an exchange of the same currency interest rate payments between two parties interested to either express an interest rate view and or hedge interest rate risk. The more commonly referred convention is the exchange of fixed vs floating rate payments. Risk on an IRS therefore can be decomposed as being a long (short) floating rate bond and short (long) fixed rate bond. Other salient features below:
 Counterparties that trade par interest rate swaps enter into a swap with 0 Present Value adjusted of course for bidoffer slippage i.e. execution mid denotes that the PV of projected floating rate cash flows is equal to PV of fixed rate cash flows through the tenor of the swap.
 Markets also trade swaps with a fixed strike (usually different from the par swap level) if one is trading the same currency for the same dates so as to minimise the number of line items on the risk blotter. For instance a 3.5% September IMM 5y KRW IRS can be initiated by trading the bid or the offer side of the par swap quote with an exchange of fee between the two counterparties offsetting the present value on the swap.
 Paying a fixed rate swap = short rates or selling a bond, receiving a fixed rate swap = long rates or buying a bond
 There is no principal notional exchange as in the same currency IRS the fixed and the floating payments net off. Only net cash flows at the time of reset/unwind/maturity are exchanged.
 Pricing of a swap – fixed rate can be thought of as the IRR of the expected floating rate for the underlying swap tenor. In other words the fixed rate of a swap for a tenor T is the implied average floating rate for that tenor such that the PV of the expected floating rate payments is equal to the PV of the fixed rate payments.
Where C is the par fixed swap trading in the market,
DF_{n} is the discount factor for the nth time period
FLR_{n} is the Expected/Projected floating rate for the nth time period
Mathematically then the observed par fixed swap rate C can also be derived by equating the two notations above and plugging in the expected floating rates in progressive time periods.
Importantly,
assuming for the sake of simplicity the cash flows are funded at the floating rate. We discuss a more nuanced concept of the discount curve being different from the projected floating rate curve later in the section.
 Duration and DV01 of a swap – worth repeating here that if you think of duration as the time taken to receive the promised cash flows then the concept is incongruous for a swap with 0 PV (when it’s initiated). To assess the interest rate sensitivity for a portfolio of bonds and swaps we end up using the modified duration of just the fixed leg of the swap (same as that for a fixed rate bond with same fixed leg cash flows) as the duration of the swap; rate sensitivity of the floating leg is much smaller as discussed in the duration of a floating rate bond. Dollar value of a basis point (DV01) for swap is a more astute measure for assessing the interest rate sensitivity (and the more widely followed duration metric) of a swap.




 DV01 of a swap is the sum of the DV01 of the fixed and floating legs (remember that DV01 captures the direction of risk to assess the portfolio’s pnl swings with movement in rates) of the swap or is the sum of the DV01 of a long (short) fixed rate vs short (long) floating rate bond being exchanged as implied in the swap cash flows. DV01 of the floating leg is a function of the swap reset frequency. Also until the first reset – the interest rate sensitivity of the floating rate bond doesn’t come into play i.e. the DV01 of the swap for a tenor T is equal to that of a fixed rate bond for the same tenor. After the first reset however the floating leg would acquire a finite DV01 albeit small, equivalent to that of a bond with a tenor of the floating rate period.



 If you recall the Bond DV01 for a Bond’s dollar duration notation below where B is the bond price:
The same for a swap would be denoted as
 The positive convexity of being long a bond in the same manner would also apply on a long swap (received rates) position.
 Breaking down risk buckets in a swap – risk in a swap is crucially dependent on the way the curve is constructed. Usually market traded tenors are used to construct (bootstrap) the curve, hence risk of any tenor of a swap would be deemed sensitive to the movement in swap tenors that are used to interpolate its price. The process of recognising the risk sensitivity across tenors is called risk bucketing. For instance, the present value of a 4 year swap would be impacted by moves in the 2y and 5y par swaps (both benchmark/more liquid tenors) and a linear interpolation effectively would have 1/3rd risk in 2y vs 2/3rd risk in 5y. In other words, the size of component risk on a certain benchmark tenor is inversely proportional to its distance from the nonbenchmark tenor.
 The risk sensitivities are dependent on the way intermediate (and generally less traded) points on the curve are constructed. Different methods of interpolation could use regression analysis, cubic spline interpolation or for that matter try and fit any form of a polynomial interpolation depending on what best depicts the curvature of that rates market.




 Leverage on a swap – linking the concept of a forward starting swap and risk bucketing – a 1y forward 1y swap has twice as much risk on 2y as it has on 1y as per its bucket components and in the opposite directions. Hence a 1bp move in 2y with no change to 1y would amount to a 2bps move in the total pnl of the swap i.e. that’s a 2x sensitivity which can be understood as leverage of the swap to the 2y point. Defined simply when a 1bp move in any component tenor (all else constant) causes a pnl move in excess of the overall DV01 of the swap, it is understood to have leverage with respect to that tenor.



Table 1 – Bucketing and Leverage of spot and forward starting swaps
Tenor bucket  Spot 6y  Spot 8y  1y forward 1y  4y forward 1y  3y forward 2y 

1y  +10K  
2y  20K  
3y  +15K  
4y  +40K  
5y  5k  50K  25K  
7y  5k  6.6k  
10y  3.3k  
Leverage  2x  5x  2.5x 
* Standard DV01 risk assumed to be $10K, swap direction is received fixed rate
Source: Pandemonium.
 As an example – for a 1y forward 1y swap a 1bp move in just the 2 year tenor will move the swap pnl by 2x, hence the 2x gearing to the 2y point. Similarly in a 3y forward 2y swap a 1bp move in the 5y bucket only would move the swap pnl by 2.5x hence leverage (gearing) is 2.5x to the 5y point. A simple notation can be:
Leverage multiple on Rxy swap w.r.t (x+y) year tenor =
 Street likes to trade forward starting swaps for the following reasons: a) to avoid exposure to fixings and instead express views on it b) depending on the conviction of the trade (outrights and curvature) take leverage to express it c) administrative ease of managing risk for those who’d like more outright risk.