# Carry calculation on an interest rate swap

I was hoping that I can get help on a simple yet not so straight forward topic :

Looking at valuing the costs of holding an IRS in the books this would entail marketed-to-market due to price movements in addition to Carry & roll down.

My question is specific to Carry of an interest rate swap.

On an IRS there would a fixed leg and a floating leg, assume that we are running a 5 year IRS where we are paying a USD fixed rate quarterly and receiving 3m Libor floating quarterly .Assume 5y spot rate is 2% & 3m libor is 1.3%

Intuitively the 3 month carry would be (spot rate - libor ) , in our case 2%-1.3% quarterly

My question is why is it to calculate carry the following is used instead:

Carry = forward rate - spot rate

In our case (4.75 years IRS starting in 3 months) - ( 5y spot rate )

please explain to me like I'm a 6 year old

Any links or txt that you can provide would be appreciated

Kind Regards

## 3 Answers

It turns out that the two things are the same, appropriately scaled. Proof: we can construct a 5 year swap using 3 month libor combined with a 3mo-4.75yr forward swap, weighted by the dv01s of each part. Thus, ignoring discounting, we have

5yr swap rate = (0.25*3mo libor + 4.75*forward rate)/5.

This can be rewritten as

0.25*(5yr swap rate - 3moLibor) = 4.75*(forward rate - 5yr swap rate)

Thus the two methods are equivalent, when each is multiplied by its relevant weighting. Note: if you do this with discounting, the 4.75 gets replaced by the dv01 of the forward swap.

• I did not quite understand how the first equation can be re-written as the second – Alex C Aug 27 '17 at 17:02
• if a = (0.25b +4.75c)/5 then 0.25*(a-b)=4.75*(c-a) is obtained by multplying both sides by 5, then subtract 4.75a+0.25b from both sides. – dm63 Aug 27 '17 at 20:38
• Hi both yes indeed I didn't get how the equation was rewritten – user29352 Aug 29 '17 at 19:15
• Thanks dm63 for you answer , but still kinda confused around this – user29352 Aug 29 '17 at 19:16
• Looking at the carries for USD IRS(running) for example and these can be found on Bloomberg carry function .. one would notice that 10yr carry is close to 5y carry .. and these are calculated using carry=fwd yield -spot yield . But if just using (5y IRS yield -3mlibor) compared to (10y IRS yield-3m libor) one would assume that 10y would have a higher carry given the steep nature of the curve – user29352 Aug 29 '17 at 19:30

I will attempt to summarise the content included in this book, which has a specific chapter dealing with carry and roll-down.

There, two concepts are made completely separate.

1. Costs-of-carry are defined as costs relating to holding a trade that are not directly related to market movements. For example, funding a margin requirement for an IRS facing a clearing house, or funding the regulatory capital charge imposed by regulators for transacting IRSs. These costs are not necessarily consistent from one user to another.
2. Roll-down is defined as the expected profit-and-loss (PnL) if over a period of time the interest rate swap curve remains the same as its current state (shifted in time) as opposed to evolving to its future predicted values.

Personally I have traded IRSs for over 11 years and have never used carry and roll in the way you describe. Why? A mid-market swap is precisely that; a swap expected to not gain or lose any value given the future forecast evolution of rates. If, over the first 3 months you acquire 0.7% (2%-1.3%) but rates evolve exactly as forecast you are left with cash in your pocket and a swap liability of precisely the opposing amount of cash. If you wanted to exit the swap at that point you would be left with no P nor L, since your cash would have to fund its exit.

On the other hand, if the interest rate curve had moved so that the future curve reflected the starting curve (shifted in time), this would represent a genuine PnL event. This movement is described as 'roll-down'. Since the first fixing is known, the only part of your 5Y swap that can change is the 3-mth fwd 4.75-Yr part. The roll-down is calculated by evaluating the difference in rate between the current 4.75Y swap and the 3M4.75Y swap (delta adjusted for just that portion of the swap).

I recognise this isn't a direct answer to the specific question but I hope it elucidates the concept nonetheless.

I think that there is really a bit more to this calculation that hasn't been answered yet.

How to compute carry (and rolldown) depends on your view of realized forward scenario. In particular, Forward(t, n) - Spot(n) would be the answer if you assume that tomorrow realized spot rate is the same as today's spot rate.

• Assuming tomorrow realized spot rate is the same as today's spot rate

Under this assumption, the return we make as carry is ((1+Spot5y)^5/(1+Spot4y)^4 - 1) - Funding rate(i.e., spot 5y). Since we know (1+Spot5y)^5 = (1+Fwd4y5y) * (1+Spot4y)^4, then carry = Fwd4y5y - Spot5y (assuming we pay fixed). The corresponding roll down is Spot5y-Spot4y on the rest 4y swap. So in total, we make Fwd4y5y - Spot4y.

• Assuming tomorrow realized spot rate is the same as today's forward rate

Similarly, the return we make as carry is ((1+Spot5y)^5/(1+Fwd1y2y)/(1+Fwd2y3y)/(1+Fwd3y4y)/(1+Fwd4y5y) - 1) - Funding rate(i.e., spot 5y) = Spot1y - Spot5y. The corresponding roll down is Spot5y - Fwd1y5y. So in total, we make Spot1y - Fwd1y5y.