# MtM of interest rate swap if forward rates are realised

It might be a very simple question but for some reason I’m a bit confused.

Let’s say we enter a long SOFR vs fix interest swap at par. Say 5 year swap with annual coupons (the rfr is daily compounded and paid at each annual reset). The swap rate will be the average of forward rates weighted by the DFs (5 periods so 5 sets of DF*forward rate)

Zero coupon curve (and swap curve) is upward sloping.

At initiation the NPV of the swap is zero. If we assume the curve stays what it is then we would lose in MtM due to the rolldown as in 1 year for example our now 4y swap would be MtM at a lower swap rate than our fixed coupon rate.

Now if assume the forward rates are actually realised, the final NPV of the swap should end up being zero by definition. But what does ‘forward rates realised’ actually mean? Does that mean a swap rate constant as we go through time? Spot 5y swap rate = 4y swap rate in 1y = 3y swap rate in 2y, etc.?

If that’s the case, for example in 1 year, our now 4y swap rate would still be the same as our fixed coupon rate -> no mtm impact but if we look at the past cash flows and the first reset, we most likely paid more than we received. So the NPV (past cash flows + future mtm) should be negative at this time.

How would it go up to 0? It seems to me the swap rate of the smaller time-to-expiry swaps should go progressively up to compsentate.

Forward rates realized means if today the 1y forward 4y swap rate is $$X$$ then in one year the 4y spot starting swap rate will be $$X$$. In your example, let's say at inception the 5y spot swap rate is $$Y$$ and the 1y fwd 4y is $$X$$. Let's also set $$Z$$ to be the spot 4y swap rate today (so $$Y>Z$$ for an upward sloping curve). Now you enter a pay fixed 5y swap today at $$Y$$ (so 0 MtM). For a non-flat curve $$X \neq Y$$ (in fact for an upward sloping curve $$X>Y>Z$$). So, if forward rates are realized, after one year you will be paying $$Y$$ on a 4y spot swap while the market is $$X$$, so your MtM on the residual swap will be $$X-Y>0$$.

It's important to distinguish between "nothing happens" and "forwards are realized": The former means the curve stays exactly the same as it is today (so in a year the 4y spot swap rate is still $$Z$$ so your MtM is $$Z-Y<0$$). The latter means the curve changes to exactly what it was expected to as implied by today's rates.

If we let $$U$$ (for unknown) denote what the 4y swap rate ultimately ends up being, then $$Y-X$$ and $$U-Y$$ are the carry and roll, respectively. In this sense, carry is a known quantity and roll is the variable (more on this below). Crucially,

the forward value of the residual swap over a given horizon = -carry over that horizon.

The essential point is this: if you enter a pay fixed 5y swap today, hold the trade for a year, and then unwind it at market, how much money do you expect to make? The answer is 0. Because the expected value of $$U$$, $$E(U)=X$$: when you enter the swap, you lock in negative carry of $$𝑌−𝑋<0$$. When you unwind it at the prevailing 4y rate of $$𝑋$$ you make $$𝑋−𝑌>0$$.

On the other hand if, in a year's time, you unwound the swap and the 4y rate was unchanged i.e. $$U=𝑍$$, then you pocket/lose $$𝑍−𝑋$$ i.e. total carry+roll generated by your trade (mtm of your swap aka rolldown of $$𝑍−𝑌$$ + locked-in carry of $$𝑌−𝑋$$). Note that this the same as paying 1y4y at $$𝑋$$ and closing it out at $$𝑍$$ in a year with just a rolldown. This is the point Attack68 makes below (i.e. carry is a superfluous term in this context).

Finally, receiver carry/roll trades generate a profit if "nothing happens" (i.e. $$U=Z$$). But this is a subtle point because something is happening, namely the fwds are not being realized! So the market is "doing something unexpected" over that year. Any outcome other than the forwards being realized is equivalent in the sense that it indicates presence of volatility. That's a definition of volatility: deviation from expected value. So people looking to profit from putting on receiver carry/roll trades are effectively going long vol.

• Thank for you for your answer. So in the case where the forward rates are realised, in 1 year we make X-Y>0 as mtm on the residual swap. And we make Y-W<0 on the first payment (realised pnl) where W is the 1y spot starting swap. So in terms that’s (X-Y)*4*Notional + (Y-W)*1*Notional and that sum is equal to 0? May 6 at 17:08 • Also is it correct to call X-Y= 1Y carry and Y-Z = 1Y rolldown (for a spot starting swap)? May 6 at 17:10 • I would advise against using the term "carry" with IRS. Rolldown has a meaning, it projects a market movement where the forward rates are projected to move to align with 'todays' observation of the curve. The value of the rolldown is as commented (X-Z). There is no carry, it is a derivative, after 1Y any cashflow(s) you have received are equal and offset to the value of the residual derivative (also as commented here) (unless there is a market movement, such as rolldown, in which case the only acquired value is the rolldown amount) – Attack68 May 6 at 21:41 • In the case of a forward starting swap 1y4y for example, I agree that for a 1y horizon there’s no carry and the rolldown would be X-Z. In the case of a spot starting swap, isn’t the rolldown Y-Z instead for the mtm of the residual swap after 1 year? And on top that we need to add the realised cash flow after the first reset? May 7 at 4:37 • Correct.Y-X$and$Z-Y$are the carry and roll (since we're paying fixed in our example these are both negative). I also think my earlier answer was not helpful so some clarification is warranted. The forward value of the residual swap over a given horizon = -carry over that horizon. The essential point is this: if you enter a pay fixed 5y swap today, hold the trade for a year, and then unwind it at market, how much money do you expect to make? The answer is 0. When you enter the swap, you lock in negative carry of$Y-X<0$. When you unwind it at the prevailing 4y rate of$X$you make$X-Y>0\$. May 7 at 11:04