# Swap curve construction

I am new to this area so my question might be basic to many but please answer.

For valuing interest rate swaps how will we define which curve to take, like sometimes we use usd 3m curve, or usd 3mv 6m curve.

1.whats the logic behind selection of curve

2 how these curves are constructed specially 1mv3m or 3mv6m curve

3 any reading material u can suggest to read about these curves.

• If the IRS swap you are valuing is '3m Libor vs fixed' you need the 3m curve, that is clear yes? In additional you will need a separate discounting curve. Are you familiar with the "two curve framework" for valuing swaps? – noob2 Jul 8 '20 at 17:25
• Yes estimation and discounting curve – Novice Jul 8 '20 at 17:48
• When we will use 3mv6m or 1mv6m – Novice Jul 8 '20 at 17:52
• @Novice: could you pls accept one of the answers below (by clicking on the tick mark), so that this thread is marked as "answered" ? – Jan Stuller Sep 28 '20 at 10:53

## 2 Answers

I think I understand the question, but maybe not.

In USD market, the most liquid IR swaps have floating leg reset quarterly from 3Mo LIBOR. (The fixed leg is semi-annual. Ths will change when LIBOR is discountinued; it looks like the most common SOFR floaters will have annual frequency both for fixed and floating legs). (Market conventions differ for other currencies.) If you build your swap curve from such swap rates and ED futures (whose underlying is 3MO LIBOR) and all you want from this curve is to project 3Mo LIBORs (e.g. to project the cash flow of a vanilla IR swap), then you don't need the tenor basis.

But if you intend to use this swap curve to project other LIBOR tenors, e.g. 1Mo, 6Mo, or 12Mo, perhaps because you have some cash flows that reset from these, then, for better accuracy, you should include some more quotes among your fitting instruments, to specify the spreads for float-for-float swaps, where one pays 3Mo LIBOR and receives 1Mo (or 6Mo or 12Mo) LIBOR + $$n$$ basis points for various swap maturities.

If you're using a good rates library that supports multi-curves, then you just pass the 3-6 etc tenor bases to your curve fitter, and later specify the tenor when you ask for a discount factor. But if your old library limits you to single cuvres, then the easiest and (I believe) most common workaround is to have multiple USD swap curves:

• the vanilla one uses swap rates for "3mo libor v ssemi-annual fixed".

• the "6mo" and "12mo" ones, for which you may need to solve for the fixed rates that you will pass to the curve fitter. (I believe 6mo or 12mo libor v fixed swaps are less liquid than v 3mo libor.)

I just searched and found this paper, which seems to explain float-for-float swaps pretty clearly.

Also this paper has a good discussion of tenor swaps.

I think your question can be split into two parts: (i) how to value a swap mathematically and (ii) how swaps actually work as a traded product.

Part (i):

As noob2 pointed out, "theoretically", a swap is valued with the help of two curves: one "forward" curve and one "discounting" curve. Say you want to "value" a 10-year swap, fixed against 6-m floating. The formula is straight forward:

$$\sum_{i=1}^{10}r*Df(t_i)= \sum_{j=0}^{19}\tau r_f\left(t_{j/2}\right) Df(t_{j/2})$$

Above, on the LHS: $$r$$ is the fixed annual rate for which you need to solve, $$Df(t_i)$$ is the discount factor between time $$t_0$$ and time $$t_i$$, where the unit of $$i$$ is years (so $$t_{i=10}$$ is 10 years from now). On the RHS: $$\tau$$ is the annual fraction, $$r_f(t_j)$$ is the forward rate at time $$t_j$$, where the unit of $$j$$ is again 1 year: to make the notation clear, $$t_{1/2}$$ would denote a point in time six months from today, $$r_f(t_0)$$ would be today's value of the (spot) 6-month rate and $$r_f(t_{1/2})$$ would be the 6-month rate 6 months from today.

The equation is easy to solve as:

$$r= \frac{\sum_{j=0}^{19} \tau r_f\left(t_{j/2}\right) Df(t_{j/2})}{\sum_{i=1}^{10}*Df(t_i)}$$

Where does the discount curve come from? From OIS Swaps. Where does the 6-m forward curve come from? Up to about 2 to 3 years, the FRAs tend to be very liquid so the 6-m forward rates can be extracted directly from the FRAs. How do you build the 6-m forward curve from the 3-year mark onwards? That brings me to part (ii):

Part (ii):

Pure quant would think of the fixed rate $$r$$ as the rate we need to "solve for" in the equation in part (i) above. However, in practice, it actually doesn't quite work like that. The swaps are one of the most liquid products and the swap rate $$r$$ itself is actually the traded and quoted product: market makers constantly update the value of $$r$$ as they get asked for quotes.

The 6-m forward curve actually isn't liquid or even traded beyond the 3-year mark (depending on the currency we're talking about). So actually, the 6-m forward curve (beyond - say - the 3-year mark) would be constructed via bootstrapping from the traded $$r$$!! Not the other way around!!

One might wonder how that can be done, because the granularity of the 6-m forward curve is 2 points per year, but the granularity of $$r$$ tends to be only 1 point per year: yes, you guessed it, some sort of interpolation would have to be used to get the two-forward 6-m points per each annual $$r$$.

If the $$r$$ is the traded rate, why would you then ever need to construct the 6m or the 3m forward curve? The asnwer is: to value more exotic swaps, such as "3-month forward-starting, 6.5 years swap".

With regard to your question about 3m or 6m floats versus fixed: for liquid currencies such as EUR or USD, there are two separate swap curves (i.e. $$r$$ curves): one is against 3m float, and the other is against 6m float.

Edit - Libor Transition: it is interesting to note that the Libor quoting system will undergo a transition in the near future: (https://en.wikipedia.org/wiki/Libor#LIBOR_transition)

• Regarding your last comment I'm not sure that going back to single-curve architecture would be possible now. For USD, the projection curve for SOFR-linked coupons will be the SOFR curve; but there are swaps reset from Fed Funds or SIFMA Muni index or other exotics, which may be in the multicurve . The discount curve should be whatever the collateral earns / funding costs, which may be approximately OIS. For other currencies, the multicurve may include cross-currency basis. There are also advantages in including treasury yield curves in multicurves. – Dimitri Vulis Jul 10 '20 at 15:53
• @DimitriVulis: good points, Dimitri: I removed the comment from the edit. Thanks for your inputs! – Jan Stuller Jul 11 '20 at 7:45