Hello and welcome back to my channel. Today's video is my third video in a
series on swaps. Today we'll learn about two different approaches to pricing
interest rate swaps. Hi guys welcome back to my YouTube channel if this is the
first one of my videos you've watched, you should check out some of the others.
We're trying to learn about quantitative finance and derivatives.
Make sure you subscribe if you'd like to see more content like this. Anyhow let's
get on with it and learn about how we price interest rate swaps. We'll work
through the example from my book, which is called "Trading and Pricing Financial
Derivatives. There's a link to it in the description below. As explained in my
first video on swaps, which is linked to above, swaps are usually structured such
that they can be valued at zero at inception. This allows the two parties to
the swap to get going without any cashflow. A swap can be priced at zero
when the NPV or net present value of both cash flows is equal. Once the swap
agreement has been entered into, it can take on a positive or a negative
value, and that just occurs once the market rates for whatever the two cash
flows are changes. Swaps are a zero-sum game,
meaning that one counterparties gain is equal to the other counterparties loss.
If you receive fixed in a swap, the swap value equals the present value of the fixed
cash flows, minus the present value of floating cash flows. If you pay fixed in
a swap, the swap value equals the present value of floating cash flows, minus the
present value of fixed cash flows. So basically what I'm saying is that the
value of the swap equals the present value of whatever you're receiving, less
the present value of whatever it is that you have to pay out. Hopefully that makes
sense to you and it's not very complicated.
For this reason, swaps are amongst the easiest derivatives that there are to
value. There's no complicated formula like with the Black Scholes Merton model.
There's no stochastic calculus, no drawing of lattices like in the
Binomial Tree and no generating price paths like with the Monte Carlo Method. All you
have to do is lay out the expected cash flows and then present value them.
That's really all there is to it, and then the swap is worth to you the
present value of what you get less the present value what you have agreed to
pay. Whatever profit you've made on the swap your counterparty has lost and
whatever loss that you might have made on the swap your counterparty will have
gained. That's it, once we know that, we can use whichever method we prefer to
present value the cash flows. If you've ever priced a bond you can
price a swap. Some people get a little bit confused when valuing a swap
because we sometimes lay out the cash flows a little bit differently to how
you might lay them out when you're pricing a bond, but essentially it's
still the same thing we just kind of group it all together into a table with
a swap because you're able to see it all at once. but in truth you can just price
one bond, price the other bond and you know the the value of the swap. It is what
you're receiving less what you're paying. So you can do this however you want but
In my book I cover two approaches so let's work through them here. A plain
vanilla interest rate swap can be priced as either a combination of a long
position in one bond and a short position in another bond, or as a
portfolio of forward rate agreements. All of the discounting is done using LIBOR zero
coupon interest rates, as it's assumed that the risk on the cash flows is
equivalent to a loan in the interbank market. Let's start out with a
description of a swap and then we will value it using each of our different
approaches suppose that a swap has been in existence for some time already and
that we now are looking to value it assume that it pays floating six month
LIBOR versus 5% fixed semi-annually it's on a hundred million dollars notional
and has 1 and 3/4 years remaining to maturity suppose we know that LIBOR
continuously compounded is 6% six 1/2 percent seven percent and 7.3
percent for the three-month nine month 15 month and 21 month period we are also
aware that on the last payment date six month LIBOR price was 6.2 percent what
is the value of this swap so that's what we have to solve there are two methods
to do this calculation one calculates the equivalent bond values one fixed and
one floating and takes the difference between these present values the other
method involves calculating the value of the swap as a portfolio of fr-s our
forward rate agreements for each of the upcoming payment periods and then taking
the present value of the differences in fixed versus floating cash flows from
each upcoming payment period on screen right now you can see the calculations
of the bond valuation method which is what we're going to work through first
the value to the receive fixed counterparty is the fixed bond valuation
- the floating bond valuation we concede that the swap is worth minus 4.2 -
million dollars. The opposing counterparty will have the opposite
value so they will see the value as plus four point two two million dollars so
what have we done in there we just put together a table of the expected cash
flows for each bond and the timing of the payments remaining on the swap the
fixed payments are easy enough they're just two and a half million dollars each
period and because we're valuing this as if it was a bond a hundred and two point
five million dollars at the last payment we use the LIBOR rates from the question
to present value those cash flows so the column labeled discount factor is just e
to the discount rate which in the first payment period three months is 6% so
that is e to the minus 6% times three over 12 which is three months out of the
twelve months in a year that gives us 0.9851.
We do the same calculation subbing in the different applicable interest rates
and holding periods and we calculate all of the different discount rates we then
just multiply those discount rates by the 2.5 million dollar cash flows and
the final 102.5 million dollar cash flow, sum them all up and
that is the value of the fixed-rate bond which you can see in our example that
it's $97.3419 million the next step is to present
value the floating rate bond. How do we do that? Well a basic semi-annual coupon
floating rate bond has the coupon indexed to LIBOR each coupon date the
coupon is equal to the par value of the note, times one-half the six-month rate
quoted six months earlier at the beginning of the coupon period. So each
coupon is based on the previous six month rate only the next coupon is known
at the current date the later ones are random thus a floating rate bond is
always work par on the next coupon date with certainty and that's just because
we have a floating rate being discounted at a floating rate so it just comes to
par value for the bond so it's actually quite easy to price a floating rate bond
all we have to do is discount the upcoming coupon which we know with
certainty at the appropriate discount rate now you'll note that in the
question the very last line in the question said we are also aware that the
last payment date six month LIBOR price was 6.2%t so to price
that floating rate bond we know that that payment is coming in three months
so we must PV it or present value it at the three month interest rate after
which the bond will be worth par so that's all that's happening in that
column that's labeled floating bond value we have a hundred million plus a
hundred million times six point two percent over two as it's a semi-annual
coupon giving us $103.1 million. We then multiply that by the
discount factor which we calculated earlier as 0.9851 and that gives us
101.565 million dollars the value to the receive fixed counterparty
is the fixed bond valuation minus the floating point value asian we can then
see that the swap is worth -4.22 million dollars. The
opposing counterparty will have the opposite value so that's it that's all
we have to do to price our swap next let's look at valuing the same swap but
this time using a different method we're going to the swap as a portfolio of F or
a's which stands for forward rate agreement we draw up a table with a lot
of the same elements in it as in the last table as you can see the fixed cash
flows are the same the time periods and discount factors are the same and
they're calculated in the same method the first floating cash flow we know in
advance and the remaining ones are calculated as shown in row two where
we're taking the F for a continuously compounded and converting it into an F
or a semi annually compounded in the next column we then work out the
floating cash flows subtract them from the fixed cash flows multiplied that by
the discount rates to get the present value of the net cash flows we sum up
the present values of the net cash flows and get minus four point two two million
dollars the same as the bond method that we just did earlier if you're having
difficulty with that method you should watch my video from a few days ago on
forward rate agreements which will explain the idea of forward rate
agreements to you so that's it the two easiest ways to value a swap you should
probably just learn off one method the one that you find easiest remember that
all we're doing here there's nothing complicated happening
we're just present valuing two cash flow and looking at the difference between
them and that's how we value a swap if you can price a bond you can price a
swap I haven't yet done any videos on bond
valuation maybe I will at some point but if you're having difficulty with bond
valuation you can probably just google it and there's a lot of good information
that you can find online you've made it to the end of the video and the rules of
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