In this module we're going to introduce
you to swap contracts, introduce you to the ideas why these swap contracts are constructed, what is the advantage of swap
contracts. And also show you how to price a very simple interest rate swap using a
no-arbitrage principle. Swaps are contracts that transform one
kind of cash flow into another. A plain vanilla swap transforms a fixed
interest rate cash flow into a floating interest rate cash
flow. A commodity swap, swaps or exchanges
floating price of, for the commodity for a fixed price for the
same commodity. Examples include gold swaps, oil swaps,
and so on. Currency swaps allows you to swap a cash
flow in one currency for a cash flow in another
currency. So why do companies construct or entities
construct swaps? They do because you want to change the nature of
cash flows. The example is fixed interest rate versus
floating interest Another possibility is to construct swaps to
leverage strengths in different markets. Here's an example. Company A, were it to borrow in the fixed interest rate market would be charged 4%
per annum. If it were to borrow in the floating rate
market, it would have to pay LIBOR plus 0.3%. LIBOR stands for the London Interbank
Offer Rate, and that's the rate which is used as the base for
floating, floating interest rates. Company B, were it to borrow in the fixed
interest-rate market, would be charged 5.2 percent, and in the floating interest-rate
market, it could borrow at LIBOR plus one percent. So company A is clearly superior to
company B, both in the fixed interest rate market as well as the
floating interest rate market. However, company B is relatively stronger
in the floating rate market. The difference between the rate for
company A and B is only 0.7% in the floating rate
market. Whereas it's one point two percent in the
fixed rate market. So these companies, could take advantage
of the difference of their relative strengths
in the true markets to create an, an
instrument which lets them borrow at the better rate than they could have
individually borrowed in either of the two markets. Company A, which is stronger in the
fixed-rate market, borrows in the fixed-rate market, company
B, which is stronger in the floating-rate market, borrows in
the floating-rate market, and then they construct a swap in order to
make an additional product or a derivative
product. Which is going to be better than each of these individual deals that are available
to this company. So here's company A, it borrows at 4%, which means that it's going to pay out 4%.
Here's company B, which borrows at libor plus 1%.
And then we construct a swap, we have company A, B is
company B LIBOR and company B pays company A
3.95%. And if you see what the net effect of this
swap is to each of these companies. Company A now ends up paying LIBOR plus 0.05%, which is better than what it could
have borrowed in the floating rate market, and company B ends
up paying 4.95%. Which is better than what it could have
gotten in the fixed-rate market. So by constructing this swap, these two companies are able to leverage their
relative strength to get a deal which is better than what
they could have achieved in, either the fixed-rate
market or the floating-rate market. Both of them end up gaining. The details of how to the 3.95% gets set
depends on supply and demand. But there is an inplicit assumption that
is being made in this particular example, and that's that
company A and company B continue to exist. That neither of them is going to default. If one of these companies were to default then this entire superstructure breaks
down and company A or company B, depending upon which one has
a default, will be exposed to a big risk. So most of the times when swaps are
constructed. You don't make a swap with a counterparty
directly because you don't want to be exposed to the
counterparty default risk. You would rather make it with an intermediary, a
financial intermediary, that is able to take on the counterparty
risk, and guarantees you, that you will get the
swap cash flow that you're expecting to get
from the counterparty. So here's how, these swaps get, set up. The same two companies, A and B, and now there's a financial intermediate that
contracts the swap. Company A borrows in the fixed market at 4%, swaps
with an intermediary and pays LIBOR and receives 3.93%.
So, 0.02% less. Company B borrows in the floating rate
market at LIBOR plus 1%.
Constructors swap with an intermediary, receives LIBOR and pays, 3.97%.
So 0.02% more, or this is the same thing as saying two
basis points less, two basis points more. The financial intermediary in the middle
makes this 0.04%. It receives two basis points from company
A, and receives two basis points from company
B. Why does it get that? This is the compensation for taking on the
counter party risk. If either of these two companies default,
the financial intermediary's on the hook, to provide the cash flow
necessary for the surviving party. And it also constructs a service in the sense that, typically, in
the market, company A and company B don't know
that they exist, and their relative strengths are
different so, by creating a swap, they would be able to better
position themselves. So financial intermediary is able to bring
these two parties to the table and construct a swap that is going to mutually beneficial,
and ends up getting paid for providing this
service. So let's see how an interest rate swap is priced. Let rt denote the floating unknown
interest rate of time t. And let's consider a swap whose cash flows
at time little t equal to 1 through capital T is given as
false. Company A, which takes on the long
position in the swap, receives a notional principal n, times the random
interest rate prevailing at time t minus 1. So the cash flow that it receives at time t is given by the notional principal n.
Times the interest rate at time T minus 1. And it pays the same notion of principal n
times a fixed interest rate x. Company B, which nominally takes on a
short position on the swap, receives the fixed
interest rate payment n times capital X, and pays
the floating payment n times rt minus 1. Now what we want to do is compute the
value of the swap to company A. So there are two pieces to the cash flow
to company A. It receives the cash flow, the principle,
n times r 0, r 1, r 2 and so on up to r T minus 1 at times 1, 2, 3, 4 up
to capital T minus 1. This is precisely the cash flow associated
with the floating weight bond, minus the face value. So in a floating weight bond, at the
expiration capital T, in addition to the coupon payment, you have
received the notional principle back. This time you do not get the notional
principle. And therefore, the value of the swap to
company A consists of two elements. This is the value, of the floating rate, payments received, this is the value of fixed rate payments, paid to company B. We know that the value of a floating rate
bond is directly equal to the principle. But we don't get the face value back, so I
have to subtract from that value n times d, 0, t, which is the discounted value of the principal which was received at time
capital T. What happens to the value of the
fixed-rate payment? Every period, I get n times x.
I have to discount that from time t equal to 1 through capital T.
So this is simply the discount values. How is this x set? This is this x set, in a similar manner as
we had done for former contracts. We set at, at a value such that the va is
exactly equal to 0 at time t equal to 0. If you set it up, p equal to 0 implies
that x must be equal to 1 minus d, 0, t divided by the sum of little t going from 1 to
capital T to 0 T. This is the interest rate that you would
have to set up, so that the value of the swap is exactly equal to
company, zero for company A. And it's also equal to zero for company B. So, the two companies going into this
swap, are eq, are, indifferent between taking a long
position or a short position.