Interest Rates and Discount Bonds

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Do curso por École Polytechnique Fédérale de Lausanne

Interest Rate Models

76 classificações

École Polytechnique Fédérale de Lausanne

Interest Rate Models

76 classificações

This course gives you an easy introduction to interest rates and related contracts. These include the LIBOR, bonds, forward rate agreements, swaps, interest rate futures, caps, floors, and swaptions. We will learn how to apply the basic tools duration and convexity for managing the interest rate risk of a bond portfolio. We will gain practice in estimating the term structure from market data. We will learn the basic facts from stochastic calculus that will enable you to engineer a large variety of stochastic interest rate models. In this context, we will also review the arbitrage pricing theorem that provides the foundation for pricing financial derivatives. We will also cover the industry standard Black and Bachelier formulas for pricing caps, floors, and swaptions. At the end of this course you will know how to calibrate an interest rate model to market data and how to price interest rate derivatives.

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Interest Rates and Related Contracts

We learn various notions of interest rates and some related contracts. Interest is the rent paid on a loan. A bond is the securitized form of a loan. There exist coupon paying bonds and zero-coupon bonds. The latter are also called discount bonds. Interest rates and bond prices depend on their maturity. The term structure is the function that maps the maturity to the corresponding interest rate or bond price. An important reference rate for many interest rate contracts is the LIBOR (London Interbank Offered Rate). Loans can be borrowed over future time intervals at rates that are agreed upon today. These rates are called forward or futures rates, depending on the type of the agreement. In an interest rate swap, counterparties exchange a stream of fixed-rate payments for a stream of floating-rate payments typically indexed to LIBOR. Duration and convexity are the basic tools for managing the interest rate risk inherent in a bond portfolio. We also review some of the most common market conventions that come along with interest rate market data.

Interest Rates and Discount Bonds11:45

Conheça os instrutores

Damir Filipović
EPFL
The Swissquote Chair in Quantitative Finance and Swiss Finance Institute Professor

In this first part

I will introduce the various notions of interest rates and related contracts.

Starting from the basic definition of interest paid on a loan

we derive simple and continuously compounded rates,

also called LIBOR, and yield.

They depend on the length of the lending period

and at the short hand coincide

and are equal to the so called short rate.

Investing and continuously reinvesting at the short rate

gives us the money market account

and we will see that this is a risk free asset.

In turn, the zero coupon bond

is the securitized form of a loan.

And they can be used to discount future cash flows

which is why they are also called discount bonds.

Interest refers to the rent paid by a borrower of money

to a lender of money over a period of time.

Here we have a period of time.

It starts at calendar date t,

and it ends at the maturity date T.

The lender will pay a loan of 1

unit of currency to the borrower.

And the borrower in turn will pay back this principal amount

plus interest at the maturity T.

Interest is expressed in annualized interest rates.

With simple compounding, we arrive at a simple rate

that we call L.

Which is defined by this equation here

which simply states that L is the interest R

divided by the length of the lending period.

There's also a continuously compounded version

which is also called the yield

and we label it by the letter y.

And is defined by this equation here.

Here is a market example, the LIBOR.

LIBOR stands for London InterBank Offered Rate.

This is the rate at which high credit financial institutions can borrow

on the London InterBank money market.

This rate however

has become an important reference rate for many interest rate contracts

beyond the London InterBank money market.

It is quoted daily as a simple rate

and this is why I choose the letter L

to denote the simple rate.

And it is quoted for various maturities

ranging from 1 day to 1 year,

and it is quoted for various currencies.

Interest rates depend on the lending period

this is why we also use the argument t and T

when we notate for simple and continuously compounded rates.

If we freeze the current calendar date t

and we vary the maturity T.

We obtain what is called the term structure of interest rates

prevailing at t.

We see the curve here in black

the simple interest rate.

And in red, we see the term structure of yields

which we also call the yield curve.

It is a mathematical fact that

for the same interest,

the simple rate is larger than the yield.

While at the short end

both curves coincide

and we will see later this is what we call the short rate.

If we vary the calendar date t

we obtain a time series of terms structures

that is plotted in this graph overhear.

So what we saw on the previous slide

was a snapshot for a given calendar date t,

we saw the term structure of interest rates.

It is clear as calendar time

proceeds, economic conditions change

and this term structure will fluctuate.

As you can see from this plot, which is taken from empirical data.

These fluctuations of the term structures are manifold.

Modeling this time series,

is the aim of this course on interest rate models.

Such models are useful then

for financial risk management and pricing interest rate derivatives

in a consistent and arbitrage free way.

Here I come back to what we saw on the previous slide

with term structure of interest rate and the yield curve.

They both coincide at the short end.

This common short end is called the short rate.

The short rate is the rate earned

on a loan over the short period

[ t, t + dt ].

Investing 1 unit of currency

at the short rate, and continuously reinvesting it

at the short rate

gives the money market account.

The return on the money market account

over the period B(t + dt)

is given by 1 + r(t) dt.

with the initial normalization, this equation is equivalent

to this expression here.

So what we see on the plot on the right hand side

is, for a given trajectory

of the short rate evolving over time.

We have it on that scale here.

The red line shows us how the money market evolves.

At each time

the rate of return on this money market account

is given by the respective short rate.

The money market account

is a risk free asset.

Risk free in the sense that,

it's return

over a time interval

is known at the beginning of the time interval.

This is very much in difference

to risky assets

where the return is only known at the end of the time period.

And where mathematically we would have to add

a noise term here.

A zero coupon bond with maturity T

is a contract that pays its holder 1 unit of currency

at the maturity.

This zero coupon bond with maturity T

for short is also called a T-bond.

Its price at time t

is denoted by

P(T, t).

A zero coupon bond is the securitized

that is, tradable form of a loan.

It is related to a loan in the following sense.

At t,

the lender buys a zero coupon bond from the borrower.

and pays the price P.

At the maturity T,

the lender receives 1 from the borrower.

Buying a T-bond at time t

and holding it until maturity

generates a simple rate of return

which is given by the simple rate L.

Formally speaking,

the zero coupon bond price is related to the simple rate

by this equation.

Buying a T-bond and holding it until maturity

generates a logarithmic rate of return

which is given by the yield.

Again this is expressed by this relation down here.

A zero coupon bond. is also called a discount bond.

P is the price

at calendar time t.

over cash flow of 1 at maturity T.

Now for varying maturities T

we obtain the term structure of zero coupon bond prices

prevailing at t.

This is also called the discount curve.

This discount curve is typically decreasing and smooth.

It reflects a positive time value of money.

At the short end.

The discount curve equals 1

because the cash flow of 1

at calendar date t

has value 1 at T.

In reality, interest rates depend on various factors.

They can depend on the credit worthiness of the borrower.

The less credit worthy the borrower,

the more the lender will charge for a loan.

They also depend on the liquidity needs of the lender.

The more liquidity the lender needs,

the more the lender will charge for a loan.

Interest rates also depend on regulatory requirements

and the monetary policy by the central banks.

And interest rates also depend on the market micro structure.

Various interest rate contracts

are traded on various markets.

And these markets are not always perfectly integrated.

In theory however, for this course

we will assume that there is no credit risk;

which means buying a bond

will give the holder of the bond

the face value at the end of the maturity.

And we also assume that there exists a frictionless market.

where all T-bonds for all maturities are traded.

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