financial products markets introduction definitions

7/30/2019 Financial Products Markets Introduction Definitions

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Products and Markets

1 In this lecture. . .

G the time value of money

G an introduction to equities, commodities, currencies and indices

G fixed and floating interest rates

G futures and forwards

G no-arbitrage, one of the main building blocks of finance theory


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2 Introduction

This lecture is just a collection of

G definitions,

G notations and

G specifications

concerning the financial markets in general.

The one technical issue that we see, the time value of money, isextremely simple.

We will also have our first example of no arbitrage. This is very

important, being one part of the foundation of derivatives theory.


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3 The time value of money

The simplest concept in finance is that of the time value of money.

$1 today is worth more than $1 in a years time.

There are several types of interest:

G There is simple and compound interest. Simple interest is when

the interest you receive is based only on the amount you initially

invest, whereas compound interest is when you also get interest on

your interest.

G Interest comes in two forms, discretely compounded and contin-

uously compounded.


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GInvest $1 in a bank at a discrete interest rate of

U, assumed for the

moment to be constant, paid once per annum.

At the end of one year your bank account will contain

@ U

Example: If the interest rate is 10% you will have one dollar and tencents.

After two years you will have

@ U @ U U

or one dollar and twenty-one cents.

AfterQ

years you will have

U

3


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GNow suppose you receive 2 interest payments at a rate of 7 2

per annum.

After one year you will haveK

7

2

L

2

(1)

Imagine that these interest payments come at increasingly frequent

intervals, but at an increasingly smaller interest rate: we are going

to take the limit P . This will give a continuously paid rate

of interest.

Expression (1) becomesK

7

2

L

2

0

2 O R J

7

2

z 0

7

That is how much money you will have in the bank after one year ifthe interest is continuously compounded. And similarly, after a time

W you will have an amount

H

7 9


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Lets look at this another way.

GYou have 0 W in the bank at time W , how much does this increase

with time?

If you look at your bank account at time W and then again a short

while later, time W G W , the amount will have increased by

0 W G W b 0 W {

G 0

G W

G W c c c

(Taylor series expansion).

The interest you receive must be proportional to the amount youhave, 0 , the interest rate, U , and the timestep, G W . Thus

G 0

G W

G W U 0 W G W

Dividing by G W gives the ordinary differential equation

G 0

G W

U 0 W


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If you have $0 initially then the solution is

0 W 0 H

7 9

This equation relates the value of the money you have now to the

value in the future.

Conversely, if you know you will get one dollar at time 7 in thefuture, its value at an earlier time W is simply

H

> 7 % > 9

We can relate cashflows in the future to their present value by mul-tiplying by this factor.

Example: U is 5% i.e. U , then the present value of

to be received in two years is

@ H

> @


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Dow Jones Industrial Average

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5000

25-May-79 18-Feb-82 14-Nov-84 11-Aug-87 7-May-90 31-Jan-93 28-Oct-95

1.A time series of the Dow Jones Industrial Average.

Prices have a large element of randomness.

This does not mean that we cannot model stock prices, but it does

mean that the modeling must be done in a probabilistic sense.


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One of the simplest random processes is the tossing of a coin.

Simple experiment: Start with the number 100 (think of this as

the price of your stock), and toss a coin.G If you throw a head multiply the number by 1.01, if you throw a

tail multiply by 0.99. After one toss your number will be either 99

or 101.

G Toss again. If you get a head multiply your new number by 1.01 or

by 0.99 if you throw a tail. You will now have either @ ,

@ @ @ @ or @ .

G Continue this process and plot your value on a graph each time you

throw the coin.


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Results of one particular experiment are shown below.

Instead of physically tossing a coin, the series used in this plot was

generated on a spreadsheet.

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102

104

0 20 40 60 80 100

2.A simulation of an asset price?


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This spreadsheet uses the Excel function RAND() to generate a

uniformly distributed random number between 0 and 1. If this num-

ber is greater than one half it counts as a head otherwise a tail.


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Simple coin tossing experiment.

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A B C D E

Initial stock price 100 Stock

Up move 1.01 100Down move 0.99 101

Probability of up 0.5 102.01

100.9899

101.9998100.9798

101.9896103.0095

104.0396

105.08106.1308

105.0695106.1202

105.059

106.1096105.0485

103.998

105.038106.0883

105.0275106.0777

107.1385108.2099

107.1278

106.0565

104.996103.946102.9065

101.8775

=B1

=D6*IF(RAND()>1-$B$4,$B$2,$B$3)


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4.1 Dividends

The owner of the stock theoretically owns a piece of the company.

However, to the average investor the value in holding the stock

comes from the dividends and any growth in the stocks value.

G Dividends are lump sum payments, paid out every quarter or everysix months, to the holder of the stock.

When the stock is bought it either comes with its entitlement to the

next dividend (cum) or not (ex).


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4.2 Stock splits

Stock prices in the US are usually of the order of magnitude of $100.

In the UK they are typically around 1. There is no real reason for

the popularity of the number of digits. Nevertheless there is some

psychological element to the stock size.

GEvery now and then a company will announce a stock split.

Example: The company with a stock price of $900 announces a

three-for-one stock split. This simply means that instead of holding

one stock valued at $900, you hold three valued at $300 each.


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5 Commodities

G

Commodities are usually raw products such as precious metals,

oil, food products etc.

The prices of these products are unpredictable but often show sea-

sonal effects. Scarcity of the product results in higher prices.

Commodities are usually traded by people who have no need of

the raw material. For example they may just be speculating on the

direction of gold without wanting to stockpile it or make jewellery.

GMost trading is done on the futures market, making deals to buy or

sell the commodity at some time in the future.


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6 Currencies

GThe exchange rate is the rate at which one currency can be ex-

changed for another.

This is the world offoreign exchange or FX for short.

Some currencies are pegged to one another, and others are allowed

to float freely.

There must be consistency throughout the FX world: If it is pos-

sible to exchange dollars for pounds and then the pounds for yen,

this implies a relationship between the dollar/pound, pound/yen and

dollar/yen exchange rates.

If this relationship moves out of line it is possible to make arbi-

trage profits by exploiting the mispricing.


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The fluctuation in exchange rates is unpredictable.

However, there is a link between exchange rates and the interestrates in the two countries. If the interest rate on dollars is higher than

the interest rate on pounds sterling we would expect to see sterling

depreciating against the dollar.

GCentral banks can use interest rates as a tool for manipulating ex-

change rates, but only to a degree.


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7 Indices

For measuring how the stock market/economy is doing as a whole,

there have been developed the stock market indices.

GA typical index is made up from the weighted sum of a selection

or basket of representative stocks.

The selection may be designed to represent the whole market, such

as the Standard & Poors 500 (S&P500) in the US or the Financial

Times Stock Exchange index (FTSE100) in the UK, or a very special

part of a market.


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EMBI Plus

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12/31/96 2/13/97 3/31/97 05/12/97 6/24/97 08/06/97 9/18/97 10/31/97 12/16/97

3.JP Morgans EMBI+

The JP Morgans Emerging Market Bond Index (EMBI+) is an in-

dex of emerging market debt instruments, including external-currency-

denominated Brady bonds, Eurobonds and US dollar local mar-ket instruments.

The main components of the index are Argentina, Brazil, Mex-

ico, Bulgaria, Morocco, Nigeria, the Philippines, Poland, Russia and

South Africa.


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8 Fixed-income securities

In lending money to a bank you may get to choose for how long you

tie your money up and what kind of interest rate you receive.

GIf you decide on a fixed-term deposit the bank will offer to lock

in a fixed rate of interest for the period of the deposit, a month,six months, a year, say. The rate of interest will not necessarily be

the same for each period, and generally the longer the time that the

money is tied up the higher the rate of interest.

GOften, if you want to have immediate access to your money then

you will be exposed to interest rates that will change from time to

time.

These two types of interest payments,

G

fixed and

G

floating,

are seen in many financial instruments.


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Coupon-bearing bonds pay out a known amount every six months

or year. This is the coupon and would often be a fixed rate of interest.At the end of your fixed term you get a final coupon and the return

of the principal, the amount on which the interest was calculated.

Interest rate swaps are an exchange of a fixed rate of interest fora floating rate of interest.

Governments and companies issue bonds as a form of borrowing.

The less creditworthy the issuer, the higher the interest that they will

have to pay out. Bonds are actively traded, with prices that continu-

ally fluctuate.


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9 Inflation-proof bonds

A very recent addition to the list of bonds issued by the US govern-

ment is the index-linked bond. These have been around in the UK

since 1981

G They are a very successful way of ensuring that income is noteroded by inflation.

In the UK inflation is measured by the Retail Price Index or RPI.

This index is a measure of year-on-year inflation, using a basket

of goods and services including mortgage interest payments. Thecoupons and principal of the index-linked bonds are related to the

level of the RPI.


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10 Forwards and futures

GA forward contract is an agreement where one party promises

to buy an asset from another party at some specified time in the

future and at some specified price. No money changes hands until

the delivery date or maturity of the contract. The terms of thecontract make it an obligation to buy the asset at the delivery date.

GA futures contract is very similar to a forward contract. Futures

contracts are usually traded through an exchange, which standard-

izes the terms of the contracts. The profit or loss from the futuresposition is calculated every day and the change in this value is paid

from one party to the other. Thus with futures contracts there is a

gradual payment of funds from initiation until maturity.


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Forwards and futures have two main uses:

GSpeculation

G

Hedging

If you believe that the market will rise you can benefit from this by

entering into a forward or futures contract. Speculation is very risky.Hedging is the opposite, it is avoidance of risk.

Example of Hedging: You are expecting to get paid in yen in six

months time, but you live in America and your expenses are all in

dollars. You could enter into a futures contract to lock in a guaran-teed exchange rate for the amount of your yen income. Once this

exchange rate is locked in you are no longer exposed to fluctuations

in the dollar/yen exchange rate. (But then you wont benefit if the

yen appreciates.)

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10.1 A first example of no arbitrage

Forwards provide us with our first example of the no-arbitrage prin-

ciple.

Example: Consider a forward contract that obliges us to hand over

an amount $)

at time%

to receive the underlying asset. Todays date

is 9 and the price of the asset is currently $$ 9 .

When we get to maturity we will hand over the amount $ and

receive the asset, then worth $$ % . How much profit we make

cannot be known until we know the value $$ %

, and we cant knowthis until time % .

We know all of , $ 9 , 9 and % , is there any relationship between

them? By entering into a special portfolio of trades now we can

eliminate all randomness in the future. This is done as follows.

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GEnter into the forward contract. This costs us nothing up front

but exposes us to the uncertainty in the value of the asset at matu-rity.

GSimultaneously sell the asset. It is called going short when you

sell something you dont own. This is possible in many markets,

but with some timing restrictions. We now have an amount $ 9

in cash due to the sale of the asset, a forward contract, and a short

asset position. But our net position is zero.

GPut the cash in the bank

, to receive interest.

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When we get to maturity we

Ghand over the amount and receive the asset,

G

this cancels our short asset position regardless of the value of$ % .

GAt maturity we are left with a guaranteed > in cash as well as

the bank account.The word guaranteed is important because it emphasises that it is

independent of the value of the asset.

The bank account contains the initial investment of an amount $ 9

with added interest, this has a value at maturity of

$ 9 0

7 % > 9

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Our net position at maturity is therefore

6 9 0

7 % > 9

> )

Since we began with a portfolio worth zero and we end up with a

predictable amount, that predictable amount should also be zero. We

can conclude that

) 6 W H

7 % > 9

This is the relationship between the spot price and the forward

price.

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Holding Worth Worth at

today (W ) maturity (7 )

Forward 0 6 % >

-Stock> $ 9 > $ %

Cash $ 9 $ 9 0 7 % > 9

Total 0 $ 9 0 7 % > 9 >

1.Cashflows in a hedged portfolio of asset and forward.

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0

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0 0.2 0.4 0.6 0.8 1

Forward

Spot asset price

Maturity

t

4.A time series of an asset price and its forward price.

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If this relationship is violated then there will be an arbitrage oppor-

tunity.Example: Imagine that is less than $ 9 0 7 % > 9 . To exploit this

and make a riskless arbitrage profit, enter into the deals as explained

above.

At maturity you will have $ 9 07 % > 9

in the bank, a short asset anda long forward. The asset position cancels when you hand over the

amount , leaving you with a profit of $ 9 0 7 % > 9 > .

If is greater than $ 9 0 7 % > 9 then you enter into the opposite

positions.

GThe standard economic argument then says that investors will act

quickly to exploit the opportunity, and in the process prices will

adjust to eliminate it.

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