Programme in Mathematical Finance - Introduction

8/9/2019 Programme in Mathematical Finance - Introduction

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Programme in Mathematical Finance 2010-11Finance I

Basic Concepts

Mathematical Sciences FoundationNew Delhi

www.msfonline.in

Copyright c2010 Mathematical Sciences Foundation.

This handout forms part of a Mathematical Sciences Foundation programme.Further details of this and other MSF programmes can be obtained fromthe MSF website (www.mathscifound.org). You can also call MSF at 011-29230401 and 65182616, or email [email protected].

All rights reserved; no part of this publication may be reproduced, stored,

transmitted or utilised in any form or by any means, without written per-mission from the Mathematical Sciences Foundation.

Contents

1 Introduction to Finance 2

2 Arbitrage 7

3 Return and Interest 9

4 The Time Value of Money 16

5 Bonds, Shares and Indices 18

A Solutions to Exercises 21

1


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2 Mathematical Sciences Foundation: Programme in Mathematical Finance

Basic Concepts

In this handout, we shall explore basic features of finance such as the notionsof risk and profit, interest rates, present and future value, inflation, bonds,shares, and indices. We shall also encounter a fundamental principle knownas the No Arbitrage Principle that will underlie much of our subsequentwork.

Some standard references for Finance are listed at the end of this handout,and we encourage you to refer to them for additional details and insights.For example, relevant sections for the material in this handout are 1.2, 1.3,

2.1 and 2.2 of Luenberger [1].

We emphasize that this handout gives a brief introduction to concepts and

products that form the basis of finance. We shall revisit all of these repeatedlyand in much greater detail as the course progresses.

Study Plan

Solutions to most exercises are provided at the end of the unit, but do try tosolve them yourself first!

Mathematically, you will need to be familiar with the exponential functionex, limits, and derivatives. You should also review basic Probability: theconcept of a random variable and its expectation and standard deviation.What is needed here is an intuitive idea of what these concepts represent.Soon, however, you will need their precise definitions and you may as wellstart reviewing them now.

1 Introduction to Finance

The aim of Finance is to explore how money should be invested. Imaginethat you have inherited a large sum of money from a rich uncle. The sum isso large that even when you have satisfied your immediate needs and desires,you still have a considerable amount left over. What can you do with it? Welist some typical responses below:

1. Put it in a savings account.


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Finance I: Basic Concepts 3

2. Put it in a fixed deposit.

3. Buy bonds.

4. Buy shares.

5. Invest in a Mutual Fund.

6. Buy gold.

7. Buy real estate.

Of course, there are many other possibilities, but let us start with just these.The question which arises is in which of these should we put our money?This naturally depends on how the nature of these investments matches withour requirements.

Fixed Deposits

For example, the advantage of a fixed deposit as opposed to a savings accountis that the former pays a higher rate of interest. The savings account, on theother hand, allows you constant access to your money while a fixed deposit

requires the money to be with the bank for a set time such as three monthsor a year.

Bonds

A bond provides regular payments over a set time period in return for aninitial payment, and is thus rather like a fixed deposit. Many bonds canbe traded during their lifetime, and thus provide additional flexibility to theinvestor. Bonds are also issued by companies, not only banks, and typically

offer higher gains than fixed deposits. But there is a downside - if the com-pany hits sufficiently bad times it may not be able to meet its obligationsand the investor may not receive the promised payments or even get theinitial payment back. In other words, the investor faces default risk, thoughonly in exceptional circumstances. Risk also enters the picture if the investorwishes to sell the bond before its expiry, since the price would be affected byprevailing market conditions.


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Shares

Risk comes even more into prominence when we consider the remaining pos-sibilities for investing. Share prices, for instance, vary greatly from day today. Even if we invest in a company with an excellent record, there is noguarantee that we will gain by owning its shares over the next few months.On the other hand, if we hold on to the shares for many years we have agood chance of making a handsome profit.

Mutual Funds

It used to be thought that bonds provide the optimal way to do well in thelong run say over 20 or 30 years. Current opinion, however, is in the favourof shares, provided one invests in a diverse collection of stable companiesand thus reduces the possible loss due to one or more of them doing badly.Mutual funds, which distribute the investors money over such a collection,cater to the investor who wants steady long-term growth. The investor whowishes to make money quickly would invest in just a few shares that hebelieves are going to do exceptionally well in the immediate future. Such aninvestor would naturally be exposed to high levels of risk.

Risk and Profit

Our discussion has brought forth some aspects of risk and profit. In thiscourse we shall investigate these in greater detail. The main task is to quan-tify the relationship between risk and profit, so we can make well-informedand precise decisions.

The initial problem is to figure out the correct price for a product, bywhich we mean a price that satisfies both buyer and seller. We will refer to

it as the value of the product. The products, by the way, could be anythingfrom commodities like cars or wheat, to bonds and shares, or even contractsabout future transactions. We will use the generic term assetfor the productsbeing traded. A collection of assets will be called a portfolio.


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Asset Choice

Beyond pricing, the main decision is what assets to invest in. Naturally, wewould like to invest in ones whose value seems likely to increase at a fasterrate. It is almost a law of Nature, however, that bigger promises are also lessreliable. In fact, less reliable promises must be bigger, if they are to have anytakers. Thus there is a trade-off between expected profit and risk: to aim forhigher profit, the investor must undertake greater risk.

The Meaning of Risk

The word risk is used in Finance in a special way. It refers to uncertainty, anddoes not necessarily have a negative connotation. Thus, consider the choicebetween putting money in a bank account or using it to buy shares in acompany. The second investment is riskier because it has more uncertainty,but it is not obvious how its worth compares to that of the the first one.Lotteries provide an extreme instance of high-risk investments which arenevertheless popular.

Probability

This discussion leads us to the role of Probability in Finance. The relativeworth of an investment depends on the probabilities of the possible pay-offs. If higher pay-offs are perceived as more likely, its value will increase.For example, if we can model the fluctuations in prices of a stock, we canassign probabilities to the possible pay-offs from this investment, and thusestimate its value to the investor.1 Specifically, we treat the future profit asa random variable. Its expectation then represents the expected profit, whileits standard deviation represents fluctuations and hence risk. (See Figure 1)2

1

We shall look at specific stock price models later in this course.2You can ignore this discussion for now if you are unfamiliar with the terms random

variable, expectation, and standard deviation. However, you should start your study ofbasic Probability and familiarize yourself with these concepts as they will soon becomeessential.


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0.02 0.04 0.06 0.08

-0.005

0.005

0.01

0.015

0.02

Figure 1: This diagram considers the 65 stocks making up the Dow Jones Com-posite Index, and their weekly profits over the one year period ending November 6,2006. The mean profit (per dollar invested) is plotted on the vertical axis, and thestandard deviation of the profits (representing risks) is plotted on the horizontalaxis. The curve has been drawn to emphasize that higher mean profit requiresgreater risk.

Risk-Free Assets

Some assets can be viewed as free of risk. For instance, deposits in banks andbonds bought from governments are typically treated as risk-free. Of course,both banks and governments can collapse, but such instances are rare. Weshall soon see that there is good reason to expect that all risk-free assets willgain in value at the same rate, and we may therefore talk of the risk-freerate of growth. This rate is not universally fixed, but varies with market and

time.

Portfolios

So far, we have considered individual assets. To create a portfolio, we needto consider not only the individual characteristics (regarding profit and risk)of the assets, but also their relationships with each other. Two assets couldbe linked together in certain ways for instance, they may show a tendency


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to rise or fall in value together. Alternately, one may tend to move in the

opposite direction to the other. In the latter case, a rise in one would be offsetby the fall in the other, and a portfolio consisting of both these assets wouldbe less risky than a portfolio consisting of only one of them! By combiningassets in various ways, one can tailor a portfolio to satisfy the risk preferencesof any investor.

Hedging

The process of reducing risk by combining assets appropriately is called hedg-

ing. By hedging, we reduce risk, and therefore also lower our expected profit.One of the goals we will pursue in this text is to see how to hedge againstspecific risks to which a portfolio is exposed, for instance fluctuations in theprices of stocks or in interest rates. If the hedging is complete, no risk willremain, and the portfolio will grow slowly at the risk-free rate. Therefore,we will also consider how to hedge to the right extent, so that the remainingrisk just falls within acceptable levels and the portfolio is able to grow at afaster rate.

2 Arbitrage

Arbitrage is the making of profit without undertaking risk. It can be earned,for instance, when a product is being sold at different prices in differentmarkets. Then risk free profit can be made by buying it where it is cheaperand selling it where it is costlier. A variation is when the different pricesare at different times, so that it is possible to buy today at a low price andsell some days later at a higher price. For this profit to qualify as arbitrage,however, you must be certain beforehand that the price will go up.

Deciding whether a profit has in fact been made can be tricky. If we investRe 1 and after a while it becomes Rs 2, we may feel we have made a profitof Re 1. Suppose however, that we are based in the US and therefore countour gains in dollars. In the given time period, if the value of the rupee interms of dollars falls sufficiently far we will perceive a loss rather than a gain.Yet again, suppose that in the same period a rupee put in a savings accountwould have more than doubled. Then it would be difficult to see the firstinvestment as truly representing a profit.

A simple way to resolve these ambiguities is to demand that arbitrage must


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be carried out without investing your own money (essentially, this means you

start by borrowing some money and pay off the loan by the end). If you startwith zero and end up with something, you have definitely made a profit.

A basic principle is that arbitrage opportunities are short-lived: Pricesevolve in such a way as to eliminate them. For as soon as it is realized thata product is under-valued and is creating an arbitrage opportunity, investorswill rush to buy it. This will drive up its price, reducing and ultimatelyeliminating the arbitrage opportunity. Similarly, if the opportunity arisesfrom an over-priced product, there will be a rush to sell it and this will driveits price down.

Reflecting on this process, we are led to the formal definition of arbitrage.We start by noting that the amount of profit does not have to be known be-forehand: it is enough to know that it cannot be negative and has a chanceof being positive. This will suffice to attract investors and initiate the sta-bilization process described above. Therefore, an investment strategy is saidto lead to arbitrage if:

1. It does not involve an initial investment of the investors own money.

2. It is known that at some future time the investment will have a value

which is definitely non-negative and additionally has a non-zero prob-ability of being strictly positive.

Exercise 2.1 Which of the following situations provides an arbitrage op-portunity?

1. A guarantee that in return for Rs 10 paid now, Rs 20 will be returnedafter ten years.

2. A guarantee that in return for Rs 10 paid now, Rs 20 will be returnedtomorrow.

3. A lottery ticket.

4. A free lottery ticket.

5. Bank A loans money at an annual interest rate of 10%, while Bank Bpays 15% interest annually on deposits.

6. Bank A loans money at an annual interest rate of 15%, while Bank Bpays 10% interest annually on deposits.


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How long an arbitrage opportunity lasts depends on the communication

within the market. The better it is, the faster investors will react to thesituation and eliminate the opportunity. Thus, in the idealized situation ofan efficient market, in which communication is instantaneous and complete,arbitrage opportunities will die immediately:

No Arbitrage Principle: In an efficient market, there are no arbitragepossibilities.

The No Arbitrage Principle is a surprisingly powerful tool for establishing

the correct price of a product and underlies every important result inthis course. It seems to have been first mentioned by Louis Bachelier in1900 when he described operations in which one of the traders would profitregardless of eventual prices and also noted that these are never found inpractice.3 Its systematic use in modern Finance, however, was initiated byFranco Modigliani and Merton Miller in the 1950s.4

3 Return and Interest

Consider an asset whose value evolves from V0 at an initial time t = 0 to VTat a later time t = T. Then the return from this asset over the time interval[0, T] is defined to be

Return = VT V0.

The rate of return is defined by

Rate of Return =VT V0

V0.

Commonly, one also writes return for rate of return. Confusion isavoided by noting that return has a currency as unit, while the rate of returnis unit-free. We will further express rate of return in percentages. Thusthe phrase The return was Rs 10 refers to the first definition, while Thereturn was 10% refers to the second and conveys that the rate of return was0.1.

3Bachelier worked on the Paris stock exchange before doing his PhD on modelingthe price fluctuations of various securities. His work led to striking innovations in bothmathematics and economics, and he is now considered the father of modern Finance.

4Modigliani was awarded the Nobel Prize in Economics in 1985, Miller in 1990.


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10Mathematical Sciences Foundation: Programme in Mathematical Finance

Interest

We will call the income from an investment interest if it is earned regularlyand in a predetermined manner, without risk.

Interest can be calculated according to different conventions. Consider astarting amount P (called the principal) on which interest is earned over atime period T. The amount of interest earned is given by the rate of interest,denoted r, in accordance with the adopted convention. The rate r is givenrelative to some time interval, called its period. The most commonly usedperiod is one year, in which case the rate is called annual. If the period is

a half-year (six months) the rate is called semi-annual. Other periods usedare monthly, weekly and daily.

Interest rates are commonly given as percentages, which have to be con-verted to fractions for calculations.

Now we shall consider the various ways of computing interest.

Simple Interest

In simple interest, the interest earned over one period is not added to theprincipal (e.g., it may be returned to the investor), and further interest isagain earned on the principal alone. Thus, ifP is invested at a rate of interestr, the amount after one period is

A = P + P r = P(1 + r).

During the second period, interest is again earned on P alone, so that theamount after two periods is

A = P(1 + r) + P r = P(1 + 2r).

In general, the final amount after earning simple interest over n periods is:

A = P(1 + nr)

Example 3.1 A common example of simple interest is the provision of fixeddeposits by banks. The interest earned on the money in a fixed depositaccount is returned to the investor, so that future interest is earned on theoriginal amount alone. 2


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Discrete Compound Interest

In compound interest, interest earned over one period is added to the prin-cipal, and earns interest in subsequent periods. If an amount P is investedat a rate r, then the amount after one period is

A = P(1 + r),

just as for simple interest. However, in the second period P(1 + r) serves asthe principal, so that the amount after two periods is

A = P(1 + r)(1 + r) = P(1 + r)2.

After n periods, the final amount is:

A = P(1 + r)n

Sometimes the period for which the rate is quoted is not the same as theinterval at which interest is compounded. For instance, the rate may begiven as an annual one, while the interest is calculated every 6 months. Inthis situation, the rate is adjusted linearly. If interest is compounded m times

during the period of the rate, then the rate per compounding interval is setto r/m and so the final amount over n periods is calculated by:

A = P

1 +r

m

mnExample 3.2 Savings accounts in banks provide an example of compoundinterest, since the interest earned on the amount in the account is fed backinto the account. 2

Exercise 3.3 Suppose you take a loan of Rs 1000, and have to pay it backin two equal and equally spaced instalments over a year. The annual rateof interest applied to this loan is 15% and the interest is compounded semi-annually.

(a) What will be the size of each instalment?

(b) How much of each instalment will go toward the principal and howmuch toward the interest? (Assume that each payment has to pay offthe outstanding interest at the time.)


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Continuous Compound Interest

Consider a bank offering interest compounded annually at a rate r. Supposeit allows an investor who withdraws his money at a time t before one yearto earn interest at a linearly adjusted rate of rt. For example, an investorcan withdraw his investment of P after 6 months, together with the interestearned. It would total P(1 + r/2). He can then immediately reinvest it foranother 6 months. This strategy nets him a final amount of

A = P(1 + r/2)(1 + r/2) = P(1 + r + r2/4),

which is slightly better than the P(1 + r) he would have had if he had justlet the money sit in the bank for the whole year. An investor who can createthis strategy, will certainly think of pushing it further by using smaller andsmaller investment periods. In general, if he withdraws and reinvests mtimes, he will end up with

A = P

1 +r

m

m.

The larger the value of m, the greater is his profit. This naturally leads toconsidering the limit m . If we recall that limh0+(1 + h)

1/h equalsEulers Number e 2.71, we can easily calculate:

limm

P

1 +r

m

m= P lim

m

1 +

r

m

(m/r)r= P lim

h0+(1 + h)(1/h)r = P er.

This suggests creating a new kind of interest, calculated by A = P er for asingle period. Over n periods, the growth is given by

A = P enr

Interest calculated according to this formula is said to have been con-

tinuously compounded, and r is called the continuously compounded rate ofinterest.

Interest at Arbitrary Times

We have been considering the interest earned when money is invested for a fulltime period, or for n full time periods. Now we look at what happens whenan investor withdraws his money at some intermediate time. In particular,


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(a) simple

(b) compounded annually

(c) compounded continuously

Continuous compounding is mathematically pleasant in another way.Withdrawing and reinvesting becomes just the same as making a single longinvestment, since

erT1erT2 = er(T1+T2).

Moreover, if continuous compounding is used, the same r can be used for

borrowing and lending without creating arbitrage opportunities. This is notthe case with discrete compounding.

Exercise 3.5 Suppose a bank has fixed an annual 5% discretely com-pounded interest rate for both deposits and loans. Show that this creates anarbitrage opportunity.

Exercise 3.6 Given that continuous compounding has nicer behaviour thandiscrete compounding, can you explain why financial institutions use thelatter?

Let us now consider the possibility of different interest rates being availablein the market. The differences could be of various types:

1. Use of different types of interest.

2. Different rates offered by different financial institutions.

3. Different rates for deposits and loans.

4. Different rates for investments of different time durations.

Typically, institutions use discretely compounded interest. The differencewould be in the frequency of compounding. The same value of r, but withmore frequent compounding, leads to more interest being earned. Thus in-stitutions doing more frequent compounding would also use slightly lowervalues of r. Continuous compounding is used more in mathematical mod-elling, and these models would use a value of r that is essentially equivalentto that being used for discrete compounding in the real world.


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Effective Rate of Interest

One way to reduce the confusion from different kinds of interest, is to cal-culate for each the amount of interest it earns over one year. This is calledits effective rate. Thus, suppose that a principal P has grown to an amountA by earning interest over a year. Then the interest earned is A P. Theeffective rate of interest is defined to be the interest earned per unit invested:

reff =A P

P.

We can expect that the effective rates of different available interest earning

schemes would be the same.

Example 3.7 Consider an r of 10% annually. If this is used with annualcompounding, then the corresponding effective rate is again 10%. If thecompounding is semi-annual (every 6 months), the effective rate becomes(1 + 0.1/2)2 1 = 0.1025, i.e. 10.25%. Finally, if continuous compoundingis used, the effective rate is e0.1 1 = 0.1052 or 10.52%. 2

Exercise 3.8 A credit card offers a cash withdrawal facility at a lowmonthly rate of 2%. What is the corresponding effective annual rate?

Exercise 3.9 Consider two investments A and B of the same amount, andat the same effective annual interest rate. Suppose A earns semi-annuallycompounded interest and B earns continuously compounded interest.

(a) Which one earns more interest if the period of the investment is: 6months, 9 months, 1 year?

(b) Suppose the invested amount is Rs 1000, and the common effective rateis 10%. What is the maximum difference in the interests earned by Aand B at any point during the first 6 months? (The answer is quite

small so use a good number of decimal places in your calculations.)

The second kind of variation can be expected to be negligible due to com-petition. A bank offering lower interest on deposits than its competitorswould soon start losing customers, and would have to raise its rates.

The third kind certainly exists: thus, a bank will offer lower interest ondeposits than it will exact on loans. However, it should be noted that onlythe first rate can be reasonably seen as risk-free. The second rate involves arisk taken by the bank, which explains why it is higher.


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Exercise 3.10 Show that the No Arbitrage Principle rules out a bank of-

fering higher interest on deposits as compared to loans.

The fourth kind of difference is quite important, and we will consider itin detail later. As a general rule, investments for longer time durations aregranted higher interest rates. The idea is that such an investment is exposedto more risk over its life and, to compensate, it must promise a higher profit.

Example 3.11 In April 2005, 6 month investments in Reserve Bank of Indiabonds were earning 5.4% interest, while 12 month investments were earning5.6%. 2

In sum, we can expect the same interest rates for investments of the sameduration. Thus, we may (and do) talk of a common risk-free rate that appliesto all risk-free investments over the same time period.

4 The Time Value of Money

Consider two offers: the first promises you one rupee right away, and theother after a month. Assuming both the offers are from trustworthy sources,do you have any reason to prefer one to the other? The simple answer isthat it is better to get the money early, as you can put it in a bank and startearning interest on it. This example illustrates the important idea that thevalue of a transaction involves not only an amount of money but also thetime at which it is undertaken.

Present and Future Value

We have observed that holding a rupee now is not the same as holding it a

year from now. Well then, what is the precise difference between the two?It depends on how much the rupee could have earned in a year by means ofinterest.

Example 4.1 Suppose a rupee can be invested at a simple annual rate of5%. Then, after a year, it has become 1.05 rupees. In this situation, earninga rupee now is equivalent to earning 1.05 rupees after a year. 2


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In the above example, Rs 1.05 is the future value of Re 1. Conversely, Re

1 is the present value of Rs 1.05.

In general, consider two amounts P and F that exist at times t1 and t2respectively, with t1 < t2. Let the risk-free rate of return over the interval[t1, t2] be r. Then we call P the present value of F (at t1) if

P =F

1 + r.

Conversely, F is the future value of P (at t2). The factor C = 1/(1 + r) iscalled the discount factor for this time interval.

If the risk-free rate is given in terms of interest, then the correspondingdiscount factor can be calculated as follows:

1. Suppose the annual interest rate is r, with compounding m times a year(after equal periods of time). Then the discount factor over n periodsis

C =1

(1 + (r/m))n.

2. If the interest is compounded continuously, the discount factor over T

years is C = erT.

It cannot be over-emphasized that amounts of money existing at differenttimes must not be compared or combined without taking into account therelevant discount factors.

Inflation

Another way that the value of money changes through time is with respect toits purchasing power. Typically, the same amount of money can buy less andless as time progresses this phenomenon is known as inflation. Inflationshows up as a general increase in prices. By averaging the rise in prices overvarious commodities, one can arrive at a single number the rate of inflationf which represents the annual decrease in purchasing power of a unit ofcurrency:

Purchasing Power after 1 year =Original Purchasing Power

1 + f.


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If an amount A is invested at the risk-free rate r for a year, then the effective

amount one has after a year is

1 + r

1 + fA,

and so we may talk of the real risk-free rate r defined by

1 + r =1 + r

1 + f, or r =

r f

1 + f.

Exercise 4.2 If continuous rates are used for inflation as well as risk-free

growth, show that the real risk-free rate is given by r

= r f.

Exercise 4.3 The table below shows estimates of annual inflation rates inIndia during 2001-2004.

Year Inflation Rate2001 3.82002 4.32003 3.82004 3.8

If an investment earned an annually compounded 8% interest throughoutthis period, what would be the return from it in real terms (i.e., in terms ofpurchasing power)?

In this course, we will not worry about inflation. The above discussionshows that, if necessary, inflation can be taken into account by a suitablemodification of the risk-free rate.

5 Bonds, Shares and Indices

Bonds and shares are the principal means by which institutions raise moneyfor their operations, and hence they provide the chief avenues for investment.

A bond is a contract written by a company or government (called its issuer).The purchaser of the bond makes an immediate payment to the issuer and,in return, is entitled to a certain number of regular payments in the future.Thus, a bond is essentially a loan. Institutions use bonds to raise money


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when the amount needed is too large to be obtained from a single source.

Investors use bonds as relatively safe investments providing a higher rate ofreturn than a simple deposit in a bank. Judiciously used, they can provideinsulation from interest rate changes as well.

A share represents part of the capital of a company. Its holder is thus apart-owner of the company and takes part in its fortunes. The share mayoffer him certain voting rights in the affairs of the company. Most companiesalso release regular payments, called dividends, to their shareholders out oftheir profits. The term stock is also used for a share. Another usage ofthe word stock is the total capital represented by the shares or the totalmarket capitalization (TMC) of the company.

A stock index is a hypothetical portfolio which is used to keep track ofgeneral trends in the market. Suppose a stock index has stocks labelled1, 2, . . . , n and it has ai shares of the stock labelled i. Also, let Si be the priceof one share of the stock i. Then the total value of the index is

ni=1 aiSi.

The weight of a stock is the proportion invested in it:

wi =aiSi

nj=1 ajSj

Indices change with time. If the composition (the collection ofais) is fixedthen the weights will vary. If the weights are fixed, the composition will vary.

Exercise 5.1 In an index with fixed composition, weight rises with a rise inthe corresponding stock price.

Exercise 5.2 In an index with fixed weights, a rise in the price of a stockleads to a fall in its amount in the index.

Some examples from India5:

1. BSE Sensex, is based on 30 stocks forming a sample of large, liquid andrepresentative companies listed on the Bombay Stock Exchange (BSE).

2. S&P CNX Nifty, consists of the 50 best stocks (in terms of TMC) onthe National Stock Exchange of India (NSE). It represents about 60%of the TMC on the NSE.

5For more information, consult the websites of the National Stock Exchange of India(www.nse-india.com) and the Bombay Stock Exchange (www.bseindia.com).


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3. S&P CNX 500, consists of 500 stocks and covers 98% of the total

turnover on NSE and 94% of TMC.

4. CNX Midcap 200, covers 77% of the TMC of the midcap universe(TMC of Rs 75 to 750 Cr).

Some examples from the US:

1. Standard and Poors 500 Index (S&P 500), is based on 500 stocksdistributed as follows: 400 industrials, 40 utilities, 20 transport, 40financial institutions. S&P500 covers about 70% of the total TMC and

78% of the total traded value.

2. Dow Jones Industrial Average (DJIA) consists of 30 of the largest publiccompanies in the US. It was created in 1896, when it consisted of 12companies.

3. NASDAQ 100 consists of 100 of the largest companies (including non-US ones) listed on the NASDAQ exchange. It is relatively heavy in ITcompanies. Infosys is one of the current components of this index.

4. NASDAQ Composite (or just the NASDAQ) includes every companylisted on the NASDAQ exchange currently more than 3000.

5. NYSE Composite includes each of the over 2000 stocks listed on theNew York Stock Exchange.

Other international examples:

1. FTSE 100 consists of the top 100 companies, in terms of TMC, on theLondon Stock Exchange. It represents about 80% of the TMC on theLondon Stock Exchange.

2. Nikkei 225 is based on the Tokyo Stock Exchange.3. Hang Seng Index consists of 39 companies listed on the Hong Kong

Stock Exchange and comprising 65% of its TMC.

Of these various stock indices, the smaller ones (with 30-50 constituents)give a summary of some particular aspect of the economy perhaps of itslargest companies, or of those belonging to a particular sector. Larger onesattempt to portray the national economy as a whole. Recently, stock indiceshave been created that track entire continents or the whole world.


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A Solutions to Exercises

Exercise 2.1 Let us compare the given situations with the requirements ofan arbitrage opportunity: zero initial investment and zero probability of losstogether with a positive probability of profit.

The first two situations can be arbitrage if interest rates are low enough.For we can start by borrowing Rs 10, thus avoiding an initial investment ofour own money. This will earn us Rs 20 by the end of the set time (ten yearsor one day), which we can use to pay off the loan. If the interest on the loanis low enough, we will still have money left over and this will constitute our

risk-free profit. In the second situation, this is almost certain to happen.

The lottery ticket does not constitute arbitrage since there is no guaranteeof profit (indeed, a loss is almost certain).

The free lottery ticket does provide an arbitrage opportunity. There is nopossibility of loss, and a very small one of success.

In the fifth situation, we can loan some amount from Bank A and depositit in Bank B for a year. At the end we will earn 15% interest on it, whichcan be used to pay off the 10% interest on the loan. The remaining 5% isour arbitrage profit.

The last situation does not, by itself, provide an arbitrage opportunity.

Exercise 3.3 Let each instalment be of an amount I.

(a) After 6 months, the debt is 10, 000 1.075 = 10, 750, since the interestrate for 6 months is 15/2=7.5%. On payment of the first instalment, a debt of10, 750Iremains. Over the final 6 months, this becomes (10, 750I)1.075.For the debt to be paid off, this amount must equal the last instalment andwe get the equation

I = (10, 750 I) 1.075.The solution is I = 5569.28.

(b) Of the first instalment, Rs 750 goes towards the interest and Rs 4819.28towards the principal. The last instalment pays off the remaining principalamount of Rs 5180.72 as well as interest amounting to Rs 5569.285180.72 =388.56.

Exercise 3.4 Let us base our calculations on an investment of Rs 1. Let the


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rate of interest be r. Then the three cases yield the following equations:

(a) 2 = 1 + 10r

(b) 2 = (1 + r)10

(c) 2 = e10r

The respective solutions are r = 0.1, 0.072 and 0.069, or 10%, 7.2% and6.9%.

Exercise 3.5 The arbitrage strategy is to borrow an amount X for a year anddeposit it in the same bank. Withdraw it after 6 months and immediatelyreinvest it in that bank. After 1 year, the invested amount becomes

1.0252X = 1.050625X.

We use 1.05X to pay off the loan and are left with a risk-free profit of0.000625X.

Exercise 3.8 The annual growth factor is 1.0212 = 1.268, so the effectiveannual rate is 26.8%.

Exercise 3.9

(a) Let the invested amount be Rs 1, the discrete rate be rd, and thecontinuous rate be rc. Since the interest earned over 1 year is the same forboth A and B, we find that

1 +rd2

2= erc.

Hence1 +

rd

2

= erc/2,

which shows that the interest earned over 6 months is also the same for bothA and B. If we graph the interest earnings against time, we get the followingdiagram.


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6 9 12

In particular, at the 9 month mark, A has earned more interest than B.

(b) Since the effective rate is 10%, we obtain the following equations:

1 + r

d

2

2

= erc = 1.1.

This yields rd = 0.098 and rc = 0.095. Over the first 6 months, using yearsas units, the difference between the interests earned by A and B is

f(t) = 1000(1 + 0.098t e0.095t), 0 t 0.5.

To locate the maximum we use the first derivative test:

0 = f(t) = 1000(0.098 0.095e0.095t),

which gives t = 0.33. Hence the maximum gap is f(0.33) = 0.49, or a mere49 paise!

Exercise 4.2 When continuous rates are used, the relationship between theoriginal and final purchasing power is

Purchasing Power after 1 year =Original Purchasing Power

ef.


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If an amount A is invested at the risk-free rate r for a year, at the end we

have the effective amount (in terms of purchasing power):

erA

ef= erfA.

Therefore the real risk-free rate is r f.

Exercise 4.3 The growth in purchasing power is

1.084

1.0383 1.043= 1.166.

Therefore the return in real terms is 16.6%.

Exercise 5.1 Let us verify the statement for the first stock. We have therelation

w1 =a1S1ni=1 aiSi

. (1)

Differentiate with respect to S1:

w1S1

=a1n

i=1 aiSi a21S1

(

ni=1 aiSi)

2=

a1n

i=2 aiSi(

ni=1 aiSi)

2> 0.

Hence a rise in S1 causes a rise in w1.

Exercise 5.2 Equation (1) can be rearranged into

a1 =

w1

1 w1

ni=2

aiSi

1

S1.

Therefore an increase in S1 (with everything else untouched) causes a de-crease in a1.

References

[1] David G. Luenberger. Investment Science. Oxford University Press India.

[2] Zvi Bodie, Alex Kane, Alan J. Marcus and P. Mohanty. Investments. 6thEdition. Tata McGraw-Hill.

[3] William F. Sharpe, Gordon J. Alexander and Jeffery V. Bailey. Invest-ments. 6th Edition. Prentice-Hall India.

Documents

Programme in Mathematical Finance - Introduction