13.5Summary

The nancial services industry has always been heavily regulated. This has been

particularly true of banking because of the vulnerability of banks to a loss of public

condence. The collapse of a single bank arouses fears of contagion, causing prob-

lems for the banking industry as a whole and hence for the provision of the eco-

nomy’s medium of exchange. Important consumer protection issues also arise in the

failure of banks.

The general case for regulation is based upon the existence of various types of

market failure, notably the existence of asymmetric information; while strong argu-

ments against regulation are centred on the ideas of moral hazard, agency capture

and compliance costs. Regulation increases the costs of entry and exit for new rms

and thus may inhibit competition. In some circumstances, regulation may even

increase the instability of an industry. A compromise position is to support regula-

tion but to argue in favour of self-regulation on the grounds that practitioners have

an interest in maintaining the reputation of their industry and are in the best posi-

tion to understand the impact of regulation on the industry. Nonetheless, several

problems have emerged in self-regulatory schemes. Self-regulation was tried for

the nancial services industry in the UK from 1986 to 1997 but did not operate well

and the industry was beset by scandals. Consequently, there has been a move back

towards statutory regulation, with the establishment in June 1998 of the FSA. From

that time, the FSA has also become responsible for the supervision of banking under

the Bank of England Act 1998, which gave the Bank full control of UK monetary

policy.

The attempt to develop a single nancial market across the EU also presented

regulatory problems as the different regulatory regimes in member countries were a

major obstacle to the integration of the markets. The slow progress in the harmon-

isation of regulatory procedures led to the acceptance of the mutual recognition and

home country regulation principles. Extra problems were caused by the different

banking traditions in member countries.

Other serious difculties have arisen as a result of globalisation and the rapid

development of complex derivatives markets. A major outcome of concerns over these

issues has been the Basel Accord, which has led to the wide acceptance of capital

adequacy rules that at the time of writing are being modied to take account of off-

balance-sheet business and market risk.

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Further reading

Questions for discussion

1	What have been the impacts on nancial markets of: (a) internationalisation of themarkets, and (b) technological change?

2	Consider the arguments for and against self-regulation of nancial markets asopposed to statutory regulation.

3	Discuss the extent to which consumers can be held to be responsible for their owndifculties in cases such as split capital investment trusts, endowment mortgagesand the Lloyd’s of London ‘names’.

4	How important is moral hazard as a determinant of people’s behaviour? Provideexamples of moral hazard related both to everyday life and to the nancial servicesindustry.

5	Under what circumstances might regulation decrease rather than increase the stabilityof an industry?

6	List the arguments in favour of host country regulation and discuss them. Why didthe European Commission favour home country regulation?

7	Explain the basis of the distinction between the types of capital in the BaselConcordat.

8	Why is it thought that simple capital adequacy ratios are insufcient as a basis forsupervising the activities of rms engaged in securities trading?

9	What aspects of the regulatory problem were highlighted by the collapse of Baring’s Bank in 1995?

10	What is meant by systemic risk in connection with the banking industry?
11	Why is consumer protection such an important issue in insurance?

Further reading

Basel Committee on Banking Supervision, Supervisory Recognition of Netting for Capital

Adequacy Purposes, Consultation Proposal (Geneva: Bank for International Settlements,

April 1993)

Basel Committee on Banking Supervision, Risk Management Guidelines for Derivatives

(Geneva: Bank for International Settlements, July 1994)

R Dale, Risk and Regulation in Global Securities Markets (Chichester: John Wiley & Sons, 1996)

FSA, ‘Endowments – consumers act but FSA keeps close eye on complaints’, FSA Press

Release, FSA/PNO8I/2005, 15 July 2005, www.fsa.gov.uk/pages/Library/communication/

PR/2005

FSA, ‘Mortgage endowments – shortfalls and consumer action’, prepared for the FSA by IFF

Research Ltd, July 2005, www.fsa.gov.uk/pubs/consumer-research

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Chapter 13 • The regulation of nancial markets

D Gowland, The Regulation of Financial Markets in the 1990s(Aldershot: Edward Elgar, 1990)

HS Houthakker and PJ Williamson, The Economics of Financial Markets(Oxford: Oxford

University Press, 1996) ch. 11

SA Rhoades, ‘Banking acquisitions’, in P Newman, M Milgate and J Eatwell (eds), The New

Palgrave Dictionary of Money and Finance(London: Macmillan, 1992)

Financial Services Authority at http://www.fsa.gov.uk

Financial Ombudsman Service at http://www.nancial-ombudsman.org.uk

http://www.ex.ac.uk/~Rdavies/arian/scandals for details of past nancial scandals

http://www.ft.com for up-to-date information on regulatory practices and problems

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APPENDIXI

Portfolio theory

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Appendix I • Portfolio theory

Objectives

What you will learn in this appendix:

l	How to nd the present value of a xed sum of money due for payment at a future date

l	How to nd the present value of a future stream of payments
l	How to make allowance for risk in the valuing of these payments
l	How to calculate a price for risky and risk-free assets

In section 1.1.1 we noted that choosing a nancial product was rather different from

choosing everyday consumer goods because the benets from a nancial product

often stretch quite a long way into the future and, since the future is uncertain, those

benets would also have a degree of uncertainty attached to them. In this appendix

we set out some rules which enable us to make decisions about the value of assets

which yield futureand uncertainbenets.

Because the benets of most nancial assets are uncertain, the assets are said to

be risky: the return that we actually get may be different from the return that we

expect. In nancial analysis we assume that people are rational, risk-averse wealth

maximisers. ‘Risk-averse’ means that they prefer less risk to more. Other things

being equal, a high-risk asset is less attractive than a low-risk one. When it comes to

pricing assets, therefore, we would expect to nd, as a general rule, that in a range

of assets differentiated solely by risk, the higher the risk associated with an asset, the

lower its price is likely to be. This is an important piece of common sense that we

should keep close to hand throughout the following discussion.

Risk:The probability that a future outcome will differ from the expected outcome.

Notice that in practice ‘time’ and ‘risk’ go together. Outcomes are uncertain

(and may therefore differ from the expected) precisely because they lie in the future.

However, for the purposes of understanding how to value uncertain future income

payments, we shall proceed in a series of steps. In section A1.1 we deal with the

valuation of future payments (both lump sum and a series) as though the payments

were known with certainty. In section A1.2 we look at how risk affects the return that

we require from an asset and thus at how risk affects our valuation of assets.

Note that the techniques we are about to discuss can be applied to the valuation

of anyasset generating a future stream of benets. The nance director of a large

corporation, for example, who may be considering a major investment in new plant

and equipment, needs to put a value on the likely future earnings from that equip-

ment in order to compare those earnings with the present cost of the project. The

following techniques are entirely appropriate for that purpose. There is nothing

special about the techniques that restricts their use only to risky nancial assets.

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A1.1	Appendix I • Portfolio theory Discounting future payments
	It is a fundamental principle of nance (and economics) that:

Any payment received at some point in the future is worth less than the same payment

received today.

A simple way of appreciating this from a personal point of view is to consider

the offer of a nancial gift from a generous aunt. Would we prefer to receive £1,000

now or wait until next birthday, which might involve waiting for 364 days? Most

people would opt for the gift now, but if we think it makes little difference, we

have only to think how we would feel if we had to wait ve years! If we agree that

we would rather have the payment now, we are in effect saying that we value the

payment less highly if we have to wait for it. This raises the crucial question of why

we prefer payment now, and it is crucial because the answer may give us some way of

quantifying the amount of ‘sacrice’ involved in waiting. What is it about ‘waiting’

that we do not like?

The fundamental reason is that having the payment nowgives us the use of it now.

For example, if we have the payment now, instead of in two years’ time, we can buy

goods (for example) which will give us two years of their benets, which we would not

otherwise have. The ‘sacrice’ involved in waiting is the two years’ worth of benets.

If we now think not about goods but about nancial assets, we can say that the reason

for wanting payment now is that we could use the payment nowto buy other nancial

assets which give us their advantages right away. Once again, the sacrice would be

the forgoing of those advantages, for two years in the present example.

If we understand this, we can begin to see how we might put a value upon the

cost or sacrice involved in waiting for further payment and thus how we might

adjust our valuation of a future payment to take account of the waiting. Consider

again the loss of benets in not being able to buy our nancial assets now if we

have to wait for payment in two years’ time. What are the benets forgone? Rather

obviously, it depends upon the assets which we would have bought. But let us con-

sider the very least that we would have lost.

If we had the use of the payment now, we could use the payment to buy interest-

earning assets. What is forgone, therefore, is the interest on those assets. So far, so

good, but there are many rates of interest. High rates are paid on risky assets while

lower rates are paid on less risky ones (remember that investors are assumed to be

risk-averse and so they need a higher reward to induce them to hold the riskier assets).

Which of these many interest rates should we use?

In the present case we are assuming that the future payments are known with

certainty. In effect, therefore, we are saying that the future payments are risk-free: it

is only the delay (not risk) that causes us to adjust their value downwards. If we want

to nd the cost of waiting for payments from any source, we have to ask what return

we could have had from investing the payment (if it were available now) in assets of

similar risk. In the present case, therefore, the appropriate rate of interest is that which

could be earned on some zero-risk asset. (Assets which have strictly zero risk are, in

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Appendix I • Portfolio theory

practice, hard to nd, but we might take something like a three-month treasury bill.

The interest payable on it is xed, the government is unlikely to default and, if the

bill is held to redemption, its maturity value is also certain.) If we are trying to put

a price on waiting for risk-free payments it makes no sense to assume that, if we had

the payment now, we could invest it at the very high rate of return which might be

available on very risky assets. This is just not comparing like with like. Let us assume

that the two-year risk-free rate of interest is 5 per cent. Then, if the payment were

available to us now, we could have earned 5 per cent on it over the next year and

then another 5 per cent on that (already larger) sum.

Notice how the benets forgone mount up over time. The loss in the second year

of waiting is bigger than the loss in the rst year because we calculate it not on

the original payment but on the original payment already increased by 5 per cent.

The effect is to make the losses increase quite sharply as the length of time increases.

Remember, though, that we are not concerned to calculate the losses as such; only

as a means of making a downward adjustment to our valuation of a payment if we

have to wait for it. Put like this, we can say that time and interest rates can have a

dramatic (downward) effect on our valuation of a future payment if it lies a long way

in the future. When we make this adjustment to the value of a future payment we

are said to be calculating its present value.

Present value:The value now of a sum of money due for payment at some point in

the future.

How do we do it? Remember that the cost of waiting involves two elements: a

rate of interest which we could have earned if we had the payment available now

andthe length of time which we have to wait. The simplest adjustment is required

when we wish to nd the present value of a single payment (a ‘lump sum’) due for

payment at a date in the future. If we let Astand for the amount, ifor the rate of

interest that could be earned, and nfor the number of periods for which we have to

wait, the present value of this amount can be found as follows:

	A
PV		(A1.1)
	(1 i)n

Notice how the rate of interest and the length of waiting combine to reduce the

present value of the future payment. Both appear in the denominator so that as they

get bigger, the denominator gets larger and present value is reduced. The value of

i)ⁿcan be found from discount tables such as those in Appendix II. The table

A(1

assumes that A£1. Each row of gures gives us the present value of £1 paid at a

given time in future, for different interest rates; reading down a column gives us the

present value of £1 at a given rate of interest, but for different periods of waiting. The

table clearly shows how the present value diminishes both as interest rates increase and

as the waiting term increases. (To see the most dramatic effect, read along a diagonal

from top left to bottom right. The present value of £1 diminishes very quickly!)

Alternatively, we can nd the present value by using a calculator. Exercise A1.1 demon-

strates the effect of changing both the rate of interest and the term.

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Appendix I • Portfolio theory

Exercise A1.1

(a)Find the present value of £1,000 to be received in two years’ time if the current rate of interest is 5 per cent.

(b)	Repeat the calculation for a rate of interest of 6 per cent.

(c)	Now calculate the present value if the rate of interest were still 5 per cent but the pay-ment were to be received in four years’ time.

Notice the effect of both the level of interest rates and time.

Answers at end of appendix

What we have just seen, then, is that any payment received in future is worth less

than the same payment received now. Furthermore, we can place a value on the cost

of waiting and we can use this cost to calculate the present value of the future payment.

This process is known as discountingand the rate, i, used in the calculation is known

as the rate of discountor discount rate. If we ignore any risk that might be attached

to the future payments we might discount them simply by using a risk-free rate of

interest currently available on assets whose life is similar to the period for which we

have to wait. The reason for choosing this rate is that it represents the return that

we could have had by investing the payment if it had been available to us now.

Discounting:The process of determining the present value of a sum of money promised

at some point in the future.

We have seen that the rate of discount which we use represents the cost of wait-

ing for the payment(s). The rate that we choose must be a reasonable approximation

of what we could have earned by investing the payment if it were available now in

an asset of similar risk. But consider this: if the discount rate is the rate we could have

earned by investing the payment if it were available now, then the discount rate

must also be the rate of return which we require from the asset that generates the

payment – otherwise we would not hold it. In other words, the rate at which we discount

future earnings from an asset is also the rate of return which we require from that asset to

make it worth holding. Consider Exercise A1.1 again. We assumed there that if the future

payment were available to us now, we could earn 5 per cent. The cost to us, in other

words, of holding the asset and waiting for payment reduces the present value of the

£1,000 payment to £907. In the circumstances, we are saying, we would be happy

to pay £907 to acquire an asset which would pay us £1,000 in two years’ time (or,

alternatively that we would be happy to hold that asset if we had already bought it).

This is because the return we could earn on £907 over two years would make its

future value equal to £1,000, the sum that we do expect to get in two years. Suppose,

however, that it became possible to earn 6 per cent interest on some other asset.

Then we could sell (or not buy) the rst asset and invest the £907 elsewhere to earn:

£907 1.06 1.06 £907 1.06²£1,019

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Appendix I • Portfolio theory

We should be better off in two years’ time (by £19) if we were to sell (or not buy)

the rst asset for £907 and reinvest £907 at 6 per cent. In the circumstances, there

would be no demand at £907 for an asset yielding £1,000 in two years’ time (and

existing holders would want to sell). The consequence would be that the price would

fall. To what level would it fall?

The answer is that it would fall to the level such that its future value will be £1,000

when it is invested for two years at 6 per cent. The general formula for nding the

future value of a present sum is:

FVA(1 i)n

(A1.2)

where A, iand nall have the same meanings as before. The process of nding the

future value of a present sum is known as compounding. Exercise A1.2 gives us the answer.

Compounding:The process of nding the future value of a present sum beneting from

regular interest payments.

What we have now established is that we can nd the present value of a lump

sum payable to us at some time in the future. Furthermore, the present value will

always be less than the face value of the payment because if we had the sum avail-

able now we could reinvest it. The rate at which we could reinvest is the rate at

which we discount the future payment in order to nd its present value and because

the reinvestment and discount rates are the same thing it follows that the rate

requiredto induce us to hold the asset in question is also the rate at which we dis-

count the earnings from the asset.

Exercise A1.2

What sum of money, invested now, will yield £1,000 in two years’ time if the rate of return is 6 per cent?

Hint: Rearrange eqn A1.2, remembering that the future value £1,000.

Answer at end of appendix

This equivalence between the rate of discount and the required rate of return on

assets is fundamental to understanding why an asset’s present value is also the price

which rational investors should be willing to pay for the asset – this is the main

theme of section A1.2 below.

Required rate of return:That rate of return on an asset which is just sufcient to

persuade investors to hold it.

Finding the present value of a future single payment is fairly straightforward. Finding

the present value of a streamof payments, made at regular intervals, is a little more

awkward since we have to apply the appropriate adjustment to each payment. Such

a series of payments is known as an annuity.

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Appendix I • Portfolio theory

Annuity:A series of xed payments received at set intervals.

Nonetheless, the principle remains the same. Each of the payments has to be

adjusted to take account of the level of interest rates and time. Imagine that we are

to receive four equal payments at one-yearly intervals. Then each must be adjusted

appropriately and the present value of the series of payments will be the sum of the

adjusted values. Thus:

AAAA
PV	(A1.3)
(1 (1 i)2(1 i)3(1 i)4
i)

Notice again how the effect of higher interest rates would be to bring down the

present value and notice too how the more distant payments are more severely

affected by the time spent waiting for them. Exercise A1.3 illustrates this.

Exercise A1.3

Find the present value of a series of four annual payments of £1,000, when interest ratesare 7 per cent.

Compare the present value of the rst and last payments and notice the effect of time.

Answer at end of appendix

With a long series of payments, eqn A1.3 becomes very cumbersome to write out.

Thus we often use an abbreviated form:

	n
		A
PV		t	(A1.4)
	∑	t
		(1 i)
	t1

Equation A1.4 says that the present value is the sum of (∑means ‘summation of’)

anyseries of payments each made at time t(where ttakes values from 1 to n), each

payment adjusted to take account of the value of t. Writing eqn A1.3 in terms of

eqn A1.4 would simply involve changing nto 4.

Unfortunately, whether we use eqn A1.3 or eqn A1.4, nding the present value of

a series of payments can be tedious if we are doing it by calculation rather than using

discount tables. The rst step, dividing Aby (1 i), is quick and easy, but next we

have to square (1 i) before dividing it into A, and then we have to nd (1 i)³

before dividing and so on. Depending upon our calculator and the total number of

periods in the series, it could be a very slow job! The good news is that there is an

alternative method which enables us to nd the value of a series of payments with

fewer steps (especially if the series is long). Its disadvantage is that it looks complicated

and, except to mathematicians, it does not so obviously describe the adjustments

that we are making. In this version we nd the PVas follows:

AA	1D
PV1		(A1.5)

iC	(1 i)nF

The advantage here is that the term (1 i) has to be calculated only once, though

it has to be calculated for n, the total number of periods.

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Appendix I • Portfolio theory

Exercise A1.4

Using a calculator and eqn A1.5, nd the present value of a series of four annual pay- ments of £1,000 when the rate of interest is 7 per cent p.a.

(The answer should be the same as in Exercise A1.3!)

A1.2Allowing for risk

In the last section, we saw that we can assign a present value to future payments

(lump sum or periodic) by the process of discounting. Discounting is necessary,

we said, because we have to wait for payment and this involves a cost which is the

income or benets that we could have earned from the payment if we had it avail-

able now. In all our examples we chose a risk-free interest rate as the rate of discount.

We chose this rate because we assumed that the onlyproblem associated with future

payments was the cost of waiting. We assumed that the payments were known with

certainty and thus that the asset producing those payments was a risk-free asset.

Since our discount rate has to represent the rate of return which could be earned

by having the payments now and investing them in an asset of similar risk, the

discount rate in this case must be the risk-free rate.

In practice, however, as we said at the beginning of this appendix, future pay-

ments are not likely to be risk-free. On the contrary, the very fact that they lie in the

future makes them uncertain and thus risky. This means that they will be subject to

a higher rate of discount. (Remember we said that people are risk-averse and there-

fore the value of a risky asset will be lower, and the return required on it will be

higher, than on a risk-free asset.) The question is, how do we nd a rate of discount

(or required rate of return) which takes account of this risk?

The simple answer is that we add a ‘risk premium’ to the risk-free rate. In practice,

we may not have to do much calculation. We might just look at the return generated

in the recent past by an asset of similar characteristics. If we are concerned about

the required return on a bond we may nd that one of the international risk-rating

agencies has placed the bond in one of its many categories, each of which has a risk

premium associated with it.

The theorybehind the risk premium, however, is usually derived from the capital

asset pricing model. This says:

the required rate of return on a risky asset is equal to the risk-free rate of return plus a

fraction (or multiple) of the market risk premium where the fraction (or multiple) is

represented by the asset’s -coefcient.

Formally, this is often written as:

A K rf (Km Krf )	(A1.6)
A

where Kis the required return on asset A, Kis the risk-free rate of return (what

Arf

we called i in our discussion above), is the ‘beta-coefcient’ of asset A, and Km is

A

the rate of return on a portfolio consisting of all risky assets.

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Appendix I • Portfolio theory

The basic idea is really very simple and quite easy to understand. Notice rstly

that the model says that the required return on a risky asset is equal to the risk-free

rate (K) plus some mark-up. That is exactly what we have been saying all along. The

rf

risk-free rate (ior K) must set a minimum threshold level of return. Since this can

rf

be earned on risk-free assets, it follows that the return on a risky asset must be equal

to this plus something further. The expression (Km Krf ) is the ‘something further’.

A

KKis sometimes called the market risk premium. This is the difference between

mrf

the rate of return on a risk-free asset and the return on the ‘whole market portfolio’,

a portfolio consisting of all the risky assets in the market. The return on ‘whole

market portfolio’ is a difcult idea (which we look at in a moment) but it may be

taken as representing a ‘benchmark’ return on risky assets in general. The market risk

premium therefore shows the compensation for risk required by the investment

community as a whole to persuade it to hold ‘representative’ risky assets compared

with risk-free ones. The market risk premium might therefore be said to measure the

investment community’s degree of risk aversion. is then an index of the asset’s

A

risk compared with that of the whole market portfolio. It enables us to say what pre-

mium should be paid on Aby comparing its risk with that of a whole market portfolio.

Depending on that comparison, the asset may need to earn less, the same, or even

more than the market risk premium. If, for example, the asset has the same risk as

the whole market portfolio, then 1 (and the required return on the asset will be

A

equal to K, the return on the whole market portfolio). If asset Ais less risky than

m

the whole market portfolio, then 1, and if it is riskier, then 1. Exercise A1.5

provides examples.

Exercise A1.5

Assume that the risk-free rate of interest is 5 per cent p.a. and that the return on a whole market portfolio of risky assets is 15 per cent p.a. Find the required return on three assets whose -coefcients are 0.8, 1.1 and 1.3 respectively.

Answers in Figure A1.1

The message of the capital asset pricing model can be shown in a simple diagram.

Figure A1.1 shows that if we know the risk-free rate and the return on the whole

market portfolio, the required return on a risky asset will depend upon its -coefcient.

The security market line (SML) is simply a visual representation of eqn A1.6. Figure A1.1

also shows the answers to Exercise A1.5.

The idea that we should compare the risk of an individual asset with some bench-

mark and then demand a premium based upon that comparison is sensible enough.

But why do we choose the ‘whole market portfolio of risky assets’ as the benchmark?

The answer is that rational risk-averse investors will never hold a risky asset in

isolation because they can eliminate a lot of risk by diversication, and the whole

market portfolio represents the extreme case of the most fully diversied portfolio.

In order to understand this, we need to look in more detail at the benets of

diversication.

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Appendix I • Portfolio theory

Figure A1.1

Consider an individual risky asset, A. We could take an ordinary company share

as an example. The essential characteristics of this asset, from an investor’s point of

view, can be described by its return and its risk. Unfortunately, we cannot know the

return on an asset in advance (if we did, there would be no risk!) and so we have

to form a judgement about the return that we expect. We cannot know exactly how

people form expectations in practice, but it seems reasonable perhaps to suggest that

people are guided by what they have observed to happen in particular circumstances

in the past, together with some assessment of the probability of those circumstances

occurring again. On this basis, the expected return on an asset, which we write as C,

can be calculated as follows:

C∑PK	(A1.7)
ii

where Pis the probability of a particular state of affairs and Kis the return

ii

expected in those circumstances.

To make an estimate of its riskiness, recall how we dened risk above. Risk, we

said, is the probability that an outcome (here the return on A) differs from what

we might expect. One obvious way of measuring the riskiness of an asset, therefore,

is to look at the variance or dispersion of possible returns around the value that we

expect. If all the possible outcomes are closely grouped around the expected value,

then the probability of being disappointed is low, while if the dispersion is large, the

probability of getting a result different from the expected is quite high. The formula

2

for calculating the variance, , is:

∑i (Ki C) 2P		(A1.8)
2
	i

Box A1.1 shows how we can use eqn A1.7 and eqn A1.8 to calculate the expected

mean return and the variance. In Box A1.1, share Ais expected to produce annual

returns ranging from 8 to 20 per cent p.a. depending on the state of the economy.

The probability of the economy being in a state of ‘boom’, ‘normality’ or ‘slump’ in

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Appendix I • Portfolio theory

the next time period varies between 30 per cent, 50 per cent and 20 per cent respect-

ively. The expected return on the share is 14.6 per cent and its variance is 17.64.

Sometimes it is convenient to express the risk as the standard deviation rather than

the variance of returns. The standard deviation is merely the square root of the

variance (here the standard deviation, , is 4.2 per cent (17.7)).

Box A1.1Mean and variance

Share A

State	P	K%	PK	(KC)	P(KC)2
Boom	i 0.3	i 20.0	ii 6.0	i 5.4	ii 8.8
Average	0.5	14.0	7.0	0.6	0.2
Slump	0.2	8.0	1.6 C14.6 A	6.6	8.7 2 17.7 A

However, rational investors are not likely to be interested in holding a single

risky asset in isolation. This is because they are risk-averse and a great deal of risk

can be eliminated, very easily, by holding a diversied portfolio of assets. The risk-

reduction effect of diversication can be seen easily if we only diversify from holding

one asset to holding two. Imagine that we are presented with the possibility of

holding two assets, asset Awith the characteristics just described, and asset B, whose

characteristics have to be found by completing Exercise A1.6.

Exercise A1.6

Share B

	P	K%	PK	(K		C)	P(K		C)2
State Boom	i 0.3	i 12.0	ii	i			ii
AverageSlump	0.50.2	8.0 8.0

	2
C	
B	B

Answers in text below

If we have done the calculations correctly, then Cshould be 9.2 per cent, while

B

2

should be 3.36. Notice immediately that while asset A has a higher return than B,

B

it is also riskier than B(we should be familiar with this relationship by now). As

investors we now have a choice of putting all our wealth into Aor, if we feel Ais too

risky, putting all our wealth into B. But why put all our wealth into either? Clearly

we could put any proportion of our wealth into A(call the proportion X) and the

balance (1 X) into B. Suppose we opt for an equal balance of Aand B(X0.5 and

1 X0.5). What happens to the return and risk that we can expect?

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Consider rstly the effect on return. This is simply the weighted average of the

two asset returns, where the weights are a proportion of the asset allocated to Aand

to B. If we let pindicate values for our portfolio, then in general terms:

C∑wK	(A1.9)
pii

where wis the ‘weight’ or proportion of the portfolio allocated to each security

i

and Kis the return on each asset. In this particular case, where the weight in each

i

case is 0.5, then:

C0.5(14.6) 0.5(9.2) 11.9%	(A1.10)
p

Consider now the effect upon risk. This is more complicated and is not just the

weighted average of the two variances (though the variances of each asset and the

weight of each asset do play a part). Why is portfolio risk not just the weighted

average of the assets comprising it? We can get a partial answer if we imagine an

extreme case of two assets whose returns were variable (and therefore risky) but

whose returns varied in exactly the opposite direction. In practice, such an extreme

case is impossible, but we might play with the idea that the return on shares in an

umbrella company varies in direct opposition to the return on shares in a company

making sunglasses. Given such an extreme case, we could put together a portfolio

of the two shares in which the portfolio return never varied. Movements in the

return on umbrella shares would be exactly cancelled by movements in the return

on sunglasses shares. Although the individual assets might each have a very large

variance, the variance of returns on the portfolio would be zero!

This is an extreme case. A more realistic one might be the returns to shareholders

in an oil producing company and the returns to shareholders in an airline company.

The dramatic rise in oil prices in 2005 certainly suggested that these could move

in opposite directions. However, both the extreme and realistic cases show why

we cannot just add up the individual variances. In the extreme case, we supposed

here that movements in return are perfectly opposed – a movement in one is exactly

cancelled by the movement in the other. However, a similar, albeit smaller effect

will always operate provided only that movements in return are less than perfectly

correlated. Provided that there is some independence in the movements in returns,

there will be some risk-reduction effect from combining risky assets in a portfolio.

More formally:

Provided that the returns on assets are less than perfectly correlated, the risk attaching

to a portfolio of risky assets will be less than the weighted average of the risk of the com-

ponent assets.

Bearing this in mind, we can produce an illustration based on our two assets A

and B. Using standard deviation as the measure of risk, the general expression for a

two-asset portfolio is:

22 )	(A1.11)
( AB cov , AB
XX2XXKK
22
pAA
BB

The interesting part of eqn A1.11 comes at the end and features the ‘covariance

of returns’ between asset Aand asset B. The covariance itself is made up of the

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Appendix I • Portfolio theory

individual standard deviations and the correlation coefcient of returns between A

and B, .

AB

cov K A,K B · · 	(A1.12)
ABAB

It is this correlation coefcient that plays the crucial role in risk reduction. If this

has the value 1, then portfolio risk is simply the weighted average of the risks of

component assets. If it has the value 1, then we can construct a two-asset portfolio

of zero risk. For any value in between, there will be some risk-reduction effect from

diversication. Since the returns on most risky assets move together to some degree,

in practice we would expect values for ranging from 0 to 1.

AB

Looking at eqn A1.11 we can see that we have most of the information we need

in order to illustrate the risk-reduction effect of combining our two assets. What we

lack is the covariance of returns, and this we can nd as follows:

covK,K∑P(KC)(KC)	(A1.13)
ABiAABB

In our case:

CovK,K0.3(20 14.6)(12 9.2)

AB

0.5(14 14.6)(8 9.2)

0.2(8 14.6)(8 9.2)

4.53 0.36 1.584

6.47 ≈6.5

Putting everything together (including the variance of share B) we can now solve

eqn A1.11 and nd the standard deviation of our two-asset portfolio as follows:

517705336205

 0.(.) .(.) (.)(.). 5 6 5 0

22

p

	. . . 443084325 

	.  . 852292

Let us now summarise our three investment possibilities: putting all our funds

into A, or all into B, or dividing them equally between Aand Bin a portfolio, C.

C	ABC 14.609.2011.90
	4.201.832.92

Investing in A, we can expect a return of 14.6 per cent p.a. if we are prepared to

accept a level of risk (shown by standard deviation) of 4.2. If this seems too risky

we could opt for asset Band earn 9.2 per cent p.a. for a risk of 1.83. Alternatively we

could opt for the portfolio of two assets, C. This would give us a return halfway

between Aand B. But look at the risk. If combining the two assets Aand Bequally

were to give us half of the risk of each asset, we would expect our portfolio to have

a standard deviation of 3.015 (0.5(4.2) 0.5(1.83)). In fact, however, the risk is

only 2.92. This is what we promised earlier: provided that the correlation coefcient

of returns is less than 1, portfolio risk will be less than the weighted average of the

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Appendix I • Portfolio theory

Figure A1.2

risk of the component assets. Out of interest we could use eqn A1.12 to check just

what the correlation coefcient of returns is in this case. Since the risk-reduction

effect is not especially dramatic, we might anticipate a gure closer to 1 than to 0.

Rearranging eqn A1.12 we have:

AB cov K A,K B · 6.5 (4.2 1.83) 6.5 7.7 0.84

AB

The benet of diversifying from one to two assets is shown in Figure A1.2. It

shows the risk and return available on each of assets Aand B, held individually. It

also shows that combining the two assets equally gives us a rate of return (shown on

the vertical axis) which is halfway between the return on Aand the return on B,

while the risk (shown on the horizontal axis) is less than halfway between the risk

of Aand the risk of B. Cis a portfolio made up of equal proportions of Aand B.

But we could, of course, combine Aand Bin any proportions at all between 0 and

100 per cent. The smooth curve joining Aand Bin Figure A1.2 shows the risk–return

possibilities resulting from all those combinations. Notice that the curve is concave

to the horizontal axis, showing that every combination of Aand Bwill produce a

return which is the weighted average of the two assets, while the risk is less than the

weighted average.

But Aand Bare not the only assets available. We could combine Awith Dor D

with E. The results of these combinations are also shown – by the dotted curves in

the gure.

More importantly, however, the benets which we can get from combining two

assets are even greater if we combine three, and greater still if we combine four assets,

and so on. The dashed line, the ‘envelope curve’, in Figure A1.2 shows the possible

risk–return combinations available if we constructed a series of portfolios using

different combinations of allassets, Ato E.

Notice that as we combine increasing numbers of assets, the curve moves up and

to the left. What this means is that for any given level of risk, we can increase the

available return by diversifying into more assets or, alternatively, for any given level

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Figure A1.3

of return we can decreasethe level of risk by adding more assets. This obviously raises

the question of whether there is any limit to the benets of diversication.

There are two answers to this. Firstly, for most investors there is a practical limit

since the purchase and sale of each asset involves dealing costs. Although the total

benets of diversication increase with each asset, the increases are very small once

a portfolio contains fteen to twenty assets. This is shown in Figure A1.3. Beyond

this point, for most investors the additional benets do not match the additional

dealing costs.

Secondly, even if we ignore dealing costs, there is a theoretical limit which is

set by the total number of risky assets available – the whole market portfolio. If we

did construct a fully diversied portfolio, we should still nd that we had to face

some risk. (This can also be seen in Figure A1.3.) To understand why, we need to dis-

tinguish between specic risk and market risk.

We know that the reason why risk diminishes as we combine risky assets together

is that their variations in return are not perfectly correlated: the uctuations tend

to cancel. But the cancelling effect – the imperfect correlation – is due to the fact

that each individual asset is subject to events which affect only that asset. If we are

thinking of company shares, for example, returns on one company’s shares will

be affected by the launch of its latest product, while returns on another might be

affected by a major strike. These events, which have their effects only on individual

assets, are said to be specic eventsand give rise to what is called specic risk. It is the

effects of specic risks that tend to cancel when we combine assets. However, there

are some events which will affect allassets to some degree. A rise in interest rates,

for example, or the movement of the economy into recession will tend to depress

the returns on all assets. Returns will all move in the same direction as a result, albeit

to a greater or lesser extent, depending upon the nature of the asset. Such events,

affecting all risky assets, are said to be market eventsand give rise to what is called

market risk. Market risk cannot be eliminated by diversication. Even a fully diversied

portfolio will still contain market risk.

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Specic risk:The variation in return on an asset which results from events peculiar to

that asset alone.

Market risk:The variation in return on assets which results from (market-wide) events

which affect all assets.

We can now see why the whole market portfolio provides a benchmark return

against which we can establish the return required on an individual asset. The

reason is that rational risk-averse investors will never hold a risky asset in isolation

because they can eliminate a lot of risk (specic risk) by diversication. Since specic

risk can be avoided cheaply, there is no reason why investors should be rewarded for

experiencing it. The only risk for which people should expect a reward is unavoid-

able or market risk. A whole market portfolio exposes investors onlyto market risk

and thus the return on that portfolio puts a price on the risk for which investors can

expect to be rewarded.

Deciding on the return which should be earned on an individual asset is then

relatively simple. As we saw above, we look at each asset in order to see how much

risk it carries when compared with the whole market portfolio. Since the whole

market portfolio is subject only to market risk, looking at how the return on an

individual asset moves with the return on the whole market tells us how sensitive

the asset is to market risk, the only type of risk that is relevant when it comes to

being compensated by a risk premium. It is market risk (and only market risk) for

which we can expect to earn a reward.

Figure A1.4 shows how we make the comparison between individual assets

and the whole market. The return on the whole market portfolio is shown on the

horizontal axis and the return on the individual asset is shown on the vertical axis.

The gure contains observations for two assets. The lower group plots the return on

asset Bagainst the return on the whole market; the upper group does the same for

asset A. It can readily be seen that when the whole market return increases by one

	Figure A1.4
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Appendix I • Portfolio theory

unit, the return on asset Balso increases, but by rather less, while for asset Aa unit

increase in the whole market return produces a larger increase in the return on A.

Fitting a line through the observations (either by eye or using standard regression

techniques) enables us to nd the slope coefcient which relates the change in the

market return to the return on the asset. For A, it can be seen that this coefcient

exceeds 1, while for Bthe coefcient is less than 1. The slope coefcient for each

asset is its -coefcient.

In section A1.1, we saw that the present value of the payment(s) produced by

an asset could be found by discounting the future payment(s) to take account of the

time for which we had to wait and the return which we could have earned had we

had use of the payment(s) immediately. We noted that the rate at which we dis-

counted was therefore the rate of return required to persuade us to go on holding

the asset. Since we assumed that the payment was due from a risk-free asset, the

discount rate which we chose was the risk-free rate of interest. Given that we now

know how to calculate a risk premium for the return required on a risky asset, we

can go ahead and use the pricing formulae in section A1.1, but using a risk-adjusted

rate of discount. Take, for example, eqn A1.1 (the formula for valuing a single, lump-

sum payment). Use this in Exercise A1.7 to see how the addition of a risk premium

to the discount rate affects the present value of an asset yielding a future lump sum

payment of £5,000.

Exercise A1.7

(a)What price would you be prepared to pay today for a non-interest-bearing asset which matures for £5,000 in three years’ time if you are assured that the maturity value is guaranteed, and the three-year risk-free interest rate is 7 per cent p.a.?

(b)

What price would you be prepared to pay for a similar asset where the payment in three years’ time was expected to be £5,000 but where the asset’s past behaviour relative to the market suggested a -coefcient of 0.7? Assume that the risk-free rate remains at 7 per cent p.a. while the return on a whole market portfolio is 17 percent p.a.

Answers in text below

In the rst part of Exercise A1.7, we are simply repeating what we did in

Exercise A1.1. We are nding the present value of a risk-free asset by discounting by

the risk-free rate of interest. We discount by the risk-free rate because this is the rate

of return which we require. And when we calculate the present value (£4,081.63),

we are saying that this is the price which we (and other rational investors) would be

prepared to pay. In equilibrium the asset will be priced at £4,081.63 in order to yield

the required return of 7 per cent p.a.

In the second part, we are doing the same, except that the discount rate incoporates

a risk premium. The CAPM tells us that the required return on this (risky) asset is

0.07 0.7(0.17 0.07) 0.07 0.07 0.14. The rate at which we discount the £5,000

is 14 per cent p.a. which tells us that the present value of this asset is £3,374.96. This

is the price which we (and others) would be prepared to pay. In equilibrium, the

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asset will be priced at £3,467.41 in order to yield 14 per cent p.a. which is the rate

of return which we require, given its risk. We can see now that if we are particularly

interested in the pricing of assets, we could modify our earlier statement of the CAPM

to read:

The market will price risky assets in such a way that the return on a risky asset will be

equal to the risk-free rate of return plus a fraction (or multiple) of the whole market risk

premium.

Notice the relationship between price and rate of return that is emerging here

and remember what we said right at the beginning of this Appendix about investors’

attitude to risk and its effect upon price. In Exercise A1.7 we have two assets which

are very similar except in their degree of risk. The rst (risk-free) asset has a price of

£4,081.63. The second, whose earnings are riskier, must yield a higher rate of return.

When we discount by this higher rate we nd the price is substantially lower at

£3,374.96.

Answers to exercises

A1.1£907.03; (b) £890; (c) £822.71.

A1.2£890.

A1.3£934.57; £873.44; £816.33; £763.36.

A1.4£3,387.70.

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APPENDIXII

Present and future value tables

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Appendix II • Present and future value tables

(1i) n

£1



Table 1 Present value of a £1 lump sum, paid in n-years’ time, discounted at i. PV 