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M.Sc. Finance

M.Sc. International Banking and

Finance and

M.Sc. International Accounting and

Finance

FINANCE II

UNIT 1

RISK AND UNCERTAINTY IN

FINANCIAL DECISION TAKING

• Aim Page
• Objectives
• INTRODUCTION
• RISK AND RETURN
• RISK AVERSION
• MEASURING RISK AND RETURN - Expected Return and Risk
• PORTFOLIO ANALYSIS - Naive Diversification - Modern Portfolio Theory - Perfect Positive Correlations - Perfect Negative Correlations - Zero Correlation - Imperfect Correlation - Portfolio Risk and Return - Covariance and the Correlation Coefficient - Covariance and Scatter Diagrams - Covariance and the Correlation Coefficient - Portfolio Variance Equation
• PORTFOLIO RISK AND PERFECTLY CORRELATED RETURNS
• DEVELOPING A TWO ASSET PORTFOLIO - Minimum Risk Portfolio
• RISK REDUCTION IN LARGER PORTFOLIOS
  • Equally Weighted Portfolios
  • Uncorrelated Returns
• EFFICIENT PORTFOLIOS
• CONCLUSION
• ANSWERS TO SELF ASSESSMENT QUESTIONS

PREVIEW

Aim

The aim of this unit is to consider the role of risk and uncertainty in financial decision taking

and to develop an understanding of how portfolios can be used in the management of risk and uncertainty.

Objectives

After completing this unit you will be able to:

• define and measure risk;
• explain how diversification reduces exposure to risk;
• identify the determinants of portfolio risk; and
• determine the risk and return of a portfolio.

INTRODUCTIO

N

As it is impossible to forecast with complete accuracy what the future will bring most investment and financing decisions are characterised by risk and uncertainty. Exposure to risk is usually perceived to be unwelcome, and will only be accepted by investors if they are offered some inducement in the form of a higher expected rate of return, the average rate of return that could be anticipated if the proposed investment could be repeated on a large number of occasions. In this unit, we will consider the nature of risk and uncertainty, and the relationship between risk and return. Investors can reduce their risk exposure by diversifying and investing in a portfolio of risky assets. We will examine the principles of portfolio construction, and the way in which the widespread use of portfolios has changed the way risk needs to be viewed for decision taking purposes. This leads in the next module to the consideration and evaluation of the nature of the risk-return trade off determined by the capital market. This risk-return trade off provides a benchmark for decision taking by management attempting to maximise the wealth of their shareholders. Unless an investment proposal can offer a return comparable to that available to shareholders in the capital market on similar risk investments it should be rejected.

We will focus on financial investments (securities), but the analysis has important

implications for investments in real assets (plant and machinery etc.). The trade off between risk and return established in the financial or capital market provides the benchmark for the appraisal of proposed investments in real assets. No manager acting in the interests of shareholders should invest in a risky real asset unless it promises an expected return comparable to that available on investments of similar risk in the financial markets. It is also considerably easier to measure the risk of financial than real assets. Extensive data is available on the average returns earned by different types of financial investments, the dispersion of these returns, and the relationship between return and their risk. The outcomes of the less standardised investments in real assets tend to be far more difficult to specify, measure and evaluate.

RISK AVERSION

A justification for using the dispersion of possible outcomes as a measure of risk, and the proposition that investors are risk averse, is found in the notion that utility is a concave function of wealth. Such a function, illustrated in Figure 1.1(a), reflects the assumption that utility increases with the level of wealth, but at a diminishing rate i.e., the marginal utility of wealth declines as the level of wealth increases. Diminishing marginal utility of wealth implies that the loss of utility resulting from a given reduction in wealth is greater than the gain of utility which would follow from an increase of wealth of equal magnitude.

Consider an individual with a wealth of £1,000 who is offered an even money bet of £200: a win would lead to wealth increasing to £1,200 and a loss to a fall in wealth to £800. The bet will leave the expected value of wealth unchanged at £1,000, the potential gain and loss offsetting each other:

E ( W ) = .5 × 800 + .5 × 1,200 = 1,

The loss of utility, indicated by the distance A – B in figure 1.1(a), reflects the greater change in utility that will occur with a change of wealth from £800 to £1000 as compared to that implied by a change from £1000 to £1200. To compensate the individual for the loss of utility from the introduction of risk it is necessary to improve the terms of the bet, offering better than even odds. This implies increasing the expected wealth above the initial level of £1000. The increment in expected wealth required to induce an individual to accept the greater dispersion of possible outcomes can be interpreted as a risk premium.

If the terms of the bet implied an even greater dispersion of possible outcomes, with the individual facing a fifty-fifty chance of ending up with £600 or £1400, for example, the loss of utility would be even higher. This is illustrated in Figure 1.1(b) where the distance AE representing the loss of utility is seen to be greater than the distance AB. As the loss of utility increases with the dispersion of outcomes it is reasonable to assume that an individual with a

wealth function reflecting diminishing marginal utility is averse to a greater dispersion of

outcomes. Interpreting risk as the dispersion of possible outcomes it seems reasonable to assume that individuals tend to be risk averse.

A

B

C

Wealth (W)

MEASURING RISK AND RETURN

The risk of an investment in the sense of the dispersion of its possible outcomes, may be measured in various ways. Among the statistical concepts that can be employed are:

• the range of outcomes;
• the mean absolute deviation;

Recession 0.3 10

Slow Growth 0.4 20

Boom 0.3 30

The expected return on the investment, E(R), is calculated as a weighted average of the possible outcomes, the weights being provided by the probabilities of these outcomes occurring:

E(R) = 0.3 x (10) + 0.4 x (20) + 0.3 x (30)

= 3 + 8 + 9 = 20

so that the expected percentage return is 20 per cent. This is the return that could be expected to occur on average if such an investment was repeated on a large number of times, with the exactly the same range of possible outcomes being anticipated on each occasion.

More generally, we can write the formula for the expected return on investment in the

following way

E

R

j ) = P 1 R j

+ P 2 R j2 n + .... + P n R jn

i=

Pi R ji

P i = probability of a state i occurring

Ri^ =^ return expected from the investment when the economy is in state i

E( R j ) = expected return on investment j

n = number of possible states

i = one possible state of the n feasible outcomes

j = the particular investment being considered

The variance is defined as the expected value of the square of the difference between the possible values of a random variable and its expected value. This is given by the weighted average of the squares of the deviations from the expected outcome that will occur in each of the possible states that may arise, the weights being provided by the probabilities of these different states arising.

VAR ( R j ) = P 1 ( R j 1 − E ( R j )) n

• K + Pn ( R jn − E ( R j ))

= ∑ Pi ( R ji − E ( R j ))

i =

and the standard deviation is the square root of the variance:

SD ( R j ) =

VAR ( R j ) = [ VAR ( R ) ] 2

The probability distribution of possible rates of return on the investment considered in the

example is shown graphically in Figure 1.3.

Figure 1.3 PROBABILITY DISTRIBUTION OF POSSIBLE RETURNS

Probability of Occurrence (%)

10 20 30 Expected

Returns

The probability distribution illustrated in Figure 1.3 is described as a discrete distribution as it is based on a limited number of possible outcomes and their associated probabilities. If we had been able to specify a large number of possible states, their associated probabilities, and

returns expected in each of these states we would have generated a probability distribution

Figure 1.4 NORMALLY DISTRIBUTED RETURNS

Probability

-9% 16% 41% Expected

Returns

+ 1(SD)

+ 2(SD)

+ 3(SD)

(As the maximum possible loss on any investment is 100% whilst in principle the potential gains are unlimited, the distribution can be expected to be skewed to the right. On this basis the log normal distribution tends to provide a better approximation of returns on shares than does the normal distribution. The empirical evidence also suggests that there are both more returns recorded that are closer to the expected value and more that are significantly different from the expected value than would be generated by a normal distribution. Nevertheless, for many practical purposes it continues to be reasonable to assume that stock returns are

normally distributed.)

As we have already seen, investors are generally assumed to be risk averse. This implies that higher risk investments will only be acceptable if they offer higher "expected returns". This leads to a notion of a trade-off between risk and return. To get some idea of the nature of this trade-off it is helpful to look at the historical rates of return earned by investors, and the distribution of these returns over time.

The average annual return by investors in the London Stock Exchange from 1945 to 1993 was about 16% while the standard deviation was 25%. Assuming that the returns are normally distributed, and historical returns are a good guide to the distribution of future returns, we can say that in any one year there is a 68% probability that the return on the London Stock Exchange will fall in the range -9% to +41%, ie. from (16-25)% to (16 + 25)%, ie. within one standard deviation of the expected outcome. (Clearly anyone who is not prepared to accept a loss in any one year, should not contemplate investing in the stock market!). The possibilities are summarised in Figure 1.4.

In choosing between investments we assume that an investor will consider both the expected return and the level of the risk as measured by the standard deviation of the expected return. Consider the following information on investments A, B and C: