The impact on the choice of A and B if an investor is forced to choose between the two investments.

Every investor has objectives or goals that need to be achieved and this influences his decision making. These objectives could be to obtain the best return that is possible, among others.

However, these investors also have risk thresholds or tolerances which impacts on their investment objectives. Risk thresholds refer to the level of risk that investors are willing to bear in order to achieve their objectives.

This indicates that both risk and return dovetail to influence the investor.

Faced with the choice between two investments, an investor would ideally like to manager their risk by dividing her outlay between the two with the aim of maximising her return. However, if forced to choose between the two, it is believed that other factors, in addition to expected risk and return will influence their decision making under uncertainty and these are:

• Utility
• Attitude to risk

This paper will consider each of these influences on decision making in turn – expected return, risk, utility and attitude to risk to see what part they play in the decision making process and how effective they are. It will conclude with criticisms of the utility theory and how another theory Prospects Theory provides a more realistic theory to understand how individuals make investment decisions.

Before we start, it would be useful to consider the definitions of uncertainty and risk.

McLaney (2006, 151) defines uncertainty as ‘the position where we simply are not able to identify all, or perhaps not even any, of the possible outcomes and we are still less able to assess their likelihood of occurrence’. The investment returns for both Assets A and B are uncertain, based on this definition. However, the investor, whom we shall name Charles, will need to use some sort of management information to analyse and thus choose. The historic data spanning 24 years showing the distributions of returns for Assets A and B is what Charles has, by way of management information, to make his decision.

Risk, on the other hand, is a position where it is possible to identify probable outcomes and their likelihood of occurrence (McLaney, 2006). It is a position where the possibility of an undesired outcome. For example, if by tossing a coin one expects a head, there’s equally a 50% chance or risk of obtaining a tail.

The main basis of making the decision between the two investments is Charles’ assumption that the past 24 years results are an indication of future results and there are no new developments like product competition or new technologies that may render the past results unreliable and therefore, unsafe to use as a basis to make the investment decision.

2.0 Average/Expected Return

Charles would initially be interested to know over a 24 year period, which asset has provided the better return on average. The way he can do this is by multiplying each return by the number of times each occurred (to obtain the total return over 24 years) and dividing by the number of years, in which these returns were achieved – this is illustrated below:

Return (x) £000 Frequency (f) (fx) £000
1 1 1
3 1 3
4 3 12
5 2 10
6 1 6
7 1 7
8 1 8
9 1 9
10 2 20
11 2 22
12 3 36
13 1 13
15 1 15
18 1 18
19 1 19
20 2 40
Total 24 239
Table 1 Asset A Returns over a 24 year period

Table 1 illustrates the distribution of returns for Asset A over a 24 year period. The average or expected return is calculated as: Σfx/Σf, i.e. £239000/24 which when rounded gives an average/expected return of £10k.

Similarly, the distribution of returns for Asset B is represented below:

Return (x) £000 Frequency (f) fx £000
-12 1 -12
3 1 3
5 1 5
7 1 7
8 1 8
10 2 20
11 2 22
12 8 96
13 5 65
17 2 34
Total 24 248
Table 2 Asset B Returns over a 24 year period

The average/expected return for Asset B is £248k/24 when rounded is also £10k

The average return or mean as is it also referred to tells Charles the point about which the values in the distribution cluster – i.e. the sum of the deviations from each of these equals zero.

Taken in isolation, Charles could select either of the options. It is worth noting at this point, that the average or mean is highly sensitive to extreme values. If Asset B had not returned a loss in its first year, 24 years ago, and even assuming the return in that year was zero, the average return would have been closer to £11k than £10k, for example.

Charles could therefore assume that over a 24 year period in the future both Assets A and B would yield £10k a year. If the probability of achieving either of these were the same, Charles could choose either.

However, looking at the historical data and assuming that Charles’ investment objective is to achieve at least a £10k return on investment every year, with both investments costing the same, he would want to choose the investment that is more probable to achieve the objective. Therefore, probability in this case is a measure of risk.

To explain further, let us analyse historical data for both investments to see which is more likely to achieve Charles’ objective. The probability of obtaining £10k or greater is approximately 50% for Asset A (this has happened 13 out of the last 24 years), while the corresponding probability for Asset B is approximately 80% (£10k or more returned for 19 out of the last 24 years). Based on this, Charles would opt for Asset B. We shall look into this bit of analysis when we come to discuss criticism of Utility theory.

Another measure of risk, the standard deviation, is useful to understand the risk of both investments and we shall look at this now.

3.0 Standard Deviation

Standard deviation is ‘a statistical measure of the dispersion of individual outcomes about their mean’ (McLaney, 2006, 503). In other words, the standard deviation is a measure of risk that, in Charles’ case, indicates the average deviation from the average return of £10k for each return. The greater the deviation, the greater the risk. Let us now calculate the standard deviation for each investment.

Return (x) £000 Frequency (f) Fx £000 fx² £000
1 1 1 1
3 1 3 9
4 3 12 48
5 2 10 50
6 1 6 36
7 1 7 49
8 1 8 64
9 1 9 81
10 2 20 200
11 2 22 242
12 3 36 432
13 1 13 169
15 1 15 225
18 1 18 324
19 1 19 361
20 2 40 800
Total ∑f = 24 ∑fx = 239 ∑fx² = 3091
Table 3 Asset A Standard Deviation

Return (x) £000 Frequency (f) fx £000 fx² £000
-12 1 -12 144
3 1 3 9
5 1 5 25
7 1 7 49
8 1 8 64
10 2 20 200
11 2 22 242
12 8 96 1152
13 5 65 845
17 2 34 578
∑f = 24 ∑fx = 248 ∑fx² = 3308
Table 4 Asset B Standard Deviation

Standard deviation is calculated as follows:

√(∑fx²/∑f – (∑fx/∑f)²)

There is a slightly greater deviation from the average/expected return for Asset B than there is for asset A as rounding reveals that standard deviation is £6k and £5k respectively.

The average return does not tell the full story – for example, if for each of the past 24 years, an asset C yielded £10k per annum, expected/average return will be equal to that of Assets A and B and basing decision of expected return alone would be impractical. This is because the average return does not reveal that for Investment B, for example, there is a possibility (albeit a small one of 4%) of a £12k loss, while Asset C almost guarantees a £10k return if historical data is the used as a guide.

It also hides the fact that there is a possibility that Asset A could yield as much as £20k (an 8% chance) or that there’s a greater chance of obtaining greater than the expected return for Asset B than for Asset C. effectively, Charles needs to consider the risks.

Based on the calculations of average returns and standard deviations for both Assets A and B, Charles should choose Asset A.

It is now time to turn our attention to Utility theory and investors’ attitude to risk. The following section will show that Charles should choose between Assets A and B based on his preference and also on his attitude to risk.

4.0 Utility Theory

Utility is a measure of satisfaction gained from, in Charles’ case increase in wealth as a result of investing in either Asset A or B. The theory indicates that the investment that Charles chooses, should be the one with the greater expected utility.

The expected utility model was proposed by Daniel Bernoulli in the 18th century and has been applied by Neumann and Morgenstern in formulating Game theory, which is based on the premise that different actions are chosen to maximise returns.

Before deciding which investment Charles should choose based on Utility theory and attitude to risk, it is worth providing an understanding of utility theory first.

The theory is concerned with the preferences of individual investors when it comes to making investment decisions, rather than the expected returns.

In other words, instead of making decisions based on expected returns, the theory proposes that the decisions will be influenced by the increasing or decreasing utility or satisfaction to be gained from consuming or investing in goods or services.

Utility is normally represented as indifference curves. For each indifference curve, a consumer is indifferent or has no preference for one combination of goods/products when compared with another. The indifference curve effectively shows all consumption combinations which yield the same utility.

Let us assume that an individual has two preferences – going to the cinema or going to watch football, with the stronger preference for watching football.

Cinema

Figure 1 Trade off between cinema and football Football

Let the curve closest to the horizontal and vertical axes be UU, the middle curve VV and the curve furthest away from both axes WW.

Curve UU shows the least amount of utility - any combination of cinema and football on UU will give the same level of satisfaction. Moves from UU to VV or from VV to WW indicate increased utility.

The shape of the curves indicates that the individual is prepared to sacrifice a lot of going to the cinema in favour of watching little more football. Towards the top of the curves, the individual is hardly keen to give up any more football for cinema.

This assumes three things about consumer preferences, i.e. an individual:

• can rank the different combinations of goods/products according to their utility
• prefers more to less when it comes to wealth
• has preferences that satisfy a diminishing marginal rate of substitution (Begg et al, 1997)

The marginal rate of substitution can be defined as the quantity of one variable that an individual is willing to give up in order to increase the quantity of another, without changing total utility or satisfaction.

In the example, above, this implies that the marginal rate of substitution of football for cinema is the amount of cinema the individual is willing to give up to increase the amount of times he can attend football matches, which is his preference.

It is worth noting that utility is individual specific, meaning that different individuals, depending on their preferences will have different levels of utility for football or cinema or a combination of these. The higher curves, VV and WW showing greater utility to UU could indicate that: both football and cinema are cheaper or that the individual has more disposable income than the UU proposition.

Let us consider how an individual’s attitude to risk influences his investment decision.

Supposing for example an individual has 3 investment options:

• a 50% chance of making £1k and equally a 50% chance of losing £1k. This is known as a fair investment. A fair investment is similar to the fair gamble theme, in that it is one in which on average, there is an equal chance of a profit or loss
• a 40% chance of making £1k and a 60% chance of losing £1k. This is an unfair investment, whereby on average the individual will make a loss
• a 60% chance of making £1k and a 40% chance of losing £1k. This is a favourable investment

Let us now compare the fair investment (equal chance of profit or loss) with another fair investment, which is a 50% chance of either making or losing £2k. Even though both are fair investments, the latter investment is the riskier as the range of possible outcomes is more than the former.

An individual’s attitude to risk or risk preference can be classified as risk averse, risk neutral or risk loving.

McLaney (2006) helps to provide definitions of the risk attitudes or states:

• a risk averse individual is an individual that is only prepared to take a risk where the expected return is greater than the cost of the project or investment in question, at entry stage

• a risk neutral individual is an individual who is prepared to take a risk where the expected return is equal to the cost of the project or investment

• a risk loving individual is one who is prepared to take a risk even were the expected return from the investment or project is less than the cost of the investment or project at entry stage, provided that at least one possible outcome has a value greater than the cost of the investment or project at entry stage

An individual’s attitude to risk can be determined by analysing whether he will be willing to invest in a fair investment, i.e. one that has an equal chance of making or losing money.

A risk averse individual will not do this and will only invest if the probability of making a profit outweighs the probability of making a loss. This type of individual dislikes risk and will need to be compensated more, in terms of return for the risk he is willing to take. The more risk averse the individual, the more that individual requires, by way of return for him to invest.

A risk neutral investor would not make his investment decision based on the probability of making or losing money. He should focus himself on how much can be made from the investment. Effectively, the risk neutral investor is more concerned about achieving his investment objective in terms of return. When faced with a choice between two investments, the risk neutral investor will choose the one with the higher expected return. If expected returns for both investments are equal, the risk neutral investor could choose either, regardless of the individual asset’s risk.

A risk loving investor will make a fair investment. In fact, a risk lover will invest in an asset where the chances of making a profit are less than the chances of making a loss. Her motivation is based on her analysis that there is greater potential for increased return, the higher the risk. The more risk loving the individual, the more unfavourable must be the chances of the investment yielding a profit for her not to invest. For example, some people will choose to bet on a non-league club, e.g. Nuneaton Borough FC defeating Liverpool in the FA Cup instead of the other way round because the payout returns if their bet materialises will be considerably more than if they were to bet on a Liverpool victory – these are risk lovers.

We shall now turn our attention back to Charles who has an investment decision to make. Remember that Assets A and B both had the same expected return of £10k while the risk of Asset B was greater than that of Asset A. Assuming that the costs of both investments were the same and that utility is based on the satisfaction to be gained from returns, Charles should make the following decisions based on his attitude to risk.

4.1 Risk Lover

Charles should opt for Asset B. Demonstrating this attitude means that Charles’ preference is influenced by the possibility of increased returns. In other words, increases in returns, in equal increments will add more and more to utility. For example, the more profits of £5k that Charles makes, the more he is satisfied or his utility increases. Charles’ attitude or risk preference means his marginal utility increases for any additional increments in returns. This is represented in the diagram below.

Utility

Figure 2 Risk Lover and Utility Returns

The extra utility to be gained from a return of £12k for example, in Asset B is more than the utility given up if Charles loses £12k. Based on this, Charles the risk lover will choose Asset B.

4.2 Risk Neutral

Because expected returns are identical, Charles will be indifferent to investing on either Asset A or B. For Charles, the risk neutral investor, successive equal increases in returns will yield the same level of utility. Similarly, the utility to be gained from a £12k return is the same as the utility sacrificed for a £12k loss, i.e. equal marginal utility.

Utility

Figure 3 Risk Neutral and Utility Returns

4.3 Risk Averse

Charles, the risk averse investor will opt for Asset A. This is because for the same expected return as Asset B, the risk of achieving this is less. In this case, increases in returns, in equal increments will add less and less to utility. As Charles attains more incremental returns, his utility decreases – decreasing or diminishing marginal utility. Additionally, he will not opt for Asset B due to the possibility of the negative return. The utility that Charles gains from a £12k return is less than the utility sacrificed for a £12k loss.

In general, most people are risk averse. To prove this, it is useful to consider human behaviour with regards to insuring their homes. People would normally insure their property against fire, for example, in the following scenario:

• Cost of home is £100k
• Probability of home catching fire is 10%, i.e. 90% chance of continuing to have home worth £100k and a 10% chance of having nothing
• Expected return is therefore 90% of £100000 + 10% of £0 = £90k
• Cost of insurance (insurance premium) = £20k whether house catches fire or not
• Insurance payout in the event of fire = £100k

Should Mr X opt for insurance, he will end up with £80k regardless of whether house catches fire or not.

Even though the insurance company is offering unfavourable odds regardless of what happens, most people will insure there home because:

• If nothing is done, average/expected outcome £90k, but the actual outcome could be £100k or zero, as there are no guarantees
• Most people would consider giving up £20k to guarantee £80k for a house worth £100k to be a much better proposition than to risk having nothing no matter how small that risk is

According to utility theory and an individual’s attitude to risk, a risk neutral person should reject the insurer’s offer as they should be more focused on preserving their £100k; a risk lover should also decline, as they would be more satisfied with the gains to be achieved by doing nothing.

5.0 Criticisms of Utility Theory and introduction to Prospects Theory

The utility theory model has been criticised for being prescriptive, rather than descriptive. The theory concerns itself with how decision making under uncertainty should be made, instead of how decisions are actually made^. Hence, the rules on how investors should behave based on their risk states.

Utility theory is based on the premise that individuals aim to maximise utility – something that is difficult to measure – by computing factors that affect that individual’s total wealth and choose accordingly. The criticism here can be overcome by Prospects Theory.

Prospects Theory states that in assessing between gambles or in this case risky investments, individuals do not look at the final levels of wealth that could be achieved (as proposed by utility theory) but at the profits or losses that can be attained relative to some reference point, which varies according to the situation. To put another way, Prospects theory replaces the notion of utility with value – while utility is defined in terms of net wealth, value is determined in terms of gains and losses. In Section 2.0 we stated that Charles’ investment objective was to achieve at least a £10k return on his investment (this is the reference point from which he will make his decision). Accordingly, Charles would opt for Asset B rather than Asset A as he has an 80% chance of obtaining his objective with the former as opposed to a 50% chance for the latter – this is not linked to utility but some measurable objective.

Prospects Theory attests to the real world in that individuals are risk averse and therefore, when faced with two risky investments, they will display the risk averse attitude. For example, an investor may be less prone to sell off a loss making investment than he is to cash in on a profitable one.

Prospects theory is descriptive and reveals how people actually behave in the real world as opposed to Utility theory which proposes how people should behave.

It proposes that preferences depend on how problem is presented, for example, if the expected outcome is a positive return, the value function will mean that individuals are risk averse. Conversely, if the expected outcome is a negative return, then the value function will imply that individuals are risk seeking.

Before concluding, it is worth pointing out that in an ideal world Charles would be looking for a combination of Asset A and B instead of just investing all in one or the other. This is in order to maximise his returns while minimising his risk, because as has been said before most rational human beings are risk averse.

6.0 Conclusion

This paper has shown how expected/average return, standard deviation (as a measure of risk), utility theory and the attitude of individual investors influence their decision making under uncertainty.

It concluded with criticisms of utility theory can be justified by considering Prospects theory.