# Study On The Equity Premium Puzzle For UK

This paper tries to explain the equity premium puzzle using behavioural finance explanations following the explanations given for myopic loss aversion (Benartzi and Thaler 1995) and disappointment aversion (Ang, Bekaert and Liu, 2005). This paper is a replicate of the paper of Fielding and Stracca (2007) calculating United Kingdom’s (UK) myopic loss aversion and disappointment aversion instead of United States (US). The data used in this paper is from the Barclays Equity Gilt study 2009. Following the simple specification about myopic loss aversion and disappointment aversion produced at the main paper and applying it for the different set of data the conclusion is that under loss aversion, a highly short-sighted investment horizon is required to explain the historical UK equity premium, while for disappointment aversion long investment horizons are required. So, although stocks lose in the short run, in the long run they can disappoint.

Introduction

The equity risk premium puzzle introduced by Mehra and Prescott (1985) is the rate at which average equity returns outperforms average risk-free returns, such as Treasury Bills, by an average of 6% per annum. Financial economic theory failed in explaining the historical 6 percent equity premium observed in the United States (US) from 1889-1978. Under capital asset pricing model (CAPM), where the return of an asset is measured by taking into consideration a portfolio including expected risk free return, expected market return and the systematic risk of the asset, this value of equity premium cannot be proved. Standard economic models used to resolve this puzzle imply that individuals have high risk aversions leading to this premium. Some models used to explain the puzzle are an Arrow-Debreu set-up used by Mehra and Prescott (1985) which ignores transaction costs as fixed costs, borrowing constraints and other real economy frictions. Making these assumptions under an Arrow-Debreu model it is given that the asset’s market will come to equilibrium. Numerous literature researches by academics on equity premium criticize these basic assumptions and come up with other plausible solutions. Some of this explanations include transaction costs (Bansal and Coleman 1996), borrowing constraints (Constantinides et al 2002), habit formation (Constantinides 1990), regulatory constraints, taxes and diversification costs (McGrattan and Prescott 2003), even the possibility of disasters (Rietz 1988) and survival bias (Brown et al., 1995) as well as myopic loss aversion (Benartzi and Thaler 1995, BT for the remaining paper) and disappointment aversion (Ang et al., 2005,). After every endeavour by the academics, they did not reach a consensus for a solution to the large equity premium. Standard constant relative risk aversion (CRRA) preferences are unable to solve this puzzle since non-participants are not included whatever the risk aversion is, given that the transaction costs are low. Habit formation which is a combination of standard and behaviour approaches needs further research but according to some researches such as Athanasoulis and Sussman (2007) the possibilities of habit formation to be the solution are limited. With this background I concluded that a behaviour approach could be the most plausible solution.

This paper, following Fielding and Stracca (2007), will check the financial behavior of investors and try to explain the equity premium. The two ‘behavioral finance’ explanations used are myopic loss aversion (BT) and disappointment aversion (Ang et al., 2005). These two psychological explanations of the equity premium will be used together because the agent’s preferences in both cases are defined against a reference point (reference dependence) instead of absolute terms as in the standard approach. Furthermore, the agents focus on the reruns of assets without considering how this is related to consumption or how to allocate their portfolio between risk free and risky assets. Myopic loss aversion, with investors evaluating the returns every 1-year and are averse to losses, comprehends risk to holding stocks much more than what is predicted by expected utility. For longer time horizons for the investor’s evaluation period, disappointment aversion can explain this risk and also uses the expected utility as a special case, eliminating the criticism that loss aversion does not include this. Can myopic loss aversion in conjunction with disappointment aversion solve the puzzle for the short and the long horizon?

Myopic loss aversion refers to the higher sensitivity investors tend to have to losses than to gains and to mental accounting. Mental accounting is when investors code and evaluate economic outcomes. In short horizons, agents are too anxious to evaluate their portfolio frequently. This short evaluation period, makes the risk-free bonds more attractive since the risky asset’s average return for the long run, which is higher than risk-free bond, is not constructed. The combination of this short evaluation period and loss aversion is referred to as myopic loss aversion.

Disappointment aversion, using the axiomatic disappointment aversion framework by Gul (1991), shows the aversion to losses after the down weight of good outcomes relative to bad outcomes. The basic idea introduced by Ang et al, is that reference points (the status quo of the investor) evolve endogenously. So for an agent, the risk free return could be set as the reference point and returns of the risky asset that are in short of the reference point will disappoint the agents by more than they will be satisfied if this returns are higher than the reference point.

The purpose of this study is to empirically investigate the myopic loss aversion and at the same time the disappointment aversion on agents at different time horizons and the role they play in determining the equity premium. This time horizons are different time intervals an agent is evaluating his investment. This has been tried by Fileding and Stracca (2007, for the remaining paper they will be named FS) using S&P 500 for US economy. I find it fascinating to try to test this analysis using United Kingdom (UK) market, another developed country with well established financial system, reproducing FS’s model.

I will use a simple analytical approach, identical to the one used by FS, with UK T-bills and stock returns. The value function is different from BT’s and Ang et al since it is defined as the excess return of the risky asset instead of using the absolute returns.

I will try to contradict BT’s results using longer time horizons and a higher degree of loss aversion, so finding out if agents are really that myopic. Also I will try to find at what horizons the parameters for disappointment aversion will best explain the equity premium found in UK market.

The empirical analysis is based on data found from Barclays Equity Gilt study. Further in depth explanation for the data is discussed in the literature review of the paper. The equity premium observed for the UK is 7%, which is close to the one observed in the US market. FTSE-All Share will be used for the risky asset and UK Treasury bills will be used as the risk free asset. Regarding this empirical results, I expect to find that myopic loss aversion and disappointment aversion at short and long horizons respectively can be a possible explanation for the well-known financial puzzle.

This paper proceeds in the following sections. The research in the next section considers the literature review, where two behavioral finance explanations are discussed in depth. In Section 3 we derive expected returns under disappointment and loss aversion. Section 4 deals with the empirical analysis, and finally section 5 is the conclusion of this research paper. In section 6 the bibliography and references can be found.

Literature Review

Before becoming immersed in the empirical part of this paper, it’s interesting to look at other papers focused on the behavior explanations to provide a theoretical foundation for the observed premium. Behavior finance explanations like myopic loss aversion and disappointment aversion have been under consideration building on the frameworks of loss aversion first introduced by Kahneman and Tversky (1979) and disappointment aversion introduced by Bell (1985). The results provided thus far are promising.

The two concepts of loss aversion, measuring the sensitivity of losses to gains and to mental accounting which refers to how individuals aggregate choices i.e. how transactions are evaluated over time, were explained empirically by Samuelson (1963). To illustrate that, Samuelson run an experiment where a coin was tossed. If one wins, he would be rewarded by £200 but if he loses, he would lose £100. The individual who was taking the experiment, refused to take the bet if the coin was tossed only one time. However, he was happy to take the bet if tossed the coin 100 times. Thus, the individual is not myopic since he is willing to aggregate all the outcomes first instead of evaluating the outcome of a single toss which could be negative. This example could be an explanation of the loss aversion. Although the decision by the individual sounds irrational, losing £100 is more harmful than the gain of winning £200. Mental accounting is illustrated by this example by the time the individual would accept two or more bets given that he does not observe the results of the previous tossing. As Kahneman and Tversky (1979) showed, the utility function of an individual is steeper for losses than for gains.

The empirical results of Bernartzi and Thaler (1995), who integrate loss aversion to utility at the reference point, give a loss aversion of around 2 which is a reasonable value. The value of 2, means that the individual taking the experiment is twice as sensitive to losses as he is to gains. They based their research on looking at what were the required loss aversion and the time for evaluation (time horizon). The parameters they found to explain the historical equity premium is a loss aversion hovering around 2, a time horizon of 1 year, and an equal allocation of assets for their portfolio consisting equities and bonds. The question yet to be answered is how can agents be so illogical to reject the volatility of high stock returns and at the same time are not consuming this portfolio but instead invest it to riskless bonds for the meager one percent per year. The answer given is the agents’ high loss aversion and due to mental accounting that this agents are myopic and check their wealth regularly. So utility is affected on asset returns and when asset returns decrease, although they will not affect consumption it will still decrease utility. If the agents are myopic, they will evaluate their investment regularly, becoming aware of short term losses and the decreased utility, so they will ask for a high risk premium to accept the variations of the return.

Thaler et al (1997) based on BT paper tried to re test the previous empirical results. They check whether agents would become risk lovers if they have higher time horizons and all their payoffs were positive. What they found is that by increasing investors’ evaluation period, investors get less frequent feedback on their investment and decreasing or even eliminating the possibility of a loss they will prefer stocks more than bonds. These findings can lead to the conclusion that mental accounting and loss aversion can be found in the financial markets and strengthen the experimental results found by BT setting myopic loss aversion as the equity premium explanation.

During the same year when Thaler et al were testing myopic loss aversion, Gneezy and Potters (1997) tested the existence of myopic loss aversion carrying out a direct experimental research. Opposing Benartzi and Thaler (1995) who did not provide experimental results for their data, they run a controlled experiment where they manipulated the evaluation periods to prove the results of Benartzi and Thaler are efficient. They conclude that increasing the time horizons of the investment increases the desire for stocks. However they are cautious for their results since it is supported that financial professionals’ behaviour is different from nonprofessional behaviour. Furthermore, real market experimental research is needed for more realistic results.

Gneezy et al (2003) results reinforce the argument advanced by Benartzi and Thaler. The paper looks at the fundamental error of their previous research which was mistakenly not focused on market interactions but was observing personal decisions instead. The reason behind this is that the market is the determinant of the prices. Personal decisions do not affect the market. They argue that market interactions will not affect past experience in the same way as repeated market interactions will do. So their previous results are false. After carrying out a market experiment they found out that market interaction produces the same results with individual decisions and this consistency does not affect their previous results.

Adding to Gneezy et al (2003) questionings, Haigh and List (2005) are also concerned about the reliability of the results found using nonprofessional investors to analyze the loss aversion parameter. All past researches were focused on students, so Haigh and List (2005) carry out an experiment using professional traders. They find that there are indeed differences between professionals and students. The unexpected result is that traders have a greater behavior consistency with myopic loss aversion than students. So these real market empirical results from financial professionals’ behavior strengthen the experimental study of Bernartzi and Thaler for myopic loss aversion increasing the possibilities of myopic loss aversion to be the explanation of the equity premium. Expected utility theory may not be suitable to explain professional’s behavior and instead economic behavior and finance models can be better approached.

A further research by Bellemare et al (2005) on the behaviour concepts, examines the effects of a separation between the information feedback and their investment flexibility. It has never been clear in the prior experiments of behaviour approach which of the two behaviour tendencies influence the evaluation period. This is investigated in this experiment finding that even if the investment flexibility is constant, the myopic loss aversion has an effect on investors. Given that investment flexibility does not influence the concept, we can assume that information feedback is the only factor affecting myopic loss aversion and researches should focus their interest on this variable.

A few years later, Blavatskyy and Pogrebna (2009) explore how individual choice patterns influence myopic loss aversion. They analyse previous papers on myopic loss aversion that used aggregate choice patterns and change it to individuals. Their results are that previous experiments could be just showing other facts, and also that expected utility theory if compared with myopic loss aversion, will give more precise results.

The problem of excluding expected utility theory from myopic loss aversion is dealt with the use of disappointment aversion. This concept, firstly introduced by Bell (1985), is integrated in the expected utility theory as a special case. The results of Loomes and Sugden (1986) are compatible with Bell’s results, showing that the violations of standard expected utility can be predicted by the disappointment model. An example of the disappointment aversion is an agent winning the top prize £1000 in a lottery will leave him in a greater euphoria than when winning the lowest prize of £1000 of the lottery. The cause of the disappointment is the comparison of the actual result to what the individual was expecting as a decision for the uncertain outcome. So we discriminate between disappointment averse investors as the ones with a pessimistic view about the future and disappointment loving agents as those who are optimistic for the future. If you toss a coin and either wins £200 or £0, there is 50% chance that you will be disappointed even though it does not match up with the expectations.

After a few years, Gul (1991) uses an axiomatic model generalizing Bell’s (1985) first idea. Disappointment aversion is consistent with the Allais paradox, a behavior in consistency where real life decisions contradict the expected utility theorem. Using an alternative for the expected utility the results are that risk aversion implies disappointment aversion and provides a better understanding for the failure of the independence axiom of the expected utility theorem. Independence axiom means that when an agent prefers P1 to P2, then he will also prefer P1+k to P2+k. Gul (1991) uses as a reference point the certainty equivalent of a lottery and as a reasonable assumption many other researchers including this paper follow this idea. Unfortunately, Gul’s preferences are not successful in explaining the asset returns. So we must use a more generalized model.

Jia et al (2001) demonstrates that finance models are special cases of the disappointment model. Following Bell’s basic idea, a more generalized disappointment model is offered, which is based on the riskiness and the value function of the assets. They also extend the basic model allowing for the use of more than two outcomes. The individual’s attitudes are endogenous and change according to lotteries changes. The reference point used by Jia et al, unlike Gul, is the expected value of the lottery. Their result is that this behavior concept can explain some non-standard risky choice behavior. Also they ‘offered additional insights into some observed risky choice behaviour and the decision paradoxes that violate the independence axiom of expected utility theory’ (p 76) as Bells’ (1985).

Another study using a more generalized idea of Gul’s (1991) concept was carried by Routledge and Zin (2003). They use a one parameter extension of Gul’s utility function. This flexibility allows them to characterize outcomes in a lottery as disappointing when they lie sufficiently below the certainty equivalent. They suggest that by introducing a conditional heteroskedasticity in Mehra and Prescott’s calibration, ceteris paribus, disappointment aversion is able to capture the value of risk aversion deduced by Mehra and Prescott.

Ang et al (2005) using the same preferences with Gul (1991) check on both static and dynamic portfolio choice. They find that disappointment averse investors with a realistic equity portfolio have utility functions with low curvature. Investigating investors, who face a sufficient degree of disappointment aversion, find that their portfolio does not include a reasonable number of equities meaning that they participate in other markets than equity. They conclude with the realization that although the equity premium can be large, the stock may still disappoint.

Both loss aversion and disappointment aversion preferences take care of gains and losses asymmetrically. Both have agents’ preferences against a reference period. The main question the supporters of behavior explanation for the equity premium have to take into consideration is: Is only one of the two behavior explanations mentioned above sufficient to explain the historic equity premium in the short and the long horizon or should we use the combination of the two behavioral concepts? Since myopic loss aversion can explain the equity premium in the short horizon and disappointment aversion is a plausible explanation of the long horizon, using the conjunction of the two may give us the covetable solution to this puzzle. This idea of using both concepts was introduced by Fielding and Stracca (2007) using US data. Replicating this paper, I will use UK data so I can have a more subjective view on the topic.

Summary of the paper

Loss aversion and disappointment aversion explanations

Loss Aversion was firstly introduced by Kahneman and Tversky (1979) and is based on how people manage risk and uncertainty, called prospect theory. It checks the psychological part of the investors, the human behavior and its anomalies. Some experimental evidence is provided by Kahneman and Tversky (2000). There are three main characteristics of prospect theory. Firstly, the reference dependence, where an individual evaluates the outcome and his utility in comparison with a reference point instead of absolute values. This reference point is usually the status quo i.e. the current wealth of the individual. Second, there is a diminishing sensitivity, showing that the marginal increases or decreases from the reference point have a greater impact if they are closer to it. The third characteristic is the loss aversion. The value function is greater for losses than for gains. Finally it has a ‘non-linear weighting of probabilities thus departing from the linear weighting as in expected utility theory’ (F&S p253).

BT believed that loss aversion with the combination of myopic behavior could be a solution of the equity premium puzzle. The value function shown is for investors, to count the returns in different horizons at time t+h.

(1)

In this value function, x is the real return of equities or 5-years bonds, the reference point for the agent is the current level of wealth, and α=2.25, b=0.88 (this estimates are from Kahneman and Tversky 1979). α shows that the agent is loss averse, and he assumes losses are more than twice as bad as gains. BT using 5-year bond returns and Standard and Poor returns from 1926-1990 for the above value function computed the horizon h for which the investor is indifferent for his investment portfolio between the two assets. They found this result to be 1 year to explain the premium observed.

In this paper I will use a linear loss aversion value function:

(2)

Disappointment aversion, the second behaviour finance explanation I will analyse, was firstly introduced by Gul (1991). The idea behind this theory is that investors are disappointed when the outcome is less that the certain outcome while on the other hand they are elated if the outcome is above the certainty equivalent. The agent is disappointment averse if he dislikes disappointment by more than he likes elation. The main difference between loss aversion and disappointment aversion is that in disappointment aversion the reference point is endogenous instead of exogenous as in the case of loss aversion as stated above. So in the case of disappointment aversion the reference point can change for each lottery.

Ang et al used the generalised disappointment aversion utility function to examine the importance of disappointment aversion to the equity premium puzzle. The value function which is again in terms of returns is

(3)

As in the case of loss aversion, x is the real return of equities or bonds, e stands for elation while d stands for disappointment. d>e>0 shows that, as in the concept of loss aversion, disappointment is more important that elation. Etxt+h show the expected return. According to the value function, when this expected return is high, the stocks will disappoint.

Investors’ returns and time horizon under loss aversion and disappointment aversion

Following the behavioral finance explanations introduced by BT and Ang et al, I will construct a simple approach where both of this explanations can be analyzed together, trying to find what time horizon and thus at what loss and disappointment aversion an agent has to hold his portfolio in order to come up to the equity premium found historically. In this paper I will use the assumption and formulas introduced by FS, using different data, country and time period in order to reinforce their result they found using US data.

The assumptions of this paper is that the individual considers distributing his portfolio in two assets, risky equity and risk free bonds (UK treasury bills), and is not thinking of his consumption and utilities. The decision problem is autonomous so the allocation is not affected by any other exogenous factors.

There is a general consensus through some academics focused on the puzzle that α, representing the individual’s degree of risk aversion is around 2. This means that the individual is twice as unsatisfied for a loss as he is for a gain. This figure is reinforced by Kahnmeman and Tversky’s experimental study (2000) and since rational I will use it. On the other hand I cannot conclude to a certain time horizon the individual holds his portfolio without checking the reports of the investment, since no answer is secure to use for this matter. BT based on the myopic theory that agents are too anxious and not patient to analyze their investment rationally, states that the time horizon of the individual is 1-year. There are not any empirical proves to agree on this so different time horizons should be tried. Longer time horizon, in this paper up to 10 years, can be seen as a more rational approach by the agent.

The main differences between this and FS paper from the BT and Ang et al ones is that we concentrate on excess returns of stocks and bonds instead on the absolute returns their value function are based on. The reason behind this is that an individual has to allocate his wealth between two assets, so he will chose the one with the higher return. The relative performance of the two will realize a high or low excess return and according to the risk associated with each asset the agent can decide where to invest. I am not interested in the results an agent will have in not investing at all, but the losses and disappointment the agent will have after his decision between this two assets. Thus we can find the loss and disappointment aversion of the agent according to the riskiness of the assets. Using the combination of αLA and h, I check at a given time horizon, what loss aversion is found - using FS formulas - that produces the current equity premium. Unlike BT that finds α to be equal to 2 when h is 1 year, I will check for longer time horizons up to 10 years.

The difference of this analysis and Ang et al one is that they do not refer to the problem of the time horizons at all for the disappointment aversion. They just state that a quarter or a year of time horizon for the disappointment aversion could explain the equity premium. Again longer time horizons for the parameters of this behavior model explanation will be included and I will try to find the combinations of αDA at different time horizons h, that can be a good explanation for the equity premium.

Next I will follow and explain FS simple model for finding the loss and disappointment aversion. Deriving the final equation of the two models, they will be used for the UK data and for the empirical analysis.

First, the marginal difference between the outcome and the reference point should be found. This is given by ρj where j=AD, DA. Computing equation (4) [1] , we have the general ex post measure ρj at time t+h. Then ex post value function for loss aversion and risk-value function for disappointment aversion (5) [2] is found. We go on to find the expected values for the two aversions given by (6) [3] and (7) [4] . The expected risk value function for the disappointment aversion includes the coefficient e which is a measure of the overall risk aversion while aDA is a measure comparing the relative importance of disappointment in terms of the elation. It is assumed for simplicity and guided by other literature reviews that e=1, thus we can focus on the aDA. Simplifying equation (7), it can be rewritten in the form of (8) [5] . The excess return on equity for both specifications of preferences is considered for t to t+h, which in this case will be shown by xt+h.

After computing the expected value functions (6) and (7), it will be assumed that the investor is using this for his evaluation process. Following KT (2000) empirical evidence on decision-making under risk, it is more appropriate to form the expectations using a non-linear weighting probability. The expected value function for the risky investment is given by (9) [6] .

Having as α the proportion of the portfolio invested in the risky asset, and normalizing the expected value function for the investment in the risk free asset at zero we derive equation (9) again. So if α is not equal to 0, the expected value will be equal to zero, thus we have equation (10) [7] . Hence, recalling from our previous results, in case of the loss aversion we have (11) [8] and in the case of

disappointment aversion (12) [9] . Under the assumption that agents being rational without any systematic errors, we have (13) [10] .

Considering a sample with t=1....T where T is large, we have (14) [11] . Based on the assumption that

εt+h is stationary then the unconditional mean of the value function is an unbiased estimate of the expected value function. Thus equation (10) must have a value function of 0 on average as (15) [12] . This is the basis for our empirical study.

For loss aversion we have

(16)

And for disappointment aversion

(17)

Using the above two equations, we can easily find, using our data, the loss aversion (αLA ) and disappointment aversion (αDA) when the values of the time horizon is given. The main objective of the paper is to find which possible combinations of aversion (αj) and time horizon (h) gives reasonable values for both variables sing the above equations.

FS used the above formulas to analyse the consequences of myopic loss aversion and disappointment aversion using - for the risky asset - the annual returns, including dividends and capital gains, of equities by Standard and Poor for years 1871 to 2001. US 1-year Treasury bill is used for the risk free asset. This data range is different from BT where data from 1926-1990 is used instead. After 1990 important economic changes occurred, for example the Argentina and tequila crisis, the dotcom bubble, the housing bubble and even the current economic crisis. So using different years for the data is biased. FS examined the excess return on the standard and poor compared with the Treasury bill, for horizon between 1-year up to 10 years. They observed higher negative results for the excess returns at the 1 year horizon that what they found for the 10 year horizon. This can prove that investors are myopic and could be an explanation of the equity premium puzzle. At the short horizon the mean excess return is 6.6 percent with a standard deviation of 19.5 percent and for long horizons the excess return has a mean of 115.5 percent with 132 percent standard deviation. These results demonstrate that the positive excess returns are 2.27 times more than negative excess returns while on the long horizon it is 24.67 times more.

FS found that if h=1, then the loss aversion is close to 2.25 so they agree with BT. But at longer horizons, at any horizon above three years the parameters have unacceptable sizes. Therefore, loss aversion could be a logical explanation of the equity premium puzzle only at very short time horizons, meaning the investors having high degree of myopia.

Their findings for the disappointment aversion are that the disappointment aversion rises very gently with the time horizon. For the data range from 1871 they found the aversion to be hover around 1 at the short horizons while on the longer horizons it is around 2. These are very reasonable values. So the disappointment aversion could be a good explanation of the equity premium puzzle for both long and short horizon. These results contradict the theory of highly myopic agents. Thus the result of this paper is that stocks in the short term can both lose and disappoint but in the long run they only disappoint. FS results agree with BT’s results for the myopic loss aversion. The same is true for FS and Ang et al results for the disappointment aversion. Actually Ang et al states that disappointing averse investors should not hold many equities in their portfolio.

Empirical Analysis

In the empirical analysis of this paper I will use data from 1899-2008 taken from the Barclays Equity-Gilt Study (which I will refer to Barclays from now on). This data covers firms listed on the London Stock Exchange. Up to 1962 this study comprises the 30 largest shares based on their market capitalisation. After 1962 the study uses the FTSE All-Share Index. This sample period which gives as 110 annual observations is different from FS who uses data from back to 1871 and BT who starts much later from 1926 up to only 1990. I do not think the data prior to 1899 is significant to my study, but I will have an analysis from 1899 and one from 1926. This would let me compare with both FS and BT since they both cover this period in their analysis. The Barclays equity-price index uses proportional weights according to the firms’ market capitalisation. The income yield used is the dividend paid at the current year divided by this year’s end price while dividend yield is the dividend paid through the current year over the prior year’s price. Using the same source, the UK Treasury Bill is constructed. Furthermore, the cost of living index is used as a proxy for the inflation rate i.e. the consumer price index so the real returns instead of nominal ones are found. This will give more precise answers to the analysis.

The assumptions made for this paper are that there is one representative investor, who has £100 at the end of 1899, and may invest it either in the UK stock market (using the equity price index from Barclays and I will call RISKY) or in 1-year Treasury Bills (which I will call SAFE). Then we find the value outstanding in each year t in pounds for these two assets.

Chart 1

Using this data it is found that for the whole sample period, stocks outperformed Treasury bills by almost 7%. So an equity premium exists. As can be shown on chart 1, the same £100 invested in 1899, would worth more than £17000 if invested in equities against a slightly more than £300 if invested in treasury bills or gilts. It is interesting to observe the growth of the value share of equity and treasury bills in UK. There was a small growth during the period of decolonisation, up to 1973 where the value share had quite a sharp decrease probably because of the oil crisis, and after this there was a rapid growth up to 1999 where we experienced the dotcom bubble. Also the current crisis can be observed in the diagram as the decline after 2007.

Chart 2

An investor with such an extensive investment horizon is unrealistic and doubtful. Therefore it would be more rational to use shorter, realistic time horizons. In this analysis, the same time horizons as SF will be used, in particular from 1 year to 10 years. This will give the opportunity to check whether the investors are highly myopic, while at the same time I will test out what happens at longer horizons than those tested by BT. The upper panel of chart 3 presents the 1-year excess return on the equities compared with the Treasury bills. On the lower panel of chart 2, for a comparison, it is the excess return for a 10-year time horizon instead of 1-year.

Chart 3 (upper panel)

Chart 3 (lower panel)

As it can be observed by the charts, the excess return on equity turns out to be equally negative and positive for the 1-year horizon with the values been highly volatile. The negative excess return is below 40% (except in 1913 and 1973 where equity was affected severely by the First World War and the oil crisis respectively). The positive excess on the other hand hovers around 40% with cases where it goes up to 100%. The negative excess is hardly seen in the 10 year horizon (it is negative only in the first decade of the century and in the early sixties). Although it’s mostly positive the volatility of the excess returns is high. It goes up to 230% excess return and a negative of 80%. Thus, the first conclusions drawn from a brief observation at the evidence is that at short horizons h, myopic loss aversion and disappointment aversion can be the right parameters to explain the puzzle. Actually, 1-year excess return on equity implies the possibility of losses corresponding to the loss aversion and a high standard deviation corresponding to the disappointment aversion. On the other hand, the 10 year excess return on equity still has a high standard deviation but the probability of losses is low. The key statistics on table 1 reflects this difference for the loss aversion in the short horizon and the similarities of the standard deviation for the disappointment aversion. The average mean for the full sample period is 7.454 percent with a standard of 25.7. Looking at the long horizon, the mean is 74.6 with a standard deviation of 59.6.

Table 1

Excess returns on equity at 1 and 10–year horizon

ER1

ER10

Sample period: 1900-1999

Mean

Median

Maximum

Minimum

Std. Dev.

Skewness

Kurtosis

Jarque-Bera

Probability

Observations

Sample period: 1926-1999

Mean

Median

Maximum

Minimum

Std. Dev.

Skewness

Kurtosis

Jarque-Bera

Probability

Observations

7.454

6.25

104

-56.9

25.73084

0.817328

5.502707

37.23184

8.23e-09

100

8.854054

7.35

104

-56.9

23.90574

0.746399

5.870626

32.27924

9.79e-08

74

74.608

71.4

228.5

-86

59.66663

0.189453

3.283452

0.932981

0.6272

100

85.86081

74.65

228.5

-40.3

52.47533

0.575478

3.226338

4.242453

0.119885

74

Moving on the empirical analysis of this paper, a time horizon for 10 years is considered. The ex post excess returns at time t+h are computed as follows:

(18)

For h=1... 10 years. For each h, we then compute αLA , αDA as implied by equations (16) and (17).

First, I run an analysis for the loss aversion for two periods, the full sample period from 1899-1999 and then for the restricted sample period 1926-1999. The second restricted period is used to compare it to BT’s and FS samples. The upper panel of chart 3 shows the estimated loss aversion against the time horizon that satisfies equation (16) for the full sample period while the lower panel represents the restricted sample period.

So, using the same methodology with FS but with different data, while using different methodology than BT, there is a consensus agreement to the results of myopic loss aversion i.e. the short time horizon results. At time horizon of 1-year I find the value of 2.16 for the full sample and 2.62 for the restricted sample.

This is very close to the results found by the other two papers. The results are not that encouraging for the longer time horizons. Testing the full sample of the UK data, at horizons of more than 3 years, the loss aversion value is unacceptable with the maximum loss aversion at the 10 year time horizon been 28. Using the restricted period for the data, at a time horizon of 3 years, the value of the loss aversion is 5.29 which is too high for a loss aversion. The interesting thing for the restricted period is that at time horizons above 3 years, the amount of loss aversion soars to unacceptable and irrational values. The estimated loss aversion goes up to 160 meaning that an investor will turn down a bet where he can win £160 or lose £1, with the outcomes to have equal probabilities to occur. This cannot be true in real life. This is not the same for the US results, where for the two different samples the results for the loss aversion did not change. Further analysis is needed to check the difference in stock returns at different time intervals between UK and US. The conclusion of this analysis is that loss aversion can be a good explanation of the equity premium only in the short horizon, thus agents must be myopic. Investors are willing to accept risks only when they evaluate their investment infrequently.

Sample period: 1899-2008

Chart 4

Sample period: 1926-2008

Going on to analyze the disappointment aversion using equation (19), chart 5 shows the disappointment aversion against time horizons. The upper panel is for the full sample while the lower panel is for the restricted period. Unlike loss aversion, disappointment aversion rises very gently. Looking at the full sample, it starts from 1.76 for the 1-year time horizon up to just 4.11 for the 10-year horizon. The disappointment an investor will have - when the outcome of equities is below his expectations - is more than 4 times the euphoria he will have -when the equity’s outcome is above his expectations- during a 10-year time horizon. The interesting thing with disappointment aversion, unlike loss aversion, is that the restricted period is very similar to the full sample period. These values are very reasonable and can explain the equity premium for short and for long time horizons. As a result, based on the disappointment aversion, the assumption that investors are myopic is uncertain.

The conclusion drawn from the empirical analysis is that equities when compared with Treasury bills may lose in the short run, but can disappoint both in the short run and the long run.

Chart 5

Sample period: 1899-2008

Sample period: 1926-2008

Conclusion

In this paper I use the two behaviour explanation of equity premium, the myopic loss aversion developed by Benartzi and Thaler(1985) following Kahneman and Tversky’s (1979) idea on prospect theory, and the general disappointment aversion model of Ang et al (2005). I used these two concepts to study the differences between them and try - by using simple specification preferences - to understand the basic idea of the two. By using a combination of the two I tried to come to the much desirable solution of the equity premium puzzle. Taking care of the different reference period used by the two behaviour explanations and applying the models, using real data of the UK stock market, I examine the relationship between the myopic loss aversion and disappointment aversion at different time horizons the investors have. The reference point used is the status quo i.e. the current wealth of the individual. This is endogenous for the disappointment aversion while it is exogenous for the loss aversion.

Progressing in line with Fielding and Stracca’s paper, the results found are consistent with the theory. Under loss aversion, a highly short-sighted investment horizon is required to explain the historical UK equity premium. On the other hand, for short horizons and for long horizons lead to disappointment averse investors. So, although stocks lose and disappoint in the short run, in the long run they can only disappoint. So the results of US and UK do not have many differences.

Coming back to the main question of asking: Is only one of the two behavior explanations mentioned above sufficient to explain the historic equity premium in the short as in the long horizon? It is obvious from my empirical results that we have to use both behavior explanations in order to have reasonable values for short and long horizon.

Moreover, this study using up to date data, provides a helpful conclusion for the academics carrying out a research on equity premium. The ‘behavior finance’ explanations could be the most robust solution to the historical equity premium of UK and US.

Finally, a number of interesting avenues can be brought up for further discussion. Extending this analysis to include more periods and more countries will draw the attention of the authors of behaviour papers. Due to globalisation we could be at a point where the psychology of developed countries is very similar and investors are the same loss averse and disappointment averse as in UK and US. The use of other reference points or an international diversification will also be interesting topics for further research.

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