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PROSPECT THEORY. In 1979, Kahneman and Tversky introduced ..... Kahneman D. and Amos Tversky, “Prospect Theory: an Analysis of Decision under Risk”,.
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EQUITY RISK PREMIUM. AN ESTIMATE INSPIRED ON BEHAVIOURAL FINANCE

JOSÉ RODRIGUES DA COSTA1 1. INTRODUCTION In spite of all its shortcomings, CAPM is still the most used model to estimate the long-term return demanded by investors to put their money in an asset suffering from systematic price volatility imposed by the instability of the surrounding market. This led to a universal quest for an appropriate Equity Risk Premium (ERP) to be plugged into the formula of that model in each individual case. Unfortunately, no universal figure was yet found. However, one of the main difficulties I have frequently found when teaching Financial Futures to university students is to get them interiorizing that the current quotation F0 of a Futures Contract is not the present expected price valid for the maturity date of the underlying asset, but rather the result of an arbitrage operation between the current spot price S0 and the simultaneous “forward” price (using some riskfree rate).

Indeed, when students realise that F0 freezes the return obtained during the tenure of such a contract from buying at the initial spot price S0, they immediately concur that the expected maturity price needs to be more expensive than the Futures quotation F0. No one would ever invest S0 to obtain an uncertain return simply equal to the guaranteed return given by that Futures price F0. The average expected final has to be sufficiently above F0 to compensate the potential pain due to the unavoidable losses suffered when ST randomly terminates below F0. All students voice their internal feelings more or less in this way: “I demand from an unprotected underlying asset a final price that, on average, is sufficiently above the correspondent Futures agreed price F0 to guarantee that my statistical gains (above that Futures price) more than compensate my less likely losses”.

1 - ISCTE – Instituto Superior de Ciências do Trabalho e da Empresa, Av. das Forças Armadas, 1649-026 Lisboa, Portugal and Euronext Lisbon. [email protected]. I THANK MARIA EUGÉNIA MATA FOR HER KIND COMMENTS AND JOHN HUSTOFF FOR CORRECTING MY ENGLISH. REMAINING ERRORS ARE MINE ONLY.

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It is this approach that may lead to a new form of estimating the ERP and to a better understanding of the role played by the different relevant market and psychological variables upon the size of that premium. 2. THE STATE OF THE ART ON THE EQUITY RISK PREMIUM Ideally, economic science should have long ago provided a method to estimate the ERP of each domestic market based on its particular characteristics. Mehra and Prescott tried exactly that approach by making use of the traditional expected utility concept of investor´s wealth, but unfortunately their first results suggested either a very low ERP or a very large risk-free rate, both cases completely out of the reality of all markets. Subsequent improvements made by them and by other authors did not solve conveniently this puzzle because, in essence, they maintained the basic concept of an expected utility calculated from the final wealth of the investor at the different states of nature, along with justifying that premium from the risk aversion characteristic (non-linearity) of that utility function. Due to these insufficient responses of theoretic models, another group of scholars decided to look at history in order to measure the past average behaviour of different domestic equity markets, hoping that those statistic samples were representative enough to guarantee that the future returns would not be much different from their past. Here R. G. Ibbotson played a leading role for the US market using a sample starting in 1926 that produced the initially much heralded 8,4% premium (Brealey and Myers, 1996) that became a temporary universal yardstick due to the lack of alternatives estimated for other countries. But, in 2002 Dimson, Staunton and Marsh, following a similar route,

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published the famous book “The Triumph of the Optimists” covering 16 different countries including USA. These were the first estimates of average returns per country with the advantage of a longer and common sampled time window (all national series begin at the very end of 1899). The fact that they found significant different premiums from country to country – from as low as 2% p.a. for Belgium to 12% p.a. for Australia – confirmed the previous suspicion that the former US premium could not be applicable to every economy as the 20th century happened to be a very success story for USA but not for some other countries. A third line of reasoning, on the opposite, has been focusing on the future rather than on the past, since what is needed in most applications is the rate to discount future cash-flows, not the average return realised in history. For that purpose, I. Welch (2000) decided to make a worldwide survey where he asked scholars (later repeated updated) what was their current value of ERP in use in projects under analysis. Similar samples have been obtained subsequently, the most recent one having been produced by Pablo Fernandez in 2011. Although benefiting from the most recent opinions from the different contributors, all those figures suffer from a common deficiency: they are mere opinion percentages which might be too much influenced by very recent events affecting the sampled respondents, as detected by Welch. My experience with students suggested also a new approach looking into the future but now based on how humans decide under uncertainty. In this respect Prospect Theory and its subsequent developments seem to offer a new method to make estimates for the future based on a behavioural model of humans and on the most recent statistical parameters for that very model estimated from empirical tests implemented in different populations.

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3. PROSPECT THEORY In 1979, Kahneman and Tversky introduced this theory after a number of field experiments with different audiences suggested them that Utility Theory and its tenets did not exactly describe our intuitive workings under uncertainty. For the purpose of this paper the relevant components of their proposal are: a) Certainty Effect2: people overweight outcomes that are considered certain; for example, people prefer to receive €50 with 100% certainty instead of €100 with 50% probability and €0 otherwise; this is also called risk aversion. b) Reflection Effect3: as an image in a mirror, people prefer gambling on a game with 50% probability of losing €100 and 50% of loosing nothing instead of suffering a sure loss of €50; that is, for losses, humans are risk seekers not risk avoiders as in a). c) Isolation Effect4: people often disregard common components included in different probabilistic alternatives and focus instead on the components that distinguish them; that is “the carriers of value are changes of wealth rather than final asset positions”; for example, the response to the problem of starting with €95 in pocket and deciding between a guaranteed final outcome of €100 – this

2- Econometrica, March 1979, pg 265 3- Econometrica, March 1979, pg 269 4- Econometrica, March 1979, pg 271 5- Econometrica, March 1979, pg 278 6- Econometrica, March 1979, pg 279 7- Journal of Risk and Uncertainty, 1992, pg 311

€5 sure gain is preferred – and a game that may end evenly in €95 or in €105, is different from the response to a similar problem that departs from €105 but either guarantees the same final €100 – this €5 sure loss is rejected – or has 50% probabilities of a final €95 in pocket and 50% of €105. d) Value Function v(x)5: instead of a utility function, they introduce a value v(x) associated to each gain/loss x which appears to be non-linear because there is a diminishing sensitivity to the size of x; for example, the value difference between a gain of €100 and a gain of €200 is larger than between gains of €1100 and €1200 (and similarly for losses). e) Weighting Function π(p): the total value of a prospect is the weighted sum of the values v(xi) of each possible gain/loss xi where the weights πi are a non-linear function of the respective probability pi. In 1992, Kahneman and Tversky improved somewhat their early Prospect Theory in two important points for our purposes: f) “Losses loom larger than gains”6: the value function v(x) for losses is steeper than for equal values of gains by a factor which is above 2. g) (again) Value function7 v(x): they wer e able to adjust a best fit function to their empirical results

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which indicates that the pain felt by a typical human after suffering a certain loss of size x is 2.25 times larger than the satisfaction extracted from an alternative gain of exactly the same size. It also indicates that there is some nonlinearity in the function v(x) which has a shape that falls somewhere between the square root function and full linearity (0.5 < 0.88 < 1.0). Finally, in 1995, Benartzi and Thaler, elaborating from the above two papers, introduced the term “myopic loss aversion” as the combination of: h) Loss Aversion8: the earlier finding that “individuals tend to be more sensitive to reductions in their levels of well-being than to increases” . i) Evaluation period9: different market indications suggest that individual investors and even institutional ones tend to reevaluate their investments once every (around) 12 months, even if their plan-

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ning horizon stretches over some decades as in pension funds. This paper starts by exploring loss aversion only and simplifying through a linear approach to the valuation of gains and losses. The issue on the evaluation period T is subsequently analysed. Finally, we introduce the non-linear v(x) function although using functional forms more flexible than the above model. The weighting function π(p) is left for fur ther studies. 4. FIRST APPROACH: LINEARITY Consider an asset S that is currently priced at S0 and on which there is a Futures Contract maturing at T with an equilibrium price where rf is a constant risk-free rate. As commonly, assume that S follows a as Brownian motion with annual return µ and volatility σ. Αfter Itô´s Lema,

indicating that, at the maturity T of the Futures

on S, the price ST follows a log-normal distribution

around an expected (average) price . So, if in reality S finalises above F0, an uncov-

ered investor gains more than taking a long position in the Futures

but, if it terminates below F0, he looses

8- The Quarterly Journal of Economics, Feb 1995, pg 75 9- The Quarterly Journal of Economics, Feb 1995, pg 76

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The average final gain and loss under any statistic distribution are

According to Kahneman and Tversky, people react to gains and to losses (not to the final level of wealth) and, additionally, gains provide a level of satisfaction that is larger than the

correspondent level of pain: . As stated above, in this section we assume both psychological feelings to be proportional/linear to that average gain or loss. So the expected net satisfaction is (1)

and the investor will be in equilibrium if

Noting the similitude of the first integral with the value of a Call Option on S at its maturity

, or

T, and of the second integr al with a par allel Put Option10,

10- Besides the different moment of reference – T instead of t = 0 – note that these two Options are calculated under the real return rate µ, not the risk-neutral return rf.

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Calling

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and introducing r f via F0

This expression determines the premium (µ µ – rf) for each multiple π, each level of volatility σ and for a selected maturity T (measured in years). The following table gives: • on the left half: the equity Premium

This table indicates: a) for the common volatility of 20% and for a pain 2,25 times larger for losses, a premium of about 6,5% p.a. b) that (volatility) premium grows with σ but not linearly; the curve is slightly concave; c) the premium also grows nonlinearly with

(µ µ - rf) for values of π from 1,0 to 4,0 and for volatilities from 10% to 40% selected after the empirical findings of Kahneman and Tversky and the frequent values of market volatilities11 • on the right half: the equity risk premium Density (per unit of volatility σ).

the multiple π d) if losses and gains were equally/ symmetrically valued – that is, π = 1 – than, as anticipated, there would be no volatility premium. 4.1 Equity Risk Premium Density Since CAPM can be written as

11- Mind that the VIX Index (new methodology) showed an realised average of 20.53% p.a., during the period Jan/1990 to May/2012 with a standard deviation of 8.22% p.a.

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it is interesting to introduce the concept of Density as

The table above gives this ERP Density for a horizon of 1-year and for volatilities σ around 20% p.a. and for multiples π around 2,25 times. The practical conclusion is that, for those com-

Notice that the numerical factor (0.32) remains almost constant for volatilities between 10% and 40%12, but does change with π even for values13 around 2.25. It is likely that investors adapt their feelings to their recent experiences and this suggests that, besides changing the σ of a market according to their mood, they may also: a) become more sensitive to losses after a period of heavy losses; this explains the historical tendency to find above normal

mon market and human parameters, a first approximation to the discount rate Ri of a volatile asset with volatility σi and correlation with ρi with the surrounding market is given by

returns after deep economic crises b) or, on the contrary, reduce their heavy “price” π for losses after a long period of gains which would, consequently, reduce the premium demanded for the same market volatility σ. 4.2 Evaluation Period What if the finding i) referred by Benartzi and Tahler – humans tend to adopt annual assessments of their previous investment decisions – is not true?

12- But for extreme volatilities like σ = 250% p.a. the equity risk premium density falls to 0.2554. 13- Notice that another common multiple referred in the literature is π = 2.50. Additionally, Benartzi and Thaler mention a value of π = 2.77 (page 83). Adopting a multiple 2.25 seems to be conservative (for estimates of discount rates) but not excessively so.

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According to the Browning motion adopted for S, as time to matur ity T gr ows, both the expected final value and the volatility around that value grow. However, volatility grows slower (square root) than (linear with T) and this means that, although an investor faces more absolute risk around , it is more probable for S to finish above F0 the farther is the horizon T. Ther efor e, one can anticipate smaller ERP´s the longer is the maturity T. The picture above and the table inserted in it confirm this.

slower than that rule. 5. INTRODUCING NON-LINEARITY OF V(X)

Notice that accumulated volatility grows with the square root of T, but the premium decreases

Humans seem not only to over-value the pain due to losses in comparison to the satisfaction received from gains, but they also value different gains with different amounts of marginal feelings. That is, a final gain (ST – F0) produces a positive feeling not proportional to that gain, but following a concave function. Symmetrically, for a loss (F0 – ST), there is non-linearity, as tests suggest that investors show here a riskloving attitude instead of the above riskaversion posture for gains.

One simple way to include these two nonlinearities in the above valuations of losses and gains and still maintain the simple mathematical trac-

tability of the model is to substitute, in the integrals (1) above, the two linear valuation functions by two exponential ones14

for ST ≥F0

for ST ≤F0

14- Note, however, that while the Tversky and Kahneman model – power function with exponent 0.88 – assumes a constant relative risk aversion, this exponential function has a constant absolute risk aversion.

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5.1 Case with α = 1 As in the linear simplification, the premium (µ µ - rf) for a certain evaluation horizon T is

The noticeable innovation in the table is the non -zero premium for π = 1 due to the non-linearity that translates the added risk-aversion and the added risk-loving attitudes. It is interesting to notice that this extra premium (not the density) is of the same order of magnitude of the value found initially by Mehra and Prescott which indicates that the empirical ERP found in many

again a function15 of only π and of σ, but the risk-free rf rate now plays also a role, although residual. But the final result is only slightly larger than before and closer to 7% p.a.

countries is more the price of loss aversion than of a diminishing sensitivity. Of course, one would expect that, adding nonlinearity to the large impact due to the multiplier π (loss aversion), the equity premium and the accompanying density should be slightly larger than in the linear case:

15- To integrate the exponentials multiplying the distribution function one can use the Maclaurin expansion:

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In any case and for parameters around the common market and human traditions – σ=20%, 12

CAPM estimate for the return Ri demanded by a certain asset with volatility σi and correlation

-month evaluation period and π = 2.25 – the

ρi with the surrounding market is not much different from the linear case:

5.2 Case with α ≠ 1

more or less risk-aversion/risk-loving. As ex-

With this coefficient α it is possible to enlarge

pected, both the volatility premium (µ µ – rf) and its density grow with increased curvatures (larger α´s) , but that variation is more intense

(α α = 2) or to reduce (α α = 0,5) the curvature of the exponential function for v(x) – both concavity and convexity – and therefore to express

Since Kahneman and Tversky´s empirical finding is a power value function v(x) with an exponent between 0.5 and 1.0, the above table suggests that the CAPM formula could be written

for π = 1 (equal valuation of gains and losses) than for the common “price” of pain π = 2.25.

with a numerical multiplier above 0.32 – for linear case – and below 0.35 – for exponential with α = 1. A good estimate might therefore be 0.34:

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6. CONCLUSION Using the empirical findings of Kahneman and Tversky that suggest that humans prefer certainty to uncertainty – certainty and reflection effects – care more about variations of wealth than of final values of it – isolation effect – and also suffer π times more from losses than enjoy from alternative potential gains – loss aversion effect – then, assuming a Brownian motion description for a volatile asset, the ERP should be around 7% p.a. depending on the level of the following parameters: a) First and above all, the amount of overvaluation of losses above gains – the multiplier π b) Second, the curvature α of the value function v(x) where x is the size of a gain or a loss c) Third, the market parameters of volatility σ and risk-free rate rf (only marginally sensitive). These findings suggest that Mehra and Prescott

and the subsequent other explanations of the ERP (based on traditional utility theory) failed to produce an appropriate order of magnitude because they considered only the non-linearity of the value function v(x) excluding the asymmetry between gains and losses that is the principal determinant of the size of that premium. This Equity Risk Premium could be smaller than the above percentage if humans used longer than 1-year periods to reassess the returns obtained from their investments. However, it seems that tradition among humans and some social routines – as fiscal reporting and performance appraisal of some collective management of savings – induce everyone to stick to that annual frequency. It is likely that normal herd behaviour of investors may alter that over-valuation π of losses according to their state of mind, in particular, reducing that sensitivity after a long period of accumulating gains – and so demanding a lower ERP – and the opposite after a period of witstanding heavy losses.

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REFERENCES

Barberis, N., Ming Huang and Tano Santos, “Prospect T heory and A sset Prices”, The Quarterly Journal of Economics, vol CXVI, issue 1, Feb 2001 Benartzi, S. and Richar d Thaler , “M yopic L oss Aversion and the Equity Prem ium Puzzle”, The Quarterly Journal of Economics”, Feb 1995 Brealey R. A. and Stewar t C. Myer s, “Principles of Corporate Finance”, McGr aw-Hill, 1996 Dimson E., P. Mar sh and M. Staunton, “Equity Prem ium s A round the W orld”, in “Rethinking the Equity Risk Premium” edited by P. Hammond, M. Leibowitz and L. Siegel for the CFA Institute, Dec 2011 Fernández, Pablo, “T he Equity Prem ium in 150 T extbook s”, IESE Business School, University of Navarra, 2009 Ibbotson, Roger G., “T he Equity R isk Prem ium ”, in “Rethinking the Equity Risk Pr emium” edited by P. Hammond, M. Leibowitz and L. Siegel for the CFA Institute, Dec 2011 Kahneman D. and Amos Tver sky, “Prospect T heory: an A nalysis of Decision under Risk ”, Econometrica, vol 47, number 2, March 1979 Mehra, R., and Edwar d C. Pr escott, "The Equity Premium: A Puzzle", Journal of Monetary Economics, March 1985, 15, 145-62. Mehra R., “T he Equity Prem ium : W hy is it a Puzzle?”, pr epar ed for the Financial Analysts Journal, January 2003 Tversky, A. and Daniel Kahneman, “A dvances in Prospect T heory: Cumulative Representation of Uncertainty” , Journal of Risk and Uncertainty, 5:297-323 (1992) Welch, I., “V iews of Financial Econom ists on the Equity Prem ium and on Professional Controversies” , Journal of Business, vol. 73, nº 4 (October), 2000. Welch, I., “A Consensus Estim ate for the Equity Prem ium by A cadem ic Financial Econom ists in December 2007. An update to Welch 2000” , Brown University, January 18, 2008