44 : CADERNOS DO MERCADO DE VALORES MOBILIÁRIOS
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.
45 : CADERNOS DO MERCADO DE VALORES MOBILIÁRIOS
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,
EQUITY RISK PREMIUM ..: 45
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.
46 : CADERNOS DO MERCADO DE VALORES MOBILIÁRIOS
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
47 : CADERNOS DO MERCADO DE VALORES MOBILIÁRIOS
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-
EQUITY RISK PREMIUM ..: 47
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
48 : CADERNOS DO MERCADO DE VALORES MOBILIÁRIOS
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.
49 : CADERNOS DO MERCADO DE VALORES MOBILIÁRIOS
Calling
EQUITY RISK PREMIUM ..: 49
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.
50 : CADERNOS DO MERCADO DE VALORES MOBILIÁRIOS
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.
EQUITY RISK PREMIUM ..: 51
51 : CADERNOS DO MERCADO DE VALORES MOBILIÁRIOS
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.
52 : CADERNOS DO MERCADO DE VALORES MOBILIÁRIOS
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:
53 : CADERNOS DO MERCADO DE VALORES MOBILIÁRIOS
EQUITY RISK PREMIUM ..: 53
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:
54 : CADERNOS DO MERCADO DE VALORES MOBILIÁRIOS
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.
55 : CADERNOS DO MERCADO DE VALORES MOBILIÁRIOS
EQUITY RISK PREMIUM ..: 55
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