Running head: EQUITY PREMIUM PUZZLE HYPOTHESIS
Equity Premium Puzzle Hypothesis
Chris Gilbert Waltzek
Northcentral University
1
EQUITY PREMIUM PUZZLE HYPOTHESIS
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Abstract
This cursory paper presents a proposed solution to an economic conundrum known as the equity
risk premium puzzle first presented by Mehra and Prescott (1985). It is shown that the disparity
between expected returns for stocks and bonds may be unexpectedly high due to the notion that
investors accept a large premium over Treasuries due to the tendency for a small fraction (less
than one half of one percent) of equities to yield enormous capital gains. Therefore, investors
accept higher risk for the miniscule chance of holding shares in the next Berkshire Hathaway,
Microsoft, Apple, or Tesla. Building on the methodology used by Fama and French (2002), the
present value of an equity is determined with a subtle nuances. The data corroborate the
hypothesis that long-term ERP should indeed be ≈ 7.5% beyond that of long-term Treasury
returns, as anticipated by supporting empirical studies.
EQUITY PREMIUM PUZZLE HYPOTHESIS
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Equity Premium Puzzle Hypothesis
Mehra and Prescott (1985) investigate the equity risk premium (ERP), the disparity between
expected returns for stocks and bonds, the unexpectedly high difference or premium in expected
returns of stocks over government bonds. The current consensus among economists indicates
that the premium is statistically abnormal, approximately 7% over the past century (Fama &
French, 2002). Conversely, this paper supports the counter thesis, that stocks require a premium,
because of their increased riskiness over the risk-free rate / government Treasuries, as well as the
desire by investors to hold equities with the potential for enormously large capital gains. Put
simply, investors accept higher risk for the miniscule chance of holding shares in the next
Berkshire Hathaway, Microsoft, Apple, and or Tesla. Building on the methodology of Fama and
French (2002), the investigation begins with a basic demonstration of the present value of a stock
(see Equation 1.1).
Future Amount
Present Value (PV) = (1 + expected return (r))time(t)
1.1.
$100
PV = (1 + 𝑟)𝑡
$100
= (1 + 0.05)10
$100
= (1 + 0.05)10
=
$100
1.63
= $𝟔𝟏
Although the present value takes into account the expected return of the stock over the next 10
years, the model must compensate for the next black swan like increase in price, similar to
EQUITY PREMIUM PUZZLE HYPOTHESIS
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owning shares of Microsoft in 1992. The model is enhanced with a new expression, accounting
for the very rare odds of such a fortuitous event (see Equation 1.2).
Future Payment
Present Value (PV) = (1 + expected return (r))time(t) +
[Probability(x)
PV =
=
=
$100
(1 + 𝑟)𝑡
]
$1000
$100
1.63
(1 + expected return (r))time(t)
+ 𝑃(0.005) [(1 + 𝑟)𝑡 ]
$1000
(1 + 0.05)10
$100
Future Payment (10 times normal)
1.2.
+ 𝑃(0.005) [(1 + 0.05)10 ]
$1000
+ 𝑃(0.005) [(1 + 0.05)10 ]
= $61 + [𝑃(0.005) ∗ $613]
= $61 + [$3.07]
= $64.07
Although the odds of holding a stock with an oversized capital gain (1000% return) is very low,
less than 1 in 200 stocks (0.005) the PV increases by $3.07 or 5%. Thus, the ERP for a nondividend paying stock is a direct result of the probability of an oversized return, despite the
incredibly low probability.
In the next example, the model is further enhanced with two more black swan events, a
10,000% return, as well as a 100,000% (Berkshire Hathaway shares held from IPO to present)
(see Equation 1.3).
PV =
=
$100
(1 + 𝑟)𝑡
$1000
$10000
$100000
+ 𝑃(0.005) [(1 + 𝑟)𝑡 ] + 𝑃(0.0001) [(1 + 𝑟)𝑡 ] + 𝑃(0.00001) [ (1 + 𝑟)𝑡 ]
$100
(1 + 0.05)10
$1000
$10000
1.3.
$100000
+ 𝑃(0.005) [(1 + 0.05)10 ] + 𝑃(0.0001) [(1+0.05)10 ] + 𝑃(0.00001) [(1+0.05)10 ]
= $61 + 𝑃(0.005)[$613] + 𝑃(0.0001)[$6130] + 𝑃(0.00001)[$61300]
= $61 + [$3.07] + [$0.61] + [$0.61]
EQUITY PREMIUM PUZZLE HYPOTHESIS
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= $61 + [$4.30]
= $𝟔𝟓. 𝟑𝟎
Clearly, the PV increases by $4.30 or 6.6%. Thus, the ERP is a direct result of the very low
probabilities of oversized returns, despite the incredibly low probability, less than one percent
𝑃(0.001; 0.00001)
Finally, the sigma symbol accounts for the potential for an infinite return, albeit exponentially
non-probabilistic (see Equation 1.4).
PV =
$100
(1 + 𝑟)𝑡
$1000
$10000
∞
[∑−∞
𝑖=0.1 𝑃(0.000001)𝑖 ∑𝑖=10 [
=
$100000
+ 𝑃(0.005) [(1 + 𝑟)𝑡 ] + 𝑃(0.0001) [(1 + 𝑟)𝑡 ] + 𝑃(0.00001) [ (1 + 𝑟)𝑡 ] +
$100
(1 + 0.05)10
$1000000𝑖
(1+𝑟)𝑡
1.4.
]]
$1000
$10000
$100000
+ 𝑃(0.005) [(1 + 0.05)10 ] + 𝑃(0.0001) [(1+0.05)10 ] + 𝑃(0.00001) [(1+0.05)10 ] +
∞
[∑−∞
𝑖=0.1 𝑃(0.000001)𝑖 ∑𝑖=10 [
$1000000𝑖
(1+𝑟)𝑡
]]
= $61 + 𝑃(0.005)[$613] + 𝑃(0.0001)[$6130] + 𝑃(0.00001)[$61300]
= $61 + [$3.07] + [$0.61] + [$0.61] + [≈ $0.61]
= $61 + [$4.91]
= $𝟔𝟓. 𝟗𝟏
Clearly, the PV and by proxy the ERP increases by $4.91 or 7.5% beyond that of the long-term
Treasury, representing a close match with the empirical data (ERP ≈ 7% over the past century).
Discussion
This cursory paper presents a proposed solution to the equity risk premium puzzle first
presented by Mehra and Prescott (1985). It is shown that the disparity between expected returns
for stocks and bonds may be unexpectedly high due to the notion that investors accept a large
premium over Treasuries due to the tendency for a small fraction (less than one half of one
EQUITY PREMIUM PUZZLE HYPOTHESIS
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percent) of equities to yield enormous capital gains. Therefore, investors accept higher risk for
the miniscule chance of holding shares in the next Berkshire Hathaway, Microsoft, Apple, or
Tesla. Building on the methodology used by Fama and French (2002), the investigation includes
the present value of equities with a subtle nuance. The data corroborate the hypothesis that longterm ERP should indeed be ≈ 7.5% beyond that of long-term Treasury returns, as anticipated by
supporting empirical studies.
EQUITY PREMIUM PUZZLE HYPOTHESIS
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References
Fama, E. F. & French, K. R. (2002). The equity premium. The Journal of Finance, 57: 637–659.
doi: 10.1111/1540-6261.00437
Mehra, R., & Prescott, E. C. (1985). The equity premium. Journal of Monetary Economics,
15(2), 145–161