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1V908
The Equity Risk Premiumand other things
Craig Ansley
November 2009
2V908
Outline
What is the ERP?
Historical values
Estimation
The ERP puzzle
Failure of financial theory
A new model for investment returns
A solution to the ERP puzzle
Solutions to other puzzles in finance
Implications for investment strategy
3V908
What is the Equity Risk Premium?
Based on index returns
Commonly long-term government bonds
Let
RE = return on equities
RB = return on government bonds
Then
ERP = E[RE] – E[RB]
B. Cornell “The Equity Risk Premium” 1999
4V908
Historical ERP
S&P 500 US Treasury 20yrs
ERP
1926-2007 12.3% 5.8% 6.5% 1946-2007 12.7% 6.2% 6.5% 1926-Sep 2009 11.8% 5.9% 5.8% 1946-Sep 2009 12.0% 6.3% 5.7%
Typical forecasts around 4%
5V908
Outline
What is the ERP?
Historical values
Estimation
The ERP puzzle
Failure of financial theory
A new model for investment returns
A solution to the ERP puzzle
Solutions to other puzzles in finance
Implications for investment strategy
6V908
Utility and Risk Aversion
0.0
0.5
1.0
1.5
2.0
2.5
0 5 10 15 20 25 30 35 40
Consumption
Uti
lity
More risk-averse
Less risk-averse
1
1)1(CU
7V908
Consumption Asset-Pricing ModelLucas 1978
= risk aversion c = change in per capita consumption
R = asset return Risk premium for asset = ),()()( RcRc
Maximise utility:
8V908
The Equity Premium PuzzleMehra & Prescott 1985
Sharpe ratio = Asset risk premium
σ(Asset return)
= γ x σ(Δc) x ρ(Δc)
= 4 x 0.01 x 0.2
= 0.008
If equity volatility is 15%, then ERP should be 0.008 x .15 = 0.12%
Observed ERP not explained by economic theory
9V908
Outline
What is the ERP?
Historical values
Estimation
The ERP puzzle
Failure of financial theory
A new model for investment returns
A solution to the ERP puzzle
Solutions to other puzzles in finance
Implications for investment strategy
10V908
Disaster model for economic outputBarro 2005
vt = 0 large probability = large loss small probability
111 )log()log( tttt vuAA
Drift iid N(0,σ2)
Output At evolves as random walk + drift
Disaster model
11V908
What’s a disaster?
Natural disaster
Credit crisis
Wars
Bubbles
Agricultural disaster
12V908
0
2
4
6
8
10
12
14
16
17% 22% 27% 32% 37% 42% 47% 52% 57% 62% 67%
Contraction
Nu
mb
er o
f E
ven
tsCalibrating the model--GDP
Source: Barro, NBER 2005 Based on 60 economic disasters in 35 countries 1900-2000
Probability of disaster = 1.7%
13V908
A new model for asset returnsRiesz (1988), Barro 2005
Return over a given period = Expected return + Normal deviation + Disaster return
Disaster return = 0 large probability = large loss small probability
Predicted ERP close to historical values
14V908
Time varying probability of disasterGabaix 2008
Equity premium puzzle
Excess volatility puzzle
Value-growth puzzle
Corporate bond spread puzzle
Correlations between asset classes close to 1 in bad times
High price of out-of-the money puts
Uncovered interest parity puzzle
If the probability of disaster varies over time, several puzzles in finance are explained:
15V908
Equity premium puzzle
Economic theory predicts ERP of 0.1%(Mehra & Prescott, 1985)
Average ERP since 1880 has been 7%
ERP predicted by Barro’s model 7.1%
16V908
Uncovered interest parity puzzle
Country A has interest rate 3%Country B has interest rate 1%Country A’s currency should depreciate by 2%
But FX rates of high interest rate countries do not trend down!
Carry traders subject to crash risk (Brunnermeier, Nagel & Pedersen, 2008)
Disaster model predictions:For countries with high disaster probabilities High interest rates
Appreciating currencies
Currency crash risk (Farhi & Gabaix, 2009)
17V908
Disaster model
Explicit allowance for unusually bad events
Explains many problems with conventional theory
Can be calibrated from historical data
High ERP is here to stay
18V908
Outline
What is the ERP?
Historical values
Estimation
The ERP puzzle
Failure of financial theory
A new model for investment returns
A solution to the ERP puzzle
Solutions to other puzzles in finance
Implications for investment strategy
19V908
A simple example
Target fund $100 in T years
Contributions Ct at t = 0,1,…,20
Ct set each year by valuing at rate i
1
tT
ttT
t a
FAvC
20V908
Penalty function
j
T
tj
tjtt CdEL
)1(
tL = penalty at time t
tE = expectation at time t d = rate of time preference
tC = contribution at time t = risk aversion
21V908
Power penalty
0
2000
4000
6000
8000
10000
0 1 2 3 4 5 6 7 8 9 10
Contribution
Pen
alty
CL
22V908
Assumptionsfrom Barro (2005)
Expected Return Volatility
Equities 3.7% 1.0%
Bonds 9.6% 14.3%
with Barro’s disaster model
23V908
Equity Return Density
0
0.5
1
1.5
2
2.5
3
3.5
-75% -50% -25% 0% 25% 50% 75%
Return
Den
sity
Disaster No disaster
24V908
Comparison of model allocationsi = 4.0%, d = 4.5%
0%
20%
40%
60%
80%
100%
1.0 1.5 2.0 2.5 3.0 3.5 4.0
Risk Aversion (gamma)
Op
tim
al e
qu
ity
allo
cati
on
Disaster No disaster
25V908
Dynamic Asset Allocation
Conventional: constant asset allocation
Alternative: change in response to performance
Dynamic programming problem
e.g. Dempster et. al., British Actuarial Journal, 2002
26V908
0%
20%
40%
60%
80%
100%
20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
Time Period
Eq
uit
y A
llo
cati
on
Dynamic Asset AllocationQuartiles of simulated strategies γ = 3, i = 4.0%, d = 4.5%
Optimal constant strategy 29% equities
27V908
Advantage of Dynamic Asset Allocation
Optimal penalty with constant strategy 641.7
Optimal penalty with DAA 494.7
If all returns raised by 1.2%, optimal constant strategy penalty drops to 494.7
DAA is worth an increase of 1.2% in returns
28V908
Conclusions
Standard model can’t explain ERP (or other things)
Disaster model solves many puzzles in finance
Historical disaster experience consistent with ERP
Disaster model requires lower equity allocations
DAA outperforms conventional approach
29V908
Effect of Valuation Rate γ = 2.5, d = 4.5%
0
50
100
150
200
250
300
350
400
1.0% 2.0% 3.0% 4.0% 5.0% 6.0% 7.0% 8.0%
Actuarial Valuation Rate
Min
imu
m P
enal
ty
No disaster Disaster
30V908