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The Equity Risk Premiumand other things

Craig Ansley

November 2009

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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

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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

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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%

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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

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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

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Consumption Asset-Pricing ModelLucas 1978

= risk aversion c = change in per capita consumption

R = asset return Risk premium for asset = ),()()( RcRc

Maximise utility:

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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

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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

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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

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What’s a disaster?

Natural disaster

Credit crisis

Wars

Bubbles

Agricultural disaster

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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%

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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

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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:

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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%

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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)

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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

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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

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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

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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

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Power penalty

0

2000

4000

6000

8000

10000

0 1 2 3 4 5 6 7 8 9 10

Contribution

Pen

alty

CL

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Assumptionsfrom Barro (2005)

Expected Return Volatility

Equities 3.7% 1.0%

Bonds 9.6% 14.3%

with Barro’s disaster model

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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

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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

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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

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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

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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

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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

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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

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