1. V908 1 The Equity Risk Premium and other things Craig Ansley November 2009

2. V908 2 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

3. V908 3 What is the Equity Risk Premium?  Based on index returns  Commonly long-term government bonds B. Cornell “The Equity Risk Premium” 1999

4. V908 4 Historical ERP Typical forecasts around 4%

5. V908 5 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

6. V908 6 Utility and Risk Aversion More risk- averse Less risk- averse

7. V908 7 Consumption Asset-Pricing Model Lucas 1978 Maximise utility:

8. V908 8 The Equity Premium Puzzle Mehra & 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

9. V908 9 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

10. V908 10 Disaster model for economic output Barro 2005 v t = 0 large probability = large loss small probability Drift iid N(0,σ 2 ) Output A t evolves as random walk + drift Disaster model

11. V908 11 What’s a disaster?  Natural disaster  Credit crisis  Wars  Bubbles  Agricultural disaster

12. V908 12 Calibrating the model--GDP Source: Barro, NBER 2005 Based on 60 economic disasters in 35 countries 1900-2000 Probability of disaster = 1.7%

13. V908 13 A new model for asset returns Riesz (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

14. V908 14 Time varying probability of disaster Gabaix 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:

15. V908 15 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%

16. V908 16 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)

17. V908 17 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

18. V908 18 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

19. V908 19 A simple example  Target fund $100 in T years  Contributions C t at t = 0,1,…,20  C t set each year by valuing at rate i

20. V908 20 Penalty function

21. V908 21 Power penalty

22. V908 22 Assumptions from Barro (2005) with Barro’s disaster model

23. V908 23 Equity Return Density

24. V908 24 Comparison of model allocations i = 4.0%, d = 4.5%

25. V908 25 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

26. V908 26 Dynamic Asset Allocation Quartiles of simulated strategies γ = 3, i = 4.0%, d = 4.5% Optimal constant strategy 29% equities

27. V908 27 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

28. V908 28 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

29. V908 29 Effect of Valuation Rate γ = 2.5, d = 4.5%

30. V908 30