Persistence, Predictability and Portfolio Planning M.J. Brennan - - PowerPoint PPT Presentation

Persistence, Predictability and Portfolio Planning

M.J. Brennan – UCLA Yihong Xia – University of Pennsylvania

Stocks are sometimes cheap and sometimes dear. Important for long run investors? Price/10 year Average Earnings 1880-2003

5 10 15 20 25 30 35 40 45 50

1860 1880 1900 1920 1940 1960 1980 2000 2020

Year Price-Earnings Ratio

1901 1929 1966 2000

Background

Academic studies have found:

• stock returns predictable by such variables as

Dividend yield, B/M, interest rates etc

• But virtually no out of sample return predictability

Does this mean that investors should ignore time

variation in returns and behave as though expected returns constant?

NO!

We show that:

Return predictability that is of first order

importance to long run investors will be:

• associated with large price variation.
• hard to detect using standard regression framework even

when a perfect signal is available

• hard to estimate for portfolio planning purposes

A promising alternative to popular academic

predictors is forward looking forecasts of long run returns from DDM

• Convert long run forecasts to short run

A Simple Model of Return Predictability

μ μ

σ μ μ κ μ σ μ dz dt d dz dt P dP

P P

+ − = + = ) (

Mean reverting expected return

Parameters for simulations chosen so that:

Unconditional distribution of μ is fixed at

• varies a lot: 1 sigma interval (5% to 14%)

Consistent with a 14% annual stock return volatility Risk free rate is constant at 3%, implying 6% equity premium. Nine scenarios from the combination of

%) 4 %, 9 ( = = ν μ N

( )

9 . , 5 . , . , and , 5 . , 10 . , 02 . − − = = =

u P dz

dz corr ρ κ

Strategy

Use this (simulation) model to show this amount

• f expected return variability
• Implies big variability in prices
• Little short run return predictability an dis hard to

detect

• Possibly strong long run return predictability

Later we will show:

• The data consistent with this amount of predictability
• How to exploit it

P/D Ratios implied by the scenarios

Dividend Process: Expected Rate of Return: Differential equation allows us to solve for price as a

function of dividend growth rate: P(D,μ) = Pv(μ)

dD D gdt dz

D D

= + σ

E Ddt dV V dt + ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = μ

μ μ

σ μ μ κ μ dz dt d + − = ) (

Dividend yields can vary a lot as μ changes even though dividend growth assumed constant (g = 1.85%)

0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 1 2 3 4 5 6 7 8 9

mubar-sig mubar mubar+sig

c

Under our scenarios

• Prices vary a lot
• Expected returns vary a lot (5%-13%)

Are we likely to detect this predictability by

regressing returns on (or proxies for )?

R t t a b

t t t

( , )

,

+ = + +

+

τ μ ε

τ

Distribution of corrected t-ratios on the predictor using 70 years of simulated monthly returns

0.5 1 1.5 2 2.5 3

1 2 3 4 5 6 7 8 9

Scenario

25% Median 75%

R2 (%) in an Annual Return Predictive Regression (70-years simulated returns)

2 4 6 8 10 12 14

1 2 3 4 5 6 7 8 9

25% Median 75%

R2 as a function of horizon for different values of κ and ρ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Years

k=0.02,rh0=-0.9 k=0.10,rh0=-0.9 k=0.50,rh0=-0.9 k=0.02,rh0=-0.5 k=0.10,rh0=-0.5 k=0.50,rh0=-0.5 k=0.02,rh0=0.0 k=0.10,rh0=0.0 k=0.50,rh0=0.0

Short run predictability is hard to detect and

measure.

Would it be valuable if we could detect it?

e.g observe

Economic Value of Market Timing

Investor is assumed to maximize expected

CRRA utility (RRA = 5) over terminal wealth.

Measure economic value using certainty

equivalent wealth ratio between different strategies (CEWR)

• Optimal dynamic strategy
• Myopic strategy
• Unconditional strategy

Value of (optimal) dynamic strategy relative to unconditional strategy: CEW(O)/CEW(U) (T=20, σP=0.14, σμ=0.04, μ = 9%=mubar)

• 0.9
• 0.5

0.02 0.1 0.5 1 1.1 1.2 1.3 1.4 1.5 1.6

C E W R

rho kappa

Scenario (vi) ρ=-0.9, κ=0.1

Value of (optimal) dynamic strategy relative to myopic strategy: CEW(O)/CEW(M) (T=20, σP=0.14, σμ=0.04, μ = 9%=mubar)

• 0.5
• 0.9

0.02 0.1 0.5 1.00 1.10 1.20 1.30 1.40 1.50 1.60

CEW Ratio rho kappa

Scenario (vi) ρ=-0.9, κ=0.1

Value of Market Timing (CEWRou) for Investors with 20-year horizon

1 year R2 tells us nothing about value of timing

1.0 1.1 1.2 1.3 1.4 1.5 1.6

0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075 0.08

One-Year R2

C E W R

• u

κ=0.5 ρ=-0.9 κ=0.5 ρ=-0.5 κ=0.5 ρ=0.0 R2=4.8% R2=7.4% κ=0.1 ρ=-0.9 κ=0.02 ρ=-0.9

Market timing valuable if we could observe . But in practice we can only rely on proxies for

(dividend yields) and badly estimated regression coefficients.

A better approach is to rely on direct estimates of

that do not rely on regression estimates

A Forward-Looking Method: DDM Model

DCF approach

• Forecasts of future dividends provided by analysts

yield current estimates of long run expected returns on stocks, kt

• Problem: How to map k into short run expected

rate of return μ

[ ] ( )

P E D k

t t t t

= +

+ = ∞

∑

τ τ τ

1

1

μ μ

σ μ μ κ μ dz dt d + − = ) (

DDM approach to estimating

• If we know the parameters of the Vasicek interest rate model

we can infer the short rate, r, from the long rate, l.

• In same way, if we know the parameters of

we can infer from kt

• Iterative procedure for inferring and updating parameters
• Also estimate model in which dividend growth rate follows O-

U process:

d dt dz

μ κ μ μ σ μ

μ

= − +

( )

• Arnott & Bernstein (A&B) (2002), and Ilmanen (IL)

(2003)

• 1950.1 - 2002.2 quarterly
• real, ex-post (back-casted)
• based on smoothed GDP growth rate
• Barclays Global Investors (BGI) and Wilshire

Associates (WA)

• 1973.1 to 2002.2 monthly – converted to quarterly
• nominal, ex-ante (real time)
• using I/B/E/S consensus estimates

Four DDM k series

Estimates of μ from the WA DDM k series (1973.Q1 to 2002.Q2)

• 5%

0% 5% 10% 15% 20% 25% 30%

1973 1976 1979 1982 1985 1988 1991 1994 1997 2000 200303

k4 μ4,1 μ4,2

Estimated μ process parameters

Similar across 4 models and Close to scenario (vi)

Real Nominal Scenario (A&B) (IL) (BGI) (WA) (vi) κμ 0.085 (0.083) 0.122 (0.115) 0.091 (0.085) 0.122 (0.095) 0.1 σμ 0.017 (0.017) 0.0196 (0.022) 0.024 (0.021) 0.034 (0.027) 0.018 νμ 0.042 (0.042) 0.040 (0.047) 0.056 (0.052) 0.068 (0.061) 0.04 ρμP

• 0.98

(-0.98)

• 0.88

(-0.89)

• 0.81

(-0.66)

• 0.68

(-0.71)

• 0.9

Statistical Significance: In-Sample Quarterly Predictive Regressions

Regression: Theoretical value: a1=1.0

(WA) 4 , (BGI) 3 , (IL) 2 , B) & (A 1 . 1 ) 25 . , (

2 , 25 . 1

= + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − + = +

−

i e a a t t R

t i t

ε μ κ

κ

In-Sample Quarterly Predictive Regressions Results

Predictor a_1 H0: a_1=1 R2 (%) N μ1,2 0.874 [1.60] 1.43 209 (1950.Q2 – 2002. Q2) μ2,2 0.701 [1.27] 1.12 209 (1950.Q2 - 2002. Q2) μ3,2 1.026 [1.69] 2.89 117 (1973.Q2 to 2002.Q2) μ4,2 0.924 [1.66] 2.74 117 (1973.Q2 to 2002.Q2)

Economic Importance: Simulation of Market Timing and Unconditional Strategies

RRA = 5 Unconditional Strategy: Optimal Market Timing: Risky asset: S&P500 Riskless asset: 30 day T-Bill

x r

P * =

− μ γσ

2

x r plus hedging terms

t P * =

− μ γσ

2

Proportion of Wealth Invested in Stocks

0.0 0.2 0.4 0.6 0.8 1.0 1.2 Mar-73 Mar-75 Mar-77 Mar-79 Mar-81 Mar-83 Mar-85 Mar-87 Mar-89 Mar-91 Mar-93 Mar-95 Mar-97 Mar-99 Mar-01

Unconditional (based on full sample mean) Unconditional (based on gradually updated sample mean) Optimal

Wealth under optimal and unconditional Strategies for a 20-year horizon constrained investor using μ4,2 (RRA = 5, 1973.Q2 -1993.Q1)

2 4 6 8 10 12

Sep-73 Sep-75 Sep-77 Sep-79 Sep-81 Sep-83 Sep-85 Sep-87 Sep-89 Sep-91

uncondtional

• ptimal

Wealth under optimal and unconditional Strategies for a 9-year horizon constrained investor using μ4,2 (RRA = 5, 1993.Q2 - 2002.Q2)

0.0 0.5 1.0 1.5 2.0 2.5 Jun-93 Jun-95 Jun-97 Jun-99 Jun-01

Unconditional Optimal

Wealth under optimal and unconditional Strategies for a 29-year horizon constrained investor using μ4,2 (RRA = 5, 1973.Q2 – 2002.Q2)

5 10 15 20 25

Jun-73 Jun-76 Jun-79 Jun-82 Jun-85 Jun-88 Jun-91 Jun-94 Jun-97 Jun-00

uncondtional

• ptimal

Conclusion

Time-varying expected returns economically

important, even though

• Hard to detect, measure

Substantial benefit from the optimal strategy

DDM discount rates are a useful input for long

run investor.

Long run investors (pension, insurance) should

hedge against changes in investment

• pportunities.