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
50
45
2000
40
1929
Price-Earnings Ratio
35
30
1966
1901
25
20
15
10
5
0
1860
1880
1900
1920
1940
Year
1960
1980
2000
2020
Background
z
Academic studies have found:
•
stock returns predictable by such variables as
Dividend yield, B/M, interest rates etc
•
z
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?
z
NO!
We show that:
z
Return predictability that is of first order
importance to long run investors will be:
•
•
•
z
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
dP
= μdt + σ P dzP
P
dμ = κ ( μ − μ )dt + σ μ dzμ
Mean reverting expected return Parameters for simulations chosen so that:
z
Unconditional distribution of μ is fixed at
•
N ( μ = 9%,ν = 4%)
varies a lot: 1 sigma interval (5% to 14%)
z
Consistent with a 14% annual stock return volatility
z
Risk free rate is constant at 3%, implying 6% equity premium.
z
Nine scenarios from the combination of
κ = 0.02,0.10,0.5, and ρ = corr (dz P , dzu ) = 0.0,−0.5,−0.9
Strategy
Use this (simulation) model to show this amount
of 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
z
z
dD
= gdt + σ D dz D
D
Expected Rate of Return:
Dividend Process:
⎡ Ddt + dV ⎤
= μdt
E⎢
⎥
V
⎣
⎦
z
d μ = κ ( μ − μ ) dt + σ μ dz μ
Differential equation allows us to solve for price as a
function of dividend growth rate:
P(D,μ) = Pv(μ)
Dividend yields can vary a lot as μ changes even though
dividend growth assumed constant (g = 1.85%)
10%
9%
mubar-sig
mubar
mubar+sig
8%
7%
6%
5%
c
4%
3%
2%
1%
0%
1
2
3
4
5
6
7
8
9
z
Under our scenarios
• Prices vary a lot
• Expected returns vary a lot (5%-13%)
z
Are we likely to detect this predictability by
regressing returns on (or proxies for )?
R(t , t + τ ) = a + bμt + ε t ,t +τ
Distribution of corrected t-ratios on the predictor
using 70 years of simulated monthly returns
3
25%
Median
75%
2.5
2
1.5
1
0.5
0
1
2
3
4
5
Scenario
6
7
8
9
R2 (%) in an Annual Return Predictive
Regression (70-years simulated returns)
14
25%
12
Median
75%
10
8
6
4
2
0
1
2
3
4
5
6
7
8
9
R2 as a function of horizon for different
values of κ and ρ
0.8
k =0.02,r h0=-0.9
k =0.10,r h0=-0.9
0.7
k =0.50,r h0=-0.9
k =0.02,r h0=-0.5
k =0.10,r h0=-0.5
k =0.50,r h0=-0.5
0.6
k =0.02,r h0=0.0
k =0.10,r h0=0.0
k =0.50,r h0=0.0
0.5
0.4
0.3
0.2
0.1
0.0
1
2
3
4
5
6
7
8
9
10
11 12
Years
13 14 15
16 17
18 19 20
z
Short run predictability is hard to detect and
measure.
z
Would it be valuable if we could detect it?
e.g observe Economic Value of Market Timing
z
z
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)
Scenario (vi)
ρ=-0.9, κ=0.1
1.6
1.5
CEWR
1.4
1.3
1.2
1.1
1
0.5
-0.9
rho
0.1
-0.5
0
0.02
kappa
Value of (optimal) dynamic strategy relative
to myopic strategy: CEW(O)/CEW(M) (T=20,
σP=0.14, σμ=0.04, μ = 9%=mubar)
Scenario (vi)
ρ=-0.9, κ=0.1
1.60
1.40
1.30
1.20
1.10
1.00
-0.9
rho
0.5
-0.5
0.1
0
0.02
kappa
CEW Ratio
1.50
Value of Market Timing (CEWRou) for Investors
with 20-year horizon
1 year R2 tells us nothing about value of timing
1.6
κ=0.5
ρ=-0.9
1.5
R2=4.8%
CEW
R
ou
1.4
κ=0.1
ρ=-0.9
κ=0.5
ρ=-0.5
1.3
κ=0.02
ρ=-0.9
R2=7.4%
1.2
κ=0.5
ρ=0.0
1.1
1.0
0.04
0.045
0.05
0.055
0.06
One-Year R
0.065
2
0.07
0.075
0.08
z
Market timing valuable if we could observe .
z
But in practice we can only rely on proxies for (dividend yields) and badly estimated regression
coefficients.
z
A better approach is to rely on direct estimates of
that do not rely on regression estimates
A Forward-Looking Method:
DDM Model
z
DCF approach
∞
Pt = ∑
τ =1
Et [ Dt +τ ]
(1 + k )
τ
t
• 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 dμ = κ (μ − μ )dt + σ dz
μ
μ
DDM approach to estimating z
If we know the parameters of the Vasicek interest rate model
we can infer the short rate, r, from the long rate, l.
z
In same way, if we know the parameters of
dμ = κ ( μ − μ )dt + σ μ dzμ
z
z
we can infer from kt
Iterative procedure for inferring and updating parameters
Also estimate model in which dividend growth rate follows OU process:
Four DDM k series
• 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
Estimates of μ from the WA DDM k series
(1973.Q1 to 2002.Q2)
30%
25%
k4
μ4,1
μ4,2
20%
15%
10%
5%
0%
1973
-5%
1976
1979
1982
1985
1988
1991
1994
1997
2000
200303
Estimated μ process parameters
z
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
z
Regression:
⎡1.0 − e −0.25κ ⎤ i , 2
R(t , t + 0.25) = a0 + a1 ⎢
⎥ μt + ε t
κ
⎣
⎦
i = 1 (A & B), 2 (IL), 3 (BGI),4 (WA)
z
Theoretical value: a1=1.0
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
z
RRA = 5
Unconditional Strategy:
z
Optimal Market Timing:
z
x* =
x* =
μ −r
γσ
t
2
P
z
z
Risky asset: S&P500
Riskless asset: 30 day T-Bill
μ −r
γσ P2
plus hedging terms
Proportion of Wealth Invested in Stocks
1.2
Unconditional (based on full sample mean)
Unconditional (based on gradually updated sample mean)
Optimal
1.0
0.8
0.6
0.4
0.2
0.0
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
Wealth under optimal and unconditional
Strategies for a 20-year horizon constrained
investor using μ4,2 (RRA = 5, 1973.Q2 -1993.Q1)
12
10
uncondtional
optimal
8
6
4
2
0
Sep-73
Sep-75
Sep-77
Sep-79
Sep-81
Sep-83
Sep-85
Sep-87
Sep-89
Sep-91
Wealth under optimal and unconditional
Strategies for a 9-year horizon constrained
investor using μ4,2 (RRA = 5, 1993.Q2 - 2002.Q2)
2.5
2.0
Unconditional
Optimal
1.5
1.0
0.5
0.0
Jun-93
Jun-95
Jun-97
Jun-99
Jun-01
Wealth under optimal and unconditional
Strategies for a 29-year horizon constrained
investor using μ4,2 (RRA = 5, 1973.Q2 – 2002.Q2)
25
20
uncondtional
optimal
15
10
5
0
Jun-73
Jun-76
Jun-79
Jun-82
Jun-85
Jun-88
Jun-91
Jun-94
Jun-97
Jun-00
Conclusion
z
Time-varying expected returns economically
important, even though
•
Hard to detect, measure
z
Substantial benefit from the optimal strategy
z
DDM discount rates are a useful input for long
run investor.
z
Long run investors (pension, insurance) should
hedge against changes in investment
opportunities.