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Investing with a Stock Valuation Model
Zhiwu Chen, Yale University
Ming Dong, Ph.D. candidate, OSU
Purpose
Models:
The stock valuation model developed by Bakshi & Chen (1998)
and extended by Dong (1998)
The residual-income model implemented in Lee-Myers-
Swaminathan (1997)
To compare their performance to traditional stock-selection
measures: book/market, P/E, momentum, size, and so on
Motivation: why not expected-return models?
The CAPM, APT and other multi-factor models all focus on EXPECTED FUTURE RETURNs
Stock-Selection Idea: if the actual expected return on IBM is higher than its deserved expected return, then IBM is a buy (hence, Jensen’s Aplha)
But, what is IBM’s actual expected 1-yr-forward return today? ----- You cannot observe it!
Conclusion: you cannot really apply such expected-return models.
Motivation: why stock-valuation models?
There is always a market price for each stock !
Stock-Selection Idea: if IBM’s market price is lower than its model price (fair value), then IBM is a buy (hence, undervalued stocks)
Conclusion: stock valuation modeling is the way to go.
But, is there a “good” equity-valuation model?
Motivation: existing stock valuation models
Variants of the Gordon model: too many unrealistic assumptions (e.g., a constant and flat term structure, constant dividend growth forever)
Multi-stage dividend/earnings/cashflow discount models:
No structural parameterization of the firm’s business
No attention paid to how the stock has historically been valued by market
Fair values determined by these models are too often below market price.
The Bakshi-Chen-Dong (BCD) Model
Fundamental Variables: current EPS, expected future EPS, and 30-yr bond yield
Firm-specific parameters: EPS growth volatility
Long-run EPS growth rate
Duration of business-growth cycle
Systematic or beta risk of the firm
Correlation between the firm's EPS and the interest-rate environment
30-yr Treasury yield’s parameters: Its long-run level
Interest-rate volatility
Duration of interest-rate cycle
Comparison
• The BCD Model
– Detailed parameterization of EPS processes and interest-rate processParameters to be estimated from past data
– Closed-form stock valuation formula
– Past data are used to estimate parametersSo, valuation reflects both past valuation standard for the stock and the stochastic discounting of future prospects
• The Residual-Earnings Model (e.g., Lee, Meyer and Swaminathan (1998))
– Two parameters: beta and dividend-payout ratio
– No closed-form valuation formula. Requires ad hoc approximation of the stock’s future price at end of forecasting horizon
– Valuation is independent of past valuation standard for the stock
Data
I/B/E/S, CRSP, and Compustat
Future EPS forecasts: consensus analyst estimates
Period covered: Jan. 1979 - Dec. 1996
Stock universe: about 2500 U.S. stocks (mostly large cap)
What Constitutes a Good Stock-Selection Measure?
Mean-reverting, so that if too low, you can buy the stock, counting on the measure to go back to its norm.
Not too persistent, e.g., if book/market ratio is too persistent, you will not want to buy a stock just because it has a high B/M ratio. You would like fast mean-reversion
High predictive power of future stock performance
Behavior of Book/Market Ratio over Time
• This figure shows the average B/M ratio path for each quartile obtained by sorting all stocks according to their B/M ratios as of January 1990.
Average B/M by Quartile
0
1
2
7901
7912
8011
8110
8209
8308
8407
8506
8605
8704
8803
8902
9001
9012
9111
9210
9309
9408
9507
9606
Date
B/M
Q1 (low )Q2Q3Q4(high)
Behavior of LMS Value/Price over Time
• This figure shows the average Lee-Myers-Swaminathan V/P ratio path for each quartile obtained by sorting all stocks according to their V/P as of January 1990.
Part A: Average V/P Ratio by Quartile
0
1
2
7902
8001
8012
8111
8210
8309
8408
8507
8606
8705
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9002
9101
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9211
9310
9409
9508
9607
Date
V/P
Q1(low)
Q2
Q3
Q4(high)
Part B:V/P Autocorrelation for the Lowest Quartile
-0.5
0
0.5
1
1 5 9
13
17
21
25
29
33
37
41
45
49
53
57
Number of Months Lagged
Aut
ocor
rela
tion
Behavior of E/P Ratio
• This figure shows the average E/P ratio path for each quartile obtained by sorting all stocks according to their E/P ratios as of January 1990.
• You would like to see the qartiles crossing each other over time. Yes, they do to some extent.
Part A: Average E/P by Quartile
-0.1
-0.05
0
0.05
0.1
0.15
0.2
7901
8002
8103
8204
8305
8406
8507
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8709
8810
8911
9012
9201
9302
9403
9504
9605
Date
E/P
Rat
io
Q1(low )Q2Q3Q4(high)
BCD Model Mispricing
Step 1: use past 2-yr data to estimate model parameters for the stock
Step 2: use current EPS, 1-yr-forward EPS forecast and 30-yr yield, plus the estimated parameters, to compute the stock’s current model price (out of sample)
Mispricing = [market price - model price] / model price
Thus, a negative mispricing means an undervalued stock, and so on.
Behavior of BCD Model Mispricing
• This figure shows the average BCD Model mispricing path, for each quartile obtained by sorting all stocks according to their mispricing levels as of January 1990.
• The quartiles switch from over- to undervalued, and vice versa, every few years!
Figure 2: Reversals of Mispricing Across Quartiles
-25
-15
-5
5
15
25
35
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01
79
11
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09
81
07
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01
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11
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96
07
Date
Mis
pri
cin
g (
%)
Q1 (undervalued)Q2Q3Q4 (overvalued)
Persistence of BCD Model Mispricing
Part A: Mispricing Autocorrelation for the Most Undervalued Quartile
-0.6
-0.2
0.2
0.6
1
1 5 9 13 17 21 25 29 33 37 41 45 49 53 57
Number of Months Lagged
Au
toco
rre
lati
on
Part B: Distribution of Mispricing Mean-Reversion Time Full Sample
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1
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4
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6
7
8
9
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3 5 7 9 11
13
15
17
19
21
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25
27
29
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33
35
37
39
41
43
Mean-Reversion Time in Months
Perc
ent
of S
tocks
A Small Summary
BCD Model mispricing is the least persistent over time and mean-reverting the fastest
It takes about 1.5 years for a group of stocks to go from most over- to most underpriced, or the reverse
P/E ratio is the second least persistent.
High P/E stocks do not always have the highest P/E.
B/M and V/P are the most persistent. Stocks with the highest B/M seem to be always so. Low B/M
stocks seem to always have low B/M.
Try to Understand the Measures Again
Panel A: Mispricing portfolios (based on Misp)
MP1 MP2 MP3 MP4 MP5 All StocksMisp (%) -19.63 -4.96 2.58 10.59 30.67 3.86
V/P 1.00 1.00 0.96 0.90 0.78 0.93ME ($Millions) 1118.6 1703.9 1975.4 1966.0 1450.8 1643.3
B/M 0.89 0.81 0.75 0.71 0.69 0.77Ret-6 (%) -7.51 3.03 9.26 15.61 27.86 9.65Ret+1 (%) 2.04 1.83 1.53 1.31 1.18 1.67Ret+6 (%) 9.21 10.20 9.44 8.96 10.12 9.59
Beta 1.25 1.05 1.02 1.05 1.22 1.12
Panel B: V/P portfolios
VP1 VP2 VP3 VP4 VP5 All StocksV/P 0.41 0.69 0.89 1.11 1.54 0.93
Misp (%) 9.92 5.78 3.11 1.49 -0.97 3.86ME ($Millions) 1189.4 1841.8 2187.1 1958.2 1343.8 1643.3
B/M 0.58 0.61 0.70 0.84 1.03 0.77Ret-6 (%) 15.74 11.31 9.21 7.74 5.18 9.65Ret+1 (%) 1.33 1.27 1.50 1.59 1.87 1.67Ret+6 (%) 9.10 8.66 9.16 9.48 10.60 9.59
Beta 1.50 1.31 1.14 0.93 0.70 1.12
Try to Understand the Measures One More Time
Panel E: Momentum portfolios (based on Ret-6)
MO1 MO2 MO3 MO4 MO5 All StocksRet-6 (%) -18.79 -1.95 7.66 18.00 43.32 9.65Misp (%) -8.92 -1.41 3.24 8.10 18.26 3.86
V/P 0.93 0.98 0.97 0.92 0.82 0.93ME ($Millions) 1020.9 1681.4 1975.6 2084.1 1452.8 1643.3
B/M 0.94 0.82 0.77 0.71 0.60 0.77Ret+1 (%) 1.51 1.56 1.52 1.44 1.86 1.67Ret+6 (%) 7.64 9.02 9.36 9.70 12.22 9.59
Beta 1.25 1.06 1.02 1.04 1.21 1.12
Panel D: B/M portfolios
BM1 BM2 BM3 BM4 BM5 All StocksB/M 0.25 0.45 0.66 0.89 1.61 0.77
Misp (%) 9.86 4.52 2.89 1.72 0.30 3.86V/P 0.67 0.83 0.97 1.09 1.11 0.93
ME ($Millions) 2357.1 1924.9 1512.5 1386.9 1036.3 1643.3Ret-6 (%) 19.42 12.48 9.01 6.28 1.11 9.65Ret+1 (%) 1.52 1.48 1.37 1.56 1.95 1.67Ret+6 (%) 9.41 9.38 8.91 9.39 10.84 9.59
Beta 1.29 1.21 1.10 0.97 1.02 1.12
Predictive Power for Future Returns
From the regression tables,
BCD Model Mispricing has the highest predictive power (for future 1-month, 6-month and 12-month returns)
Momentum comes second (defined on past 6-month or 12-month returns)
Size is the third most significant (the smaller the firm, the higher the future return)
Last comes B/M & V/P
Regressions of 1-month-forward Stock Returns on predictive variables
No. Intercept Misp V/P Size B/M Ret-6 Ret-12 Adj-R2 No.Obs.
1 2.404(4.82)
-0.029(-8.97)
-0.142(-2.79)
0.130(1.16)
0.021(5.91)
0.051 216
2 2.357(4.62)
-0.138(-2.69)
0.162(1.42)
0.009(2.48)
0.042 216
3 2.475(4.92)
-0.031(-9.17)
-0.151(-2.96)
0.275(2.53)
0.019(7.99)
0.054 216
4 2.485(4.81)
-0.152(-2.96)
0.292(2.68)
0.012(4.90)
0.044 216
9 2.278(4.78)
-0.029(-7.71)
0.211(2.21)
-0.126(-2.62)
0.175(1.72)
0.018(7.77)
0.059 215
10 2.356(4.81)
0.319(3.45)
-0.135(-2.79)
0.157(1.51)
0.012(4.84)
0.048 215
11 1.629(5.29)
0.291(2.49)
0.010 215
Do they perform differently across months: Month-of-the-Year Effect
Month Intercept Misp Size B/M Ret-12 Adj-R2 No.Obs
January 8.961(5.91)
-0.062(-6.14)
-0.811(-9.16)
0.440(1.89)
0.011(1.22)
0.076 18
February 4.229(1.82)
-0.034(-2.03)
-0.208(-1.07)
0.544(1.11)
0.019(2.25)
0.065 18
March 3.727(2.61)
-0.026(-2.44)
-0.357(-2.69)
0.580(1.62)
0.022(3.15)
0.050 18
April 2.571(1.46)
-0.019(-1.84)
-0.170(-0.77)
0.301(1.82)
0.021(2.93)
0.049 18
May 3.792(2.69)
-0.039(-3.44)
-0.317(-1.88)
0.249(0.73)
0.013(2.23)
0.044 18
June 2.231(1.91)
-0.017(-1.73)
-0.060(-0.49)
0.564(1.59)
0.022(2.28)
0.046 18
July 1.389(0.98)
-0.029(-2.33)
-0.083(-0.50)
0.160(0.44)
0.023(3.50)
0.055 18
August 1.980(0.60)
-0.048(-3.69)
0.125(0.61)
0.101(0.22)
0.012(1.62)
0.060 18
September 2.042(1.41)
-0.023(-3.02)
-0.221(-1.65)
0.042(0.09)
0.008(0.69)
0.057 18
October -0.417(-0.26)
-0.013(-1.13)
0.092(0.68)
0.163(0.47)
0.032(4.16)
0.046 18
November -0.036(-0.01)
-0.031(-2.43)
-0.067(-0.25)
-0.019(-0.04)
0.022(2.30)
0.062 18
December 0.226(0.18)
-0.028(-3.21)
0.127(0.94)
0.173(0.56)
0.024(2.64)
0.041 18
Forming 2-dimensional Portfolios
Take mispricing - size quintile portfolios as an example
Step 1: for each month, sort all stocks into 5 quintiles according to their Mispricing levels. Independently, sort all stocks into 5 firm-size quintiles.
Step 3: intersections of the 5 Mispricing and 5 size quintiles result in 25 portfolios, for each month.
Step 3: average monthly return and volatility are then calculated for each Mispricing-size sorted portfolio.
All sorting and portfolio formations are out of sample.
Investment Performance by Mispricing & Size
0
0.5
1
1.5
2
2.5
Mo
nth
ly R
etu
rn (
%)
Monthly Returns on Mispricing--Size Sorted Portfolios
Investment by Mispricing & Book/market
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0.5
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Mo
nth
ly R
etu
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Monthly Returns on Mispricing--Book/Market Sorted Portfolios
Investment by Mispricing & Momentum
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3.5
Mo
nth
ly R
etu
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%)
Monthly Returns on Mispricing--Momentum Sorted Portfolios
Alpha & Beta: for Mispricing & Momentum portfolios
• All the portfolios here are same as in preceding chart, based on Mispricing & Momentum.
-2
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-1
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0
0.5
1
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Mo
nth
ly A
lph
a (
%)
LMS Mispricing & Momentum• Fair value in the V/P ratio is determined by the LMS residual-income model, where
book value, EPS estimates and CAPM-based expected returns are used as the basis.
0
0.5
1
1.5
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2.5
3
Mo
nth
ly R
etu
rn (
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Monthly Returns on LMS V/P Ratio--Momentum Sorted Portfolios
Investment by Mispricing & Sharpe Ratio
0
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3
3.5
4
Mo
nth
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etu
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Monthly Returns on Mispricing--Sharpe Ratio Sorted Portfolios
Sharpe ratio is based on the stock’s past-5-yr average return divided by its volatility. It measures the risk-return tradeoff offered by the stock, hence representing “quality”. Not shown in this figure is that in each given Mispricing group, the higher the Sharpe ratio, the lower the
portfolio’s volatility.
Forecasting the Stock MarketThe “% of Undervalued Stocks” path indicates the then-current percentage of stocks that were undervalued at the time, relative to the entire stock universe. The other path is the then-1-yr-forward return on the S&P 500 index.
-50%
-20%
10%
40%
70%
100%79
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Date
% of Stocks Undervalued
1-Yr. Forw ard S&P 500 Return
Concluding Remarks
BCD Mispricing is strongly mean-reverting
overvalued => undervalued => overvalued => undervalued …..
BCD Mispricing shows persistent winner-loser reversals (once every 1.5 years or so)
The winning strategy:
“ BCD Valuation + Momentum + Size ”