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 • The Residual-Earnings Model
(e.g., Lee, Meyer and Swaminathan (1998))

– Detailed parameterization of EPS

– Two parameters: beta and
processes and interest-rate process
dividend-payout ratio
Parameters to be estimated from
past data

– Closed-form stock valuation – No closed-form valuation formula.

formula Requires ad hoc approximation of
the stock’s future price at end of
forecasting horizon
– Past data are used to estimate
parameters
So, valuation reflects both past
valuation standard for the stock – Valuation is independent of past
and the stochastic discounting of valuation standard for the stock
future prospects
 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

2
Q1 (low )
Q2
Q3
B/M

1 Q4(high)

0
8407

8902
7901
7912

8011
8110

8209
8308

8506
8605

8704
8803

9001
9012

9111

9210

9309

9408

9507
9606
Date
 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.
Q1(lo w)
Part A: Average V/P Ratio by Quartile Q2
2
Q3

Q4 (hig h
)
V/P

0
8001
8012

8210

8507

8705
8804

9002

9508
9607
7902

8111

8309
8408

8606

8903

9101
9112
9211
9310
9409
Part B:V/P Autocorrelation for the Lowest Quartile Date
1
Autocorrelation

0.5

0
13
17

25
29
33
37
21

41
45
49
53
57
1
5
9

-0.5
Number of Months Lagged
 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

Q1(low )
0.2
Q2
0.15 Q3
Q4(high)
0.1
E/P Ratio

0.05
0
-0.05
-0.1
7901

8103

8204
8305

8406

8507

8709

8810
8911

9201

9302

9403
9504

9605
8002

8608

Date 9012
 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

Q1 (undervalued)
Q2
35 Q3
Q4 (overvalued)
Mispricing (% )

25

15

-5

-15

-25
7901
7911
8009
8107
8205
8303
8401
8411
8509
8607
8705
8803
8901
8911
9009
9107
9205
9303
9401
9411
9509
9607
Date
 Persistence of BCD Model Mispricing

Part A: Mispricing Autocorrelation

for the Most Undervalued Quartile
1

0.6
Autocorrelation

0.2

-0.2

-0.6
1

13

17

21

33

37

41

57
25

29

45

49

53
Number of Months Lagged Part B: Distribution of Mispricing Mean-Reversion Time
Full Sample

10
9
Percent of Stocks 8
7
6
5
4
3
2
1
0
11

15

17
19

25

29

33

37

41

43
13

21

23

27

31

35

39
3

5
7
9

Mean-Reversion Time in Months

 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 Stocks

Misp (%) -19.63 -4.96 2.58 10.59 30.67 3.86
V/P 1.00 1.00 0.96 0.90 0.78 0.93
ME ($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 0.77
Ret-6 (%) -7.51 3.03 9.26 15.61 27.86 9.65
Ret+1 (%) 2.04 1.83 1.53 1.31 1.18 1.67
Ret+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 Stocks

V/P 0.41 0.69 0.89 1.11 1.54 0.93
Misp (%) 9.92 5.78 3.11 1.49 -0.97 3.86
ME ($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 0.77
Ret-6 (%) 15.74 11.31 9.21 7.74 5.18 9.65
Ret+1 (%) 1.33 1.27 1.50 1.59 1.87 1.67
Ret+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 Stocks

Ret-6 (%) -18.79 -1.95 7.66 18.00 43.32 9.65
Misp (%) -8.92 -1.41 3.24 8.10 18.26 3.86
V/P 0.93 0.98 0.97 0.92 0.82 0.93
ME ($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 0.77
Ret+1 (%) 1.51 1.56 1.52 1.44 1.86 1.67
Ret+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 Stocks

B/M 0.25 0.45 0.66 0.89 1.61 0.77
Misp (%) 9.86 4.52 2.89 1.72 0.30 3.86
V/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 1643.3
Ret-6 (%) 19.42 12.48 9.01 6.28 1.11 9.65
Ret+1 (%) 1.52 1.48 1.37 1.56 1.95 1.67
Ret+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 -0.029 -0.142 0.130 0.021 0.051 216
(4.82) (-8.97) (-2.79) (1.16) (5.91)

2 2.357 -0.138 0.162 0.009 0.042 216

(4.62) (-2.69) (1.42) (2.48)

3 2.475 -0.031 -0.151 0.275 0.019 0.054 216

(4.92) (-9.17) (-2.96) (2.53) (7.99)

4 2.485 -0.152 0.292 0.012 0.044 216

(4.81) (-2.96) (2.68) (4.90)
9 2.278 -0.029 0.211 -0.126 0.175 0.018 0.059 215
(4.78) (-7.71) (2.21) (-2.62) (1.72) (7.77)

10 2.356 0.319 -0.135 0.157 0.012 0.048 215

(4.81) (3.45) (-2.79) (1.51) (4.84)

11 1.629 0.291 0.010 215

(5.29) (2.49)
 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 -0.062 -0.811 0.440 0.011 0.076 18
(5.91) (-6.14) (-9.16) (1.89) (1.22)

February 4.229 -0.034 -0.208 0.544 0.019 0.065 18

(1.82) (-2.03) (-1.07) (1.11) (2.25)

March 3.727 -0.026 -0.357 0.580 0.022 0.050 18

(2.61) (-2.44) (-2.69) (1.62) (3.15)

April 2.571 -0.019 -0.170 0.301 0.021 0.049 18

(1.46) (-1.84) (-0.77) (1.82) (2.93)

May 3.792 -0.039 -0.317 0.249 0.013 0.044 18

(2.69) (-3.44) (-1.88) (0.73) (2.23)

June 2.231 -0.017 -0.060 0.564 0.022 0.046 18

(1.91) (-1.73) (-0.49) (1.59) (2.28)

July 1.389 -0.029 -0.083 0.160 0.023 0.055 18

(0.98) (-2.33) (-0.50) (0.44) (3.50)

August 1.980 -0.048 0.125 0.101 0.012 0.060 18

(0.60) (-3.69) (0.61) (0.22) (1.62)

September 2.042 -0.023 -0.221 0.042 0.008 0.057 18

(1.41) (-3.02) (-1.65) (0.09) (0.69)

October -0.417 -0.013 0.092 0.163 0.032 0.046 18

(-0.26) (-1.13) (0.68) (0.47) (4.16)

November -0.036 -0.031 -0.067 -0.019 0.022 0.062 18

(-0.01) (-2.43) (-0.25) (-0.04) (2.30)

December 0.226 -0.028 0.127 0.173 0.024 0.041 18

(0.18) (-3.21) (0.94) (0.56) (2.64)
 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
Monthly Returns on Mispricing--Size Sorted Portfolios

2.5

Monthly Return (%)

1.5

0.5

0
Investment by Mispricing & Book/market

Monthly Returns on Mispricing--Book/Market Sorted Portfolios

2.5

Monthly Return (%)

2

1.5

0.5

0
Investment by Mispricing & Momentum
Monthly Returns on Mispricing--Momentum Sorted Portfolios

3.5

Monthly Return (%)

2.5

1.5

0.5

0
 Alpha & Beta: for Mispricing & Momentum portfolios
• All the portfolios here are same as in preceding chart, based on Mispricing &
Momentum.

1.5

Mon th ly Alp ha (% )
1

0.5

-0.5

-1

-1.5

-2
 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.
Monthly Returns on LMS V/P Ratio--Momentum Sorted Portfolios

2.5

Monthly Return (% )
2

1.5

0.5

0
 Investment by Mispricing & Sharpe Ratio
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.
Monthly Returns on Mispricing--Sharpe Ratio Sorted Portfolios

3.5

Monthly Return (%)

2.5

1.5

0.5

0
 Forecasting the Stock Market
The “% 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.

% of Stocks Undervalued
100%
1-Yr. Forw ard S&P 500 Return

70%

40%

10%

-20%

-50%
8109
8208
8307
8406

8604
8703

8901
8912
9011
9110

9407
9506
7911
8010

8505

8802

9209
9308

9605
Date
 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 ”