Quantitative Finance authors titles recent submissions
A simple model for forecasting conditional return
distributions
Stanislav Anatolyev∗
Jozef Barunı́k†
arXiv:1711.05681v1 [q-fin.ST] 15 Nov 2017
November 16, 2017
Abstract
This paper presents a simple approach to forecasting conditional probability distributions of asset returns. We work with a parsimonious parametrization of ordered
binary choice regression that quite precisely forecasts future conditional probability
distributions of returns, using past indicator and past volatility proxy as predictors.
Direct benefits of the proposed model are revealed in the empirical application to 29
most liquid U.S. stocks. The forecast probability distribution is translated to significant economic gains in a simple trading strategy. The model can therefore serve as
useful risk management tool for investors monitoring tail risk, or even building trading strategies based on the entire conditional return distribution. Our approach can
also be useful in many other applications where conditional distribution forecasts are
desired.
JEL Classification: C22, C58, G17
Keywords: returns, predictive distribution, conditional probability, forecasting, ordered binary choice
∗
CERGE-EI, Politických vězňů 7, 11121, Prague, Czech Republic; New Economic School, 100A Novaya
Street, Skolkovo, 143026, Moscow, Russia.
†
Institute of Economic Studies, Charles University, Opletalova 26, 11000, Prague, Czech Republic; Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Pod Vodarenskou
Vezi 4, 18200, Prague, Czech Republic.
1
1
Introduction
“Those who have knowledge, don’t predict. Those who predict, don’t have
knowledge.”
- Lao Tzu, (c. 604 – 531 B.C.)
Several decades of research provides overwhelming evidence about predictability of first
two moments of stock return distributions. Expected values of stock returns are predictable
to some extent using economic variables (Keim and Stambaugh, 1986; Fama and French,
1989; Ang and Bekaert, 2006; Viceira, 2012), while the conditional second moment can
be well characterized by simple time series models, or even measured from high frequency
data (Bollerslev, 1986; Andersen et al., 2003). While volatility forecasting quickly became
central to the financial econometrics literature due to its importance for risk measurement
and management, research focusing on the entire return distribution is still very rare in the
literature.1
One of the main reasons why researchers may not be focused on characterizing the entire
return distribution is a prevailing practice of convenient mean-variance analysis that is still
central to modern asset pricing theories. Unfortunately, investor’s choices guided using
first two moments are restricted by binding assumptions, such as multivariate normality of
stock returns or a quadratic utility function. More importantly, an investor is restricted to
have classical preferences based on von Neumann-Morgenstern expected utility. In contrast
to this, Rostek (2010) recently developed a notion of quantile maximization and quantile
utility preferences. This important shift in decision-theoretic foundations provokes us to
depart from the limited mean-variance thinking, and work with entire distributions.
Specification and estimation of the entire conditional distribution of future price changes
is useful for a number of important financial decisions. Prime examples include portfolio
selection when returns are non-Gaussian, (tail) risk measurement and management, market
timing strategies with precise entries and exits reflecting information in tails. Despite its
importance, forecasting of the conditional distribution of future returns has so far attracted
little attention in contrast to point forecasts and its uncertainty mentioned before. In this
paper, we present a simple approach to forecasting a conditional distribution of stock returns
using parameterized ordered binary choice regression. Although focusing on financial returns,
we note that our approach may be useful to many other applications where the conditional
distribution forecasts are of interest.
The majority of studies focusing on prediction of conditional return distributions characterize the cumulative conditional distribution by a collection of conditional quantiles (Engle
1
Few studies focused on directional forecasts, or threshold exceedances (Christoffersen and Diebold, 2006;
Chung and Hong, 2007; Nyberg, 2011).
2
and Manganelli, 2004; Cenesizoglu and Timmermann, 2008; Žikeš and Barunı́k, 2014; Pedersen, 2015). In a notable contribution, Foresi and Peracchi (1995) focus on a collection of
conditional probabilities and describe the cumulative distribution function of excess returns
using a set of separate logistic regressions. To be able to approximate the distribution function, Foresi and Peracchi (1995) estimate a sequence of conditional binary choice models over
a grid of values corresponding to different points of the distribution. In addition to a need
to estimate a sequence of separate equations to capture individual points of the distribution,
one has to ensure monotonicity crucial for distribution forecasts. Peracchi (2002) argues
that the conditional distributions approach has numerous advantages over the conditional
quantile approach, and Leorato and Peracchi (2015) continue further their comparison. The
approach has been also considered by Fortin et al. (2011), Rothe (2012), Chernozhukov et al.
(2013), Hothorn et al. (2014), and Taylor and Yu (2016).
In this paper, we develop further the ideas set forth by Foresi and Peracchi (1995) and
present a related simple model for forecasting conditional return distributions. The proposed
model is based on the ordered binary choice regression that is able to forecast the entire
predictive distribution of stock returns using fewer parameters than the set of separate binary
choice regressions. To achieve this substantial reduction in the degree of parameterization,
we tie the coefficients of predictors via smooth dependence on quantile levels. As a side
effect, our approach allows one to impose the desirable monotonicity property more easily.
Our specification can be motivated in a semiparametric way, as we approximate the smooth
quantile functions by low-order polynomials.
The probability forecasts are conditional on past information contained in returns, as
well as their volatility proxy. The main reason of choosing volatility as one of explanatory
variables is that the cross-sectional relation between risk and expected returns generally
measuring a stock’s risk as the covariance between its return and some factor is well documented in the literature. In the laborious search for proper risk factors, volatility plays a
central role in explaining expected stock returns for decades. Although predictions about
expected returns are essential for understating of classical asset pricing, little is known about
a potential of the factors to precisely identify extreme tail events of the returns distribution.
In our illustrative empirical analysis, we estimate the conditional distributions of 29 most
liquid U.S. stocks and compare their generated forecasts with those from the buy-and-hold
strategy and the benchmark collection of separate binary choice models. The benefits of our
approach are translated to significant economic gains in a simple trading strategy that uses
conditional probability forecasts.
The paper is organized as follows. Section 2 describes the model and emphasized its differences with the collection of separate binary choice models. Section 3 contains information
about the data we use and lays out details of particular specifications. In Section 4, empirical
results are presented. Section 5 concludes.
3
2
The Model
We consider a strictly stationary series of financial returns rt , t = 1, ..., T . Our objective is
to describe, as fully as possible, the conditional cumulative return distribution F (rt |It−1 ) ,
where It−1 includes the history of rt as well as possibly past values of other observable variables. To that end, we model jointly the conditional probabilities for a number of thresholds
that correspond to p > 1 quantile levels
0 < α1 < α2 < ... < αp < 1.
The higher p, the more precise is the description of the conditional distribution. The p
quantile levels correspond to empirical cutoff levels, or empirical quantiles
c1 < c2 < ... < cp .
That is, cj is empirical αj -quantile of returns, j = 1, ..., p. For convenience, define c0 = −∞
and cp+1 = +∞.
Let Λ : u 7→ [0, 1] be a (monotonically increasing) link function. Both unordered and
ordered binary choice models are represented by a collection of conditional probabilities
Pr{rt ≤ cj |It−1 } = Λ (θt,j ) ,
j = 1, ..., p,
for some specification of the driving processes θt,j , j = 1, ..., p. For convenience, define
Λ (θt,0 ) = 0 and Λ (θt,p+1 ) = 1.
Let xt−1,j , j = 1, ..., p be a vector of predictors for I{rt >cj } that may depend on j via
dependence of some of them on cj . For instance, one of predictors may be I{rt−1 >cj } , the
past indicator (dependent on j), while another may be rt−1 , the past return (independent of
j), yet another may represent some volatility measure (also non-specific to j). Suppose for
simplicity that the number of predictors in xt−1,j is the same for all j and equals k.
In the unordered model, the specification for the underlying process θt,j is
θt,j = δ0,j + x′t−1,j δj .
There are no cross-quantile restrictions, and each binary choice problem is parameterized
separately. This results in a flexible but highly parameterized specification. In addition,
no monotonicity, in the language of Foresi and Peracchi (1995), holds in general. That is,
Pr{rt ≤ cj−1 |It−1 } may well exceed Pr{rt ≤ cj |It−1 } with positive probability even though
cj−1 < cj .
In the proposed ordered model, there are cross-quantile restrictions placed on the parameters. In particular, the coefficients of predictors are tied via smooth dependence on
the quantile levels. This leads to a substantial decrease of a degree of parameterization,
4
and in addition allows one to impose the desirable monotonicity property more easily. The
specification for the underlying process θt,j is
θt,j = δ0,j + x′t−1,j δ (αj ) ,
where δ (αj ) are coefficients that are functions of the quantile level. Each quantile-dependent
slope coefficient vector is specified in the following way: δ (αj ) = (δ1 (αj ) , ..., δk (αj ))′ , where
for each ℓ = 1, ..., k,
qℓ
X
2i (αj − 0.5)i · κi,ℓ ,
δℓ (αj ) = κ0,ℓ +
i=1
and qℓ ≤ p − 1. Note that each intercept δ0,j is j-specific and represents an ‘individual effect’
for a particular quantile level, while the slopes δ’s do not have index j, i.e. they depend on
j only via dependence on αj ’s. The motivation behind such specification is semiparametric:
any smooth function on [0, 1] can be approximated to a desired degree of precision by the
system of basis polynomials {αj − 0.5, (αj − 0.5)2 , ..., (αj − 0.5)q } by taking q big enough.
Because all αj ∈ (0, 1), the polynomial form behaves nicely even for large q; the basis
polynomials are uniformly bounded on [0, 1] . The additional weights 2i are introduced to
line up the coefficients κi on a more comparable level.
Let us compare the degrees of parameterization of the unordered and ordered binary choice
models. Denote
k
X
q=
qℓ .
ℓ=1
In the unordered model, the total number of parameters is
KU O = (1 + k) p
(namely, one intercept δ0,j and k slopes δj in each of p equations for θj ) while in the unordered
model, the total number of parameters is
KO = p + k + q
(namely, p intercepts δ0,j and k slopes δ (αj ) each parameterized via 1 + qℓ parameters). The
difference
KU O − KO = k (p − 1) − q
is larger the larger is p, the fineness of the partition by thresholds. The resulting difference
is also positively related to the number of predictors used.2
In the unordered model, the composite loglikelihood corresponding to observation t is
ℓUt O
=
p+1
X
I{rt ≤cj } ln (Λ (θt,j )) ,
j=1
2
In our empirical illustration, p = 41, k = 2 and q1 = 2, q2 = 3. Hence, KU O = 123 while KO = 48, and
the difference is KU O − KO = 75.
5
PT U O
and the total composite likelihood
can be split into p independent likelihoods
t=1 ℓt
PT (j)
t=1 ℓt , where
(j)
ℓt = I{rt ≤cj } ln (Λ (θt,j )) ,
to be maximized over the parameter vector (δ0,j , δj )′ . In the ordered model, the loglikelihood
corresponding to observation t is
ℓO
t
=
p+1
X
I{cj−1 cj }
.
xt−1,j =
ln (1 + |rt−1 |)
The first predictor is a lagged indicator corresponding to the quantile level αj . The second
predictor is a proxy for a volatility measure, with the absolute return dampened by the
logarithmic transformation. Note that the first predictor is specific for a specific quantile,
while the second predictor is common for all quantiles. In the ordered model, we set, after
some experimentation, q1 = 2 and q2 = 3. That is, the polynomial is quadratic in the quantile
level α for the past indicator and cubic for the past volatility proxy.
The full specification of the model for j = 1, ..., p empirical quantiles is
Pr{rt ≤ cj |It−1 } =
exp(−θt,j )
,
1 + exp(−θt,j )
θt,j = δ0,j + δ1 (αj ) I{rt−1 >cj } + δ2 (αj ) ln (1 + |rt−1 |) ,
with the coefficient functions
δ1 (αj ) = κ0,1 + 2(αj − 0.5) · κ1,1 + 22 (αj − 0.5)2 · κ2,1 ,
δ2 (αj ) = κ0,2 + 2(αj − 0.5) · κ1,2 + 22 (αj − 0.5)2 · κ2,2 + 23 (αj − 0.5)3 · κ3,2 .
In total, there are KO = 48 parameters: p = 41 individual intercepts δ0,j and k = 2 slopes
δ (αj ), one parameterized via 1+q1 = 3 parameters, the other via 1+q2 = 4 parameters. This
parametrization is parsimonious enough and approximates the distribution quite well, and
additional terms do not bring significant improvements. Hence, estimating seven parameters
κi,ℓ in addition to individual intercepts is enough to approximate the conditional return
distribution.
Oracle Corporation (ORCL), PepsiCo, Inc. (PEP), Pfizer Inc. (PFE), Procter & Gamble Co (PG), QUALCOMM, Inc. (QCOM), Schlumberger Limited. (SLB), AT&T Inc. (T), Verizon Communications Inc. (VZ),
Wells Fargo & Co (WFC), Wal-Mart Stores, Inc. (WMT), Exxon Mobil Corporation (XOM).
7
4
Empirical Findings
We now present the results of estimating the conditional distribution function of returns
using the ordered binary choice model. We consider forecasts for 29 stocks, so first we
present individual estimates of three illustrative stocks, namely Intel Corporation (INTC),
QUALCOMM, Inc. (QCOM), and Exxon Mobil Corporation (XOM), and then we present
the results for all 29 stocks in an aggregate form.
4.1
Parameter estimates
Table 1 shows estimates of quantile specific intercepts δ0,j in the ordered logit model for the
three illustrative stocks. The intercepts have expected signs, and exhibit expected monotonic behavior decreasing from the left tail to the right tail thus generating an increasing
cumulative distribution function (assuming zero predictors). The intercepts are statistically
significant in most cases except for few cutoff points near the median. The intercept values
are quite similar across three stocks, though there is some variability.
The left plot shown in Figure 2 collects the intercept estimates for all 29 stocks, and
reveals that indeed the intercepts are similar across the stocks. The plot has a shape similar
to an inverse logit link function Λ (u) defined earlier, with a stronger effect in tails. A
flexible j-specific intercept allows for controlling individual quantile effects, and shows that
unconditional expectations are an important part of the predicted distribution.
The estimates of the seven coefficients κi,ℓ driving the slopes on the predictors for the
three illustrative stocks are shown in Table 2. The coefficients κi,1 correspond to the lagged
indicator predictor I{rt−1 >cj } . All parameter estimates for the lagged indicator are of a small
magnitude, and, except of the parameter driving the quadratic term κ2,1 of XOM, they
are not highly significantly different from zero. The left plot in Figure 1 complements these
findings with estimates of the lagged indicator coefficients for all 29 stocks shown by the boxand-whisker plots. The estimates document a heterogeneous effect of the lagged indicator
on the future probabilities for different stocks. While we document zero coefficients for some
stocks, the quadratic term seems to play role in others.
The second predictor, the past volatility proxy ln (1 + |rt−1 |), carries more important information about future probabilities. The estimated coefficients κi,2 corresponding to the
volatility predictor reveal that the cubic polynomial has almost all coefficients significant,
except that of κ0,2 for all three stocks (see Table 2). The right plot of Figure 1 shows estimated coefficients for all 29 stocks as box-whisker plots. We can see that κ1,2 and κ3,2
are significantly different from zero for most of the stocks, and so the past volatility proxy
contributes significantly to prediction of return distributions.
Figure 2 plots the functions δ1 (αj ), and δ2 (αj ), in addition to the intercepts. Corresponding to heterogenous parameters κi,1 , the function δ1 (αj ) is also heterogeneous in 29 stocks,
8
exhibiting a mixed effect of the lagged indicator predictor (shown in the middle plot of Figure 2). Overall, we can see that the effect is small. In contrast, the coefficient function
δ2 (αj ) implied by the past volatility proxy shows a much stronger and more robust impact.
The probability distribution is predicted strongly in both tails with the expected signs of
coefficients for all 29 stocks alike.
4.2
Economic Significance of Predicted Distributions
To understand the economic usefulness of the proposed model, we study a simple profit rule
for timing the market based on the model forecasts. The idea is to explore the information
from the entire distribution. In the spirit of previous literature (Breen et al., 1989; Pesaran
and Timmermann, 1995; Anatolyev and Gospodinov, 2010), we evaluate the model forecasts
in terms of profits from a trading strategy for active portfolio allocation between holding a
stock and investing into a risk-free asset.
To build a trading strategy, we use a simple rule exploring the difference between the
predicted conditional probability, and unconditional probability Pr{rt ≤ cj } = αj . We sum
the differences over the interval of empirical quantiles [a, b] as
St =
b
X
cj =a
c
Pr{rt ≤ cj |It−1 } − αj .
If we sum all p available quantiles, we are using the information from the entire distribution.
If we want to use the information only about positive returns, we sum only a half of available
empirical quantiles corresponding to the cutoffs at positive returns. For example, in case
the positive returns are predicted with a higher probability than the negative returns for
all corresponding empirical quantiles, the sum St will be positive. Further, it may be useful
to compare St computed for the empirical quantiles corresponding to negative returns, and
to positive returns. After some experimentation, we obtain threshold values for each stock
depending on the shape of conditional distributions, generating consistent profits. Hence
we build the trading strategy on St exceeding these thresholds, but we note that this could
further be optimized for maximum profits.
Starting with a one dollar investment at the beginning of the sample, our investor decides
to hold the stock depending on whether the predicted probability is favorable or not. We
compare the cumulative returns from this simple market-timing strategy using predictions
from the ordered logit, separate logistic regressions, and a standard benchmark of the buy
and hold strategy for all 29 stocks separately.
Table 3 summarizes the results of running the trading strategies in terms of a mean and
median return, volatility, as well as Sharpe ratio. While the benchmark strategy earns 0.929
return with 0.235 volatility, when investor uses predictions about the entire distribution
from separate logit models, a similar mean return of 0.955 is obtained with a much lower
9
volatility. In the meantime, our proposed ordered logit model generates 1.481 mean return
with volatility similar to that for separate logits. The Sharpe ratio showing a risk-adjusted
return, or average return per unit of volatility, reveals that the ordered logit model generates
twice as high returns when we take risk into account. The figures for median values are even
more impressive.
Figure 4 shows the returns, volatilities, and Sharpe ratios for all 29 stocks by box-andwhisker plots. The figure reveals that the naive buy and hold strategy yields lowest returns
with highest volatility. In terms of risk-adjusted Sharpe ratios, the separate logistic regressions yield a similar result, while the ordered logit model makes a marked improvement. One
can see positive Sharpe ratios for almost all considered stocks.
Figure 3 looks closely at the cumulative returns and drawdowns of the three illustrative
stocks we used earlier. It can be seen that returns from the ordered logit strategy are
consistent over time, with a lowest drawdown. This holds even in the case of XOM that
has been growing for the whole period, hence making it difficult to beat the buy and hold
strategy.
Finally, we compare the relative performance of all three strategies in Figure 5. The left
plot in the Figure compares the ordered logit with the ‘market,’ or naive buy and hold
strategy, while the right plot compares the ordered logit with separate logits. We document
a consistently better performance of the proposed ordered logit model in comparison to both
the unordered logit as well as naive strategy for all 29 stocks.
5
Conclusion
This paper investigates predictability of stock market return distributions. We propose a
relatively parsimonious parametrization of an ordered binary choice regression that precisely
forecasts the conditional probability distribution of returns. In order to see how useful the
model is economically, we use distributional predictions in a simple market timing strategy. Using 29 liquid U.S. stocks we find significant economic gains when compared to the
benchmarks.
Our findings are useful for risk management and measurement, especially focused on tails,
or building trading strategies using the entire conditional distribution of returns. The model
nevertheless has a much wider potential use in any application that exploits distribution
forecasts including forecasting of interest rates, term structures, and macroeconomic variables.
10
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analysis. Journal of American Statistical Association 90 (430), 451–466.
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of labor economics 4, 1–102.
Hothorn, T., T. Kneib, and P. Bühlmann (2014). Conditional transformation models. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 76 (1), 3–27.
11
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Rostek, M. (2010). Quantile maximization in decision theory. Review of Economic Studies 77 (1), 339–371.
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12
6
Tables
Table 1. Estimates of intercepts δ0,j in the ordered logit specification θt,j = δ0,j + x′t−1,j δ (αj )
for the three illustrative stocks.
INTC
QCOM
XOM
INTC
QCOM
XOM
1%
5.420
5.123
5.402
99%
−5.248
−5.433
−5.759
(0.243)
(0.238)
(0.274)
2.5%
4.367
4.160
4.367
97.5%
−4.270
−4.429
−4.715
(0.169)
(0.156)
(0.183)
5%
3.542
3.396
3.566
95%
−3.472
−3.591
−3.852
7.5%
3.040
2.925
3.073
92.5%
−2.990
−3.083
−3.309
10%
2.658
2.574
2.698
90%
−2.610
−2.694
−2.905
12.5%
2.354
2.287
2.386
87.5%
−2.315
−2.390
−2.574
15%
2.062
2.052
2.131
85%
−2.064
−2.122
−2.291
(0.078)
(0.082)
(0.082)
17.5%
1.824
1.837
1.905
82.5%
−1.839
−1.893
−2.042
(0.075)
(0.077)
(0.078)
20%
1.614
1.645
1.697
80%
−1.647
−1.681
−1.820
22.5%
1.429
1.473
1.515
77.5%
−1.465
−1.484
−1.627
25%
1.257
1.316
1.334
75%
−1.298
−1.308
−1.440
27.5%
1.092
1.165
1.177
72.5%
−1.153
−1.149
−1.257
30%
0.938
1.014
1.029
70%
−1.012
−1.000
−1.082
(0.065)
(0.067)
(0.066)
32.5%
0.800
0.880
0.881
67.5%
−0.876
−0.855
−0.933
(0.065)
(0.066)
(0.065)
35%
0.664
0.751
0.738
65%
−0.746
−0.721
−0.791
37.5%
0.535
0.627
0.611
62.5%
−0.625
−0.593
−0.658
40%
0.413
0.503
0.482
60%
−0.506
−0.463
−0.530
42.5%
0.293
0.383
0.361
57.5%
−0.392
−0.335
−0.405
45%
0.176
0.263
0.233
55%
−0.277
−0.215
−0.275
(0.061)
(0.061)
(0.060)
47.5%
0.062
0.147
0.111
52.5%
−0.165
−0.096
−0.139
(0.060)
(0.061)
(0.059)
0.50%
−0.074
0.020
−0.018
(0.280)
(0.194)
(0.146)
(0.123)
(0.107)
(0.096)
(0.087)
(0.083)
(0.078)
(0.075)
(0.072)
(0.069)
(0.067)
(0.066)
(0.064)
(0.063)
(0.062)
(0.061)
(0.061)
(0.060)
(0.060)
(0.262)
(0.189)
(0.153)
(0.131)
(0.116)
(0.104)
(0.094)
(0.087)
(0.082)
(0.075)
(0.071)
(0.068)
(0.066)
(0.065)
(0.063)
(0.062)
(0.062)
(0.061)
(0.061)
(0.060)
(0.061)
(0.276)
(0.192)
(0.152)
(0.127)
(0.112)
(0.101)
(0.091)
(0.084)
(0.080)
(0.076)
(0.072)
(0.069)
(0.067)
(0.065)
(0.064)
(0.062)
(0.061)
(0.061)
(0.060)
(0.059)
(0.059)
13
(0.127)
(0.103)
(0.092)
(0.084)
(0.071)
(0.069)
(0.070)
(0.067)
(0.064)
(0.063)
(0.062)
(0.061)
(0.121)
(0.102)
(0.094)
(0.086)
(0.075)
(0.073)
(0.071)
(0.067)
(0.064)
(0.063)
(0.062)
(0.062)
(0.136)
(0.114)
(0.100)
(0.089)
(0.074)
(0.071)
(0.068)
(0.067)
(0.064)
(0.062)
(0.062)
(0.061)
Table 2. Estimates of slope coefficients κi,ℓ in the ordered logit specification δℓ (αj ) =
Pℓ i
κ0,ℓ + qi=1
2 (αj − 0.5)i · κi,ℓ for the three illustrative stocks.
INTC
QCOM
XOM
κ0,1
−0.006
0.016
0.025
κ1,1
0.059
0.007
κ2,1
−0.057
−0.100
(0.024)
(0.042)
(0.095)
(0.024)
(0.045)
(0.096)
INTC
QCOM
XOM
κ0,2
4.940
−2.960
0.988
−0.019
κ1,2
21.27
32.40
57.69
−0.164
κ2,2
−5.78
14.81
17.73
κ3,2
29.29
12.52
17.74
(0.025)
(0.044)
(0.093)
(3.920)
(6.26)
(4.64)
(7.93)
(3.822)
(5.20)
(4.65)
(5.87)
(4.869)
(7.10)
(5.73)
(9.65)
Table 3. Mean and median return-volatility characteristics from three strading strategies
for all 29 stocks.
Market
Logit
Ordered Logit
return
0.929
0.955
1.481
mean
volatility
0.235
0.179
0.177
Sharpe
0.059
0.162
0.313
14
return
0.005
0.203
0.608
median
volatility
0.217
0.170
0.166
Sharpe
0.002
0.158
0.311
Figures: Parameter Estimates
20
40
●
●
−0.2
−20
0
coefficients for δ2(α)
−0.1
0.0
coefficients for δ1(α)
0.1
60
0.2
7
κ0,1
κ1,1
●
κ2,1
κ0,2
κ1,2
κ2,2
κ3,2
−6
0.01
0.2
0.4
0.6
α
0.8
0.99
0
−100
−0.2
−50
−0.1
δ2(α)
0.0
δ1(α)
0
−4
−2
δ0,j
2
50
0.1
4
6
0.2
100
Figure 1: Parameter estimates: ordered logit parameters estimated for all 29 stocks shown
by box-and-whisker plots.
0.01
0.2
0.4
0.6
α
0.8
0.99
0.01
0.2
0.4
0.6
0.8
0.99
α
Figure 2: Parameter estimates: coefficient functions implied by parameters estimated for all
29 stocks. Minimum and maximum values shown as light grey area, 50% of distribution as
grey area, and median as black line.
15
Results: Economic Evaluation
Performance of QCOM
Performance of XOM
3.0
2.5
2.0
Cumulatuve Return
2.0
1.5
Cumulatuve Return
1.0
0.5
1.0
1.4
1.2
1.0
0.8
0.6
0.2
0.4
Cumulatuve Return
3.5
1.6
Performance of INTC
1.5
8
2012
2014
2016
2006
2008
2010
2012
2014
2016
2008
2010
2012
2014
2016
2006
2008
2010
2012
2014
2016
2008
2010
2012
2014
2016
2006
2008
2010
2012
2014
2016
−0.1
0.0
2006
−0.4
−0.5
−0.8
−0.6
−0.6
−0.4
Drawdown
Drawdown
−0.2
0.0
−0.2
−0.4
Drawdown
2006
−0.2
2010
−0.3
2008
0.0
2006
1.0
6
0.5
Figure 3: Performance: cumulative return and drawdown of three typical stocks. Trading
strategy based on the probability predictions from ordered logit model as bold black line,
strategy from separate logits as black line, and benchmark buy and hold strategy as grey.
●
0.3
Volatility
0.1
0
−0.5
0.2
2
●
●
0.0
●
●
Sharpe Ratio
4
●
●
●
Return
0.5
0.4
●
●
−2
0.0
−1.0
●
logit
ordered logit
market
●
logit
ordered logit
market
logit
ordered logit
market
Figure 4: Performance: return, volatility and Sharpe ratio for three strategies for all 29
stocks shown by box-and-whisker plots.
16
3
1
2
Relative Performance
3
2
1
Relative Performance
4
Ordered Logit vs. Logit
4
Ordered Logit vs. Market
2006
2008
2010
2012
2014
2016
2006
2008
2010
2012
2014
2016
Figure 5: Relative performance: trading strategy based on the probability predictions from
ordered logit model relative to separate logits (right) as well as benchmark ‘market’ (left).
Median value from all 29 stocks in black line surrounded with 90% of the distribution in
grey area. Note that value of one shows equal performance of both strategies compared.
17
distributions
Stanislav Anatolyev∗
Jozef Barunı́k†
arXiv:1711.05681v1 [q-fin.ST] 15 Nov 2017
November 16, 2017
Abstract
This paper presents a simple approach to forecasting conditional probability distributions of asset returns. We work with a parsimonious parametrization of ordered
binary choice regression that quite precisely forecasts future conditional probability
distributions of returns, using past indicator and past volatility proxy as predictors.
Direct benefits of the proposed model are revealed in the empirical application to 29
most liquid U.S. stocks. The forecast probability distribution is translated to significant economic gains in a simple trading strategy. The model can therefore serve as
useful risk management tool for investors monitoring tail risk, or even building trading strategies based on the entire conditional return distribution. Our approach can
also be useful in many other applications where conditional distribution forecasts are
desired.
JEL Classification: C22, C58, G17
Keywords: returns, predictive distribution, conditional probability, forecasting, ordered binary choice
∗
CERGE-EI, Politických vězňů 7, 11121, Prague, Czech Republic; New Economic School, 100A Novaya
Street, Skolkovo, 143026, Moscow, Russia.
†
Institute of Economic Studies, Charles University, Opletalova 26, 11000, Prague, Czech Republic; Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Pod Vodarenskou
Vezi 4, 18200, Prague, Czech Republic.
1
1
Introduction
“Those who have knowledge, don’t predict. Those who predict, don’t have
knowledge.”
- Lao Tzu, (c. 604 – 531 B.C.)
Several decades of research provides overwhelming evidence about predictability of first
two moments of stock return distributions. Expected values of stock returns are predictable
to some extent using economic variables (Keim and Stambaugh, 1986; Fama and French,
1989; Ang and Bekaert, 2006; Viceira, 2012), while the conditional second moment can
be well characterized by simple time series models, or even measured from high frequency
data (Bollerslev, 1986; Andersen et al., 2003). While volatility forecasting quickly became
central to the financial econometrics literature due to its importance for risk measurement
and management, research focusing on the entire return distribution is still very rare in the
literature.1
One of the main reasons why researchers may not be focused on characterizing the entire
return distribution is a prevailing practice of convenient mean-variance analysis that is still
central to modern asset pricing theories. Unfortunately, investor’s choices guided using
first two moments are restricted by binding assumptions, such as multivariate normality of
stock returns or a quadratic utility function. More importantly, an investor is restricted to
have classical preferences based on von Neumann-Morgenstern expected utility. In contrast
to this, Rostek (2010) recently developed a notion of quantile maximization and quantile
utility preferences. This important shift in decision-theoretic foundations provokes us to
depart from the limited mean-variance thinking, and work with entire distributions.
Specification and estimation of the entire conditional distribution of future price changes
is useful for a number of important financial decisions. Prime examples include portfolio
selection when returns are non-Gaussian, (tail) risk measurement and management, market
timing strategies with precise entries and exits reflecting information in tails. Despite its
importance, forecasting of the conditional distribution of future returns has so far attracted
little attention in contrast to point forecasts and its uncertainty mentioned before. In this
paper, we present a simple approach to forecasting a conditional distribution of stock returns
using parameterized ordered binary choice regression. Although focusing on financial returns,
we note that our approach may be useful to many other applications where the conditional
distribution forecasts are of interest.
The majority of studies focusing on prediction of conditional return distributions characterize the cumulative conditional distribution by a collection of conditional quantiles (Engle
1
Few studies focused on directional forecasts, or threshold exceedances (Christoffersen and Diebold, 2006;
Chung and Hong, 2007; Nyberg, 2011).
2
and Manganelli, 2004; Cenesizoglu and Timmermann, 2008; Žikeš and Barunı́k, 2014; Pedersen, 2015). In a notable contribution, Foresi and Peracchi (1995) focus on a collection of
conditional probabilities and describe the cumulative distribution function of excess returns
using a set of separate logistic regressions. To be able to approximate the distribution function, Foresi and Peracchi (1995) estimate a sequence of conditional binary choice models over
a grid of values corresponding to different points of the distribution. In addition to a need
to estimate a sequence of separate equations to capture individual points of the distribution,
one has to ensure monotonicity crucial for distribution forecasts. Peracchi (2002) argues
that the conditional distributions approach has numerous advantages over the conditional
quantile approach, and Leorato and Peracchi (2015) continue further their comparison. The
approach has been also considered by Fortin et al. (2011), Rothe (2012), Chernozhukov et al.
(2013), Hothorn et al. (2014), and Taylor and Yu (2016).
In this paper, we develop further the ideas set forth by Foresi and Peracchi (1995) and
present a related simple model for forecasting conditional return distributions. The proposed
model is based on the ordered binary choice regression that is able to forecast the entire
predictive distribution of stock returns using fewer parameters than the set of separate binary
choice regressions. To achieve this substantial reduction in the degree of parameterization,
we tie the coefficients of predictors via smooth dependence on quantile levels. As a side
effect, our approach allows one to impose the desirable monotonicity property more easily.
Our specification can be motivated in a semiparametric way, as we approximate the smooth
quantile functions by low-order polynomials.
The probability forecasts are conditional on past information contained in returns, as
well as their volatility proxy. The main reason of choosing volatility as one of explanatory
variables is that the cross-sectional relation between risk and expected returns generally
measuring a stock’s risk as the covariance between its return and some factor is well documented in the literature. In the laborious search for proper risk factors, volatility plays a
central role in explaining expected stock returns for decades. Although predictions about
expected returns are essential for understating of classical asset pricing, little is known about
a potential of the factors to precisely identify extreme tail events of the returns distribution.
In our illustrative empirical analysis, we estimate the conditional distributions of 29 most
liquid U.S. stocks and compare their generated forecasts with those from the buy-and-hold
strategy and the benchmark collection of separate binary choice models. The benefits of our
approach are translated to significant economic gains in a simple trading strategy that uses
conditional probability forecasts.
The paper is organized as follows. Section 2 describes the model and emphasized its differences with the collection of separate binary choice models. Section 3 contains information
about the data we use and lays out details of particular specifications. In Section 4, empirical
results are presented. Section 5 concludes.
3
2
The Model
We consider a strictly stationary series of financial returns rt , t = 1, ..., T . Our objective is
to describe, as fully as possible, the conditional cumulative return distribution F (rt |It−1 ) ,
where It−1 includes the history of rt as well as possibly past values of other observable variables. To that end, we model jointly the conditional probabilities for a number of thresholds
that correspond to p > 1 quantile levels
0 < α1 < α2 < ... < αp < 1.
The higher p, the more precise is the description of the conditional distribution. The p
quantile levels correspond to empirical cutoff levels, or empirical quantiles
c1 < c2 < ... < cp .
That is, cj is empirical αj -quantile of returns, j = 1, ..., p. For convenience, define c0 = −∞
and cp+1 = +∞.
Let Λ : u 7→ [0, 1] be a (monotonically increasing) link function. Both unordered and
ordered binary choice models are represented by a collection of conditional probabilities
Pr{rt ≤ cj |It−1 } = Λ (θt,j ) ,
j = 1, ..., p,
for some specification of the driving processes θt,j , j = 1, ..., p. For convenience, define
Λ (θt,0 ) = 0 and Λ (θt,p+1 ) = 1.
Let xt−1,j , j = 1, ..., p be a vector of predictors for I{rt >cj } that may depend on j via
dependence of some of them on cj . For instance, one of predictors may be I{rt−1 >cj } , the
past indicator (dependent on j), while another may be rt−1 , the past return (independent of
j), yet another may represent some volatility measure (also non-specific to j). Suppose for
simplicity that the number of predictors in xt−1,j is the same for all j and equals k.
In the unordered model, the specification for the underlying process θt,j is
θt,j = δ0,j + x′t−1,j δj .
There are no cross-quantile restrictions, and each binary choice problem is parameterized
separately. This results in a flexible but highly parameterized specification. In addition,
no monotonicity, in the language of Foresi and Peracchi (1995), holds in general. That is,
Pr{rt ≤ cj−1 |It−1 } may well exceed Pr{rt ≤ cj |It−1 } with positive probability even though
cj−1 < cj .
In the proposed ordered model, there are cross-quantile restrictions placed on the parameters. In particular, the coefficients of predictors are tied via smooth dependence on
the quantile levels. This leads to a substantial decrease of a degree of parameterization,
4
and in addition allows one to impose the desirable monotonicity property more easily. The
specification for the underlying process θt,j is
θt,j = δ0,j + x′t−1,j δ (αj ) ,
where δ (αj ) are coefficients that are functions of the quantile level. Each quantile-dependent
slope coefficient vector is specified in the following way: δ (αj ) = (δ1 (αj ) , ..., δk (αj ))′ , where
for each ℓ = 1, ..., k,
qℓ
X
2i (αj − 0.5)i · κi,ℓ ,
δℓ (αj ) = κ0,ℓ +
i=1
and qℓ ≤ p − 1. Note that each intercept δ0,j is j-specific and represents an ‘individual effect’
for a particular quantile level, while the slopes δ’s do not have index j, i.e. they depend on
j only via dependence on αj ’s. The motivation behind such specification is semiparametric:
any smooth function on [0, 1] can be approximated to a desired degree of precision by the
system of basis polynomials {αj − 0.5, (αj − 0.5)2 , ..., (αj − 0.5)q } by taking q big enough.
Because all αj ∈ (0, 1), the polynomial form behaves nicely even for large q; the basis
polynomials are uniformly bounded on [0, 1] . The additional weights 2i are introduced to
line up the coefficients κi on a more comparable level.
Let us compare the degrees of parameterization of the unordered and ordered binary choice
models. Denote
k
X
q=
qℓ .
ℓ=1
In the unordered model, the total number of parameters is
KU O = (1 + k) p
(namely, one intercept δ0,j and k slopes δj in each of p equations for θj ) while in the unordered
model, the total number of parameters is
KO = p + k + q
(namely, p intercepts δ0,j and k slopes δ (αj ) each parameterized via 1 + qℓ parameters). The
difference
KU O − KO = k (p − 1) − q
is larger the larger is p, the fineness of the partition by thresholds. The resulting difference
is also positively related to the number of predictors used.2
In the unordered model, the composite loglikelihood corresponding to observation t is
ℓUt O
=
p+1
X
I{rt ≤cj } ln (Λ (θt,j )) ,
j=1
2
In our empirical illustration, p = 41, k = 2 and q1 = 2, q2 = 3. Hence, KU O = 123 while KO = 48, and
the difference is KU O − KO = 75.
5
PT U O
and the total composite likelihood
can be split into p independent likelihoods
t=1 ℓt
PT (j)
t=1 ℓt , where
(j)
ℓt = I{rt ≤cj } ln (Λ (θt,j )) ,
to be maximized over the parameter vector (δ0,j , δj )′ . In the ordered model, the loglikelihood
corresponding to observation t is
ℓO
t
=
p+1
X
I{cj−1 cj }
.
xt−1,j =
ln (1 + |rt−1 |)
The first predictor is a lagged indicator corresponding to the quantile level αj . The second
predictor is a proxy for a volatility measure, with the absolute return dampened by the
logarithmic transformation. Note that the first predictor is specific for a specific quantile,
while the second predictor is common for all quantiles. In the ordered model, we set, after
some experimentation, q1 = 2 and q2 = 3. That is, the polynomial is quadratic in the quantile
level α for the past indicator and cubic for the past volatility proxy.
The full specification of the model for j = 1, ..., p empirical quantiles is
Pr{rt ≤ cj |It−1 } =
exp(−θt,j )
,
1 + exp(−θt,j )
θt,j = δ0,j + δ1 (αj ) I{rt−1 >cj } + δ2 (αj ) ln (1 + |rt−1 |) ,
with the coefficient functions
δ1 (αj ) = κ0,1 + 2(αj − 0.5) · κ1,1 + 22 (αj − 0.5)2 · κ2,1 ,
δ2 (αj ) = κ0,2 + 2(αj − 0.5) · κ1,2 + 22 (αj − 0.5)2 · κ2,2 + 23 (αj − 0.5)3 · κ3,2 .
In total, there are KO = 48 parameters: p = 41 individual intercepts δ0,j and k = 2 slopes
δ (αj ), one parameterized via 1+q1 = 3 parameters, the other via 1+q2 = 4 parameters. This
parametrization is parsimonious enough and approximates the distribution quite well, and
additional terms do not bring significant improvements. Hence, estimating seven parameters
κi,ℓ in addition to individual intercepts is enough to approximate the conditional return
distribution.
Oracle Corporation (ORCL), PepsiCo, Inc. (PEP), Pfizer Inc. (PFE), Procter & Gamble Co (PG), QUALCOMM, Inc. (QCOM), Schlumberger Limited. (SLB), AT&T Inc. (T), Verizon Communications Inc. (VZ),
Wells Fargo & Co (WFC), Wal-Mart Stores, Inc. (WMT), Exxon Mobil Corporation (XOM).
7
4
Empirical Findings
We now present the results of estimating the conditional distribution function of returns
using the ordered binary choice model. We consider forecasts for 29 stocks, so first we
present individual estimates of three illustrative stocks, namely Intel Corporation (INTC),
QUALCOMM, Inc. (QCOM), and Exxon Mobil Corporation (XOM), and then we present
the results for all 29 stocks in an aggregate form.
4.1
Parameter estimates
Table 1 shows estimates of quantile specific intercepts δ0,j in the ordered logit model for the
three illustrative stocks. The intercepts have expected signs, and exhibit expected monotonic behavior decreasing from the left tail to the right tail thus generating an increasing
cumulative distribution function (assuming zero predictors). The intercepts are statistically
significant in most cases except for few cutoff points near the median. The intercept values
are quite similar across three stocks, though there is some variability.
The left plot shown in Figure 2 collects the intercept estimates for all 29 stocks, and
reveals that indeed the intercepts are similar across the stocks. The plot has a shape similar
to an inverse logit link function Λ (u) defined earlier, with a stronger effect in tails. A
flexible j-specific intercept allows for controlling individual quantile effects, and shows that
unconditional expectations are an important part of the predicted distribution.
The estimates of the seven coefficients κi,ℓ driving the slopes on the predictors for the
three illustrative stocks are shown in Table 2. The coefficients κi,1 correspond to the lagged
indicator predictor I{rt−1 >cj } . All parameter estimates for the lagged indicator are of a small
magnitude, and, except of the parameter driving the quadratic term κ2,1 of XOM, they
are not highly significantly different from zero. The left plot in Figure 1 complements these
findings with estimates of the lagged indicator coefficients for all 29 stocks shown by the boxand-whisker plots. The estimates document a heterogeneous effect of the lagged indicator
on the future probabilities for different stocks. While we document zero coefficients for some
stocks, the quadratic term seems to play role in others.
The second predictor, the past volatility proxy ln (1 + |rt−1 |), carries more important information about future probabilities. The estimated coefficients κi,2 corresponding to the
volatility predictor reveal that the cubic polynomial has almost all coefficients significant,
except that of κ0,2 for all three stocks (see Table 2). The right plot of Figure 1 shows estimated coefficients for all 29 stocks as box-whisker plots. We can see that κ1,2 and κ3,2
are significantly different from zero for most of the stocks, and so the past volatility proxy
contributes significantly to prediction of return distributions.
Figure 2 plots the functions δ1 (αj ), and δ2 (αj ), in addition to the intercepts. Corresponding to heterogenous parameters κi,1 , the function δ1 (αj ) is also heterogeneous in 29 stocks,
8
exhibiting a mixed effect of the lagged indicator predictor (shown in the middle plot of Figure 2). Overall, we can see that the effect is small. In contrast, the coefficient function
δ2 (αj ) implied by the past volatility proxy shows a much stronger and more robust impact.
The probability distribution is predicted strongly in both tails with the expected signs of
coefficients for all 29 stocks alike.
4.2
Economic Significance of Predicted Distributions
To understand the economic usefulness of the proposed model, we study a simple profit rule
for timing the market based on the model forecasts. The idea is to explore the information
from the entire distribution. In the spirit of previous literature (Breen et al., 1989; Pesaran
and Timmermann, 1995; Anatolyev and Gospodinov, 2010), we evaluate the model forecasts
in terms of profits from a trading strategy for active portfolio allocation between holding a
stock and investing into a risk-free asset.
To build a trading strategy, we use a simple rule exploring the difference between the
predicted conditional probability, and unconditional probability Pr{rt ≤ cj } = αj . We sum
the differences over the interval of empirical quantiles [a, b] as
St =
b
X
cj =a
c
Pr{rt ≤ cj |It−1 } − αj .
If we sum all p available quantiles, we are using the information from the entire distribution.
If we want to use the information only about positive returns, we sum only a half of available
empirical quantiles corresponding to the cutoffs at positive returns. For example, in case
the positive returns are predicted with a higher probability than the negative returns for
all corresponding empirical quantiles, the sum St will be positive. Further, it may be useful
to compare St computed for the empirical quantiles corresponding to negative returns, and
to positive returns. After some experimentation, we obtain threshold values for each stock
depending on the shape of conditional distributions, generating consistent profits. Hence
we build the trading strategy on St exceeding these thresholds, but we note that this could
further be optimized for maximum profits.
Starting with a one dollar investment at the beginning of the sample, our investor decides
to hold the stock depending on whether the predicted probability is favorable or not. We
compare the cumulative returns from this simple market-timing strategy using predictions
from the ordered logit, separate logistic regressions, and a standard benchmark of the buy
and hold strategy for all 29 stocks separately.
Table 3 summarizes the results of running the trading strategies in terms of a mean and
median return, volatility, as well as Sharpe ratio. While the benchmark strategy earns 0.929
return with 0.235 volatility, when investor uses predictions about the entire distribution
from separate logit models, a similar mean return of 0.955 is obtained with a much lower
9
volatility. In the meantime, our proposed ordered logit model generates 1.481 mean return
with volatility similar to that for separate logits. The Sharpe ratio showing a risk-adjusted
return, or average return per unit of volatility, reveals that the ordered logit model generates
twice as high returns when we take risk into account. The figures for median values are even
more impressive.
Figure 4 shows the returns, volatilities, and Sharpe ratios for all 29 stocks by box-andwhisker plots. The figure reveals that the naive buy and hold strategy yields lowest returns
with highest volatility. In terms of risk-adjusted Sharpe ratios, the separate logistic regressions yield a similar result, while the ordered logit model makes a marked improvement. One
can see positive Sharpe ratios for almost all considered stocks.
Figure 3 looks closely at the cumulative returns and drawdowns of the three illustrative
stocks we used earlier. It can be seen that returns from the ordered logit strategy are
consistent over time, with a lowest drawdown. This holds even in the case of XOM that
has been growing for the whole period, hence making it difficult to beat the buy and hold
strategy.
Finally, we compare the relative performance of all three strategies in Figure 5. The left
plot in the Figure compares the ordered logit with the ‘market,’ or naive buy and hold
strategy, while the right plot compares the ordered logit with separate logits. We document
a consistently better performance of the proposed ordered logit model in comparison to both
the unordered logit as well as naive strategy for all 29 stocks.
5
Conclusion
This paper investigates predictability of stock market return distributions. We propose a
relatively parsimonious parametrization of an ordered binary choice regression that precisely
forecasts the conditional probability distribution of returns. In order to see how useful the
model is economically, we use distributional predictions in a simple market timing strategy. Using 29 liquid U.S. stocks we find significant economic gains when compared to the
benchmarks.
Our findings are useful for risk management and measurement, especially focused on tails,
or building trading strategies using the entire conditional distribution of returns. The model
nevertheless has a much wider potential use in any application that exploits distribution
forecasts including forecasting of interest rates, term structures, and macroeconomic variables.
10
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6
Tables
Table 1. Estimates of intercepts δ0,j in the ordered logit specification θt,j = δ0,j + x′t−1,j δ (αj )
for the three illustrative stocks.
INTC
QCOM
XOM
INTC
QCOM
XOM
1%
5.420
5.123
5.402
99%
−5.248
−5.433
−5.759
(0.243)
(0.238)
(0.274)
2.5%
4.367
4.160
4.367
97.5%
−4.270
−4.429
−4.715
(0.169)
(0.156)
(0.183)
5%
3.542
3.396
3.566
95%
−3.472
−3.591
−3.852
7.5%
3.040
2.925
3.073
92.5%
−2.990
−3.083
−3.309
10%
2.658
2.574
2.698
90%
−2.610
−2.694
−2.905
12.5%
2.354
2.287
2.386
87.5%
−2.315
−2.390
−2.574
15%
2.062
2.052
2.131
85%
−2.064
−2.122
−2.291
(0.078)
(0.082)
(0.082)
17.5%
1.824
1.837
1.905
82.5%
−1.839
−1.893
−2.042
(0.075)
(0.077)
(0.078)
20%
1.614
1.645
1.697
80%
−1.647
−1.681
−1.820
22.5%
1.429
1.473
1.515
77.5%
−1.465
−1.484
−1.627
25%
1.257
1.316
1.334
75%
−1.298
−1.308
−1.440
27.5%
1.092
1.165
1.177
72.5%
−1.153
−1.149
−1.257
30%
0.938
1.014
1.029
70%
−1.012
−1.000
−1.082
(0.065)
(0.067)
(0.066)
32.5%
0.800
0.880
0.881
67.5%
−0.876
−0.855
−0.933
(0.065)
(0.066)
(0.065)
35%
0.664
0.751
0.738
65%
−0.746
−0.721
−0.791
37.5%
0.535
0.627
0.611
62.5%
−0.625
−0.593
−0.658
40%
0.413
0.503
0.482
60%
−0.506
−0.463
−0.530
42.5%
0.293
0.383
0.361
57.5%
−0.392
−0.335
−0.405
45%
0.176
0.263
0.233
55%
−0.277
−0.215
−0.275
(0.061)
(0.061)
(0.060)
47.5%
0.062
0.147
0.111
52.5%
−0.165
−0.096
−0.139
(0.060)
(0.061)
(0.059)
0.50%
−0.074
0.020
−0.018
(0.280)
(0.194)
(0.146)
(0.123)
(0.107)
(0.096)
(0.087)
(0.083)
(0.078)
(0.075)
(0.072)
(0.069)
(0.067)
(0.066)
(0.064)
(0.063)
(0.062)
(0.061)
(0.061)
(0.060)
(0.060)
(0.262)
(0.189)
(0.153)
(0.131)
(0.116)
(0.104)
(0.094)
(0.087)
(0.082)
(0.075)
(0.071)
(0.068)
(0.066)
(0.065)
(0.063)
(0.062)
(0.062)
(0.061)
(0.061)
(0.060)
(0.061)
(0.276)
(0.192)
(0.152)
(0.127)
(0.112)
(0.101)
(0.091)
(0.084)
(0.080)
(0.076)
(0.072)
(0.069)
(0.067)
(0.065)
(0.064)
(0.062)
(0.061)
(0.061)
(0.060)
(0.059)
(0.059)
13
(0.127)
(0.103)
(0.092)
(0.084)
(0.071)
(0.069)
(0.070)
(0.067)
(0.064)
(0.063)
(0.062)
(0.061)
(0.121)
(0.102)
(0.094)
(0.086)
(0.075)
(0.073)
(0.071)
(0.067)
(0.064)
(0.063)
(0.062)
(0.062)
(0.136)
(0.114)
(0.100)
(0.089)
(0.074)
(0.071)
(0.068)
(0.067)
(0.064)
(0.062)
(0.062)
(0.061)
Table 2. Estimates of slope coefficients κi,ℓ in the ordered logit specification δℓ (αj ) =
Pℓ i
κ0,ℓ + qi=1
2 (αj − 0.5)i · κi,ℓ for the three illustrative stocks.
INTC
QCOM
XOM
κ0,1
−0.006
0.016
0.025
κ1,1
0.059
0.007
κ2,1
−0.057
−0.100
(0.024)
(0.042)
(0.095)
(0.024)
(0.045)
(0.096)
INTC
QCOM
XOM
κ0,2
4.940
−2.960
0.988
−0.019
κ1,2
21.27
32.40
57.69
−0.164
κ2,2
−5.78
14.81
17.73
κ3,2
29.29
12.52
17.74
(0.025)
(0.044)
(0.093)
(3.920)
(6.26)
(4.64)
(7.93)
(3.822)
(5.20)
(4.65)
(5.87)
(4.869)
(7.10)
(5.73)
(9.65)
Table 3. Mean and median return-volatility characteristics from three strading strategies
for all 29 stocks.
Market
Logit
Ordered Logit
return
0.929
0.955
1.481
mean
volatility
0.235
0.179
0.177
Sharpe
0.059
0.162
0.313
14
return
0.005
0.203
0.608
median
volatility
0.217
0.170
0.166
Sharpe
0.002
0.158
0.311
Figures: Parameter Estimates
20
40
●
●
−0.2
−20
0
coefficients for δ2(α)
−0.1
0.0
coefficients for δ1(α)
0.1
60
0.2
7
κ0,1
κ1,1
●
κ2,1
κ0,2
κ1,2
κ2,2
κ3,2
−6
0.01
0.2
0.4
0.6
α
0.8
0.99
0
−100
−0.2
−50
−0.1
δ2(α)
0.0
δ1(α)
0
−4
−2
δ0,j
2
50
0.1
4
6
0.2
100
Figure 1: Parameter estimates: ordered logit parameters estimated for all 29 stocks shown
by box-and-whisker plots.
0.01
0.2
0.4
0.6
α
0.8
0.99
0.01
0.2
0.4
0.6
0.8
0.99
α
Figure 2: Parameter estimates: coefficient functions implied by parameters estimated for all
29 stocks. Minimum and maximum values shown as light grey area, 50% of distribution as
grey area, and median as black line.
15
Results: Economic Evaluation
Performance of QCOM
Performance of XOM
3.0
2.5
2.0
Cumulatuve Return
2.0
1.5
Cumulatuve Return
1.0
0.5
1.0
1.4
1.2
1.0
0.8
0.6
0.2
0.4
Cumulatuve Return
3.5
1.6
Performance of INTC
1.5
8
2012
2014
2016
2006
2008
2010
2012
2014
2016
2008
2010
2012
2014
2016
2006
2008
2010
2012
2014
2016
2008
2010
2012
2014
2016
2006
2008
2010
2012
2014
2016
−0.1
0.0
2006
−0.4
−0.5
−0.8
−0.6
−0.6
−0.4
Drawdown
Drawdown
−0.2
0.0
−0.2
−0.4
Drawdown
2006
−0.2
2010
−0.3
2008
0.0
2006
1.0
6
0.5
Figure 3: Performance: cumulative return and drawdown of three typical stocks. Trading
strategy based on the probability predictions from ordered logit model as bold black line,
strategy from separate logits as black line, and benchmark buy and hold strategy as grey.
●
0.3
Volatility
0.1
0
−0.5
0.2
2
●
●
0.0
●
●
Sharpe Ratio
4
●
●
●
Return
0.5
0.4
●
●
−2
0.0
−1.0
●
logit
ordered logit
market
●
logit
ordered logit
market
logit
ordered logit
market
Figure 4: Performance: return, volatility and Sharpe ratio for three strategies for all 29
stocks shown by box-and-whisker plots.
16
3
1
2
Relative Performance
3
2
1
Relative Performance
4
Ordered Logit vs. Logit
4
Ordered Logit vs. Market
2006
2008
2010
2012
2014
2016
2006
2008
2010
2012
2014
2016
Figure 5: Relative performance: trading strategy based on the probability predictions from
ordered logit model relative to separate logits (right) as well as benchmark ‘market’ (left).
Median value from all 29 stocks in black line surrounded with 90% of the distribution in
grey area. Note that value of one shows equal performance of both strategies compared.
17