Predicting completion of cash acquisitions using option implied risk-neutral probabilities
National Research
University - Higher School of EconomicsCollege of Economics and Finance
Graduation Thesis
Predicting completion of
cash acquisitions using option implied risk-neutral probabilities
Author:Andreeva
Supervisor:Dr.
Sergey Gelman
. June 16, 2015
Introduction
In the year 2014 global Mergers and Acquisition activity
experienced an outstanding jump with over 40 000 transactions and total value
of approximately $ 3,5 billion. This corresponds to a 47% increase compared to
year 2013 and also is the seven-year high since 2007. The significantly
increased M&A activity might again raise demand for academic research on
the topic, in particular, on sources of risk involved in the deal. Deal failure
is widely accepted to be the key risk factor and, therefore, accurate estimates
of the probability of success could potentially be of interest to a broad
audience that is concerned with outcomes of pending transactions. The list of
interested parties includes target and the acquirer, banks, hedge funds and
asset management firms, individual investors and many others. the acquisition
offer is made target company’s stock price movements, as indicated by Samuelson
and Rosenthal (1986), collect the market expectations of the ultimate deal
outcome. However, market often misperceives and reacts with a price increase
even for failed deals. The topic that was widely studied in recent financial
literature is the reaction of options market to M&A announcements. This
paper examines the predictive power of option prices after the deal announcement
in relation to forecasting the outcome of cash corporate takeovers and compares
it with a more commonly accepted probability measure based on stock market
reaction. The key contribution of this research is the proposed method that
estimates risk-neutral probability of success from option prices of the target
company and the empirical testing of its forecasting ability. The approach is
based on the relationship between risk-neutral cumulative density function and
the first derivative of option price with respect to strike price that was
discovered by Breeden and Litzenberger (1978). Estimation of risk-neutral CDF
for specific intervals uses first-difference approximation highlighted by
Gelman (2005). To my knowledge, this is the first work that directly relates
these probability estimates to M&A deal success or failure and tests their
predictive power. study of option-implied probabilities of success leads to a
conclusion that option prices after announcement are indeed a worthy predictor
of the deal’s outcome. For the analysed sample risk-neutral probabilities
outperform stock-implied probabilities in terms of forecasting power for the
period of 3 weeks after the deal announcement. However, the empirical analysis
also reveals that despite being a good predictor, option-implied probabilities
tend to underestimate the success rate of the announced deals, which may lead
to potential arbitrage opportunities in the option market. This study also
briefly examines merger arbitrage strategies on the stock market and the
dependence of excess returns on the risk-neutral probability estimates. paper
is structured as follows: it starts with an overview of existing financial
literature on related topics that include studies of stock and option markets
behaviour after the deal announcement and its implication to deal outcome
predictability, analysis of merger arbitrage and associated excess returns and
papers that develop pricing models for options of target companies. Next
chapter is devoted to the derivation of the model used to estimate risk-neutral
probabilities and the rational for its application. The paper continues with a
summary of data selection process and key features of the obtained sample.
Methodology section that follows up provides a description of forecast
performance estimation and hypothesis testing procedures. Finally, last section
summarizes empirical results of predictive power assessment for risk-neutral
probabilities as well as their comparisons with stock-implied probabilities
defined as in Samuelson and Rosenthal (1986).
Literature overview
Mergers and acquisitions are, perhaps, the most heavily
studied topic in corporate finance. However, the literature that studies the
source of uncertainty for M&A deals is relatively scares, as most of the works
concentrate on determining value and wealth effects as well as their drivers.
the first steps in academic research that investigates M&A deal success
probability were taken in 1980s following the boom of M&A activity driven
by private equity firms. Brown and Raymond (1986) proposed a method of
estimating success probability based on a fallback price that they assumed to
be equal to the pre-announcement price (on average over a number of weeks). A
more sophisticated approach was outlined by Samuelson and Rosenthal (1986).
They focused on the study of cash corporate takeovers and started with the
empirical formula for the after announcement stock price as a function of
future stock price in case of deal’s success (offer price) and failure (some
fallback price). Then, following the assumption that success probability and
fallback price are constant for at least some time-intervals, they employ an
econometric method to forecast the fallback price and, thus, the success
probability. However, they didn’t distinguish between risk-neutral and actual
probabilities. The conclusion of their paper is that uncertainties involved in
takeovers are reflected well by the stock market and that market’s forecasts
improve monotonically with time. Both of the mentioned works allow for a
calculation of success probability using stock price data through out the
announcement period (from the announcement to the resolution day) and, thus,
can generate a sequence of probabilities. Regarding this matter they can be
contrasted with a study of Walkling (1985) that develops a multivariate technic
created for the purpose of a single estimation around the announcement date.
This paper was greatly influenced by Samuelson and Rosenthal’s work and mainly
follows its footsteps in the framework of methodology used to access
probabilistic forecasts. and Rosenthal as well as Brown and Raymond focus only
on the stock market price movements and ignore the effect of proposed
acquisitions on derivatives market, options in particular. Jayamaran, Mandelker
and Shastri (1991) were among the first to show that strong inferences
regarding M&A activity can be drawn from option prices. Conclusion they
reached is that implied volatilities for target companies increased
significantly prior to the announcement, suggesting that market anticipated a
takeover bid. Levy and Yoder (1993) reached the same conclusion, pointing out
that option-implied standard deviations for the target firm rise drastically 3
days before the deal announcement. Adesi et al. (1994) pioneered with
investigating post-announcement option volatilities to infer predictions
regarding resolution date. More recent work by Wang (2009) replicates their
approach to draw inferences regarding market’s assessment of the deal’s success
probability. In his work he constructs a volatility ratio of the observed
implied volatility to the fallback volatility that is taken as historical
average and shows that for the failed deals the ratio converges to one, while
for successful it does not.works that focus on option pricing in the time of
the expected M&A deal include the paper by Subramanian (2004) and Martinez
(2009). In their studies Subramanian develops an arbitrage-free model to price
options in stock-for-stock deals, while Martinez focuses on option pricing for
cash tender offers. Subramanian exploits a theoretically perfect correlation
between acquirer’s and target’s stock price in stock-for-stock deals and solves
the model by imposing assumptions that fallback price follows a given basket of
securities and that arrival process in a Poisson process with constant
intensity determines risk-neutral probability. The paper by Martinez is of
particular interest due to its close relation with the topic of this paper, as
the developed formula for option pricing allows
recovering both, risk-neutral success probability and the fallback price.
Moreover, Martinez compares estimated option-implied success probabilities to
the commonly used “naïve” ones obtained using the approach of Brown and Raymond (1986)
and arrives at conclusion that risk-neutral probabilities are a better
predictor of offer’s outcome. and Pulvino (2001) examined risk and return in
risk arbitrage and demonstrated that risk arbitrage returns are positively
correlated with market returns in severely depreciating markets, but
uncorrelated with market returns in flat or appreciating markets. Baker and
Savasoglu (2002) reported that a diversified portfolio of risk arbitrage
positions generate a modest abnormal return of 0,6%-0,9% per month and, most
importantly to this study, that returns to risk arbitrage increase in ex ante
prediction of completion risk. recover risk-neutral probabilities from option
prices this paper follows the approach by Breeden and Litzenberger (1978) that
exploits the relationship between second derivative of option price with
respect to strike price and risk-neutral PDF. One of the first academic works
that related this method of risk-neutral PDF derivation with approximation of
CDF for specific intervals using first and second differences was the paper by
Basset (1997). He used the above-mentioned non-parametric method to bind the
set of probability distributions. In 2005 Gelman applied the same binding
procedure to recover probability intervals for options of a target company that
was undergoing an M&A and compared them with Black-Scholes probability
distributions. Conclusion of his paper was that is worth noting that financial
literature that studies derivation of risk-neutral probability density
functions from option prices accounts for a vast amount of works. Rubenstein
(1994) used the non-parametric binomial trees technique. Risk-neutral
probabilities in this paper are estimated by minimizing the sum of squared
deviations between risk-neutral probabilities associated with binominal stock
price at maturity and the prior risk-neutral probabilities, conditioning on the
restriction that generated probabilities price options and the underlying asset
in such a way that they lie in the existing bid-ask spread. Jackwerth and
Rubinstein (1996) further extended this approach and introduced smoothens
criteria.
Model
theorem of asset pricing states that in a complete market a
derivative’s price should equal to the discounted expected value of its future
payoff under unique risk-neutral measure. Cox and Ross (1976) showed that a
European call option in continuous time could be priced as follows:
- price of the underlying asset at time t, T=t+ - expiration date, K - strike price,
-risk-free rate, - risk-neutral probability density
function. shown by Breeden and Litzenberger (1978) the risk-neutral probability
distribution can be recovered from the option price via differentiation. Taking
the first derivative with respect to strike price yields us:
PDF properties and rearranging the equation we can express
the first-order derivative as a function of the cumulative density function of
the risk-neutral distribution and get a direct dependents between “exercise
price delta” and risk-neutral CDF:
CDF is always less or equal to 1 it follows:
arbitrage condition then is that first derivative of the call
price function with respect to strike price should be negative but greater than
. In other words, price of a call
should be a decreasing function of strike price, but the fall should be less or
equal to the discounted value of the strike price increase.risk neutral CDF and
differentiating once again will yield us PDF that is proportionate to the
second derivative of option’s price to strike price:
implies that call price function is convex with respect to
strike price as PDF is non-negative. Any local non-convexity would generate
negative risk-neutral probabilities and would, therefore, violate the
no-arbitrage condition.the above-mentioned equations the risk-neutral PDF can
be easily estimated from call prices. Many different techniques have been
developed to do so. Generally, these techniques can be divided into 2 different
approaches. First approach is to assume PDF to be of some kind of functional
form and then directly use equation 1) to fit the resulting theoretical option
prices to observed ones in order to estimate the free parameters in the
distribution. However, this approach is very restrictive and relies on the
assumption regarding PDF distribution. Second, non-parametric approach is much
more preferable. It uses equation 5) to derive risk-neutral PDF. However,
strike price distribution is, in fact, not continuous. Thus, non-parametric
techniques use interpolation and extrapolation to obtain continuous option
pricing function and then differentiate it in order to derive risk-neutral PDF.
Numerous approaches to interpolation have been developed: Shimko (1993) fitted
the volatility smile with polynomials, Jackwerth and Rubenstein (1996) used
quadratic approximation, Ait-Sahalia and Lo (1998) exploited kernel
regressions, while Bliss and Panigirtzoglou (2002) fitted the volatility smile
with cubic spline. Unfortunately, none of the above listed approaches is
applicable in case of expected M&A deal due to violation of continuity of
probabilities of different states and other necessary assumptions. it up, the
most appropriate way to deal with strike price discontinuity in our case, as
mentioned by Gelman (2005), is to approximate the derivatives through first and
second differences:
that delta in strike prices is the same for both, and . This assumption is, in fact,
usually satisfied with rare exceptions for deep out-of-the-money or
in-the-money options., risk-neutral CDF can be approximated as:
PDF as:
Finally, the probability of a stock price to fall into the
interval between and at option's maturity is:
equation above provides a simple and elegant way to
approximate the risk-neutral probability of the stock price to lie in a certain
interval at a specified moment of time, maturity of the option. In application
to expected M&A deal this approach allows us to estimate risk-neutral
probability of the offer’s success. The rational behind this conclusion is as
follows: if the acquisition offer is successful, target shares will trade at a
price equal or very close to the offer bid before being delisted. If, on the
contrary, deal was unsuccessful prices will settle at the new “fallback” level.
Therefore, if we were to choose 3 options: and with maturity date that is close to,
but after deal's resolution and strike prices such that offer price per share
lies exactly between and , we would be able to forecast the success probability and
outcome of the announced deal using option-implied risk-neutral probabilities.
, this approach has some serious limitations that should be mentioned. Firstly
and most importantly, it requires the resolution date to be known in advance
which is a rare thing in M&A announcements. Thus, for it to be practically
applicable in deal outcome forecasting we will require some sort of estimate
for the resolution date. Secondly, first and second difference approximation
does not insure non-violation of no-arbitrage conditions. For some
discretionary data we can obtain probabilities that would be negative or
greater than one. In this paper this problem will be further discussed in
methodology section. Moreover, we can't get the probability estimates for
intervals other than ; which can be quite large and, thus, possibly include the
fallback price that the stock settles to in case of the deal’s failure.
Finally, this approach requires Target Company to have option with matching
maturities and strike price that are sufficiently liquid. This limits the
practical application of the method, as only a few of the potential M&A
targets satisfy this criterion.
Data Selection
risk neutral samuelson Rosenthal
This work studies cash acquisitions with the announcement
date falling in the period from January 2010 to December 2013. The sample is
restricted to cash only takeovers in order to eliminate additional influence on
the target's stock price. All of the deal data e.g. companies’ names, effective
dates and offer prices were taken from Dealogic. OptionMetrics database was
used to choose suitable options (i.e. options with needed maturities and
strikes). Then option data e.g. prices, strikes and maturity dates were
downloaded from Bloomberg. Stock prices used to
access “naïve” probabilities and excess returns were also taken from
Bloomberg.the
above-mentioned time period Dealogic reports 19 743 corporate takeover offers
where the type of payment exclusively cash. Competing offers, pending deals and
partial acquisitions i.e. those with an offer for less than 80% of the
outstanding shares were excluded. Sample size dropped to 2 179 deals. Then
sample was further restricted to only include target companies with market value
of equity higher than $ 1 bln, as they are more likely to have options traded.
Deal duration (number of days until the offer either succeeded or failed) was
insured to be more than 30 days in order to build in dynamics in risk-neutral
probabilities estimation. The resulting sample consists of 306 deals.
Significant sample size reduction shows that most of the companies acquired are
relatively small and are less likely to have options traded on their stock. the
criterion of OptionMetrics to have data on options traded for the target
company and further insuring that there are options with fitting maturities and
strike prices the sample of 164 deals was obtained that was then further used
for analysis. Out of 164 deal offers 126 succeeded while 38 failed to reach
agreement. Most of the target companies (approximately 97%) are U.S. companies
which is not surprising due to United States having the most developed
derivatives market. Median deal duration is 94 days; average duration-128 days
and the longest deal took 636 days. Table 1 reports percentiles for deal
durations. Table 2 a), b) summarizes information on 5 successful and 5
unsuccessful deals from the sample for which target companies are largest in
market size. Price before the announcement was estimated to be a 5-day average
2 weeks prior to the announcement.
1. Percentiles for deal durations
Percentile
|
5%
|
25%
|
50%
|
75%
|
95%
|
Deal Duration
|
35
|
54
|
94
|
161
|
366
|
2 a). Information on 5 largest deals.
Target Company
|
Target Ticker
|
Acquirer Company
|
Target Equity
Value, mln $
|
Anadarko
Petroleum Corp
|
APC
|
BHP Billiton Ltd
|
44 603
|
Alcoa Inc.
|
AA
|
Rio Tinto plc.
|
27 329
|
Dell Inc.
|
DELL
|
Silver Lake
Management LLC (MBO)
|
21 073
|
HJ Heinz Co
|
HNZ
|
Berkshire
Hathaway Inc.; 3G Capital Inc.
|
23 576
|
Genzyme Corp
|
GENZ
|
Takeda
Pharmaceutical Co Ltd
|
21 237
|
Goodrich Corp
|
GR
|
United
Technologies Corp
|
16 513
|
Whole Foods
Market Inc.
|
WFM
|
Kohlberg Kravis
Roberts & Co and Bain Capital
|
15 947
|
Life
Technologies Corp
|
LIFE
|
Thermo Fisher
Scientific Inc.
|
13 641
|
Sara Lee Corp
|
SLE
|
JBS SA;
Blackstone Group LP
|
13 425
|
Motorola
Mobility Holdings Inc.
|
MMI
|
Google Inc.
|
12 938
|
Table 2 b). Information on 5 largest deals.
Target Ticker
|
Announcement
date
|
Resolution date
|
Offer price,
$per share
|
Target price
before announcement, $ per share
|
Target Price,
Completion Date, $ per share
|
Offer premium, $
per share
|
|
|
Success
|
Failure
|
|
|
|
|
APC
|
30.12.10
|
|
15.03.12
|
90
|
68,5
|
|
31,4%
|
AA
|
03.05.11
|
|
11.09.12
|
25,5
|
16,438
|
|
55,1%
|
DELL
|
05.02.13
|
29.10.13
|
|
13,88
|
12,8
|
13,86
|
8,7%
|
HNZ
|
14.02.13
|
07.06.13
|
|
72,5
|
60,7
|
72,49
|
19,5%
|
GENZ
|
14.11.10
|
|
14.03.12
|
82
|
72,18
|
|
13,6%
|
GR
|
21.09.11
|
26.07.12
|
|
127,5
|
86,76
|
127,48
|
47,0%
|
WFM
|
18.08.11
|
|
18.08.12
|
90
|
31,2746
|
|
187,8%
|
LIFE
|
15.04.13
|
03.02.14
|
|
76,13
|
71,11
|
76,04
|
7,1%
|
SLE
|
18.12.10
|
|
14.03.12
|
21
|
17,43
|
|
20,5%
|
MMI
|
15.08.11
|
22.05.12
|
|
40
|
38,13
|
39,98
|
4,9%
|
our sample of 164 cash tender offers with options traded on
the target company. As we observe in Table 1, the length of the offer period
varied significantly for the chosen sample. Thus, for the sake of comparability
we consider two intervals for which risk-neutral probability forecasts will be
analysed: 3 weeks after the announcement date and 3 weeks prior to deal’s
resolution. For each target company a daily time-series of option bid and ask
prices and stock prices were constructed for post-announcement days d=1, 2, …,
15 and pre-resolution days d=-15, -14, …, -1. Then, daily risk-neutral
probability forecasts were calculated using equation 10) for the
above-mentioned time period. Risk-free rate was estimated as 90 days T-bill
rate. first difference approximation of risk-neutral CDF doesn’t insure that
probabilities lie within the interval [0,1] the following procedure was
followed: for probabilities that satisfied the [0,1] condition midpoint option
price was used in calculations, for those outside the desired range - some
weighted average of bid and ask price that would insure convexity of option
price with respect to strike price. Weights assigned to bid and ask were
conditioned to be lower than 1 to insure that chosen option price lies within
bid-ask spread. Furthermore, to insure no arbitrage option prices were checked
to satisfy the following bound:
analyse the forecasting performance of success probabilities
derived from option prices Murphy’s Partition of the Brier score was used - a
widely employed measure in probabilistic forecasting. For this purpose daily
probability estimates were averaged to weekly and grouped in discrete
categories and then observed success rate for each category was calculated. For
binary events Brier score represents the standardized measure of forecasts’
mean-square error and is defined as:
N-number of forecasts, - predicted probability of success
for i-th observation, -actual outcome (1 if offer succeded, 0 if failed). Lower
Brier score corresponds to more accurate forecasts. Murhpy's partition
decomposes the original Brier score into 3 components: Uncertainty (or Base
rate), Calibration and Resolution. The partition can be represented as:
z is the success frequency for the whole sample, - frequency of forecasts falling to
probability category j, - forecasted success frequency for category j (for example
for the category that aggregates probability estimates from 0,2 to 0,3 would be 0,25), - actual success frequency for
category j. The first term, or uncertainty component, is in fact, independent
of the probability forecasts and simply depends on the overall probability of
success. Second term measure the calibration of the forecasts, i.e. how close
the probability estimates are to the ex post observed success frequencies.
Perfectly unbiased forecasts would always generate and, thus, imply calibration
component equal to 0. Therefore, a reduction in calibration component, holding
other things constant, would improve the Brier score. Third term represents the
forecasts' resolution or, in other words, by how much the conditional
probabilities given the different forecasts deviate from the sample average.
The higher the resolution component, the lower the Brier score. However, there
is usually a trade off between calibration and resolution components. In
general, Brier score encourages forecast discrimination as long as calibration
is not offset too significantly.proceed with estimating «naive» probabilities
derived from stock prices and testing the hypothesis whether option-implied
probabilities outperform them in terms of forecasting quality. Samuelson and
Rosenthal (1986) evaluated the probability of a tender offer's as:
-price of the stock at time t, -fallback price, -offer price per share, - future value of the stock price at
the resolution date that was defined as and is defined as risk-free rate of
return for the period starting from time t to the resolution date T.price was
estimated based a sample of failed deals using OLS regression with restricted
coefficients. The proposition was that fallback price can be modelled as a
weighted average of the stock price before the announcement and the offer bid:
was restricted to 0 and =1-. To eliminate the effect of
potential information leakages or market fluctuations the pre-announcement
price was taken as a five-day average 2 weeks before the announcement, while
fallback price - five-day average 2 weeks after the resolution. both
probability forecasting approaches daily predictions were gathered into weekly
by taking the 5-day average and then probit regressions of the deal's outcome
(was viewed as a binary event: 1 in case of success and 0 in case of failure)
on both estimates jointly and separately were fitted for each week. Predictive
power of risk-neutral and naïve
probabilities was compared on the basis of pseudo- and significance of coefficients.
research is finalized with a brief evaluation of merger arbitrage and its
dependents on the option-implied probability of success. After the M&A
announcement stock of the target company generally trades at a price below the
one offered by acquiring company. The difference between target’s stock price
and the offer price is commonly known as arbitrage spread. Merger arbitrage, or
risk arbitrage, is an investment strategy that makes an attempt to profit from
this spread. For cash offers the strategy is to simply buy the target’s stock
and hold it until the deal’s resolution, expecting to sell it at the offer
price if the offer is successful. The key feature to point out is that risk of
this strategy is not linear. In case of the offer’s success investor captures
the arbitrage spread, but if the deal fails he incurs a loss that is usually
larger than profit that would have been obtained if the deal succeeded. Therefore,
inside on the probability of the deal’s success could potentially improve the
excess returns from merger arbitrage strategies. evaluate the hypothesis
whether obtained risk-neutral probability forecasts have any implications
regarding potential merger arbitrage profits on the stock market 4 different
portfolios (featuring different share of stocks depending on their respective
risk-neutral probability forecasts) were constructed using the chosen sample
and their returns were compared to each other and to the chosen benchmark
(returns on Hedge Fund Merger Arbitrage index) that was used in excess returns
estimation.
Empirical results. Risk-neutral probability forecasts
and their predictive power
For the sample of 164 deals probability of the tender offer
success was calculated using equation 10 for each trading day during the chosen
period of 3 weeks after the announcement and 3 weeks before the resolution
(denoted as week -3, week -2 and week-1). 24 of the deals didn’t have quoted
option prices through out the whole 6-week period and for 11 deals no
combination of bid and ask could guarantee convexity and, thus, they were
excluded from the sample. For the remaining 129 deals, out of which 100
succeeded and 29 failed weekly forecasts were calculated by taking five-day
average.display the option-implied probability forecasts and gain additional
inside on their predictive power each weekly forecast was assigned to one of 6
subintervals: [0.0; 0.1), [0.1; 0.2), [0.2; 0.4), [0.4; 0.6), [0.6; 0.8) and [0.8;
1). These intervals will further be referred as 0.05, 0.15, …, 0.5, 0.7 and 0.9
probability categories. Taking Statistical limitations into account, finer
partition is undesirable, as it would imply weaker statistical tests. forecast
distribution is shown in Figure 1. It is worth noting that successful offers
make up approximately 78% of the sample. Assuming that sample is
representative, we can denote this proportion as “prior” probability of success
for the typical target. Thus, unsurprisingly largest proportion of forecasts
fall into 0,7 category during week 1 and 2 after the announcement. However, for
successful offers the proportion of forecasts in top 0,9 category increase
significantly over 6 weeks (from 28% in week 1 to 43% in week -1). As expected,
the exact opposite can be observed for unsuccessful deals: the share of
forecasts for unsuccessful takeovers that fall into lowest probability category
(0,05) increase from 20,1% in week 1 to 41,4% in week -1. Another important
trend to point out is that proportion of “successes” in top 2 categories
increase from 96,9% in week 1 to 100% in week -3 and onwards. All of the above
may suggest that our forecast captures true probability quite well and its
predictive power tend to increase when closer to resolution date.
Table 4. Brier score
|
Brier Score
B=B1+B2-B3
|
P-value
|
Base rate B1
|
Calibration rate
B2
|
Resolution rate
B3
|
Week 1
|
0,165
|
0,013
|
0,174
|
0,063
|
0,072
|
Week 2
|
0,163
|
0,0096
|
0,174
|
0,069
|
0,081
|
Week 3
|
0,156
|
0,004
|
0,174
|
0,063
|
0,081
|
|
|
|
|
|
|
Week -3
|
0,135
|
0,000
|
0,174
|
0,059
|
0,099
|
Week -2
|
0,126
|
0,000
|
0,174
|
0,054
|
0,102
|
Week -1
|
0,126
|
0,000
|
0,174
|
0,052
|
0,101
|
|
|
|
|
|
|
Total 6 weeks
|
0,145
|
0,000
|
0,174
|
0,058
|
0,088
|
4 reports weekly Brier scores featuring base rate,
calibration and resolution components. Brier score monotonically decreases and
the main conclusion that can be drawn is that market’s forecasting ability
significantly increases as resolution date becomes closer. The interpretation
of the Brier score, as mentioned by Samuelson, Rosenthal (1986), can be
conveniently described as follows: a Brier score of in terms of forecasting performance
is equivalent to a probability forecast of that is right of the time. For example, week’s 1
Brier score of 0,165 is equivalent to a 79,2% forecast that is right 79,2% of
the times. Week -3 scores an 83,9% correct forecast equivalent while week -1 is
equivalent to 85,2% correct forecast.all of the weeks Brier score is
substantially lower than the base rate component. If the market had used the
base rate frequency of success to access all offers at 78% success probability,
it would have been right only 78% of the time. By conducting a chi-squared test
on a sample variance we test the hypothesis that the difference between
obtained Brier score and base rate is statistically insignificant. Table 3
provides the resulting p-values, rejecting H0 of no difference at 5%
significance level for all weeks and at 1% significance level for all weeks
except for week 1.insight can be gathered from analysis of calibration and
resolution components of Brier score, as they are the source of forecasts’
performance weekly improvement. Calibration and resolution of forecasts both
improve over time, with calibration component falling while resolution
component rising, on average. Brier score’s improvement, on average, is mainly
driven by resolution component’s increase. Resolution component is also the
source of a significant drop of the Brier score in week -3 compared to week 3.
However, we observe jumps of calibration and resolution components in a 3-week
post-announcement period. For week 2 calibration slightly worsens compared to
week 1 and a trade off occurs between resolution and calibration that was
mentioned in the methodology section. For week 3 calibration falls to the
initial week 1 level, while resolution stays unchanged compared to week 2. On
average, however, these effects balance in such a way that Brier score
monotonically improves.well are the obtained probability forecasts calibrated?
To answer this question we will analyse a break down of observed success
frequencies by probability category and week presented in Table 5. Reviewing
these frequencies yields a conclusion that they are highly correlated with our
predicted probability estimates, but considerably greater in most cases. Under
null hypothesis that is the true success probability for tender offers falling
into the j-the category number of successes follows a binominal distribution
with mean and variance . We then test this hypothesis using standard t-test against
the two-sided alternative and an Unconditional Coverage test with likelihood
ratio given by: . Table 5 reports p-values for both tests. Note that for
category 0,7 and 0,9 success rate is 1 in most of the cases and, thus,
Unconditional Coverage test is not applicable. In general, p-values are very
low, rejecting the hypothesis that that is the true probability of success
for the offer falling into the respective probability category. At 5%
significance level only 14 cells out of 36 for t-test and 12 out of 27 for
Unconditional Coverage test cannot reject H0. Note that in our case tests
provide slightly different results, rejecting H0 for different sells.
Rejections are mainly concentrated in 0,3, 0,5 and 0,7 probability categories
and suggest that market severely underestimates the success probability for
tender offers falling into these categories. However, small sample size limits
tests’ power and, therefore, we cannot treat the above-mentioned results as a
definite sign of market inefficiency. Generally, we observe that risk-neutral
probability forecasts tend to underestimate the true probability of success, as
for nearly all entries in the table observed frequencies are higher than the
option-implied probability estimates. Despite their significant predictive
power, option-implied probability forecasts appear to be poorly calibrated.
5. Calibration tests
Probability
category
|
0,05
|
0,15
|
0,3
|
0,5
|
0,7
|
0,9
|
Observed
frequency
|
|
|
|
|
|
|
Week 1
|
0,25
|
0,18
|
0,63
|
0,86
|
0,95
|
1,00
|
Week 2
|
0,22
|
0,17
|
0,64
|
0,91
|
0,94
|
1,00
|
Week 3
|
0,29
|
0,17
|
0,56
|
0,90
|
0,97
|
1,00
|
Week -3
|
0,08
|
0,14
|
0,61
|
0,83
|
1,00
|
1,00
|
Week -2
|
0,08
|
0,13
|
0,64
|
0,80
|
1,00
|
1,00
|
Week -1
|
0,08
|
0,14
|
0,68
|
0,72
|
1,00
|
1,00
|
P-value t-test
|
|
|
|
|
|
|
Week 1
|
0,01
|
0,77
|
0,00
|
0,00
|
0,00
|
0,08
|
Week 2
|
0,02
|
0,87
|
0,00
|
0,00
|
0,00
|
0,07
|
Week 3
|
0,00
|
0,87
|
0,00
|
0,00
|
0,00
|
0,06
|
Week -3
|
0,60
|
0,96
|
0,00
|
0,00
|
0,00
|
0,05
|
Week -2
|
0,60
|
0,84
|
0,00
|
0,02
|
0,00
|
0,03
|
Week -1
|
0,66
|
0,96
|
0,00
|
0,06
|
0,00
|
0,03
|
P-value LR test
|
|
|
|
|
|
|
Week 1
|
0,06
|
0,77
|
0,00
|
0,00
|
0,00
|
N/A
|
Week 2
|
0,08
|
0,87
|
0,00
|
0,00
|
0,00
|
N/A
|
Week 3
|
0,04
|
0,87
|
0,01
|
0,00
|
0,00
|
N/A
|
Week -3
|
0,63
|
0,96
|
0,00
|
0,00
|
N/A
|
N/A
|
0,63
|
0,84
|
0,00
|
0,02
|
N/A
|
N/A
|
Week -1
|
0,68
|
0,96
|
0,00
|
0,05
|
N/A
|
N/A
|
verify this proposition let’s consider the output of weekly
probit regressions of the deal outcome on the risk-neutral probability
forecasts. Table 6 reports results of these regressions, including pseudo , coefficients and p-value of the
coefficient before the probability estimate. We observe that forecasting power
of the probability estimates increase significantly closer to resolution, once
more, supporting the interpretation of Brier score improvement. Pseudo improves from 32,9% in the first week
after the announcement to 55,1% in the week prior to resolution. However, by
looking at coefficients in probit regression without intercept we can denote
that they are significantly higher than one (ranging from 2,19 to 2,43 for all
weeks). This finding suggest that option-implied probability forecasts, indeed,
under predict the probability of a cash takeover success, supporting the
insight gathered from calibration component analysis of the Brier score.
substantial gap between risk-neutral probability estimates and ex post realized
frequency suggests that options on target companies could be undervalued and
indicates a potential to earn excess returns once an appropriate investment
strategy is chosen.
6. Probit regression
|
Week 1
|
Week 2
|
Week 3
|
Week-3
|
Week-2
|
Week-1
|
Average 6 weeks
|
Risk-neutral
probabilities (with constant)
|
|
|
|
|
|
|
Pseudo
|
32,9%
|
38,1%
|
40,4%
|
53,0%
|
53,9%
|
55,1%
|
53,3%
|
|
|
|
|
|
|
|
|
P-value
|
0,00
|
0,00
|
0,00
|
0,00
|
0,00
|
0,00
|
0,00
|
Coefficient
|
3,96
|
4,41
|
4,65
|
5,55
|
5,26
|
5,61
|
6,37
|
Constant
|
-0,9
|
-1,13
|
-1,12
|
-1,48
|
-1,41
|
-1,47
|
-1,76
|
Risk-neutral
probabilities (without constant)
|
|
|
|
|
|
|
|
P-value
|
0,00
|
0,00
|
0,00
|
0,00
|
0,00
|
0,00
|
0,00
|
Coefficient
|
2,19
|
2,32
|
2,18
|
2,40
|
2,43
|
2,37
|
2,38
|
Number of
observations
|
129
|
129
|
129
|
129
|
129
|
129
|
129
|
Comparative analysis of option-implied and
stock-implied forecasts
We continue the
analysis of forecasts’ performance by comparing their predictive power with
that of “naïve” probabilities derived from stock prices. Recall that
“naïve” probabilities are defined, as in Samuelson and Rosenthal (1986)
and given by:
-price of the stock at time t, -fallback price, -offer price per share, (T-t)-time to
deal resolution, - risk-free rate for the appropriate perioda regression for a
fallback price of failed deals on pre-announcement price and offer price yields
the following result (standard deviation of the coefficient in parenthesis),
suggesting that pre-announcement price and offer bid predict fallback price
quite well:
(0,07)
the obtained fallback price estimates into
the probability formula we obtain the weekly “naïve” probability forecasts
for our sample of 129 deals and then compare them with option-implied probability estimates.
For many of the deals stock-implied probabilities were outside the desired
[0;1] range and, thus, those deals had to be excluded from the comparative
analysis. Table 7 summarizes the results (pseudo- and p-values for coefficients) of
cross-sectional probit regressions for each week and for the 6-week average.
results of the comparisons are mixed. For the first 3 weeks after announcement
and for the week that is 3 weeks before resolution risk-neutral forecasts
generate, on average, larger pseudo ,
suggesting to have higher predictive power than “naïve” probabilities.
However, the
situation is reversed, as resolution date approaches. 2
weeks before resolution “naïve” probability estimates experience a
significant jump of their predictive power and start to outperform risk-neutral
probability forecasts (pseudo of 50,3% compared to 44,4% for week -2 and 63,0% compared to
47,6% for week -1 respectively). For the average of the
6-week period risk-neutral forecasts, indeed, outperform “naïve” ones in
terms of predictive quality (pseudo of 41,8% compared to
36,8%).regression of deal outcome on both probability estimates suggest that
predictive power of the model is significantly higher when both forecasts are
used in combination. Both estimates tend to be significant, with the exception
for week -3 and week -1 for which the hypothesis of
no significance is rejected at 1% level for “naïve” and option-implied
forecasts respectively. All in all, no straightforward
answer on whether option-implied probabilities outperform “naïve” ones can
be given. For
the period right after the deal announcement option market tends to react more
wisely, implying better predictive power of risk-neutral probabilities. Closer
to resolution, however, stock market revises its expectations and stock price
movements become more informative. But option-implied probabilities still add
significant value to forecasting deal outcome, especially when used in
combination with stock-implied probabilities.
3. Probit regression output
|
Week 1
|
Week 2
|
Week 3
|
Week-3
|
Week-2
|
Week-1
|
Average 6 weeks
|
Risk-neutral
probabilities
|
|
|
|
|
|
|
Pseudo
|
23,2%
|
33,4%
|
34,7%
|
39,6%
|
44,4%
|
47,6%
|
41,8%
|
P-value
|
0,00
|
0,00
|
0,00
|
0,00
|
0,00
|
0,00
|
0,00
|
"Naïve" probabilities
|
|
|
|
|
|
|
|
Pseudo
|
24,3%
|
26,0%
|
23,7%
|
26,9%
|
50,3%
|
63,0%
|
36,8%
|
P-value
|
0,00
|
0,00
|
0,00
|
0,00
|
0,00
|
0,00
|
0,00
|
Joint
regression Pseudo
|
38,4%
|
57,6%
|
48,6%
|
46,2%
|
69,0%
|
72,3%
|
59,9%
|
P-value
"naïve"
probabilities
|
0,00
|
0,00
|
0,00
|
0,04
|
0,00
|
0,00
|
0,00
|
P-value
risk-neutral probabilities
|
0,00
|
0,00
|
0,00
|
0,00
|
0,00
|
0,05
|
0,00
|
Number of
observations
|
74
|
73
|
78
|
86
|
89
|
89
|
103
|
Deal examples
Let’s now take a closer look at some of the deals from the
sample. We first consider the bid by ConAgra to acquire Ralcorp that ultimately
failed. This deal also provides an example of divergence between option-implied
and stock-implied probability estimates and how it changed over time. The deal
was announced on 29th of April 2011 and the stock market reacted
positively, indicating 75,5% success probability for the first week after the
announcement. Ralcorp shares traded at and above $86 offer bid. However, this
finding contradicted the unsupportive reception of the offer by the Ralcorp’s
board. In the article published by the New York Times on 4th of May
it was outlined that Ralcorp commented that the offer “is not in the best
interest of shareholders” and adopted a shareholder rights plan. The option
market, on the contrast, showed little reaction to the announcement and
risk-neutral probability of success was estimated to be 25,2%. Offer was
withdrawn on 19th of September. By that time bid price was raised to
$94 dollars per share. Option-implied success probability dropped to 17,2% two
weeks before the withdrawal and then to 2,8% one week before the withdrawal.
Stock market still over predicted the success probability, estimating it to be
51,1% two weeks before the resolution. However, during one week before the withdrawal
the gap between option-implied and stock-implied probability estimates shrank
with stock market indicating probability of success to be 12,4%. Daily
forecasted success probabilities for post-announcement and pre-resolution
periods are shown in Figure 2.acquisition of Ariba, provider of cloud-based
collaborative commerce applications, by SAP AG in 2012 is the example of a
successful deal for which “naïve” probability
estimates outperformed the risk-neutral ones for the period of 3 weeks after
the announcement. On 22th of May 2012 SAP AG, the largest maker of
enterprise-applications software, announced to acquire Ariba Inc. for the price
of $45 per share. This offer corresponded to 15% premium compared to average
price of Ariba’s 2 weeks before the announcement. Market reacted with a price
increase to $45 and the stock continued to trade approximately at the offer
price for the following 3 weeks. The probability of success estimated from
stock prices was 99,6%, 92,1% and 86,4% for weeks 1,2 and 3 respectively.
Option market, on the contrary, didn’t react as sharply and estimated the
success probability only at 63,8%, 72,8% and 77,6% for the above mentioned time
periods. However, option market predictions improved significantly and
converged to those of the stock market closer
to resolution. One week before the resolution risk-neutral probability of
success equalled to 90,5% while “naïve” method forecasted 92,0%. Figure 3
represents daily probability forecasts for both methods. Another important
thing to notice is that we detect higher volatility for risk-neutral forecasts.2.
Post-announcement and pre-resolution option-implied and stock-implied
probabilities for Ralcorp.
Figure 3. Post-announcement and pre-resolution option-implied
and stock-implied probabilities for Ariba.
arbitrage and excess returns
’s now briefly consider practical application of the obtained
risk-neutral probabilities to investment decisions and merger arbitrage. Recall
that merger arbitrage (for cash deals) is a strategy associated with buying
target company’s stock as soon as possible after the announcement and selling
it at the resolution date. We define the excess return on a portfolio of stocks
as the difference between its return
and the return on Hedge Fund Merger Arbitrage index provided by HFR database.
This index aggregates the performance of merger arbitrage strategies of the
whole hedge fund industry and is assumed to be a benchmark that carries the
comparable level of risk. Table 4 summarizes information of excess returns
associated with different portfolios. Based on the chosen sample equally
weighted portfolio that is comprised of stocks that exhibited option-implied
probability of success above 0,6 during first week after announcement generated
the return of 4,2%, compared to 0,4% return of HFRX Merger Arbitrage index
(Portfolio 4). If the investor didn’t bother with analysing success probability
and simply invested equal shares in all target companies after the deal’s
announcement the return would have been 2,7% compared to 0,3% HFRX Merger
Arbitrage index return (Portfolio 1). Thus, the excess return for “high probability
strategy” exceeds the one of “simple risk arbitrage strategy” by 1,5 percentage
points. Portfolios that put weights on “high probability” stocks in proportion
of 2 to 1 and 10 to 1 compared to “low probability” stocks generate the excess
return of 2,4% and 2,6% respectively (Portfolios 2 and 3). Thus, based on the
chosen sample one can infer that the optimal strategy would be to invest in
“high probability” stocks only as this strategy generates higher excess
returns.
4. Excess returns for different merger arbitrage strategies.
|
Return, %
|
HFRX Merger Arbitrage Index
return, %
|
Excess return, %
|
Portfolio 1
|
2,7%
|
0,3%
|
2,4%
|
Portfolio 2
|
2,7%
|
0,3%
|
2,4%
|
Portfolio 3
|
2,9%
|
0,3%
|
2,6%
|
Portfolio 4
|
4,2%
|
0,4%
|
3,8%
|
Conclusion and further
remarks
paper contributes to literature by outlining an “easy to
implement” way to estimate risk-neutral probability of a M&A deal success
from option prices and conducting empirical analysis of its predictive power on
the basis of a sample of cash tender offers for the period of 4 years (2010 to
2013). In conclusion, empirical study suggests that option prices embed
significant predictive content for forecasting outcomes of cash acquisitions.
Forecasting power of the risk-neutral probabilities increases monotonically
closer to the resolution date. However, despite the above-mentioned inference
risk-neutral probability forecasts appear to be poorly calibrated. Options
market tends to under predict the probability of success, suggesting excess
returns opportunities. The above-mentioned inference provides a starting point
for further research that could focus on option market arbitrage strategies for
companies that are subject to a takeover bid.
Comparative analysis of option implied and stock-implied
probabilities suggest that both probability estimates are worth of
consideration. For the period shortly after the announcement date stock market
reacts to strongly and tends to over predict the success probability, while
option market provide more accurate forecasts. Closer to deal resolution,
however, stock market adjusts and “naïve”
stock-derived probabilities become better estimates than risk-neutral ones. Additionally, combination
of stock market and option market probability forecasts outperform models based
on isolated information from either of the markets. All of the above suggests
that proposed method for risk-neutral probability estimation could be of use in
relation to deal outcome prediction, especially when used together with other
probability estimating models. basic analysis of excess returns associated with
merger arbitrage reveals that on the basis of the chosen sample a portfolio of
high risk-neutral success probability stocks generate a slightly higher excess
return than a simple “invest in all” portfolio. This empirical finding provides
a vast area for further research. For example, the study could extend to
determine the optimal portfolio weights that should be assigned to stocks
depending on their option-implied success probability. in all, risk-neutral
probability estimates obtained in this paper could serve as a starting point
for analysis of arbitrage strategies in both, derivatives and stock market. It
should be also mentioned, however, that probabilistic forecasts’ estimation and
testing, especially of those that rely on option market, put many restriction
on the sample data and have, indeed, limited area of application.
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