Merger Arbitrage Risk Model by man15117



                            Merger Arbitrage Risk Model

                                                St´ phane Daul
                                              RiskMetrics Group

      A traditional VaR approach is not suitable to assess the risk that merger arbitrage funds carry in their
      portfolios. We propose a simple two-state or three-state model that captures the risk characteristics of
      the deals in which merger arbitrage funds invest. This model has been tested on a set of mergers and
      acquisitions between large US public companies in 2005.

1    Introduction

There are numerous types of hedge fund strategies. The most common one is to take long and short
equity positions. Some strategies invest in more complicated products such as OTC derivatives, while
others in illiquid instruments such as distressed debt. We will focus on merger arbitrage, which takes
long (and to some extent short) equity positions that are subject to a specific event risk, namely deal

These strategies bet on announced (or sometimes rumored) mergers or acquisitions concerning large
publicly traded companies. There are two main types of mergers: cash mergers and stock mergers. In a
cash merger, the acquirer offers to exchange cash for the target company’s equity. In a stock merger, the
acquirer offers its common stock to the target in lieu of cash.

Let us consider a cash merger in more detail. Company A decides to acquire Company B, for example
for a vertical synergy (B is a supplier of A). Company A announces that they offer a given price for each
share of B. The price of stock B will immediately jump to that level. However, the transaction typically
will not be effective for a number of months, as it is subject to regulator clearance, shareholder approval,
and other matters. During the interim, the stock price of B actually trades at a discount with respect to
the offer price, since their is a risk that the deal fails. Usually, the discount decreases as to the effective
date approaches and vanishes at the effective date.
130   Merger Arbitrage Risk Model

      In a stock merger, A offers to exchange a fixed number of its shares for each share of B. The stock price
      of B trades at a discount with respect to the share price of A (rescaled by the exchange ratio) as long as
      the deal is not closed.

      In the following, after a description of typical merger arbitrage investments, we propose a risk model
      capturing these deal specifics. This model is then tested on 200+ merger and acquisition deals, and leads
      us to various applications.

      2    Merger Arbitrage Description

      Merger arbitrage is a strategy attempting to capture the spread between the bid price offered by the
      acquiring company and the target company’s stock price.

      In a cash merger, the arbitrageur simply buys the target company’s stock. As mentioned above, the
      target’s stock sells at a discount to the payment promised, and profits can be made by buying the target’s
      stock and holding it until merger consummation. At that time, the arbitrageur sells the target’s common
      stock to the acquiring firm for the offer price.

      For example, on 8 August 2005, Quest Diagnostic announced that it was offering $43.90 in cash for each
      publicly held share of LabOne Inc. Figure 1 shows the LabOne share price. It can be seen that the shares
      closed at $42.82 on 23 August 2005. This represents a 2.5% discount with respect to the bid price. The
      deal closed successfully on 1 November 2005 (just over two months after the announcement), generating
      an annualized return of 10.9% for the arbitrageur.

      In a stock merger, the arbitrageur sells short the acquiring firm’s stock in addition to buying the target’s
      stock. The primary source of profit is the difference between the price obtained from the short sale of the
      acquirer’s stock and the price paid for the target’s stock.

      For example, on 20 December 2005, Seagate Technology announced that it would acquire Maxtor Corp.
      The terms of the acquisition included a fixed share exchange ratio of 0.37 share of Seagate Technology
      for every Maxtor share. Figure 2 shows the movement of both the acquirer share price and the target
      share price. On December 21, Maxtor shares closed at $6.90 and Seagate at $20.21 yielding a $0.58
      merger spread. The deal was completed successfully on 22 May 2006.

      More complicated deal structures involving preferred stock, warrants, or collars are common. From the
      arbitrageur’s perspective, the important feature of all these deal structures is that returns depend on
                                                                                               Merger Arbitrage Description   131

Figure 1
Share price of LabOne Inc.



                     Share Price




                       31−May−05        31−Jun−05   31−Jul−05       31−Aug−05   30−Sep−05    31−Oct−05

Figure 2
Share prices of Maxtor (thick line) and Seagate Technology (dotted line)
Seagate Technology share prices are rescaled by the exchange ratio.



                     Share Price





                        31−Oct−05            31−Dec−05           28−Feb−06          30−Apr−06 31−May−06
132   Merger Arbitrage Risk Model

      Figure 3
      Share price of infoUSA Inc.





                            Share Price






                                          9−Jun−05   31−Jun−05   31−Jul−05   31−Aug−05   30−Sep−05

      mergers being successfully completed. Thus the primary risk borne by the arbitrageur is that of deal
      failure. For example, on 13 June 2005, Vin Gupta & Co LLC announced that it was offering $11.75 in
      cash for each share of infoUSA Inc. In Figure 3, we see that after the announcement, the share price of
      infoUSA jumped to that level. The offer was withdrawn, however, on 24 August 2005, and the share
      price fell to a similar pre-announcement level.

      A recent survey of 21 merger arbitrageurs (Moore, Lai, and Oppenheimer 2006) found that they invest
      mainly in announced transactions with a minimum size of $100 million and use leverage to some extent.
      They gain relevant information using outside consultants and get involved in deals within a couple of
      days after the transaction is announced. They take a longer time to unwind their positions in cases where
      the deal is canceled, minimizing liquidity issues. Their portfolios consist, on average, of 36 positions.

      Finally, from Figure 1, we clearly see that the volatility of the share price before and after the
      announcement is very different. Measuring the risk with a traditional VaR approach in terms of historical
      volatility is surely wrong. Thus arbitrageurs typically control their risk by setting position limits and by
      diversifying industry and country exposures. In the following sections, we introduce a risk model
      suitable for a VaR approach.
                                                                                                         Data   133

Figure 4
Definition of parameters

                           Share Price



                         31−May−05       31−Jun−05   31−Jul−05       31−Aug−05   30−Sep−05   31−Oct−05

3     Data

We use merger and acquisition information from the SDC database. We consider deals in 2005
concerning US public companies where the target market value is larger than $100 million. The daily
stock prices are from DataMetrics. In all, we have data for 203 completed deals and 19 withdrawn deals.

4     Risk Model

4.1 Two-state model

We start by introducing the deal specific parameters (see Figure 4):

    St is the stock price at time t.

    t0 is the announcement date.
134   Merger Arbitrage Risk Model

        St0 is the stock price just before announcement.

         π is the probability of deal success.

         ∆ is the merger spread, that is, the difference between the bid price and the price at which the
           arbitrageur buys the stock.

         Λ is the elapsed time (measured in days) between the announcement and completion dates.

         V is the bid price.

      We introduce the following binomial random variable to describe the deal completion

                                                            S with probability π,
                                                   C=                                                                 (1)
                                                            F with probability 1 − π.

      The state {S} stands for deal success and {F} for deal failure. We make the hypothesis that the stock
      price dynamics prior to the announcement are captured by a Geometric Brownian Motion with zero drift
      and volatility σ. Hence the stock evolves as

                                                                St+∆t = St e∆Z ,                                      (2)
      where ∆Z follows a normal distribution with mean 0 and volatility σ ∆t.17

      If the transaction is completed successfully, then the payout is the bid price minus the price at which the
      arbitrageur bought the stock, namely the spread ∆. In case of failure, we assume that the stock price
      would have behaved as if there were no transaction put in place. Hence the payout at time of completion
                                                 ∆                   if C = S,
                                         X=           ∆Z − (V − ∆) if C = F.
                                                 St0 e
      We obtain the distribution function of X using the identity

                                      P(X ) = P(X |C = S)P(C = S) + P(X |C = F)P(C = F),                              (4)

      yielding the density
                                                                                   x − (V − ∆)
                                             f (x) = πδ(x − ∆) + (1 − π)g                      ,                      (5)
        17 Note   that this framework places no restrictions on the stock price level in the event of deal failure.
                                                                                                    Risk Model   135

where g(z) is the density function of a lognormal distribution with mean 0 and volatility σ ∆t. The
expected value of X is
                                              E[X ] = π∆ + (1 − π)St0 eΛσ
                                                                               2 /2
                                                                                      .                    (6)

The main hypothesis is that in case of deal failure the stock price would end at the same level as if no
deal was put in place originally. To test this hypothesis we compute the residuals

                                                                    0 +Λi +5
                                                           log        Sti0
                                                    ui =           √                                       (7)
                                                                 σi Λi

for all withdrawn deals. The index i = 1, . . . , 19 stands for each deal. The volatility σi is obtained using
the stock price time series prior to the announcement date.18 We take the stock price five days after
completion date to mimic the arbitrageurs behavior. The withdrawn deals, along with their residuals, are
listed in Table 1. Under our hypothesis, these residuals should be drawn from a standard normal
distribution. We see in Table 2 that our hypothesis is almost rejected at a 95% confidence level
(corresponding to a p-value ≤ 5%).

4.2 Three-state model

A closer look at our withdrawn deals reveals that a merger can also fail if a new acquirer comes in the
game with a higher bid price. The stock price behavior of Juno Lighting in Figure 5 illustrates this
m´ nage a trois behavior. At time tA , Abrams Capital attempted to acquire Juno Lighting for $40. At time
tB , Square D made an offer to acquire Juno Lighting for $44. Finally, the deal was completed by Square
D at time tC .

To capture this additional feature, we introduce the following parameters:

 ∆P is the new premium offered in addition to V by second acquirer.

  π1 is the probability that the deal is completed by the first acquirer.

  π2 is the probability that the deal is completed by the second acquirer.
  18 We   use a simple equally weighted estimate over one year of data.
136   Merger Arbitrage Risk Model

      Table 1
      Data from the nineteen withdrawn deals

       Acquirer                       Target                            St0   St0 +Λ   Λ(days)   σdaily    u
       Nanometrics                    August Technology Corp.            9.4 12.0       158      7.1%     0.28
       Qwest Communications           MCI                               16.3 20.1        80      1.8%     1.33
       Medicis Pharmaceutical Corp.   Inamed Corp.                      66.2 86.6       267      1.6%     1.02
       Goldner Hawn Johnson           ShopKo Stores                     22.9 28.5       193      2.7%     0.58
       General William Lyon           William Lyon Homes                72.4 125.6       90      2.5%     2.29
       Audax Group                    CFC International                 23.9 16.1       104      5.6%     -0.69
       Abrams Capital                 Juno Lighting                     33.8 44.0       111      2.1%     1.18
       Investor Group                 Maytag Corp.                      22.2 25.5        95      1.8%     0.79
       Vin Gupta & Co                 infoUSA                            9.4  9.9        72      3.0%     0.21
       CNOOC                          Unocal Corp.                      63.5 65.7        41      2.0%     0.26
       VNU N.V.                       IMS Health                        25.9 24.7       130      1.1%     -0.38
       Musculoskeletal Transplant     Osteotech                          4.0  3.9        97      3.7%     -0.11
       Equity One                     Cedar Shopping Centers            27.0 26.7         4      0.4%     -1.59
       Opportunity Partners           Hector Communications Corp.       26.1 35.0       292      1.5%     1.13
       FreeMySpace                    Intermix Media                    11.9 12.0         7      4.3%     0.01
       Levine Leichtman Capital       Fox & Hound Restaurant Group      10.2 16.2       121      2.7%     1.57
       Sun Capital Partners           Goody’s Family Clothing            8.8  9.5        17      2.9%     0.69
       Advanced Digital Infon Corp.   Overland Storage                   7.8  8.0        36      3.3%     0.10
       Mentor Corp.                   Medicis Pharmaceutical Corp.      27.8 30.1        78      1.8%     0.52

      Table 2
      Kolmogorov-Smirnov test on the residuals of all withdrawn deals

                                        u = −0.4846
                                        ¯                 KS-test
                                        su = 0.8759   p-value = 5.5 %
                                                                                                               Risk Model   137

Figure 5
Share price of Juno Lighting Inc.


                       Share Price



                         31−Mar−05          30−Apr−05   31−May−05          31−Jun−05   31−Jul−05   31−Aug−05

The deal completion indicator is now
                                              S with probability π1 ,
                                          C=   N with probability π2 ,                                                (8)
                                               F with probability 1 − π1 − π2 ,

where the additional state {N} represents the new acquirer completing the deal.

As before, we assume that the share price prior to the announcement is correctly captured by a lognormal
distribution, and in case of failure, the stock price returns to where it would have been had there been no
deal. Hence the payout at time of completion is
                                               ∆
                                                                 if C = S,
                                           X=   ∆ + ∆P            if C = N,                                           (9)
                                                    ∆Z − (V − ∆) if C = F.
                                                St0 e

Out of the nineteen previously withdrawn deals twelve actually ended in a m´ nage a trois situation. We
138   Merger Arbitrage Risk Model

      Table 3
                                                        e      `
      Kolmogorov-Smirnov test on residuals of all non-“m´ nage a trois” withdrawn deals

                                           u = −0.1012
                                           ¯                       KS-test
                                           su = 0.8183        p-value = 94.1 %

      use the seven remaining to compute the residuals

                                                                 0 +Λi +5
                                                        log        Sti0
                                                 ui =           √             .                             (10)
                                                              σi Λi

      The Kolmogorv-Smirnov test is shown in Table 3. We see now that we cannot reject at all the hypothesis
      that u follows a standard normal distribution.

      Though this result is impressive, it was obtained using a rather small sample—large US-based deals in
      2005. We are thus compelled to run the same analysis on a larger dataset.

      5    Probability of success

      5.1 Market-implied model

      Assuming that the market fairly prices the deals, then expected performance should be equal to the risk
      free rate r f , hence
                                                    E[X ] = r f ΛV.                                         (11)

      Using (6), we obtain the market implied probability

                                           r f Λ(V − ∆) − St0 eΛσ
                                                                       2 /2
                                                                              − (V − ∆)
                                      π=                                                  .                 (12)
                                                ∆ − St0 eΛσ           − (V − ∆)
                                                               2 /2

      We have calculated this quantity for the 90 deals (five of which ultimately failed) having clean enough
      data. We used the average completion time as the estimate for Λ and the corresponding Treasury rate for
      r f . To assess the relevance of the model, we calculate the cumulative accuracy profile. We order all deals
                                                                                                     Probability of success   139

Figure 6
Cumulative accuracy profile











                                  0   0.1   0.2    0.3    0.4   0.5    0.6   0.7     0.8   0.9   1
                                                  Fraction x (Success Probability)

by their implied probability of success, and show for a percentage x of these ordered deals the ratio of
failed deals within that set. We see in Figure 6 that the five failures lie within the 30% of the deals that
the market considered most risky. This will provide a benchmark for any empirical model predicting the
probability of completion.

5.2 Empirical Model

In addition to market-implied inference, there are two approaches to measure the probability of success
of a merger. One is based on analysis of financial statements of both target and acquirer, while the other
is a statistical approach.

In the financial statements analysis, the fundamentals considered are, among others, acquirer, leverage,
firm growth, earnings forecast, financing method, deal size, deal completion time, etc. This is the
approach employed by banking research analysts. Branch, Higgins, and Wilkens (2003) have assessed
these models indirectly by focusing on the incremental explanatory power conveyed by analyst opinions.
They show that information about analyst coverage significantly improves models of merger success.
140   Merger Arbitrage Risk Model

      The second approach consists of a statistical prediction model for merger completion. Typically, one
      gathers historical information on large number of deals, and run a stepwise logistic regression,
      eliminating those factors that are statistically irrelevant. The candidate factors used in the regression

          • target / acquirer size,

          • bid premium (difference between bid price and pre-announcement stock price),

          • resistance of target (hostile or friendly),

          • governance (particularly anti-takeover) provisions of the target,

          • overall market level,

          • type of deal (cash, equity, collar, ...), and

          • industrial sectors of target and acquirer (horizontal versus vertical merger).

      A detailed study using the statistical approach is underway.

      6    Conclusion and next steps

      We have shown that a simple model captures the specifics of a typical merger arbitrage deal. The
      parameters of the model are the probability of success, the time to completion, and the volatility of the
      target stock price prior to the announcement.

      From the perspective of a merger arbitrage fund, this model can be used to measure the risk of the
      portfolio in a VaR framework. We will describe each position as a “merger deal,” specifying the type of
      deal, the bid price (or share ratio), the expected time of completion, and (optionally) the probability of
      success. For short risk horizons, we will assume that the deal does not age, and add a risk factor
      describing the outcome of the deal. In case the probability of success is unavailable, we will use the
      statistical model to assess it.

      For investors in merger arbitrage funds we can construct a merger arbitrage benchmark as in (Mitchell
      and Pulvino 2001). The benchmark would consist of the return time series from a hypothetical merger
      arbitrage manager. As mergers are announced, he invests subject to the probability of success determined
      by the statistical model, and to adequate constraints such as capital limits.
                                                                                Conclusion and next steps   141

 Branch, B., H. N. Higgins, and K. Wilkens (2003). Risk arbitrage profits and the probabilty of
    takeover success. Worcester Polytechnic Institute. Working paper.
 Mitchell, M. and T. Pulvino (2001). Characteristics of risk and return in risk arbitrage. The Journal of
    Finance 56, 2135–2175.
 Moore, K. M., G. C. Lai, and H. R. Oppenheimer (2006). The behavior of risk aribtrageurs in mergers
   and acquisitions. The Journal of Alternatives Investments Summer.

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