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Merger Arbitrage Risk Model
St´ephane Daul RiskMetrics Group [email protected]
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 failure.
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.
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.
31−May−0534 31−Jun−05 31−Jul−05 31−Aug−05 30−Sep−05 31−Oct−05 36 38 40 42 44 46 Date Share Price Figure 2
Share prices of Maxtor (thick line) and Seagate Technology (dotted line)
Seagate Technology share prices are rescaled by the exchange ratio.
31−Oct−053 31−Dec−05 28−Feb−06 30−Apr−06 31−May−06 4 5 6 7 8 9 10 11 Date Share Price
Figure 3
Share price of infoUSA Inc.
9−Jun−058 31−Jun−05 31−Jul−05 31−Aug−05 30−Sep−05 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 Date Share Price
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
31−May−05 31−Jun−05 31−Jul−05 31−Aug−05 30−Sep−05 31−Oct−05 Date Share Price ∆ Λ S t 0 V t 0
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.
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
C=
(
S with probabilityπ,
F with probability 1−π. (1)
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 volatilityσ. Hence the stock evolves as
St+∆t=Ste∆Z, (2)
where∆Z follows a normal distribution with mean 0 and volatilityσ√∆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 is X = ( ∆ if C=S, St0e∆ Z −(V−∆) if C=F. (3)
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
f(x) =πδ(x−∆) + (1−π)g x−(V−∆) St0 , (5)
Risk Model 135
where g(z)is the density function of a lognormal distribution with mean 0 and volatilityσ√∆t. The expected value of X is
E[X] =π∆+ (1−π)St0e Λσ2/2
. (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
ui= log St0i+Λ i+5 Si t0 σi√Λi (7)
for all withdrawn deals. The index i=1, . . . ,19 stands for each deal. The volatilityσiis 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´enage `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.
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.
31−Mar−0530 30−Apr−05 31−May−05 31−Jun−05 31−Jul−05 31−Aug−05 35 40 45 Date Share Price t A tB t C
The deal completion indicator is now
C= S with probabilityπ1, N with probabilityπ2, F with probability 1−π1−π2, (8)
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
X = ∆ if C=S, ∆+∆P if C=N, St0e ∆Z−(V−∆) if C=F. (9)
Table 3
Kolmogorov-Smirnov test on residuals of all non-“m´enage `a trois” withdrawn deals
¯
u=−0.1012 KS-test su=0.8183 p-value = 94.1 %
use the seven remaining to compute the residuals
ui= log St0i+Λ i+5 Si t0 σi√Λi . (10)
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 rf, hence
E[X] =rfΛV. (11)
Using (6), we obtain the market implied probability
π= rfΛ(V−∆)− h St0eΛσ 2/2 −(V−∆)i ∆−St0eΛσ 2/2 −(V−∆) . (12)
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 forΛand the corresponding Treasury rate for rf. 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 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Fraction x (Success Probability)
CAP
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.
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 include
• 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
References
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.
