A look at arbitrage deals using arbitrage risk equation from Benjamin Graham.

There is a class of investors known as arbitrageurs who invest in arbitrage situations. There are two general types of arbitrage namely "market arbitrage" and "time arbitrage".

For market arbitrage, arbitrageurs seek to profit from price differences in the same security or investment being traded in two different markets. For example, if a company ABC's stock is concurrently being listed and traded in the London stock exchange at $10 per share and the Paris stock exchange at $12 per share, arbitrageurs will step in to sell the ABC's stock at Paris stock exchange while at the same time buy the stock at London stock exchange in order to pocket the profit of $2 per share in price difference. By the act of arbitrageurs, it is rare to see large price differences in any security or investment traded concurrently in different markets since any price differences will be immediately capitalised by arbitrageurs to profit from the price difference and thus the stock price of the same security will be brought towards approximately the same price by selling in one market and buying in another market simultaneously.

For time arbitrage, investors are arbitraging the price difference between what the stock price is today to what it will be at a future time. Many investors are already doing time arbitrage when we seek to buy a stock at today's price and hopefully sell it to another investor who will pay a higher price for the same stock in future (longing a stock) or borrowing and selling a stock at today's price and buying back the same stock at lower price in future (shorting a stock). The future time to fully transact an arbitrage in this case can be as short as one day to months or years.

Time arbitrage (longing shares - buy low sell high, 

or shorting shares - sell high buy low)

There are also special situations that will create arbitrage opportunities for the investor. These situations include mergers and acquisitions, securities buybacks or self tender offers, corporate reorganisations, corporate liquidations, corporate spin-offs and corporate stubs. With every potential arbitrage opportunity that comes on scene, there is also an element of risk that the arbitrage deal may not follow through to the end. An arbitrage deal that is created but does not follow through in the end may spell losses for investors who have invested their money into the deal to find that the deal does not work out. Thus, it is important to assess every arbitrage situation carefully to minimise the probability of entering into a losing deal.

Benjamin Graham, Warren Buffet's mentor and friend has an equation that can be used to assess the rate of return based on factoring in risk and reward of an arbitrage situation.

The first part of the equation is to determine the projected profit from an arbitrage situation (e.g. ABC company has announced a tender offer to buy all of DEF company's shares at $10 per share. DEF's shares are currently trading at $9 per share.)

Thus, the projected profit is $10 - $9 = $1 per share.

Next, we determine the probability that this tender offer deal will follow through to completion. Let's say there is a 90% chance of this deal completing. Note that the determination of the probability of the deal completing is more of an art than science since the investor has to assess all information available to him on the arbitrage situation carefully and come to a meaningful conclusion. His probability may differ from another investor's probability due to different views on the same arbitrage situation. We multiply our probability by the earlier projected profit.

Thus, the adjusted projected profit = 0.90 X $1 = $0.90 per share.

Next, we factor in the risk that the deal may fall apart and assume that the share price will return to the trading price before announcement of the tender offer. The risk is a projected loss between the current share price we pay and share price before announcement of tender offer. The current share price is at $9 per share for DEF's shares. Let's say the share price was $8 per share before the announcement of the tender offer.

Thus, the projected loss is $9 - $8 = $1 per share.

Next, we determine the adjusted projected loss. Since there is a 90% chance of the deal completing, there will be a 10% chance of the deal falling apart. We multiply this probability by the earlier projected loss.

Thus, adjusted projected loss = 0.10 X $1 = $0.10 per share.

Next, we determine the risk adjusted projected profit (after factoring in the risk involved in making the profit) by subtracting our adjusted projected loss from adjusted projected profit.

Thus, risk adjusted projected profit
         = adjusted projected profit - adjusted projected loss
         = $0.90 - $0.10
         = $0.80 per share

Lastly, we calculate our risk adjusted projected rate of return (in %) by dividing the risk adjusted projected profit over our original investment of $9 per share if we were to buy DEF's shares to enter into this arbitrage deal.

Thus, risk adjusted projected rate of return
          = ($0.80 / $9) X 100%
          = 8.9%

An 8.9% return on this arbitrage may not be too attractive over a year. However, if this arbitrage deal can be fully completed and the investor gets his profits in shorter time of six months, then the annual rate of return becomes 8.9% X 2 = 17.8%. This postulated annual rate of return is assuming that the investor can continue to keep his original capital reinvested after completion of the arbitrage deal at the same rate of return (for next six months to make up a full year). An annual rate of return of 17.8% now becomes attractive for an investor. Thus we see that an arbitrage deal becomes attractive should the arbitrage be completed in as short a time as possible.

In using this arbitrage risk equation, one is not trying to be precise in determining the rate of return on the arbitrage as no one can predict perfectly the outcome of any arbitrage situation. Therefore, the principle is to invest in arbitrage deals that have a very high probability of completion. One can be more certain of an arbitrage deal being completed if public announcements are already made and legal documents have been filed to the relevant authorities of securities exchanges.

As an arbitrage deal becomes more certain of being completed with passage of time, the gap in the traded share price to the tender offered share price (for example in the case of tender offers) closes and the traded share price in the open market will come close to the tender offer share price. The careful investor who waits for certainty before commiting his capital into buying the shares will see lesser rate of return.

However, an investor can still magnify his rate of return by using leverage (buying shares on margin) on such certain arbitrage deals. This is where the use of leverage (conventionally thought to be dangerous and destructive) works well in such certain deals to magnify returns. By combining certainty with leverage, this certainly beats uncertainty in the earlier stages of arbitrage when an investor buys on rumors and risk having the arbitrage deal falling apart and the loss on his capital.