January 17, 2004

One Hundred Interesting Mathematical Calculations, et cetera: Number 25: Understanding "Risk Arbitrage"

One Hundred Interesting Mathematical Calculations, et cetera: Number 25: Understanding "Risk Arbitrage"

One Hundred Interesting Calculations Page

The stock market is (nearly) a random walk, economists say. Price movements are unpredictable, and a large component of risk is systematic. How, then, can informed investment professionals reliably make money year-after-year in the stock market? It is possible: there are ways. Let's sketch out a finger-exercise example to give us an idea of how this might be done:

The Setup:

Let's start with a business--call it Amalgamated Airlines--and let's give the current state of its business a numerical value of zero. Each year over the next five years the business will either go well or badly. In a year in which it goes well, the state of the business will increase by one. In a year in which business goes badly, the state of the business will decrease by one. In each year, there is an independent 50-50 chance of good news and bad news. So five years from now the state of the business might be +5 (if there is good news in all five years), or +3 (if there is good news in four years and bad news in one), or +1, or -1, or -3, or -5. And let's assume that in five years a +5 business is worth $10 billion, a +3 business is worth $8 billion, a +1 business is worth $6 billion, a -1 business is worth $4 billion, a -3 business is worth $2 billion, and a -5 business is worth nothing at all--as is shown in the figure immediately below:

Now the business has two kinds of owners: bondholders and stockholders. Let's assume that the bondholders hold five-year bonds with a face value of $6 billion. That means (simplifying drastically the provisions of bankruptcy courts) that five years hence if the business index is +5, that bondholders get $6 billion of the company's value and stockholders get $4 billion; if the business index is +3, bondholders get $6 billion and stockholders get $2 billion; if the business index is +1, bondholders get $6 billion and stockholders get zero; and if the business index is negative then the company is bankrupt, the stockholders get zero, and the bondholders get $4 billion for an index of -1, $2 billion for an index of -3, and $0 for an index of -5--all as shown in the figure below.

OK. Now we know (by assumption) what the possible states of things are five years hence. What can we say about what the state of things is going to be 4 years hence? Let's ignore interest rates--assume that they are zero--and ignore extra premia for risk--assume that assets are valued at their expected values. Then four years hence the business index is either +4, +2, 0, -2, or -4, and the value of the company's bonds is the average expected value of what it might turn out to be when whatever is going to happen in year five--good news or bad news--is revealed. Thus the bonds are (or should be) worth $6 billion for an index of +4 or +2, $5 billion for an index of 0, $3 billion for an index of -2, and $1 billion for an index of -4--all as shown in the figure below.

And we can keep working backward (neglecting, as we said, interest rates--the time value of money--and risk premia) to arrive at a nice value tree showing the value of the bonds of Amalgamated Airlines (or what the value should be) in each year from now up to five years in the future, for each possible state of cumulative good or bad news about the business:

And we can construct the same kind of value tree for the stocks of Amalgamated Airlines--the stocks will be worth $4 billion in year 5 if the business index is +5, $2 billion in year 5 if the business index is +3, and $0 in year 5 otherwise. Once again, working backward (assuming a zero interest rate and no aversion to risk), we arrive at a nice value tree showing the value of the stocks of Amalgamated Airlines (or what the value should be) in each year from now up to five years in the future, for each possible state of cumulative good or bad news about the business:

All this is preliminary: just setting up the structure of the problem so that we can start to think of when and how one would want to speculate in the securities of Amalgamated Airlines.

The Payoff:

Now suppose that we are sitting in year 2, and we notice that something appears to have gone wrong with the pricing of Amalgamated Airlines's securities. We have had one year of good and one year of bad news, which means that we have to have three consecutive years of good news--a cumulative chance of 1/8--for the stocks to be worth anything in year 5 (and then they will be worth $2 billion). That means that now, in year 2, the value of Amalgamated Airlines's stocks should be 1/8 x $2 billion = $0.25 billion. But suppose that, instead, the stocks are selling for $0.4 billion.

Stocks seem to be overpriced. Selling them short seems to be a good bet. But if you do sell them short, things could go horribly wrong: suppose there is good news in year 3, and the value of stocks jumps to its (correct) fundamental value of $0.5 billion? Then if you had sold short $40 million worth of stock in year 2, you will find in year 3 that you have lost $10 million. But there is a way to hedge this risk--to make somebody else bear it. Look at the prices of Amalgamated Airlines's bonds in year 2:

Note that good news for stocks in year 3 is good news for bonds as well. If in year 2 you sell short $40 million of stock and simultaneously go long $158.33 million of bonds--investing $118.33 million--then in year 3 your portfolio will be worth $133.33 million no matter what. Either there will be bad news, and the stocks that you have sold short will be worthless while your bonds are worth $133.33 million; or there will be good news, and your bonds will be worth $183.33 million but you will have to spend $50 million to cover your short position on stock. By going short stock and long bonds, you lock in a $15 million profit from year 2 to year 3 no matter what happens to Amalgamated Airlines's business. (Ah, but you say, what happens if prices in year 3 are out of whack to. Good question. For this reason any such financial business has to be well-capitalized: you have to be able to remain solvent longer than the market can remain irrational.)

Such a hedged portfolio protects you against another form of risk as well. Suppose that stockowners in year 2 know more about the business than you do--that they have inside information that the year 3 news is going to be good not with 50% but with 80% probability. Even if you are the uninformed fool in the stock market, you still profit from shorting $40 million of stock and going long $158.33 million of bonds in year 2--the securities that are mispriced in year 2 are the bonds (which are priced too low). The key is not that you know that the stocks are too high, but that you know that the relative prices of the stocks and bond are wrong.

So this kind of short-stocks-and-buy-bonds-and-take-a-hedged-portfolio is an easy way to make riskfree money, right? Wrong. Your analysis of the fundamentals has to be correct. All this stuff about the distribution of value in five years and the news flow over the next five years and the proper gearing of your portfolio to make sure that the position taken in one asset class is of exactly the right magnitude to offset losses or gains in another asset class--that's really hard. Once you've done the fundamental analysis and are confident you've identified securities prices that are relatively out of whack, everything else is easy.

And, of course, there are competitors. Unless you have the most sophisticated and most accurate fundamental models out there, other people will short the stock of Amalgamated Airlines--and so drive down the price to a point at which it will no longer be worth your while to take a position--first.

Posted by DeLong at January 17, 2004 09:30 PM | TrackBack


Nicely put. It doesn't matter why the buying public has priced these financial instruments this way but that there is a discrepency between what is and what would be expected if all investors were rational and (more importantly) informed. Thanks.

Posted by: PT Martin on January 17, 2004 10:36 PM


Well that's easy, stocks expensive, bonds cheap !

Actually, I think the arbitrage you're thinking about is when the volatility implicit in the bond is out-of-whack with stock volatility. Lots of people do this.

Posted by: Andrew Boucher on January 17, 2004 10:46 PM


Hey! I said it was a finger exercise!

Posted by: Brad DeLong on January 17, 2004 11:03 PM


Hm. OmniGraffle?

Posted by: Jon H on January 17, 2004 11:09 PM


Finger exersize? OmniGraffle? (I'll look em up)

Random Walk?


Posted by: bulent on January 17, 2004 11:20 PM


Huf! I should have read more carefully! You did say random walk as well -- right at the beginning.

Posted by: bulent on January 17, 2004 11:50 PM


On mature reflection... ok! I thought most Risk Arbs stayed closer to equity vs. equity, but then equity vs. bonds should be doable.

My guess, though, is that the desks don't worry about a binomial lattice (which is more for option volatility). They'd look at possible scenarios (in scenarios 1, stock is worth this, bonds worth this), assign a probability to a scenario, add up expected values, and calculate a standard deviation. They'd do this for a universe of possible equity vs. bond plays and then rank them.

But I could be wrong... It just strikes me that a lattice approach means they will not be correctly evaluating unusual shocks to the system.

Posted by: Andrew Boucher on January 18, 2004 12:57 AM


I have a mathematical question that I'm sure there's an easy answer to and I am woefully ignorant of: How can I figure out quickly the implied interest rate that this company was paying on its bonds when they were issued? In other words, if, after five years of holding an investment, you are paid $6 billion on a $4.5475 investment, how do you calculate the implied annual interest rate (taking into account compounding)?

Posted by: Steve Carr on January 18, 2004 07:33 AM


Doing the fundamental analysis for Amalgamated Airlines is difficult, expensive and senstive to unpredictable exogeneous events. The anticipated returns from AA are a function of the market risk, the idiosyncratic risk for AA and the sector risk for the airlines. Market risk is as noted not just directional but also relative. To do this arb, one should short a basket of airline stocks and go long a basket of airline bonds and add hedges for general stock market exposure and interest rates. By the isolating the hedged relative value of all airline stocks vs all airline bonds magical things appear. The noise and volatility of the market falls away. The unpopularity of the president of Delta (or America) or the willingness of Northwest to fuck its customers' right to privacy becomes trivial. Nice clean trends emerge. And you should always trade the trend. Beieive me.

Posted by: Wren on January 18, 2004 08:27 AM


Interesting example - the bond vs stock thing is an interesting and important field. A simple overview with some basics for the corporate yieldcurve can be found here. A more detailed analysis on trading stocks vs bonds with a real life example and some more extensive mathematical modelling is here. If you excuse the shamless self promotion, that is.

Posted by: Mats on January 18, 2004 09:57 AM


...or maybe you don't - I was trying to point to

and here

Posted by: Mats on January 18, 2004 10:01 AM


Very interesting problem, but I thought the term "risk arbitrage" had a more narrow meaning in finance:


"Risk Arbitrage, or 'merger arbitrage', attempts to profit from selling short the stock of an acquiring company and buying the stock of the aquiree."

Is this example called "risk arbitrage" because there is uncertainty with respect to how much information has already been priced into the equity security? Or is it risky because your fundamental analysis might be flawed?

Posted by: noto on January 18, 2004 11:25 AM


Steve Carr: Given annual interest rate r, n annual compounding periods, N total compounding periods (e.g. n times #years), and initial principal P, the final nominal amount is P(N) = P*(1+r/(100*n))**N, where ** is exponentiation. Conversely, r = (rootN(P(N)/P)-1)*100*n, where rootN() is the Nth root. If your calculator doesn't have Nth-root, then rootN(x) = exp(ln(x)/N).

Note: the whole calculation is in nominal terms. I believe in the US interest compounding is daily, but I don't know whether a 30-day month, 365-day year, or actual number of days method is used, and how the fractional interest is rounded. If anybody knows, please let me know.

Posted by: cm on January 18, 2004 12:45 PM


Isn't this example unnecessarily intricate to make the following important point: If there is a random Variable X whose distribution I know and you don't (or at least there is a positive difference in our information about X), then I may make money off of you. In the case of your example, the hedger's knowledge of "fundamentals" allows her to conclude the .4 bn valuation in year 2 is an unlikely fluctuation and can construct a bet to make a very likely profit.

The reason I prefer this formulation, is that it illustrates an open-ended design technique for generating gambling games for investment firms. You don't need to know anything about stochastic processes to realize that there are an almost endless number of ways of constructing such schemes.

One possible conclusion is that investment firms will seek to build progressively more complicated structures where small differences in information can make some people lots of money. Isn't this an observed fact?

Posted by: CSTAR on January 18, 2004 02:06 PM


Is this your latest offering to your 13 year old? Is so, how did it fly?

Posted by: jml on January 18, 2004 02:44 PM


>>Hm. OmniGraffle?<<


Posted by: Brad DeLong on January 18, 2004 03:55 PM


That's an ok way to make money via aribtrage.... What I found interesting this last go around was the emergence of the toxic convertible...nothing like human ingenuity.

Posted by: William on January 19, 2004 11:34 PM


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