Contributed by Ng Kuan Luen, May 2006.

As much as risk could be a cost it is also an opportunity. Risk has been historically cast in a negative light and as far as a portfolio is concerned, we should minimize it. Is this correct? Of course, why even ask? But is it, really?Here we would like to present some alternative views to how we commonly look at risk and its effect on investment portfolios.

We will show how a portfolio in theory that will increase its own growth with an increase in the asset's volatility, yes, through maximizing risk by its own definition. However, to understand more, we have to look back a little into the history of portfolio investments, not the Nobel prize winning one, but the rarely told one.

The Other Optimal Portfolio

Optimal-growth portfolio strategy to us really began before 1960s, 1950s in fact, when it was first suggested by Claude Shannon while working on Information and Entropy Theory. Shannon, if you might not know, is one of the fathers who shaped our computing age. His idea that information does not necessarily need to have a meaning, but can rather, exists as a probability instead, has allowed all information to be coded in binary, which in turn is the beginning of the digital era.

Why information theory then, you might ask. Actually, finance has much to do with information theory, especially in the area of optimal growth rate (Kelly [1956], Breiman [1961]). Now what people might not know too, is that Shannon himself was also a very successful portfolio manager.

In the 1960s, Shannon gave a lecture in a hall packed with students and teachers alike in MIT, on the topic of maximizing the growth rate of wealth. He detailed a method on how you can grow your portfolio by rebalancing your fund between a stock and cash, while this stock stays in a random ranging market. (He used a geometric Wiener example). Essentially, you buy more when stock price is low, using the cash at hand, or sell more when stock price is high, with allocation of 50-50% value at each interval.

For example, if you start with of a portfolio of $1,000, you buy a value of $500 on Euro, and keep $500 as cash, total of $1,000. Assume now Euro goes down - say from $1.25 to $1, so the Euro exposure is now worth $400. The portfolio is of course now worth only $900. What you then do is to split the value of your current portfolio into 50-50 again, that is take $50 from your cash to buy more Euro, so that the Euro exposure is now $450 and cash at hand is $450, equal. This is of course contrary to normal view of buying when price is moving higher and vice versa. However, what you will see, if Euro is not trending and stays within a range, is that such a portfolio will grow.

The best way to see this is to simulate random white noise as an imaginary stock price and do the balancing against cash. Please see Example 1, where the portfolio begins with holding $5,000 worth of imaginary FX stock and $5,000 as cash; a total value of $10,000 worth in the portfolio. As can been seen, such portfolio actually grow despite the simulated FX prices being totally random.

At about the same period, Shannon and Samuelson (yes the Nobel Prize winner-to-be in economics) held a number of evening discussion meetings on information theory and economics (Cover [1998]). It is not clear what was said in the meetings but Samuelson became set in his views and published several papers arguing strongly against maximizing the expected logarithm of wealth (geometric mean) as an acceptable investment criterion. Samuelson [1969] even went to the extent calling it a "fallacy that has been borrowed into portfolio theory from information theory of the Shannon type" that needed to be "dispelled".

It is now clear that maximizing the expected logarithm is the prescription for the optimal growth-rate portfolio. Thorp [1969] proved among other things, that the growth rate optimal portfolio is not necessarily on the efficient frontier, thus showing the incompatibility of log optimality and the mean-variance theory of Markowitz.

The example shown previously did just that, a strategy maximizing the growth rate of a portfolio, per Shannon et al. Note that there is no maximizing of mean, because it is constant. Variance also cannot be minimized as it is given as constant standard deviation of 0.3, so you take it by investing in it, or you leave it.

On Entropy

Now if we believe that growth-rate optimization of Information theory can be applied to the world of finance, we should necessarily also understand the concept of entropy. Entropy essentially means energy with a probability, a measure of randomness in a closed system.

One might agree that there is a never-ending discussion on whether the financial market is random or not. What we really mean is: if given the stock prices today, are we able to predict tomorrow's prices? - This is of course the well-known random-walk model. Now what if we see the financial market as a huge entropic system?

We first note that entropic model is non-linear, and is not a random walk model. A random walk is a cumulative sum of a linear variable - the noise is additive. In entropic models however, noise is multiplicative (Conover [2002]). There are many proofs and examples that real-world financial market is multiplicative rather than additive, so we will not mention everything. But the easiest intuition is to look at interest rate, do you serve an additive or compound interest to your loan?

Next, if financial market is really seen as an entropic system, the implication to portfolio investments is then huge. Given that entropy means energy with a probability - and that if price is a reflection of this, energy, which is a scalar quantity without directions - then it means that it really does not matter whether prices go up or down, as long as prices exist!  Essentially, it is not about asset returns, but rather, simply if asset prices have been there given the time horizon. Without realizing it, literature discussing mean-reversion and time effects on portfolio investments are actually describing a question relating to entropy.

We Love Noise

Now in parallel, we propose to loosely take an asset's volatility as entropy's equivalent. In our case, given that our strategy is a entropic growth-optimal one, it would imply that markets having higher entropy, that is, higher volatility, would then allow our portfolio to grow faster!

In fact, one could also notice that mean-reverting or anti-trending prices provide more entropy or opportunities for the portfolio to grow, using such rebalancing strategy, compared to smooth trending markets.

This might not make sense at first since almost all trading strategies, whether technical or fundamental, involve using some form of indicators to filter out noise and identify the underlying trend, thereby creating a signal to buy or sell.

Not in our view, because if you see the market as one big "energy" system where molecules are flying all round randomly, and you are at one corner holding a paper bag trying to catch these molecules, you would imagine the higher the speed the molecules move, the higher the chances you would able to catch the molecules in your bag in shorter time. This is exactly the same with the portfolio strategy - the more volatile the prices move, the easier the portfolio will grow.

Now if you refer to Example 2 where we increased the sigma or variance of the random white noise. Using exactly the same strategy in Example 1, notice that the same portfolio actually grows faster given sigma is greater. In such cases the portfolio will actually grow exponentially.

We have shown therefore, by increasing variance, or risk, we achieve faster growth rate for the portfolio. Is this not an optimal portfolio? You may consider that financial heresy, but in short, we love noise.

Parting Words

Having established the above, the next thing is simple, look out for markets with high volatility and tendency for mean-reversion to invest - although this is perhaps just the opposite of what many may actually do! Nonetheless, it does lend some support to contrarian views of "buying when everyone is selling".

In essence, we actually felt that equating risk to be equal to volatility, and specifying that optimal portfolio should be on the "efficient" frontier by maximizing mean and minimizing variance - is really doing us a disfavor, despite probably all the good will. Our examples, although being drawn in a theoretical world, provided one case where this notion does not hold. Minimally, we showed that there is certainly more to portfolio theory than what is commonly practiced.

Risk is therefore really how you see it. For us, it does seem the more the merrier.

Reference:
[1] J. Kelly, "A New Interpretation of Information Rate", Bell System Tech. Journal, 1956.
[2] L. Breiman, "Optimal Gambling Systems for Favorable Games", Fourth Berkeley Symposium, 1:65-78, 1961.
[3] Thomas M. Cover, "Shannon and Investment", IEEE Information Theory Society Newsletter, Special Golden Jubilee Issue, Summer, 1998.
[4] P. Samuelson, "Lifetime Portfolio Selection by Dynamic Stochastic Programming", Rev. Econom. Stats. 1969.
[5] John Conover, "Quantitative Analysis of Non-Linear High Entropy Economic Systems", Feb 2002.
[6] William Poundstone, "Fortune's Formula", Hill & Wang. 2005.

Note: For readers who are interested to know more relating information theory to portfolio investments may like to refer to [3] Cover, and [6] Pounstone, where the story of this article is based upon. Author reserves all rights to the explanation of topics and examples. Email: luen@epsilonz.com