Monday, March 17, 2014

Know What You Don't Know

Over the past Christmas holidays I had some free time to pursue an intriguing project. I wanted to get some insight into the what is the best cash-stock allocation mix. In today's world of virtually zero interest yield we can think of this as the classic bond-stock allocation problem. The traditionally accepted to invest is through diversification of between stocks and bonds and also diversification within stocks. For example, Benjamen Graham devotes part of The Intelligent Investor to explain the percentage allocation of each under different circumstances. His advice is intuitive and conventional. When the market appears overvalued, allocate more to cash, up a maximum of 75%. And when the market is undervalued, put the money back in stocks. The 75% number is a bit extreme in my opinion, but I follow his advice otherwise.

As far as I have seen, though, everyone gives this advice in a heuristic manner. I want to change that. I want to find some mathematical confirmation that the conventional cash and stock mix yields a better result than just all stocks. To this end I will need to make some assumptions of the market behaviour. Under these assumptions I put in a mechanical cash-stock allocator algorithm and repeated simulate it. Each simulation is a possible realization of the stockmarket outcome over say 30 years, using my assumptions. My findings were quite surprising (to me). Try as I might, I could not beat a 100% stock strategy over the long run. So, I gave up. Though, I did not shake my belief that one must have ample cash at any time. Cash saved me in 2008-2009, and many a wise investor such as Buffett, employ this strategy. Then recently while thinking about it again, I think I answered this mystery.

Any probabilistic mathematical analysis requires some assumptions to create a model. I assumed that the stockmarket will return the same as it has for the last one or two hundred years, which is an average of about 10%. But the last two hundred years has been a resounding success for the markets. And no theory says it will continue. To assume a 7 or 10% return for the market is to put total faith in one possible scenario. We shouldn't put too much faith in the assumption because we really don't know! If we don't know what we don't know, bad things happen.

As one example consider one of the most elegant theoretical results in use in finance: the Black-Scholes formula for pricing options. It is a piece of mathematical elegance to price the option on an underlying stock assuming that stock behaves in a simplistic theoretical manner. Some blame Black-Scholes for the financial crises of 2008-2009. But I feel the problem is not the formula. The problem is with the investor who forces real world stocks to fit the Black-Scholes assumption. The Black-Scholes assumption is just one possible way of modeling stocks. This formula certainly does not factor in the human psychology that is part of every market transaction. When market participants overuse the Black-Scholes formula they can then change the behaviour stocks such that the assumption no longer holds. It is like the Truman Show. The Truman show only works if the participating is unaware of his world. If he is aware, then he will change his behaviour unpredictably.

So now back to the cash-stock mix. I realize we should always reserve some cash because we do not know the future of markets. Though I tend to think it will be somewhat like the past, the best approach given this uncertainty is to have a portion in stocks and also hedge the stocks with cash. The cash portion is like a call option on some stock at some cheap price in the future. The cost of this option is the opportunity cost. However I cannot quantify this opportunity cost because I don't know what the market will do. But I do know that the higher the market value, the less is my opportunity cost. And so I would allocate more to cash.

For me then, it is hedging for maximum benefit under all unforeseen scenarios. I know this is a bit contradictory, saying there is a optimal way to operate in a unknown world. But we have to admit we don't know something. That's a lot better than not knowing what we don't know. Investors who ignore this fact do so at their peril.

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