Get poor quick vs. get rich slow: 2-asset portfolio theory

Many people jumped in to the stock market hoping to get rich quickly, but instead of getting rich, they burned most or all of their money. Even the legendary speculator Jesse Livermore, who made millions of dollar on the stock market a century ago (he would have been a billionaire in today’s dollars), committed suicide in the end after he lost it all. The number 1 reason for bankruptcy on the stock market is not bad market analysis, but bad risk management. You can be right most of the times, and still go under if you don’t manage your risks well. On the other hand, as unbelievable as it may sound, with a proper risk management you can actually make money on the stock market  in the long term, even if you know nothing about the direction of the market !

Get poor quick

Assume that  God  talked to you, and he told you that the price of gold would be 30% higher in three  months. And God never lies. This is called “perfect insider information”. With this kind of perfect information, you were going to make a killing on the gold market. Your account had 200K USD in it, and you decided to buy gold  with it, with a 20 times leverage. So you bought 4M USD worth of gold, borrowing 3.8M USD from your brokerage (or entered into a gold futures contract worth 4M USD). Three months later, your gold was worth 5.2M USD, and you could liquidate your position with a 1.2M USD gain, making you a millionaire. So easy, isn’t it ? Unfortunately, God forgot to tell you that, before going up 30%, gold would first make a dip of 6%. So your 4M USD worth of gold, before becoming 5.2M USD, went down to 3.76M USD first. But that was more than enough to wipe out your entire equity of 200K.Your broker was forced to close your account when he saw that your equity became negative. Not only that you didn’t make your expected 1.2M USD, but now you’re broke, and moreover you owe your broker some money. What a tragedy.

The above fictional example is actually very typical of many people (or hedge funds for that matter) who made good predictions but who still got wiped out, because their positions were not strong enough to survive the unpredictable short-term movements. Even “sure things” such as “perfect” arbitrage plays can lose money, if they use too much leverage and prices diverge too much short-term leading to a forced liquidation.

Some people are lucky, and didn’t get killed in the market even though they took on very risky positions. But for every “lucky” person there are maybe 100 “unlucky” ones. And if you need to rely on your luck to make money, then sooner or later you will enter an unlucky period, where your money will be wiped out.

Get rich slow

A better strategy is to get rich slow. You probably will not become a billionaire (to become a billionaire, you really need luck, all the other things are not enough). But if you can beat the market by just a few % a year, then over a period of 20-30 years of investing, you will become quite rich, enough to have a comfortable lifestyle and to give money to your children for them to jump-start. (Many of us would wish that our parents were that rich).

Let’s say your portfolio has a real growth rate (after taking out the inflation rate) of 20% a year. Then over a period of 30 years, it will grow 237 times ! If you start at the age of 20 with 10K USD, and can keep that performance, then by the age of 50 you will get more than 2M (in constant dollars) to spare, without needing to feed the portfolio with any new money in between.

How hard or how easy is it to make 20% a year ? It depends on you. There are simple, proven methods to make more than 20%/year, year in and year out. (See the books by Joel Greenblatt for example). But these methods are “boring”. They don’t promise that you can make money fast, they ignore the noise on Wall Street. So most people don’t follow them, because it’s human nature to have too much greed or too much fear, to get agitated by the noise, to follow the herd , etc.- i.e. to have lots of bad habits which lose you money in the end.

2-asset portfolio theory, or how to make money without knowing anything

When you have reliable market analyses, then you can use them to your advantage and win big (provided that you also have a good risk management). But here I want to show something which at first may look paradoxical: with a good asset management you can make money (in the long term) even if you don’t know anything about market directions !

For simplicity assume that the market consists of just 1 currency and 1 other asset (for example USD and gold). So your portfolio consists of just two assets:

P = D + G

(D stands for dollars, G stands for Gold)

Assume for simplicity that the inflation rate is 0 (If not, you have to divide the value of P by the inflation factor to get to the value of P in constant dollars). Assume also that the interest rate is also zero (i.e. the interest rate is just equal to the inflation rate). In other words, assume that D has constant value: If P=D (you have only cash, no gold) then you will not lose anything and will not gain anything.

The price of gold changes with time:

P_n = D_n + b_n . G_n

where b_n is the amount of gold that you have at time n, and G_n is the price of gold at time n.

At each time you can either buy gold or sell gold. For simplicity, assume that transactions cost are zero, so we

have the following equality:

D_{n-1} – D_n = (b_n – b_{n-1}). G_n

(i.e. you change D_{n-1} – D_n for b_n – b_{n-1} of gold at price G_n). In other words, we have the following consistency condition:

D_{n-1} + b_{n-1}. G_n = D_n + b_n. G_n = P_n

Since the price of gold changes, sometimes it goes up and sometimes it goes down, sometimes the value of your portfio increases and sometimes it decreases. The question now is: how do you manage your portfolio (change b_n at each step) so that to have a positive average performance, i.e. to increase P_n in the long term ?

How to measure the performance ? The (composite) performance between time m and time n can be measured by the number

log (P_n/P_m)

If ther performance is positive then it means that you win, and if it is negative then you lose. A remarkable property of this performance measure is that it’s additive in time:

log (P_n/P_k) = log (P_n/P_m) + log(P_m/P_k)

The average performance from time 0 to time N is euqal to

(1/N) . (sum_1^N log (P_{k+1}/P_k) = (1/N) . log (P_N/P_0)

We want to have a positive average performance.

If you “know” the performance of G_n (i.e. know the values log(G_{n+1}/G_n) in advance) then of course you can win, buy moving all of your portfolio into G at time n when log(G_{n+1}/G_n) > 0, and moving out of G when log(G_{n+1}/G_n) < 0. Such a task would be too easy. But how can you still will if you don’t know what log(G_{n+1}/G_n) will be inadvance ?

When we know don’t about something, we call it a random variable. So here we will assume that log(G_{n+1}/G_n) is a random variable.

Even if something is a random variable, we may still have some minimal information about it. In the case of gold, what do we know for sure ?We know that the price of gold never goes to zero, and it never goes higher than the total wealth of this world. It means that log(G_N/G_0) is a bounded random variable. As a corollary,

lim_{N to infinity} (1/N). (sum_1^N log (G_{k+1})/G_k) = 0

In other words, G_n has zero average performance !

The zero average performance (or near-zero average performance) is a remarkable property of many commodities: gold, oil, rice, real estate, hard currencies, …

So here we have 2 zero-average-performance assets: dollars and gold. The diversification principle says that, by diversification you can increase the average performance of your portfolio. By combining 2 zero-average-performance assets together, you can get a positive-average-performance asset. It’s like a magic, right ? If gold is not volatile (i.e. if its price remains constant) then of course diversification doesn’t make you win anything. Here you will be able to win exatcly because of the existence of volatility !

Ok, now the concrete question: what is the optimal b_n (amount of gold) to have at time n ? Recall that

P_n = D_{n-1} + b_{n-1}. G_n = D_n + b_n. G_n

At time n, you know P_n, G_n, you don’t know G_{n+1} and you have to decide what b_n to have.

What you know is that E (log(G_{n+1}/G_n)) = 0 (the mean expected value of log(G_{n+1}/G_n) is zero, because G_n has zero average performance).

Assume, for simplicity, that there are just 2 possible outcomes for log(G_{n+1}/G_n): either log(G_{n+1}/G_n) = a or log(G_{n+1}/G_n) = -a. You have to find b in order to optimize the average performance

1/2 . (log(P1_{n+1}^1 /P_n + log(P2_{n+1}/P_n) )

where P1_{n+1} and P2_{n+1} are the two possible outcomes for P at time n+1

In other words, you have to maximize the product

P1_{n+1} . P2_{n+1} = (D_n + b_n. G1_{n+1}). (D_n + b_n. G2_{n+1})

= (D_n + b_n. G_n. exp(a) ). (D_n + b_n. G_n. exp(-a) )

= (D_n)^2 + (b_n.G_n)^2 + D_n.b_n.G_n (exp(-a)+exp(a))

= (P_n)^2 + D_n. (b_nG_n). (exp(-a) + exp(a) – 2)

Note that C = exp(-a) + exp(a) – 2 >0. You have to maximize D_n. (b_nG_n), knowing that D_n + b_nG_n = P_n. The answer is: It will be maximal when D_n = b_nG_n, i.e. you have exactly half of your portfolio in cash and the other half in gold !

The formula for b_n is: b_n = P_n/(2.G_n)

The probabiility measure of log(G_{n+1}/G_n) in practice may be much more complicated than the binary measure above, but as long as log(G_{n+1}/G_n) is random with mean value equal to zero, then D_n = b_nG_n = P_n/2 is the best asset allocation strategy.

In other words, we have proved empirically the following theorem:

In a situtation with  zero-average-performance assets, the equal-weight asset allocation is the best allocation stretegy, which gives positive average performance.

This theorem can be generalized to many-asset portfolios: if you know nothing about the assets except that they always have bounded positive prices, then the best allocation strategy is the equal weight strategy (the same amount of money for each asset)

Remark also that, if  b_n is negative (i.e. you short gold), or D_n is negative (you buy gold on margin), and the price movement is random, then your average performance will be negative ! Thats why it’s so dangerous to be a short or to play on margin.

In the binary above model for G_{n+1}/G_n, a is the volatility, and the average performance of P (which, at every time n, is accolated so that D_n = b_nG_n = P_n/2) is about a^2/8. (a is assumed to be small).For example, if a= log(1.1) (i.e. afer each period of time, gold moves by about 20%), then the average performance of P is about 0.5%, which is of course very small, not sexy at all. But that’s “the best” one can get for “knowing nothing”. A positive performance, however small, is better than zero performance or negative performance.

In order to increase the performance, you will need to

– Consider more volatile assets

– Have better models and analyses of  price momevents (they will not be entirely random, but will at times show deterministic velocity terms). I’ll discuss price movement models in some tufure posts.

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