Diversification and Correlation - Understanding Risk and Reward

Often you'll hear people quoting investment cliches: "Don't put all your eggs in one basket" "Diversify, diversify, diversify" but not really understand what they mean. Most will understand that if they only invest in one stock and it tanks that they can lose everything straight away (or hit it big). However, few understand the risk and benefits of diversification.

The CFA Level II material begins to go into the idea of correlation, one of my favourite topics in math. While I am often criticized for loving math a little too much (one colleague went so far as to say that I think math can "solve all the world's problems" whereas I would prefer to think of it as "math can describe most of the world's patterns"). I even said that "there is math to describe when math fails" and that in my opinion is statistics.

A Quick Primer on Correlation
Correlation is the idea of how closely to items move together (in finance, the most notable example is stock prices) and the strength of their linear relationship. Relationships measured in correlation can have a value between 1 (perfectly linearly correlated) and -1 (perfectly negatively linearly correlated). What does this mean in layman's terms?

With a correlation of 1, two stocks will move in perfect harmony. If one stock rises, the other stock will rise proportionally. With a correlation of -1, if one stock rises, the other stock will fall proportionally. A correlation of 0 implies no linear relationship (strictly speaking not independent, but independent variables will have a correlation of 0).

Correlations of less than 1 mean that they move in the same direction, but do not have a perfectly linear relationship (most stocks in the stock market) and do not move proportionally (sometimes one will move faster or slower than the other). I would propose that the only way to find a perfect correlation is to buy more of the stock (or short it for a -1 correlation). Obviously, correlation is a bit more complicated that this but this will do for now.

Risk and Return of a Portfolio
Now that we have a basic understanding of correlation, how can that help us understand diversification, risk and reward? Let's look at two stocks A and B with expected returns 15% and 10% and a correlation of .5. Let's say the stocks have std dev of 9% and 6% respectively and the risk free rate is 4% (therefore the Sharpe ratio is 1.22 and 1 respectively). A is riskier, but offers more marginal return per unit of investment risk.

There are four possible actions:

1. Long (buy) A - Correlation to Long A: +1
   
2. Short (sell) A- Correlation to Long A: -1
   
3. Long (buy) B - Correlation to Long A: +0.5
   
4. Short (sell) B- Correlation to Long A: -0.5
   

Note that if you only care about maximum returns you will allocate all your capital to action 1: Long (buy) A. It has an expected return of 15% so it has the highest growth potential. But note that it also has the highest risk profile (largest standard deviation). If you were more moderate, you would Long (buy) a combination of A and B (with an expected return of between 10 to 15% depending on allocation and a standard deviation between 6 to 9%).

The lower risk portfolio construction would be from some combination of stocks with negative correlation (example Long A, Short B or Short A, Long B) because if one ever went down, the negative correlation will imply that the other will go up (possibly by more, possibly by less). However, also note that if their movements are counter each other as is usually the case in a negative correlation, your profit potential becomes much less.

Diversified Portfolio
In this over simplified scenario, assume that a portfolio, evenly weighted between a Long A position and a Long A and Long B. If the both hit their growth targets their combined return is 12.5% (equally weighted average between 10 and 15% and std dev between 6 and 9%). This is less reward than just buying A, but also less risk.

Assume another evenly weighted portfolio between a Long A and Short B position has it's Long A hit +15% and it's counterpart, the Short B hits -10%. The portfolio only gains 5%. Conversely, if the Long A drops to -15% and the Short B rises to 10%, the portfolio only loses 5%. Whereas the movement in the individual stocks is much more pronounced, the portfolio is dampened from extreme gains and losses.

Implication
There are times to over diversify and there are times to cherry pick. Arguably, in this recovering economy, it's easy to pick "sprouts in scorched earth". That is to say, most stocks are undervalued so it's not hard to pick "winners". This is a decent time to over diversify, because the general trend is to go up in value.

The worst time to over diversify is at the peak of the market, when most stocks are over valued. In this case, it is better to be very specific about your investments and be extra diligent in your homework (or find another asset class like fixed income - deleverage).