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).

Burning Hot - Getting Ready for the Rotman MBA

I've recently (Sunday) returned from NYC from my internship in finance and hit the ground sprinting. I've only recently moved into my new house and the internet situation isn't what I'd like (so I apologize for not having any CFA Level II posts up yet).

I haven't quite craked open my books yet, but rest assured they are forthcoming. I've already started to received my Rotman books and course materials. Classes start on Sept 8th so I have two weeks to get sorted out (between Rotman Pre-MBA classes and events).

Expect a great deal of content to appear over the next few days as my situation improves.

Does the CFA Exam Get Harder Every Year?

This is a question I've been asked a lot recently by friends and colleagues who are looking to boost their credentials in a very competitive job market. While many are coming out of school (Undergrad or MBA) and looking for an edge, some are thinking about the timing of when to write the exam.

While competitive tendencies would suggest that writing sooner would be beneficial, school and work schedules don't necessarily permit the allocation of study time required. Conversely, potential candidates are worried that if they put it off, the CFA exam will get harder and harder.

While it is true that the curriculum changes from year to year, and possibly that the most recent exam is much different from the exams given a decade ago, the most recent exam and associated curriculum only had incremental changes from the previous year (remember, I wrote Dec 2008 and June 2009... And got sent to years worth of books for two years worth of curriculum fees).

The only real changes I've noticed between the curriculum from year to year is that they added a VERY brief section on Game theory and Nash equilibrium. If I'm not mistaken, there was only one question on the 2009 exam, and even then I'm guessing that because it is so obviously new, it was probably one of those questions which will not count and will be evaluated for future inclusion in other tests.

With the introduction of the game theory section, however, there was an incredibly large optional section (which, by the way, is incredibly interesting reading, especially for those interested in business strategy as a science, rather than an over used buzz word). It doesn't seem like the basics of financial physics has really changed all that much.

Having said that, I would not be surprised if:

• More of the "optional" material slowly starts creeping into becoming mandatory
• More of the Level II material slowly starts creeping into Level I
• More of the Level III material slowly starts creeping into Level II
• There is more material added to Level III
  

But this is no different than how our school curriculum teaches subjects earlier and earlier. Calculus used to be taught mostly at the university level but has crept into high school. In the same way, I don't think there is anything to fear, when it comes to the CFA program. However, although the physics of finance don't change, this makes a compelling argument for continuing education and keeping your skills sharp.

Capitalists being what we are, always looking for that edge, I'm sure someone will eventually be interested in providing the service to teach you how to one-up your competition.

CFA Level II Candidate

As I've been proud to mention before, I've passed my Level I CFA exam this June. Recently, I registered for the Level II exam and my books recently arrived at my home in Toronto. As I'm finishing my internship here in New York, I'll be heading home next weekend. My plan is to begin burning through as much of those books as possible as a primer for my MBA (I've never really read the "suggested reading lists" since elementary school because I always found them to be busy work to calm down nervous over achievers rather than anything useful).

There are a lot of events happening in the week I return to Toronto (which is also the week before school officially starts), however, my goal is similar to my goal in the weeks leading up to the CFA Level I exam (for those of you who remember, I was trying to post 2 to 3 questions / related topics per day).

I will be registering for a CFA prep course with Passmax again, but I'll have to decide what my time commitment and allocation can be outside of my MBA classes which promise to be quite involved (even potentially "free" time will be allocated to extracurricular activities).

Also, in the week following my departure from my internship, I will see what work is cleared for release which I can talk about and discuss what I've been doing for the past summer.

My Analyst Exchange Profile

This is a short video of me posted for the Analyst Exchange explaining what I think of the program and my experiences. There will be more profiles posted soon. Check the Analyst Exchange website for more videos and photos of speakers and participants.

Analyst Exchange - Lecture Video

This is a video uploaded on YouTube where I was showing my colleagues the underlying math behind a perpetuity, specifically as it relates to calculating terminal values using EBITDA.