# What is the theory behind Rick Van Ness's risk calculation in the video about diversification?

In the video, Rick Van Ness states the following scenario:

• Stock A has a 7% return with +/- 1% risk
• Stock B has a 3.5% return with +/- 0.5% risk
• Stock A and B are perfectly negatively correlated.

A portfolio with 1/3 of Stock A and 2/3 of stock B has:

• 4.7 % expected return calculated as E(portfolio) = (1/3) * 7 + (2/3) * 3.5
• 0% -/+ risk.

Was is the theory/calculation to get 0% risk ?

He's calculating portfolio variance. The general formula for the variance of a portfolio composed of two securities looks like this: where `w_a` and `w_b` are the weights of each stock in the portfolio and the sigmas represent the standard deviation/risk of each asset or portfolio.

In the case of perfect positive or negative correlation, applying some algebra to the formula relating covariance to the correlation coefficient (rho, the Greek letter that looks like "p"): tells us that the covariance we need in the original formula is simply the product of the standard deviations and the correlation coefficient (-1 in this case).

Combining that result with our original formula yields this calculation: Technically we've calculated the portfolio's variance and not it's standard deviation/risk, but since the square root of 0 is still 0, that doesn't matter.

The Wikipedia article on Modern Portfolio Theory has a section that describes the mathematical methods I used above. The entire article is worth a read, however.

• More simply, observe that the variance of the portfolio can be expressed as (W(a)Sigma(a) - W(b)Sigma(b))^2, or equivalently the standard deviation of the risk is W(a)Sigma(a)-W(b)Sigma(b), and so to construct the portfolio, we need to look for two numbers W(a) and W(b) such that W(a)+W(b)=1 and W(a)Sigma(a)-W(b)Sigma(b)=0. In this instance, since Sigma(a) is twice Sigma(b), we should put twice as much money in Stock B as in Stock A. – Dilip Sarwate Jun 18 '13 at 18:24
• @DilipSarwate Great point; this is a good example of how in some cases, one doesn't need the added complexity of what I included (even though the problem isn't complex in and of itself anyway). – John Bensin Jun 18 '13 at 18:34

John Bensin's answer covers the math, but I like the plain-English examples of the theory from William Bernstein's fine book, The Intelligent Asset Allocator. At the author's web site, you can find the complete chapter 1 and chapter 2, though not chapter 3, which is the one with the "multiple coin toss" portfolio example I want to highlight.

I'll summarize Bernstein's multiple coin toss example here with some excerpts from the book. (Another top user, @JoeTaxpayer, has also written about the coin flip on his blog, also mentioning Bernstein's book.)

Bernstein begins Chapter 1 by describing an offer from a fictitious "Uncle Fred":

Imagine that you work for your rich but eccentric Uncle Fred. [...] he decides to let you in on the company pension plan. [...] you must pick ahead of time one of two investment choices for the duration of your employment:

1. Certificates of deposit with a 3% annualized rate of return, or,

2. A most peculiar option: At the end of each year Uncle Fred flips a coin. Heads you receive a 30% investment return for that year, tails a minus 10% (loss) for the year. This will be hereafter referred to as "Uncle Fred’s coin toss," or simply, the "coin toss."

In effect, choosing option 2 results in a higher expected return than option 1, but it is certainly riskier, having a high standard deviation and being especially prone to a series of bad tosses. Chapters 1 and 2 continue to expand on the idea of risk, and take a look at various assets/markets over time. Chapter 3 then begins by introducing the multiple coin toss example:

Time passes. You have spent several more years in the employ of your Uncle Fred, and have truly grown to dread the annual coin-toss sessions. [...] He makes you another offer. At the end of each year, he will divide your pension account into two equal parts and conduct a separate coin toss for each half [...] there are four possible outcomes [...]:

``````Outcome   First coin toss   Second coin toss   Total return
4         Tails             Tails              -10%
``````

[...]
Being handy with numbers, you calculate that your annualized return for this two-coin-toss sequence is 9.08%, which is nearly a full percentage point higher than your previous expected return of 8.17% with only one coin toss. Even more amazingly, you realize that your risk has been reduced — with the addition of two returns at the mean of 10%, your calculated standard deviation is now only 14.14%, as opposed to 20% for the single coin toss. [...]

Dividing your portfolio between assets with uncorrelated results increases return while decreasing risk.

[...]
If the second coin toss were perfectly inversely correlated with the first and always gave the opposite result [hence, outcomes 1 and 4 above never occurring], then our return would always be 10%. In this case, we would have a 10% annualized long-term return with zero risk!

I hope that summarizes the example well. Of course, in the real world, one of the tricks to building a good portfolio is finding assets that aren't well-correlated, and if you're interested in more on the subject I suggest you check out his books (including The Four Pillars of Investing) and read more about Modern Portfolio Theory (MPT).

• I've never read Bernstein's books, but I like his way of presenting the concept. It needs more matrices, though. – John Bensin Jun 18 '13 at 14:41
• @JohnBensin "needs more matrices, though" Heh. Are you using "more matrices" as in more cowbell ? ;-) His book has lots of tables and charts, but no matrices that I can see. But, FWIW, there are a half-dozen or so "Math Details" sidebar sections in the book. :) – Chris W. Rea Jun 18 '13 at 15:04
• Yep, it was tongue-in-cheek (although I'd strangely never heard the phrase "more cowbell" before). I think a good resource is one that isn't afraid to mention the math when it's necessary, but also understands that other presentations may work better for most readers and tries to find the right balance between the math and "other", whether it's tables, coin tosses, etc. It sounds like Bernstein's book does just that. – John Bensin Jun 18 '13 at 15:11

The calculation and theory are explained in the other answers, but it should be pointed out that the video is the equivalent of watching a magic trick.

The secret is: "Stock A and B are perfectly negatively correlated." The video glasses over that fact that without that fact the risk doesn't drop to zero.

The rule is that true diversification does decrease risk. That is why you are advised to spread year investments across small-cap, large-cap, bonds, international, commodities, real estate. Getting two S&P 500 indexes isn't diversification.

Your mix of investments will still have risk, because return and risk are backward calculations, not a guarantee of future performance. Changes that were not anticipated will change future performance. What kind of changes: technology, outsourcing, currency, political, scandal.