You’re not going to understand this unless you read Part I, and you probably won’t understand it even then. That’s a criticism of our ability to explain things, not your ability to absorb them.
Last week we tried to figure out what those measures of risk under a mutual fund’s chart mean. Here’s the chart for the Dodge & Cox stock fund, updated from last week. The numbers for the fund itself haven’t changed, as Yahoo! hasn’t recalculated them since September 29 (and we can’t be bothered to do it ourselves), but the numbers for the fund category itself have changed:
α and β we handled last time out. What about Mean Annual Return, R², standard deviation, and Sharpe and Treynor ratios? Let’s see if we can do this and still retain your interest.
Mean Annual Return. “Mean” means average, right? Yes, but there are different kinds of averages. Note the time period listed in boldface on the chart. This isn’t the standard (arithmetic) average, calculated by adding up the values of a bunch of items and then dividing by the number of items. This is geometric average. Replace “adding” with “multiplying” and “dividing by” with “taking the nᵗʰ root, where n is” in the previous sentence and you’ve got it.
Here’s the difference. For instance, what’s the average of 5, 6 and 7?
That’s the arithmetic average. (Ignore the vertical line down from and to the left of the 3, it’s just a typo.) Here’s the geometric average:
Not quite the same. Geometric average is necessary when calculating returns over a period of years. If you use arithmetic average it’s confusing and paints an inaccurate picture. Anyhow, by measuring the individual annual increases (or decreases) and calculating their geometric mean, we know that over the past 3 years the Dodge & Cox Stock fund has averaged a 1.46% gain, which beats the category.
Now onto R². It’s a measure of spread, which is contingent on mean (which, for our purposes, means mean annual return.) R² is supposed to separate how much of that return is attributable to the fund itself from how much is attributable to the underlying benchmark – here, the standard index. If DODGX is just riding the market’s coattails, we want to know about it. The lower R² is, the more the fund’s performance is its own doing, for better or worse. You calculate R² by graphing the performance of both entities, the fund and the index, over time. They won’t be identical, of course. You measure the difference between the fund and the index at different points, square the differences, and in this case discover that DODGX’s movements are explained almost entirely by the index. 96.5 on a scale from 0 to 100. Some if not most people prefer funds with high R², the argument being that a rising index lifts the boats that are most in sync with it.
Generally, R² of over 90 is considered high. R² of under 70 is considered low.
What about standard deviation? Like R², this is a general statistical term that applies to all sorts of sciences, not just finance. If you didn’t learn it in high school or college, here it is in its simplest form. Take 5, 6, and 7, the data points that served us so well in the previous example. Their (arithmetic) mean is 6, we know that. Just like the mean of -40, 27 and 31 is also 6. But obviously, the first set of values is a lot tighter than the second. In other words, their standard deviation is lower. You calculate standard deviation by
- Subtracting each term from the average.
- Squaring each of those differences.
- Adding them all together.
- Dividing by the number of items.
- Taking the square root of that.
Again, we don’t expect you to calculate this yourself. But we’re not just going to throw an unfamiliar term at you and explain it without going into details. We’re writing this in such a fashion that if you want to, you can reproduce the results yourself. Otherwise we’re not really explaining anything, we’re just listening to ourselves type.
Using the 5-step process above, the standard deviation of 5, 6 and 7 is .816. The standard deviation of -40, 27 and 31 is 32.568.
But how does this help us and what does it mean? What are the quantities, the data points we’re dealing with here?
Answer: Monthly total returns. Keep in mind that the table above shows results for only the last 3 years. So we take DODGX’s return in October of 2010, its return in November of 2010, etc., until we have all 36 monthly total returns. (Well, that’s not entirely true. We then annualize them, which is to say, we take each monthly total return to the 12ᵗʰ power.) DODGX’s standard deviation is 14.67, slightly higher than that for its category of funds. That number is a percentage, by the way. It means that assuming a normal distribution – a bell curve – DODGX’s returns will be within 1467 basis points of its average return 68% of the time. It’ll be within 2934 basis points – two standard deviations – of its average return 95% of the time. And it’ll be within 4401 basis points, or three standard deviations, 99.7% of the time. The 68%, 95% and 99.7% aren’t arbitrary, either. They’re natural constants. Well, they aren’t really, but they’re close enough that we can say they are without turning this into an introductory-level probability course.
Alright, it looks like we’re going to have to stretch this out to 3 parts. Friday, we’ll do Sharpe ratio, Treynor ratio, and give you our synopsis of what the point of all this is.