Risk-return tradeoff: Going to the dogs

One of the central elements in modern finance theory is that of a risk-return tradeoff.

The idea is simply that investors are risk averse, and accordingly must be paid to incur risk. There is, then, a constant trade-off in the investment world: safe investments carry low return, high-return investments aren’t so safe. Fortunately, this conforms with almost everyone’s intuitions.

What exactly is risk, though? Yes, we have an intuitive idea. The guy jumping out of an airplane is taking a risk that the appreciative audience standing on firm ground below is not. Further, if he has neglected to check his gear properly he is taking an extra, unwarranted, risk.

But we can be a good deal more specific about what the word means in the world of investments. It means the size of the standard deviation of return. It means the width of that bell curve we've discussed in earlier posts.

Standard Deviation

A standard deviation is the “average distance from the average” here’s a simple example: there are five dogs in your neighborhood, and you weigh them. They are each conveniently named with a single letter. It turns out that the pooches weigh:

Name pounds

A 5

B 7

C 10

D 13

E 15

The average weigh of a neighborhood dog then, is 10 lbs.

The distance of each dog from that average is as follows:

Name distance

A 5

B 3

C 0

D 3

E 5

The average of those numbers is 3.2, so that would be the standard deviation of dog poundage.

In a neighborhood with a wider range of dog types, and so of dog sizes, you’d expect the standard deviation to be wider. At any rate, as you can see from the bell curve portrayed earlier this week, the curve is getting close to zero by the time it gets out to two standard deviations [or two “sigma"] from the mean. In our (small) neighborhood sample, then, it is unsurprising there are no dogs whose weight is two sigma, or in this case 6.4 lbs. away from the mean.

In a large population, randomly chosen so that the bell curve should apply, we would expect that 68 percent of the dogs would fall within one sigma of the mean, 95 percent within two sigma, and 99.7 percent within three.

It may not be intuitively clear why we should call standard deviation a good proxy for risk. One doesn’t normally speak of the “risk” of encountering a smaller-than-usual dog, after all. And only trespassers need worry about the risk of encountering a larger-than-usual dog. Even there, the size of the dog is a rather unsatisfactory proxy for what the trespasser ought to be worried about.

So why would Harry Markowitz, one of the founders of modern portfolio theory, write that “if … ‘risk’ [were replaced] by ‘variance of return,’ then little change of apparent meaning would result”?

Chiefly, because investments differ from each other according to the predictability of their return from one period to another. Those we call low-risk are designed precisely to deliver boringly consistent (but low) returns from one period to another. Corporate bonds deliver a steady contracted-for stream of interest payments. They have a variation in return (thus, a standard deviation, or risk) only to the extent that the corporation might go bust and default on the payments. The chancier the future of this corporation, the wider the standard deviation will be.

A Story About Coleridge

This is a quote from a memoir by Dorothy Wordsworth, reflecting on a trip she took with two famous poets, her brother, William Wordsworth, and their similarly gifted companion, Samuel Taylor Coleridge. We sat upon a bench, placed for the sake of one of these views, whence we looked down upon the waterfall, and over the open country ... A lady and gentleman, more expeditious tourists than ourselves, came to the spot; they left us at the seat, and we found them again at another station above the Falls. Coleridge, who is always good-natured enough to enter into conversation with anybody whom he meets in his way, began to talk with the gentleman, who observed that it was a majestic waterfall. Coleridge was delighted with the accuracy of the epithet, particularly as he had been settling in his own mind the precise meaning of the words grand, majestic, sublime, etc., and had discussed the subject with William at some length the day before. “Yes, sir,” says Coleridge, “it is a maj...

Searle: The Chinese Room

John Searle has become the object of accusations of improper conduct. These accusations even have some people in the world of academic philosophy saying that instructors in that world should try to avoid teaching Searle's views. That is an odd contention, and has given rise to heated exchanges in certain corners of the blogosphere. At Leiter Reports, I encountered a comment from someone describing himself as "grad student drop out." GSDO said: " This is a side question (and not at all an attempt to answer the question BL posed): How important is John Searle's work? Are people still working on speech act theory or is that just another dead end in the history of 20th century philosophy? My impression is that his reputation is somewhat inflated from all of his speaking engagements and NYRoB reviews. The Chinese room argument is a classic, but is there much more to his work than that?" I took it upon myself to answer that on LR. But here I'll tak...

Recent Controversies Involving Nassim Taleb, Part I

I've written about Nassim Taleb on earlier occasions in this blog. I'll let you do the search yourself, dear reader, for the full background. The short answer to the question "who is Taleb?" is this: he is a 57 year old man born in Lebanon, educated in France, who has been both a hedge fund manager and a derivatives trader. He retired from active participation from the financial world sometime between 2004 and 2006, and has been a full-time writer and provocateur ever since. Taleb's writings for the general public began where one might expect -- in the field where he had made his money -- and he explained certain financial issues to a broad audiences in a very dramatic non-technical way. Since then, he has widened has fields of study, writing about just about everything, applying the intellectual tools he honed in that earlier work. As you might have gather from the above, I respect Taleb, though I have sometimes been critical of him when my own writing ab...

Jamesian Philosophy Refreshed

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