# A Thought About Falkenstein’s Ideas, from Charles Pennington

July 16, 2010 |

Eric Falkenstein is the author of an excellent book, Finding Alpha, and of a website.

One of his big insights is that in the real world the relation "return ~ risk" is often not obeyed. He cites many examples, but a representative example is that risky stocks (whether high beta, high volatility, high idiosyncratic volatility, or whatever) have not historically outperformed less risky stocks. I'm thinking that one possible explanation for this is that when you own risky stocks, you sort of get an implied put option "for free". The market makes you pay for that put option by giving you a lower return on the riskier stocks. Here's an example to make it clear:

Suppose investor A buys the whole market, with beta=1, and gets an average return of 10% with a standard deviation 25%. Investor B instead puts just 20% of his money into a diversified portfolio of high beta stocks, with an average beta of 5. He puts the rest of his money into a "risk-free" investment, and for simplicity, I will assume that the risk-free rate is 0%. What return should investor B expect on his stocks? Well, the conventional academic view is that his stocks should have an average return of 5 times that of the market, or 50%, with a standard deviation of 125%. Since B has only 20% of his money invested, his expected average portfolio return would then be 10%, with standard deviation 25%, the same as A.

The problem though is that B has a safer portfolio than A. B has a "floor" on his losses–he can lose at most 20% of his capital. He effectively has a put option that's 20% out of the money. How much is that worth? Well, to get a ballpark understanding, a put option on SPY, expiring 1 year out, 20% out of the money, is currently going for about 6% of the SPY share price. So in a fair world, maybe B's expected portfolio return shouldn't be 10%, but rather 4%, to reflect the idea that the market makes him cough up 6% to pay for the virtual put option that he owns.

If that's all true, then beta=5 stocks should have expected average returns of 20%, not 50%, and a standard deviation of 125%.

This is only a semi-quantitative explanation, but the point is that when you own higher beta stocks, you're implicitly getting an implied put protection relative to lower beta stocks. If the market is efficient and makes you pay for that put, then the returns of the high beta stocks would be reduced as compared to what you'd otherwise expect.

Disclaimer: For all I know, probably some academic has already thought through all this and demonstrated that it's incorrect and/or insignificant, and if that's so, then maybe someone can set me on the right path.

## Stefan Jovanovich shares:

An earlier contribution from Eric Falkenstein– David Hakes' story about the risks of publication regarding the subject of risk:

When we submitted the paper to risk, uncertainty, and insurance journals, the referees responded that the results were self-evident. After some degree of frustration, my coauthor suggested that the problem with the paper might be that we had made the argument too easy to follow, and thus referees and editors were not sufficiently impressed. He said that he could make the paper more impressive by generalizing the model. While making the same point as the original paper, the new paper would be more mathematically elegant, and it would become absolutely impenetrable to most readers. The resulting paper had fifteen equations, two propositions and proofs, dozens of additional mathematical expressions, and a mathematical appendix containing nineteen equations and even more mathematical expressions. I personally could no longer understand the paper and I could not possibly present the paper alone. The paper was published in the first journal to which we submitted.

## Lars van Dort writes:

I'm not sure I have much to contribute to the main question your post raises (why is the relation risk-return often not obeyed?), but I must say I was intrigued by your example. I felt it must be flawed, but it took me quite a while to see why.

Let's consider the investment in stocks of the portfolio of B, which has an average return of 50% and a standard deviation of 125%. The following could be one of the possible return distributions, from which these numbers are derived:

-100.0%
-50.0%
-38.5%
-25.0%
0.0%
50.0%
150.0%
213.5%
250.0%

Average return = 50%
Standard deviation = (pretty close to..) 125%

We see that the worst possible result is -100%, more would not be possible for stocks anyway. Because B has invested 20% of his total portfolio in stocks and 80% risk-free against a 0% return, his worst possible total return is -20%.

We now have to decide what return distribution to assume for the portfolio of A (average return 10%, standard deviation 25%). There are two options.

Option 1:

We take the possible returns from above and divide them by 5:

-20.0%
-10.0%
-7.7%
-5.0%
0.0%
10.0%
30.0%
42.7%
50.0%

Average return = 10%
Standard deviation = (pretty close to..) 25%

Or any other distribution with a worst possible return not lower than
-20%. In this case, the portfolios of A and B can both not lose more than 20%!

Option 2:

We do allow for a worst possible return for A of lower than -20%. However, in the equivalent distribution for B this would lead to a worst possible return for B's stocks of lower than -100% (because x5). This is not possible for stocks, but even if we imagine other assets that can take a negative value, this would have the consequence that B's total portfolio loss is no longer capped at -20%.

But what if we take a distribution for A with a worst possible return of lower than -20% AND a distribution for B's stock returns with a low of -100%. In this case (and here comes the point), for all the values to still add up to the mentioned average return and standard deviation, one or more of the other possible returns in the distribution of A would have to be higher, compared (x5) to B.

So, when one wants to argue that in this situation B's portfolio includes a put option because his losses are limited, along the same lines one would have to argue that A's portfolio includes a call option, because his possible returns are also relatively higher. Although I'm not sure how to prove this, it seems logical to assume these options need to have the same value.

The numbers of the example can be changed, but I believe a reply as above can always be given.

## Tyler McClellan writes:

My quick thought is that this is not a good way to think of it.

The idea is to look at the marginal preferences of people with the same portfolio set.

In your example the relevance is not between the two portfolios you list but between what stocks the person with 80 percent in cash should chose for the remaining 20.

But I also suspect you are on to the correct way of getting insight about this, which is to show that the distribution of portfolio preferces is very correlated to specific holding within a category (for example maybe the person that owns risky stocks is highly likely never to own other stocks), such that a dynamic similar to what you describe does in fact happen. (best I can describe it is that the category of people to drive this relationship away by buying the now theoretically mispriced stocks is not big enough to overwhelm the people that continue to want volatile stocks and cash, or some other asset such as you suggest).

## Rocky Humbert shares:

There are many ways to look at this; however using a high beta subset of the index has elements of a self-referential paradox and must be avoided.

One thing to recognize is that REAL and NOMINAL interest rates greatly influence the result. In an environment of very high real and nominal rates, and low stock market volatility, one can buy a five year zero coupon bond and use the discount to buy calls on the s+p with no principal risk. At the extreme, one could achieve full index replication with no principal risk, and I'd argue that this would be the perfect baseline for analyzing the issue.

## We are honored to receive a message from Eric Falkenstein:

I appreciate Charles mentioning my name!

I think you can create such arbitrage only because the standard CAPM assumes lognormal returns, and for lognormal returns, only the first two moments (mean and standard dev) matter. So, parceling out put options is like saying there are different relations between how stdevs relate to max drawdown due to 'non-gaussian' transformations via leverage, distinctions that by definition are irrelevant within the framework of the canonical CAPM and its derivatives.

Many people, including Markowitz at the inception of the CAPM, have pointed out that returns may have important higher moments–skew, kurtosis, see here on my web site for references. Indeed, Fama did a lot of work on this in the 1960's (see my blog ),and his take-away was that these adjustments merely make second-order, intuitive changes to the base model–complications without much real add. However, downside skewness may be going thru a revival, as Cam[pbell] Harvey (editor of the JoF and mainstream finance archetype) actually  mentioned  in comment section of my blog that skewness preferences could explain a lot of these negative volatility-return empirical findings.

## Alex Castaldo adds:

As they say in China "Speak of Cao Cao and Cao Cao arrives."

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