St Petersburg lottery pricing & short investing horizons

I am a statistician (no solid background in finance). Please forward me to a book \ chapter \ paper to resolve the following general question. Suppose we have a stock with the following monthly return distribution: P(R=1%)=0.999, P(R=10000%)=0.001. The mean monthly return is about 11%, which is very good. Still, for typical investors with short horizon (say, 1-2 years), the probability to get anything over 1% per month is very small, so the stock is not that attractive as the average return would imply. That means, the price of such a stock should go down until it reaches a more acceptable profile P(R=3%)=0.99; this stock will tend to have a lower P/E because of its uncomfortable return distribution. Now suppose there is a thousand of such stocks, and their returns are independent. In this case, taken as a group, they have a good return profile (return jumps are no longer rare), so the group should not be punished with lower P/E. So, should such a stock be priced individually (based on its return profile), or in a group of similar stocks?

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Thanks for your thoughtful answers, guys. Now, let us replace "stock" with "asset" or "strategy" and consider Nassim Taleb's strategy of constantly buying puts on the whole market. The strategy has about the same profile of bleeding money almost all the time, except rare market crashes when it profits a lot. Taleb's logic (I guess) is that the market "pays" for the psychological discomfort brought by such a strategy and by the long horizon. It seem logical, but how can we quantify it? Is this particular strategy's risk diversifiable or not? – Tim Aug 29 '12 at 11:41
Hi Tim -- Taleb's reasoning is actually slightly different. His calls these impactful outliers "black swans". His option strategies take advantage of the fact that stock returns are usually assumed to be log-normally distributed, meaning such outliers are ignored in pricing options, and so they're too cheap. Therefore, it's a form of "model arbitrage". To quantify it, you could examine option prices under non-normal distributions accounting for excess kurtosis and skew. It is partially diversifiable since there are instruments correlated with the strategy -- so some risk can be mitigated. – jlowin Aug 29 '12 at 12:24
This is useful, but I seek understanding on a somewhat higher level. My formulation was not exactly what I wanted to ask. I will try to reformulate in a separate question. – Tim Aug 29 '12 at 13:24
Ok! I'll look out for your question. There are many people here who are excellent teachers and I'm sure someone will be able to help. – jlowin Aug 29 '12 at 13:35

What a great question -- it touches on many issues at the core of quantitative finance. This answer might be a lot more than you bargained for, but it's too interesting to pass up.

References

Mostly, this subject falls somewhere at the intersection of these three highly-interrelated topics: risk-neutral valuation, rational pricing and the fundamental theorem of asset pricing. In short: the prices investors are willing to pay depend on perceived risks. So, how can we decide what risks determine prices?

As your question illustrates, risk-neutral pricing is one of the most difficult areas to grasp, as it often defies intuition. As such, it is rather hard to find papers that aren't full of math or that require little knowledge of finance. However, I think these three may fit the bill:

• Risk Neutral Valuation: A Gentle Introduction (Part 1) -- This appropriately-titled but lengthy piece uses a situation similar (but not the same! see below) to the one you describe to motivate its analysis.
• What is... a Free Lunch? -- A very quick, largely non-academic introduction to the arbitrage-free pricing and the fundamental theorem of asset pricing. It's brevity may make it a tempting starting place, but it might be better to read it second, as it is more applied than the first reference. Note also that the words "risk-neutral" don't appear anywhere in it despite forming the crux of its argument; remember I said these topics are quite related.
• Risk-Neutral Probabilities Explained -- This one is probably the most complete treatment, and starts in a more traditional manner using Arrow securities. In disclosure, I wasn't familiar with this one before answering your question, but I found it paired with my first choice on the Wikipedia page for rational pricing, and after reading it thought it would be appropriate.

Some Background

Now, if you're still reading -- or still care -- I'll try to flesh out why these fields are related to your question.

You hit the nail on the head: people have subjective judgements of value, based on risk aversion or their own reads of outcome probabilities. Moreover, these opinions are swayed by context (as you set up with a portfolio, or a short horizon). That's hardly the basis for an objective pricing framework, and yet we know that risk impacts price. How can we make claims about pricing if we can't even agree on where to start?

Our jumping-off point is the idea that a security's price is the expectation of its future value. To a single investor, that value is conditional on a set of subjectively-determined probabilities. The motivation for rational or risk-neutral pricing is to find the set of probabilities under which any investor becomes indifferent to risky outcomes (hence, "risk-neutral"). While that may sound like replacing one problem with another one, there are actually very few ways to do that consistently across all securities in the market, meaning without allowing arbitrage opportunities to arise.

More explicitly, a risk-neutral probability measure makes the expected return on an asset equal to the prevailing risk-free rate. As long as a market is "complete", then the risk-neutral probability measure can be derived for all assets at once, resulting in a unified -- and objective -- pricing framework. The lightbulb really went off in 1979, when Cox, Ross and Rubenstein first used these ideas to price options.

Note -- I don't expect to have convinced anyone with these few paragraphs; please see the attached references for a complete explanation!

Now, the truth is that your question is somewhat boring, in an asset pricing sense, because you have specified the complete return distribution of the security in question. (In fact, I would argue that in many ways, the role of a quant comes down to estimating return distributions.) The fundamental theorem tells us that the asset price must therefore be its (objective) expected value, or arbitrage opportunities would result. Things become much more interesting if we didn't know the probabilities, or the future prices, or both -- and that's the reason risk-neutral pricing was developed.

A more direct answer, kind of...

So, though risk-neutral probabilities are derived from the many securities in a market, I don't consider that the same as your question's "price as a portfolio" option. It is rare to find a security that prices completely in isolation; interest rates or other underliers usually play some role. Generally, as long as those components are known to be arbitrage-free, there is no need to go through the risk-neutral exercise when pricing someone that derives from them.

Moreover, I interpret your question as using the portfolio as a means to alter perceived risk; you could have as easily increased horizon and not invoked other securities. So to be clear, I'm treating your question as asking "how do different perceived risks impact security prices" rather than "are securities priced in isolation or in a portfolio context?"

So the answer might actually be (c) none of the above: securities are priced individually, but in risk-neutral market contexts. For all intents and purposes, I believe this resolves to your "individual" pricing, but perhaps for a different reason than you expected.

...and a Little MPT

Finally, as a statistician you may be interested in how to quantify the subjective aspects of your question. This comes down to the increasingly-misnamed "modern portfolio theory", introduced in 1952 and for which Harry Markowitz won a Nobel Prize. Note that while these concepts are very important, I'd be surprised if you found a professional using them in practice today.

Let's generalize your question by saying that these securities, call them $S$, have a binomial distribution with probability $p$. If we consider multiple holding periods then they would be Bernoulli-distributed, but since that will come out to a linear scale I disregard it here. One security's expected return is therefore $p$ with variance $p(1-p)$.

Compare that to holding a portfolio of $n$ such securities, allocating $\frac{1}{n}$ of your capital to each. The expected portfolio return is $$E[ \sum^n\frac{1}{n}S] = E[S] = p$$ and its variance is $$Var[\sum^n\frac{1}{n}S] = \frac{Var[S]}{n} = \frac{p(1-p)}{n}.$$

This is diversification at work! Same expected return, but lower volatility.

You ask: if this new portfolio is obviously better, shouldn't I pay more for it? And traders immediately start lining up to sell it to you in a particularly expensive illustration of the law of no-arbitrage.

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Thanks a lot! I will have to read the papers; the main problem to me is understanding the logic of valuation \ risk, and your excellent answer helps a lot. – Tim Aug 29 '12 at 11:22

This may or may not be helpful, since I don't have anything to point you to that specifically addresses the high skewness of the distribution you mention. However, this sounds like it is probably an idiosyncratic risk, and that certainly has bearing on whether or not it would be priced.

In the standard capital asset pricing model, the marginal investor holds something close to the market portfolio and so all that matters is an asset's covariance with the market portfolio. So as long as the probability of the good event is independent of the overall performance of the market portfolio, the textbook finance answer is that an idiosyncratic risk like this would be diversified away and therefore the stock would be priced based on expected value (as long as it is small relative to the market portfolio).

In practice things aren't quite so clear, and idiosyncratic risk might be priced after all (so the stock would trade at a discount). See for instance Malkiel and Xu (2006): http://www.utdallas.edu/~yexiaoxu/IVOT_H.PDF.

Then again others have found that stocks with high idiosyncratic volatility actually have lower expected returns, as in Ang, Hodrick, Xing and Zhang (2006).

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Thank you! Will read the papers. – Tim Aug 29 '12 at 11:28