Defining Risk as Volatility in the Financial Literature

Definitions are not generally wrong. But in technical subjects they can differ greatly from common use, and it can be genuinely wrong to think the technical definition "captures" the common one when really it has simply given a word a very different meaning in the technical subject.

Consider how risk in finance is often defined as the variance of returns. This is not wrong, but is hardly synonymous with everyday use. Let's I am in a car going over a cliff. If I do nothing, I am dead for sure. But if I leap out, there is a 20% chance I will live. In ordinary talk, we'd say the first option is riskier, since we usually talk of risk as the chance a bad result will occur. But in finance theory, it is jumping out that is risky: it makes the predicted variance of return higher.

And so in auction theory: bidding higher definitely increases the chance of losing money, so in ordinary speech we'd call a bidder "cautious" who makes low bids. But the variance is higher, so auction theory calls the person who bids higher "risk averse." The most risk-averse buyer can simply ridiculously overbid for all items and guarantee huge losses but very low variance! Again, none of this is to say finance theory is wrong; we should just be careful not to think this is just a refinement of the ordinary notion of risk: it is a very different, only loosely related, concept.

4 comments:

  1. Your car example seems right to me (i.e. the commonsense meaning gets flipped in the financial economist meaning), but I still think you're not getting the auction setup. I can't be sure, since you adamantly refuse to post the actual rules of the auction in question...

    But I suspect that your prof is talking about an auction where the bidder knows his valuation of the object. So, there is 0% probability of overbidding. The issue is just, the higher I bid, the more likely I will win, but the smaller my gain.

    (It's possible in some auctions that I *don't* know how much I value the object--such as an oil field--and in those cases it's possible to overbid. This is the "winner's curse." But from the quote you gave of your prof that you were challenging, I got the impression that's not how this particular auction worked.)

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    1. Bob, I'm not adamantly refusing to do anything. PVV lecture the swimming in my house in Pennsylvania two hours away; but since you're acting up about it, I guess I'll go get my car take a drive out there rewatch the lecture and write down the rules of the auction just for you.

      I understand your point about the winner's curse; however, even in the case where I am bidding on a painting in which I'm just concerned about my personal valuation, it's still possible I can make a mistake. I know my ex ante evaluation of the object; I might get home with my Picasso and think "oh crap, this guy can't draw face right."

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  2. Bob: I gave an example in an earlier comment. The result is that in a private value auction, the equilibrium bid of risk-averse bidders will be higher than the risk-neutral case. Of course it won't be "over-bidding" - if that means bidding more than your monetary valuation.

    For example: with 2 bidders privately informed of their own valuation, where valuations for each are uniformly distributed on (0,1), the Nash equilibrium bids are half of valuation for the risk-neutral case and 2/3 of valuation for the case where bidders are risk averse with utility equal to the square root of income.

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  3. Bob: I gave an example in an earlier comment. The result is that in a private value auction, the equilibrium bid of risk-averse bidders will be higher than the risk-neutral case. Of course it won't be "over-bidding" - if that means bidding more than your monetary valuation.

    For example: with 2 bidders privately informed of their own valuation, where valuations for each are uniformly distributed on (0,1), the Nash equilibrium bids are half of valuation for the risk-neutral case and 2/3 of valuation for the case where bidders are risk averse with utility equal to the square root of income.

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