Are you concave or convex?
Are you concave or convex?
A concave strategy is one with less upside potential and/or more downside risk (relative to a reference index). A convex strategy has more upside potential and/or less downside risk.
Examples of convex strategies:
1. Not rebalancing from cash to stocks (limits downside)
2. [not correct] Buying calls (upside potential) and puts (downside protection)
3. Momentum investing
Examples of concave strategies:
1. Rebalancing (more concave: 'overrebalancing')
2. [not correct] Selling covered calls and puts
3. Contrarian investing
In another thread I suggested that maybe the famous value/small tilt works because it is a concave strategy (value underperforms in bull markets; small underperforms in bear markets). That is only a hypothesis, though; I'm not claiming it's true.
The more people use one strategy (concave or convex), the better the other strategy works.
Examples of convex strategies:
1. Not rebalancing from cash to stocks (limits downside)
2. [not correct] Buying calls (upside potential) and puts (downside protection)
3. Momentum investing
Examples of concave strategies:
1. Rebalancing (more concave: 'overrebalancing')
2. [not correct] Selling covered calls and puts
3. Contrarian investing
In another thread I suggested that maybe the famous value/small tilt works because it is a concave strategy (value underperforms in bull markets; small underperforms in bear markets). That is only a hypothesis, though; I'm not claiming it's true.
The more people use one strategy (concave or convex), the better the other strategy works.
Last edited by dumbmoney on Fri Apr 04, 2008 9:19 pm, edited 1 time in total.

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Re: Are you concave or convex?
Where are you getting these terms from?
"Ah ha! Once again, the conservative, sandwichheavy portfolio pays off for the hungry investor!"  Dr. Zoidberg
Re: Are you concave or convex?
"Dynamic strategies for asset allocation". Perold, Andre F; Sharpe, William F. Financial Analysts Journal; Jan/Feb 1995INDUBITABLY wrote:Where are you getting these terms from?
hi all,
while it's interesting to see sharpe's name on this taxonomy it personally strikes me as a little arbitrary and preachy  i.e., (unless there's a typo) who wouldn't want a bigger upside and a smaller downside? in any case, as much as i believe in the "theory" of concave investing i try to keep an open mind re the considerable number of papers and analyses posted in here re unanticiapted benefits of momentum investing, not rebalancing, etc.
all best,
pete
while it's interesting to see sharpe's name on this taxonomy it personally strikes me as a little arbitrary and preachy  i.e., (unless there's a typo) who wouldn't want a bigger upside and a smaller downside? in any case, as much as i believe in the "theory" of concave investing i try to keep an open mind re the considerable number of papers and analyses posted in here re unanticiapted benefits of momentum investing, not rebalancing, etc.
all best,
pete
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The flipside of better extreme returns is a lower median return, e.g. the "bleeding theta" discussed by Taleb.peter71 wrote:while it's interesting to see sharpe's name on this taxonomy it personally strikes me as a little arbitrary and preachy  i.e., (unless there's a typo) who wouldn't want a bigger upside and a smaller downside?
Re: Are you concave or convex?
From Bill Bernstein's Efficient Frontier:dumbmoney wrote:A concave strategy is one with less upside potential and/or more downside risk (relative to a reference index). A convex strategy has more upside potential and/or less downside risk.
I'm not convinced, dumbmoney, that your understanding of the terms is the same as Bill's. You may have the definitions confused with the implications...and I'm not at all sure you even have the implications right.You probably didn’t know this, but investors come in two shapes—convex and concave. Sharpe and Perold, in a classic piece in Financial Analysts Journal in 1985, defined the former as one who tends to buy when prices are rising, and the latter as one who buys when prices are falling: in other words, momentum players and contrarian investors.
Darin
Re: Are you concave or convex?
I listed momentum under convex and contrarian under concave...where's the mistake?Drain wrote:From Bill Bernstein's Efficient Frontier:dumbmoney wrote:A concave strategy is one with less upside potential and/or more downside risk (relative to a reference index). A convex strategy has more upside potential and/or less downside risk.
I'm not convinced, dumbmoney, that your understanding of the terms is the same as Bill's. You may have the definitions confused with the implications...and I'm not at all sure you even have the implications right.You probably didn’t know this, but investors come in two shapes—convex and concave. Sharpe and Perold, in a classic piece in Financial Analysts Journal in 1985, defined the former as one who tends to buy when prices are rising, and the latter as one who buys when prices are falling: in other words, momentum players and contrarian investors.
Re: Are you concave or convex?
Hi dumbmoney, Suppose your asset allocation target is the classic 60% equity and 40% bonds, and a bear market causes you to reach 50/50. If you then take 10% out of bonds and put it in equity, you end up with more upside potential and more downside risk than 50/50. Is that concave, convex, or both according to your definition? Best, Neildumbmoney wrote:A concave strategy is one with less upside potential and/or more downside risk (relative to a reference index). A convex strategy has more upside potential and/or less downside risk. ...
Examples of concave strategies:
1. Rebalancing (more concave: 'overrebalancing')
2. Selling covered calls and puts
3. Contrarian investing
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Re: Are you concave or convex?
I don't think one is any more concave that the other. It's like comparing a graph of x vs. 2x.docneil88 wrote:Suppose your asset allocation target is the classic 60% equity and 40% bonds, and a bear market causes you to reach 50/50. If you then take 10% out of bonds and put it in equity, you end up with more upside potential and more downside risk than 50/50. Is that concave, convex, or both according to your definition?
You could compare a strategy of 60/40 that is rebalanced against one that is allowed to drift (meaning, it's initialized at 60/40 and never rebalanced). The portfolio with rebalancing has more downside risk and less upside potential, making it more concave than the nonrebalanced portfolio.
Re: Are you concave or convex?
I said your definition was wrong, and I quoted it above. What you wrote doesn't appear to have anything to do with what the terms actually mean. See Bernstein's interpretation. Also, neither of your #2s are right.dumbmoney wrote:Drain wrote:I listed momentum under convex and contrarian under concave...where's the mistake?dumbmoney wrote:A concave strategy is one with less upside potential and/or more downside risk (relative to a reference index). A convex strategy has more upside potential and/or less downside risk.
Since I haven't read the original work, I'm going on the assumption that Bill got it right.
Darin
Re: Are you concave or convex?
Thanks, you are correct. Holding a call option isn't 'convex' because the market exposure doesn't increase with gain.Drain wrote:I said your definition was wrong, and I quoted it above. What you wrote doesn't appear to have anything to do with what the terms actually mean. See Bernstein's interpretation. Also, neither of your #2s are right.dumbmoney wrote:Drain wrote:I listed momentum under convex and contrarian under concave...where's the mistake?dumbmoney wrote:A concave strategy is one with less upside potential and/or more downside risk (relative to a reference index). A convex strategy has more upside potential and/or less downside risk.
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Well, I thought the #2s were correct. We haven't seen a rigorous definition of convex vs. concave, but look at the graphs of payoff for someone who buys calls and puts (a long straddle) vs. someone who sells them (a short straddle):
http://en.wikipedia.org/wiki/Straddle
The upward V of a long straddle is convex, while the upside down V of a short straddle is concave. It is consistent with a convex investor favoring extreme outcomes and a concave investor favoring typical outcomes.
http://en.wikipedia.org/wiki/Straddle
The upward V of a long straddle is convex, while the upside down V of a short straddle is concave. It is consistent with a convex investor favoring extreme outcomes and a concave investor favoring typical outcomes.
Re: Are you concave or convex?
I see the less upside if you rebalance, but do not see the increase downside risk. Can you explain? ( maybe I just need another cup of coffee, but that seems backwards)market timer wrote:I don't think one is any more concave that the other. It's like comparing a graph of x vs. 2x.docneil88 wrote:Suppose your asset allocation target is the classic 60% equity and 40% bonds, and a bear market causes you to reach 50/50. If you then take 10% out of bonds and put it in equity, you end up with more upside potential and more downside risk than 50/50. Is that concave, convex, or both according to your definition?
You could compare a strategy of 60/40 that is rebalanced against one that is allowed to drift (meaning, it's initialized at 60/40 and never rebalanced). The portfolio with rebalancing has more downside risk and less upside potential, making it more concave than the nonrebalanced portfolio.
We live a world with knowledge of the future markets has less than one significant figure. And people will still and always demand answers to three significant digits.

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Wouldn't you only be able to describe investment strategies as "convex" or "concave" when you're mucking about with the payoff function with derivatives? I don't see how this directly applies to, e.g., momentum investing or rebalancing.
"Ah ha! Once again, the conservative, sandwichheavy portfolio pays off for the hungry investor!"  Dr. Zoidberg
One more time...market timer wrote:We haven't seen a rigorous definition of convex vs. concave
Personally, I think those definitions are pretty clear. Sharpe and Perold may have been more rigorous, but for the purposes of this thread, Bernstein's descriptions are good enough.Sharpe and Perold, in a classic piece in Financial Analysts Journal in 1985, defined the former as one who tends to buy when prices are rising, and the latter as one who buys when prices are falling: in other words, momentum players and contrarian investors.
Darin
Strictly speaking, in mathematics anyway, a straight line is convex. I buy going up and going down, each month without fail with each pay check.Drain wrote:One more time...market timer wrote:We haven't seen a rigorous definition of convex vs. concavePersonally, I think those definitions are pretty clear. Sharpe and Perold may have been more rigorous, but for the purposes of this thread, Bernstein's descriptions are good enough.Sharpe and Perold, in a classic piece in Financial Analysts Journal in 1985, defined the former as one who tends to buy when prices are rising, and the latter as one who buys when prices are falling: in other words, momentum players and contrarian investors.
By this definition I suppose I'm on a straight line and thus convex, barely.
We live a world with knowledge of the future markets has less than one significant figure. And people will still and always demand answers to three significant digits.
I think this terminology is clear in the mathematical sense. Given a stock that is going up, if you bet that its rate of increase will increase (i.e. it will accelerate), you are making a convex investment. If you bet that that the increases will slow, you are make concave one.
Of course only one of these strategies will be right at any given time. Given the standard fluctuations in the market, you can make money with either strategy, assuming your timing is right!
Of course only one of these strategies will be right at any given time. Given the standard fluctuations in the market, you can make money with either strategy, assuming your timing is right!
Maybe that's technically right, at least according to the mathematical definition, but in the spirit of what Bernstein wrote, I'd say that automatic, mindless (in a good way ) investments are neither concave nor convex.Rodc wrote:Strictly speaking, in mathematics anyway, a straight line is convex. I buy going up and going down, each month without fail with each pay check.
By this definition I suppose I'm on a straight line and thus convex, barely.
On the other hand, since the market tends to go up more than it goes down, that would also push you towards convexity.
Darin
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 market timer
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Re: Are you concave or convex?
When the market falls, you will have more equities after rebalancing.Rodc wrote:I see the less upside if you rebalance, but do not see the increase downside risk. Can you explain? ( maybe I just need another cup of coffee, but that seems backwards)
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Yes, that one is clearer than Bernstein's. You could modify it for options strategies by looking at the portfolio's delta. Since gamma is the derivative of delta with respect to underlying price, a portfolio with positive gamma is convex and negative gamma is concave. This is the natural mathematical definition, since gamma is the second derivative of exposure with respect to underlying price.Drain wrote:One more time...market timer wrote:We haven't seen a rigorous definition of convex vs. concavePersonally, I think those definitions are pretty clear. Sharpe and Perold may have been more rigorous, but for the purposes of this thread, Bernstein's descriptions are good enough.Sharpe and Perold, in a classic piece in Financial Analysts Journal in 1985, defined the former as one who tends to buy when prices are rising, and the latter as one who buys when prices are falling: in other words, momentum players and contrarian investors.
First of all, it is Bernstein's definition.market timer wrote:Yes, that one is clearer than Bernstein's. You could modify it for options strategies by looking at the portfolio's delta. Since gamma is the derivative of delta with respect to underlying price, a portfolio with positive gamma is convex and negative gamma is concave. This is the natural mathematical definition, since gamma is the second derivative of exposure with respect to underlying price.
Second...I cannot debate your statements because they're beyond what I'm familiar with. However, I will point out that Bernstein's definition refers to investors, not to portfolios, so I'm suspicious that you, too, are thinking of something different than the terms supposedly being discussed.
Darin