Are you concave or convex?

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dumbmoney
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Are you concave or convex?

Post by dumbmoney » Fri Apr 04, 2008 6:26 am

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: 'over-rebalancing')
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?

Post by INDUBITABLY » Fri Apr 04, 2008 9:42 am

Where are you getting these terms from?
"Ah ha! Once again, the conservative, sandwich-heavy portfolio pays off for the hungry investor!" - Dr. Zoidberg

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Re: Are you concave or convex?

Post by dumbmoney » Fri Apr 04, 2008 9:56 am

INDUBITABLY wrote:Where are you getting these terms from?
"Dynamic strategies for asset allocation". Perold, Andre F; Sharpe, William F. Financial Analysts Journal; Jan/Feb 1995

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market timer
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Post by market timer » Fri Apr 04, 2008 10:29 am

Concave

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Post by peter71 » Fri Apr 04, 2008 12:59 pm

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

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Post by market timer » Fri Apr 04, 2008 2:02 pm

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?
The flip-side of better extreme returns is a lower median return, e.g. the "bleeding theta" discussed by Taleb.

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Re: Are you concave or convex?

Post by Drain » Fri Apr 04, 2008 2:15 pm

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.
From Bill Bernstein's Efficient Frontier:
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.
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.
Darin

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Post by yesosaka » Fri Apr 04, 2008 2:30 pm

I have been told I am obtuse.

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Re: Are you concave or convex?

Post by dumbmoney » Fri Apr 04, 2008 7:08 pm

Drain wrote:
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.
From Bill Bernstein's Efficient Frontier:
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.
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.
I listed momentum under convex and contrarian under concave...where's the mistake?

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Re: Are you concave or convex?

Post by docneil88 » Fri Apr 04, 2008 7:25 pm

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. ...

Examples of concave strategies:
1. Rebalancing (more concave: 'over-rebalancing')
2. Selling covered calls and puts
3. Contrarian investing
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, Neil

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Re: Are you concave or convex?

Post by market timer » Fri Apr 04, 2008 7:34 pm

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?
I don't think one is any more concave that the other. It's like comparing a graph of x vs. 2x.

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 non-rebalanced portfolio.

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Re: Are you concave or convex?

Post by Drain » Fri Apr 04, 2008 8:29 pm

dumbmoney wrote:
Drain wrote:
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 listed momentum under convex and contrarian under concave...where's the mistake?
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.

Since I haven't read the original work, I'm going on the assumption that Bill got it right.
Darin

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Re: Are you concave or convex?

Post by dumbmoney » Fri Apr 04, 2008 9:18 pm

Drain wrote:
dumbmoney wrote:
Drain wrote:
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 listed momentum under convex and contrarian under concave...where's the mistake?
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.
Thanks, you are correct. Holding a call option isn't 'convex' because the market exposure doesn't increase with gain.

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Post by market timer » Fri Apr 04, 2008 10:00 pm

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.

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Re: Are you concave or convex?

Post by Rodc » Sat Apr 05, 2008 8:50 am

market timer wrote:
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?
I don't think one is any more concave that the other. It's like comparing a graph of x vs. 2x.

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 non-rebalanced portfolio.
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)
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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Post by INDUBITABLY » Sat Apr 05, 2008 9:06 am

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, sandwich-heavy portfolio pays off for the hungry investor!" - Dr. Zoidberg

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Post by Drain » Sat Apr 05, 2008 10:11 am

market timer wrote:We haven't seen a rigorous definition of convex vs. concave
One more time...
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.
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.
Darin

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Post by Rodc » Sat Apr 05, 2008 10:16 am

Drain wrote:
market timer wrote:We haven't seen a rigorous definition of convex vs. concave
One more time...
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.
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.
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.
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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Post by dizzy » Sat Apr 05, 2008 10:51 am

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!

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Post by Drain » Sat Apr 05, 2008 10:55 am

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.
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.

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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Post by market timer » Sat Apr 05, 2008 12:21 pm

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.
A straight line is also concave.

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Re: Are you concave or convex?

Post by market timer » Sat Apr 05, 2008 12:24 pm

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)
When the market falls, you will have more equities after rebalancing.

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Post by market timer » Sat Apr 05, 2008 12:26 pm

Drain wrote:
market timer wrote:We haven't seen a rigorous definition of convex vs. concave
One more time...
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.
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.
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.

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Post by SmallHi » Sat Apr 05, 2008 12:56 pm

dumbmoney,

sounds like someone is perhaps preparing for L3 of the CFA exam. No?

sh

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Post by Drain » Sat Apr 05, 2008 1:30 pm

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.
First of all, it is Bernstein's definition.

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

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Post by dumbmoney » Sat Apr 05, 2008 7:34 pm

SmallHi wrote:dumbmoney,

sounds like someone is perhaps preparing for L3 of the CFA exam. No?
No. (I had to look up what CFA meant on Wikipeda - it's Chartered Financial Analyst).

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Post by PaPaw » Sat Apr 05, 2008 7:51 pm

I'm "flatlined" (my wife sometimes says brain dead) on the tangent to a convex sine wave of a 4th order polynomial but only in the last phase of a new moon when the outer limit of a comet streaming tail passes by a concave convoluted elipsoid. :twisted: :twisted:

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