Diversifying with a zero-correlation, zero-return volatile asset

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Oicuryy
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Re: Diversifying with a zero-correlation, zero-return volatile asset

Post by Oicuryy » Thu Oct 11, 2018 10:17 am

nisiprius wrote:
Wed Oct 10, 2018 7:45 pm
If you imagine they each start with equal amounts of VBINX (60/40 balanced index), worth exactly $100,000 at the time of the bet, then after the bet is settled one of them owns $99,000 worth of VBINX and the other owns $101,000 worth.
That is not an investment it two assets. That is an investment in one asset with random additions and withdrawals. You start with $100k in VBINX. Then each week you flip a coin to see if you will add $1k or withdraw $1k. That is not diversifying. That is market timing by coin flip.

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Re: Diversifying with a zero-correlation, zero-return volatile asset

Post by nisiprius » Thu Oct 11, 2018 10:22 am

Oicuryy wrote:
Thu Oct 11, 2018 10:17 am
nisiprius wrote:
Wed Oct 10, 2018 7:45 pm
If you imagine they each start with equal amounts of VBINX (60/40 balanced index), worth exactly $100,000 at the time of the bet, then after the bet is settled one of them owns $99,000 worth of VBINX and the other owns $101,000 worth.
That is not an investment it two assets. That is an investment in one asset with random additions and withdrawals. You start with $100k in VBINX. Then each week you flip a coin to see if you will add $1k or withdraw $1k. That is not diversifying. That is market timing by coin flip.

Ron
Is that any different from investing in a hypothetical "zero-correlation, zero-return volatile asset?" Are you simply saying that a zero-correlation, zero-return, volatile asset should not be described as "an asset" because all it does is make random additions and withdrawals from the portfolio? Or are you saying that there is a fundamental difference between my coin flip and "a zero-correlation, zero-return volatile asset?"
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Re: Diversifying with a zero-correlation, zero-return volatile asset

Post by JoMoney » Thu Oct 11, 2018 10:24 am

Oicuryy wrote:
Thu Oct 11, 2018 10:17 am
...That is not diversifying. That is market timing by coin flip...
Funny. That's my opinion on a lot of "slice and dice" portfolios where the investor believes their rebalancing machinations are adding something extra to the investment.
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Re: Diversifying with a zero-correlation, zero-return volatile asset

Post by Oicuryy » Thu Oct 11, 2018 11:30 am

nisiprius wrote:
Thu Oct 11, 2018 10:22 am
Or are you saying that there is a fundamental difference between my coin flip and "a zero-correlation, zero-return volatile asset?"
This, I think.

Imagine some sort of structured product where the annual return was determined by a coin flip. Heads and the annual return is +10%; tails its −10%. You are allowed to make a deposit or withdrawal right after the return is posted to your account.

You deposit $50k in VBINX (or whatever) and $50k in this structured product. You rebalance back to a 50:50 mix each year. (Or use any weights you want.)

Now you can use the expected means, variances and covariance of the returns of these two assets to calculate the expected mean and variance of the returns of your total portfolio. I think the math in AlphaLess' post above would apply in this case.

Ron
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Re: Diversifying with a zero-correlation, zero-return volatile asset

Post by willthrill81 » Thu Oct 11, 2018 11:40 am

vineviz wrote:
Wed Oct 10, 2018 12:48 pm
nisiprius wrote:
Tue Oct 09, 2018 8:51 pm
Which of the following is true?

a) Long-term, the coin-flip "asset" adds valuable diversification to both portfolios, and improves the risk-adjusted return for both of you. It is mutually beneficial to enter into this arrangement, and you should seriously try to find a partner willing to do it.

b) The coin-flip has exactly no effect on your long-term risk, return, or risk-adjusted return. Of course, over any short period of time, or even over the entire twenty years, one or the other of you will be an overall winner and the other will be an overall loser, by sheer luck, but over twenty years the law of averages applies and the overall win or loss will almost certainly be very small.

c) The coin-flip adds risk without adding any return, and reduces the risk-adjusted return for both of you.

For extra credit, calculate the amount of diversification the coin-flip provides, using whatever measure of diversification you like.
The answer, it turns out, is "none of the above".

I'm not sure if the answers were intentionally designed this way, or if they turned out that way due to a misunderstanding about what "diversification" means (hint: has nothing to do with risk-adjusted return).

a) is false because the coin flip DOES have an impact on risk-adjusted return (it lowers it).

b) is false because the coin flip DOES have an effect on all three criteria (long-term risk, return, or risk-adjusted return): it lowers all three.

c) is false because it lowers risk AND return for both parties, with a net effect of reducing risk-adjusted return.

The answer is "none of the above", in short, because the asset both improves diversification and reduces risk-adjusted return.
This sounds right to me. :)
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Re: Diversifying with a zero-correlation, zero-return volatile asset

Post by vineviz » Thu Oct 11, 2018 11:58 am

nisiprius wrote:
Thu Oct 11, 2018 10:22 am
Oicuryy wrote:
Thu Oct 11, 2018 10:17 am
nisiprius wrote:
Wed Oct 10, 2018 7:45 pm
If you imagine they each start with equal amounts of VBINX (60/40 balanced index), worth exactly $100,000 at the time of the bet, then after the bet is settled one of them owns $99,000 worth of VBINX and the other owns $101,000 worth.
That is not an investment it two assets. That is an investment in one asset with random additions and withdrawals. You start with $100k in VBINX. Then each week you flip a coin to see if you will add $1k or withdraw $1k. That is not diversifying. That is market timing by coin flip.

Ron
Is that any different from investing in a hypothetical "zero-correlation, zero-return volatile asset?" Are you simply saying that a zero-correlation, zero-return, volatile asset should not be described as "an asset" because all it does is make random additions and withdrawals from the portfolio? Or are you saying that there is a fundamental difference between my coin flip and "a zero-correlation, zero-return volatile asset?"
A coin flip isn't a capital asset at all, so that's a kind of fundamental difference.
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Re: Diversifying with a zero-correlation, zero-return volatile asset

Post by ThrustVectoring » Thu Oct 11, 2018 11:59 am

The correct allocation to an uncorrelated volatile asset with a return at or below the risk-free rate is zero. If your measure of "diversification" says that adding noise to a return stream helps you, your measurement is wrong.

I have a good enough understanding to explain the signal and information theory responsible for this intuition, so I basically just have to assert it. Any question that boils down to "is noise and uncertainty good or bad?" can be categorically answered with "it's bad".
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Re: Diversifying with a zero-correlation, zero-return volatile asset

Post by rkhusky » Thu Oct 11, 2018 12:07 pm

vineviz wrote:
Wed Oct 10, 2018 12:48 pm
The answer, it turns out, is "none of the above".

I'm not sure if the answers were intentionally designed this way, or if they turned out that way due to a misunderstanding about what "diversification" means (hint: has nothing to do with risk-adjusted return).

a) is false because the coin flip DOES have an impact on risk-adjusted return (it lowers it).

b) is false because the coin flip DOES have an effect on all three criteria (long-term risk, return, or risk-adjusted return): it lowers all three.

c) is false because it lowers risk AND return for both parties, with a net effect of reducing risk-adjusted return.

The answer is "none of the above", in short, because the asset both improves diversification and reduces risk-adjusted return.
I don't see how it can lower return for both parties. It seems like over a single sequence one party will come out ahead and one party will not, based on the sequence of coin flips and how the market performs. Over a large number of sequences, the two parties will end up having equal return. The only way for there to be a lower return would be if the parties were out of the market for a period of time. One could suppose that the the loser of a bet transfers the $1000 directly to the winner, such that the $1000 is never out of the market.

I also don't see how the risk is changed because there really is no risk over a sufficiently long period of time. The probabilities of coin flips with fair coins is known exactly. And over a large number of coin flip sequences, the probabilities reach fixed values.

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Re: Diversifying with a zero-correlation, zero-return volatile asset

Post by vineviz » Thu Oct 11, 2018 12:17 pm

rkhusky wrote:
Thu Oct 11, 2018 12:07 pm
I don't see how it can lower return for both parties. It seems like over a single sequence one party will come out ahead and one party will not, based on the sequence of coin flips and how the market performs. Over a large number of sequences, the two parties will end up having equal return. The only way for there to be a lower return would be if the parties were out of the market for a period of time. One could suppose that the the loser of a bet transfers the $1000 directly to the winner, such that the $1000 is never out of the market.

I also don't see how the risk is changed because there really is no risk over a sufficiently long period of time. The probabilities of coin flips with fair coins is known exactly. And over a large number of coin flip sequences, the probabilities reach fixed values.
The original problem was ambiguously specified, but conceptually you can think about whatever amount of capital is allocated to this bet (which has an expected return of zero) is being removed from the other portfolio assets (which have an expected return greater than zero).

In other words, you taking money from an asset that has a positive expected return and allocating it each period to an asset that has a 0.00% expected return.
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Re: Diversifying with a zero-correlation, zero-return volatile asset

Post by columbia » Thu Oct 11, 2018 12:19 pm

I wondering if this thread is really about currency risk. :)

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Re: Diversifying with a zero-correlation, zero-return volatile asset

Post by JoMoney » Thu Oct 11, 2018 12:20 pm

rkhusky wrote:
Thu Oct 11, 2018 12:07 pm
...
I don't see how it can lower return for both parties. It seems like over a single sequence one party will come out ahead and one party will not, based on the sequence of coin flips and how the market performs. Over a large number of sequences, the two parties will end up having equal return. The only way for there to be a lower return would be if the parties were out of the market for a period of time. One could suppose that the the loser of a bet transfers the $1000 directly to the winner, such that the $1000 is never out of the market.

I also don't see how the risk is changed because there really is no risk over a sufficiently long period of time. The probabilities of coin flips with fair coins is known exactly. And over a large number of coin flip sequences, the probabilities reach fixed values.
I think you might be at least partly right, because the $1,000 is a fixed amount and not some fraction of the two portfolios... and we don't even know how much of the portfolio $1,000 is.
There is still risk, it might be unlikely, but there's a non-zero probability of a streak of losses destroying a portfolio, and that risk gets larger depending on how large $1,000 is in proportion to the overall portfolio.
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Re: Diversifying with a zero-correlation, zero-return volatile asset

Post by Scrooge McDuck » Thu Oct 11, 2018 1:38 pm

(c), clearly. AlphaLess's post shows why.

I like this thought experiment a lot. (I actually thought of it myself a few years ago, and it helped clarify my thinking.) I do think it has relevance for how we think about international investing and "currency diversification" and such.

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Re: Diversifying with a zero-correlation, zero-return volatile asset

Post by AlphaLess » Thu Oct 11, 2018 7:16 pm

David Jay wrote:
Thu Oct 11, 2018 8:49 am
AlphaLess wrote:
Thu Oct 11, 2018 8:38 am
In expectation, a typical customer will LOSE money.
And gladly, in most cases.

Can you imagine "improving your return"? I'll schedule a major accident every year so the insurance company doesn't "make money" off of me.
Yes, you could. Moral hazards aside, if the following held true:
- insurance company was run at zero cost (e.g., by robots),
- insurance company could take more risk, than individual policy-holders could,
- insurance company could select customers, and also incentivize behavior.

With self-driving cars, lots of data, and other conditions, insurance premiums *might* make money.
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Re: Diversifying with a zero-correlation, zero-return volatile asset

Post by bhsince87 » Thu Oct 11, 2018 8:05 pm

Seems a lot like commodity "investment" to me.

If you don't use the commodity directly, what's the point? There will always be a winner and a loser every day.

I guess if you play this game out to its ultimate conclusion, one flipper will win when the other one dies, gives up, or whatever. Sort of like a cakewalk.
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Re: Diversifying with a zero-correlation, zero-return volatile asset

Post by NoRegret » Thu Oct 11, 2018 9:37 pm

AlphaLess wrote:
Tue Oct 09, 2018 11:16 pm
nisiprius wrote:
Tue Oct 09, 2018 8:51 pm
c) The coin-flip adds risk without adding any return, and reduces the risk-adjusted return for both of you.
P: random variable (portfolio)
B: random variable (bet).

Var (P + B) = Var ( P ) + 2 * Cov ( P,B) + Var ( B)

By the design of the problem, Cov ( P,B) = 0 {let's say we can muster up an iPhone app to throw a random coin, and it is truly random and does not use any known asset returns for randomness generation},

Var(B) > 0

So Var (P + B) > Var (P).

Also, notice that Std (P+B) = Sqrt ( Var (P+B)).

Thus, the risk of the (P+B) combo is higher, in variance and standard deviation terms.

E [ P + B ] = E [ P ] + E [ B ] {as expectation is additive}.
By design, E [ B ] = 0.

Thus, E [ P + B] = E [ P ]

If we use a simplest form of risk-adjusted return, ignoring interest rates:

Sharpe (P+B) = E [ P + B ] / Std (P + B) < E [ P ] / Std ( P) = Sharpe (P) , because
: numerators are equal,
: denominator Std ( P + B) is larger.

However, let me ask you this problem:
- you and a friend (let's call your friend Warren 'GEICO' Buffet) decided to meet up once a month,
- you give your friend $100 / M, for twenty years, maybe more,
- your friend Warren cashes that check, with a smirk,
- occasionally, when you have trouble with your car, of accident variety, Warren reluctantly writes a check to you, covering some part of the expenses resulting from the accident,
- it is known that the total dollar value of the checks that Warren might write to you is strictly lower than the total value of checks that you write to Warren.

Do you play this game with Warren?
E (B) = 0 only with a large number of coin tosses. Here we're "interacting" with the coin toss asset after each toss so I think you have to do a simulation.

On the other hand, let's think about this:

I have $100K in a 60/40 portfolio. I decide to give one half to each of my two daughters, to be taken possession at a future date. In the meantime, the money is invested 60/40 and they play a monthly game where they flip a coin to pass $1000 worth of their interest in Daddy's fund back and forth.

What is MY expected return and risk in MY fund from today to the day they receive their funds? What is the expected value of each daughter's share?

From the above answers what can we conclude are the expected return and risk for each daughter's share?

NR

PS note if the stipulation is that the coin-tossing asset gets a fixed allocation then the above thinking breaks down since they wouldn't be able to agree on the amount to bet.
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Re: Diversifying with a zero-correlation, zero-return volatile asset

Post by Starfish » Fri Oct 12, 2018 8:32 pm

It is obvious that adding an uncorrelated asset increseases standard deviation. If return is 0, you gain nothing for that -> risk adjusted returns go down because the risk goes up.
Adding a correlated asset can do worse or better than what an uncorrelated asset does. In can decrease or increase volatility.

Adding uncorrelated assets increases the risk! People do it because they hope that the returns grows MORE than the risk thus improving risk adjusted returns.

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Re: Diversifying with a zero-correlation, zero-return volatile asset

Post by JoMoney » Fri Oct 12, 2018 8:52 pm

Not only is the un-correlated zero-return asset adding extra risk, it has opportunity costs for the money that could have been invested in something with an expected return.
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