Risk tolerance and asset allocation with mathematics
Re: Risk tolerance and asset allocation with mathematics
So can someone briefly translate the OP's post for the common folk (like me)? Is he saying most people should just stick with a Target Retirement fund?
Re: Risk tolerance and asset allocation with mathematics
The original post might as well have been written in Chinese for me....... I feel really dumb...
Re: Risk tolerance and asset allocation with mathematics
Mathematics is ordinarily considered as producing precise and dependable results; but in the stock market the more elaborate and abstruse the mathematics the more uncertain and speculative are the conclusions we draw therefrom. In fortyfour years of Wall Street experience and study I have never seen dependable calculations made about commonstock values, or related investment policies, that went beyond simple arithmetic or the most elementary algebra. Whenever calculus is brought in, or higher algebra, you could take it as a warning signal that the operator was trying to substitute theory for experience, and usually also to give to speculation the deceptive guise of investment.
 Benjamin Graham
 Benjamin Graham
I guess it all could be much worse. 
They could be warming up my hearse.
 Steve Reading
 Posts: 2674
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Re: Risk tolerance and asset allocation with mathematics
My cliff notes:
 Don’t focus on the most efficient portfolio. Just whichever one produces the best risk and return trade off for your circumstances (maximizes utility).
 Don’t mentally account your investments (I.e 5% in Hedgefundie Excellent Adventure). What matters is the entire portfolio.
 Target Volatility doesn’t make much sense for most people.
 If you want to time the market, small changes in allocation (say from 60% to 40%) tend to be superior than full on “all stocks or no stocks”. To fully exit the market, you must believe stocks have a negative expected return. Since it’s really hard to know if that’s the case, it’s unlikely you should ever fully exit the market.
 For those of us in accumulation, you would decrease your lifetime risk, while improving lifetime returns, by following a glide path.
I’m of the thought that if this topic isn’t too exciting, just a good old Vanguard Target Date Fund, ideally with a retirement date well after your actual retirement, is good enough.
If you want to get much closer to the true, optimal glide path, then you have to do a little more work as I’ve done in this thread:
viewtopic.php?f=10&t=274390
"... so high a present discounted value of wealth, it is only prudent for him to put more into common stocks compared to his present tangible wealth, borrowing if necessary"  Paul Samuelson
Re: Risk tolerance and asset allocation with mathematics
Larry Swedroe's stomachacid test and the ability to sleep well at night tests makes the most sense to me for determining risk tolerance. You can try to mathematically quantify it all you want, but when it comes right down to it as an investor, how much of a loss can you stomach while still sleeping well at night before panic selling during a bear market is the question you have to honestly answer yourself.
Re: Risk tolerance and asset allocation with mathematics
+1BillyK wrote: ↑Fri Mar 06, 2020 2:54 pm Larry Swedroe's stomachacid test and the ability to sleep well at night tests makes the most sense to me for determining risk tolerance. You can try to mathematically quantify it all you want, but when it comes right down to it as an investor, how much of a loss can you stomach while still sleeping well at night before panic selling during a bear market is the question you have to honestly answer yourself.
Human behavioral issues are a major issue in personal finance.
If people can't be stopped from mucking up their portfolios, it doesn't really matter how good the model is.
70% Global Market Weight Equities  15% Long Treasuries 15% short TIPS & cash  RSU + ESPP
Re: Risk tolerance and asset allocation with mathematics
Waiting for the Executive Summary on this topic. In the meantime, I'll stick with my 60/40 VSMGX.
Re: Risk tolerance and asset allocation with mathematics
Yes, you are correct and hopefully Uncorrelated can fix his post to make it accurate. The general "gist" of the mathematics in the OP is correct though since stock prices are usually assumed to follow a lognormal, not a normal distribution. When the log interacts with the relative risk aversion in CRRA, you end up with the same type of formulas as when you assume normal returns and CARA. So you still end up with the mu/(gamma * sigma^2) for optimal asset allocation.acegolfer wrote: ↑Thu Mar 05, 2020 11:37 am U(X) = E(X)  1/2 * γ * Var(X)
This certainty equivalent is derived if the utility function is CARA not CRRA. See page 21 of MIT open course notes: https://ocw.mit.edu/courses/economics/1 ... _Chap3.pdf
I provide http://web.stanford.edu/class/cme241/le ... orRisk.pdf as another resource.
Re: Risk tolerance and asset allocation with mathematics
The optimal allocation formula looks similar but not identical because the gamma in CRRA is not the same gamma in CARA. But I doubt OP would understand the difference. If he knew the error, he would have fixed it by now.langlands wrote: ↑Fri Mar 06, 2020 3:24 pmYes, you are correct and hopefully Uncorrelated can fix his post to make it accurate. The general "gist" of the mathematics in the OP is correct though since stock prices are usually assumed to follow a lognormal, not a normal distribution. When the log interacts with the relative risk aversion in CRRA, you end up with the same type of formulas as when you assume normal returns and CARA. So you still end up with the mu/(gamma * sigma^2) for optimal asset allocation.acegolfer wrote: ↑Thu Mar 05, 2020 11:37 am U(X) = E(X)  1/2 * γ * Var(X)
This certainty equivalent is derived if the utility function is CARA not CRRA. See page 21 of MIT open course notes: https://ocw.mit.edu/courses/economics/1 ... _Chap3.pdf
I provide http://web.stanford.edu/class/cme241/le ... orRisk.pdf as another resource.
Re: Risk tolerance and asset allocation with mathematics
The utility I got out of this post (and Uncorrelated's previous posts), in a nutshell:
 Determining your CRRA is a very useful thing for modeling, but is tricky to get at for an individual; one can bootstrap it by relating to stock/bond portfolios. There is a chicken/egg situation if one is just looking to use it to set a 2 fund portfolio, so not directly useful if that is the only question you are asking.
 If one is choosing to go down the direction of lifecycle investing, or other more complex portfolio mixtures (leveraged, factors, etc.) then it (CRRA) provides more utility. I think this is really key as well, in that it (the whole framework described here) helps one evaluate proposed changes, and sometimes the underlying assumption behind those changes. Uncorrelated's last paragraph, "The utility loss of uninformed market timing" did a good job of presenting exemplars of this  in this case helping to quantify (given a set of assumptions) how "right" a market timer would have to be to beat buy and hold / how difficult it is  as well as highlighting pitfalls of different scenarios (all in/out).
tl;dr 
For me the biggest value was a framework for evaluating changes in a portfolio  not so much the projected returns, but rather the implicit assumptions that have to be correct for those changes to be advantageous;
I liken it in spirit to the chart vineviz posted awhile ago: viewtopic.php?t=277661
 Determining your CRRA is a very useful thing for modeling, but is tricky to get at for an individual; one can bootstrap it by relating to stock/bond portfolios. There is a chicken/egg situation if one is just looking to use it to set a 2 fund portfolio, so not directly useful if that is the only question you are asking.
 If one is choosing to go down the direction of lifecycle investing, or other more complex portfolio mixtures (leveraged, factors, etc.) then it (CRRA) provides more utility. I think this is really key as well, in that it (the whole framework described here) helps one evaluate proposed changes, and sometimes the underlying assumption behind those changes. Uncorrelated's last paragraph, "The utility loss of uninformed market timing" did a good job of presenting exemplars of this  in this case helping to quantify (given a set of assumptions) how "right" a market timer would have to be to beat buy and hold / how difficult it is  as well as highlighting pitfalls of different scenarios (all in/out).
tl;dr 
For me the biggest value was a framework for evaluating changes in a portfolio  not so much the projected returns, but rather the implicit assumptions that have to be correct for those changes to be advantageous;
I liken it in spirit to the chart vineviz posted awhile ago: viewtopic.php?t=277661
 Uncorrelated
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Re: Risk tolerance and asset allocation with mathematics
As far as I understand, the expected utility for CARA and CRRA are almost identical. Your first source only displays the derivation for CARA utility and not CRRA. The second source appears to specify µ as the geometric growth, but I use the arithmetic growth. I'm unsure if that explains the differences between the results.langlands wrote: ↑Fri Mar 06, 2020 3:24 pmYes, you are correct and hopefully Uncorrelated can fix his post to make it accurate. The general "gist" of the mathematics in the OP is correct though since stock prices are usually assumed to follow a lognormal, not a normal distribution. When the log interacts with the relative risk aversion in CRRA, you end up with the same type of formulas as when you assume normal returns and CARA. So you still end up with the mu/(gamma * sigma^2) for optimal asset allocation.acegolfer wrote: ↑Thu Mar 05, 2020 11:37 am U(X) = E(X)  1/2 * γ * Var(X)
This certainty equivalent is derived if the utility function is CARA not CRRA. See page 21 of MIT open course notes: https://ocw.mit.edu/courses/economics/1 ... _Chap3.pdf
I provide http://web.stanford.edu/class/cme241/le ... orRisk.pdf as another resource.
My sources are http://citeseerx.ist.psu.edu/viewdoc/do ... 1&type=pdf (chapter 5) and https://www.gordoni.com/lifetime_portfo ... ection.pdf (page 5), although neither sources give an explicit derivation.
It's entirely possible I'm making an error here. Despite all my talk about math, this is pretty far outside my expertise.
Re: Risk tolerance and asset allocation with mathematics
The first source is from acegolfer, not me.Uncorrelated wrote: ↑Fri Mar 06, 2020 4:26 pmAs far as I understand, the expected utility for CARA and CRRA are almost identical. Your first source only displays the derivation for CARA utility and not CRRA. The second source appears to specify µ as the geometric growth, but I use the arithmetic growth. I'm unsure if that explains the differences between the results.langlands wrote: ↑Fri Mar 06, 2020 3:24 pmYes, you are correct and hopefully Uncorrelated can fix his post to make it accurate. The general "gist" of the mathematics in the OP is correct though since stock prices are usually assumed to follow a lognormal, not a normal distribution. When the log interacts with the relative risk aversion in CRRA, you end up with the same type of formulas as when you assume normal returns and CARA. So you still end up with the mu/(gamma * sigma^2) for optimal asset allocation.acegolfer wrote: ↑Thu Mar 05, 2020 11:37 am U(X) = E(X)  1/2 * γ * Var(X)
This certainty equivalent is derived if the utility function is CARA not CRRA. See page 21 of MIT open course notes: https://ocw.mit.edu/courses/economics/1 ... _Chap3.pdf
I provide http://web.stanford.edu/class/cme241/le ... orRisk.pdf as another resource.
My sources are http://citeseerx.ist.psu.edu/viewdoc/do ... 1&type=pdf (chapter 5) and https://www.gordoni.com/lifetime_portfo ... ection.pdf (page 5), although neither sources give an explicit derivation.
It's entirely possible I'm making an error here. Despite all my talk about math, this is pretty far outside my expertise.
Yes, the second source is somewhat confusing in how it is using mu. On slide 12 it is using mu as geometric growth, but on slides 13 and 14, it is using mu as arithmetic growth. However, this does not explain the difference in results. Essentially, the utility for CRRA should be the exponential of what you wrote down (from slide 12). However, because utility functions are equivalent under monotonic transformations, this ends up not being a big deal and you can pretty much get rid of the exponential and still get all the right answers. The sources you cite essentially do this without explicitly mentioning it (indeed, why carry around an exponential when you can just take a log and be done with it). Actually, in your first source, you can see this being done on the top of page 5 for the computation of the optimal asset allocation pi.
Technicalities aside, I thought it was a great post.
Edit: I want to emphasize that the derivations for CER for CARA and CRRA are usually done under differing return distributions, simply because that's how the math comes out nicely. For CARA, the assumption is a normal distribution of outcomes while for CRRA, the assumption is lognormal distribution. It is well known that stock prices are much better approximated by a lognormal distribution than a normal distribution. I think that taking into account utility equivalences as I stated above, the formulas for CARA and CRRA do end up looking superficially similar. However, as acegolfer points out, that of course doesn't mean that CARA and CRRA are in any way similar since coefficient of absolute risk aversion and coefficient of relative risk aversion are two completely different things.
Last edited by langlands on Fri Mar 06, 2020 6:26 pm, edited 3 times in total.
Re: Risk tolerance and asset allocation with mathematics
CARA and CRRA are not almost identical.Uncorrelated wrote: ↑Fri Mar 06, 2020 4:26 pm As far as I understand, the expected utility for CARA and CRRA are almost identical. Your first source only displays the derivation for CARA utility and not CRRA. The second source appears to specify µ as the geometric growth, but I use the arithmetic growth. I'm unsure if that explains the differences between the results.
My sources are http://citeseerx.ist.psu.edu/viewdoc/do ... 1&type=pdf (chapter 5) and https://www.gordoni.com/lifetime_portfo ... ection.pdf (page 5), although neither sources give an explicit derivation.
It's entirely possible I'm making an error here. Despite all my talk about math, this is pretty far outside my expertise.
http://citeseerx.ist.psu.edu/viewdoc/do ... 1&type=pdf has typo in page 31.
"γ is the investor’s coefficient of relative risk aversion"
It should be "γ is the investor’s coefficient of absolute risk aversion"
https://www.tau.ac.il/~spiegel/teaching ... riance.pdf has a detailed derivation of CER from CARA function. I learned this when I was taking a masters level economics course in 90s. You only need to know exponential and normal distribution pdf to understand the proof.
Last edited by acegolfer on Fri Mar 06, 2020 7:09 pm, edited 1 time in total.
Re: Risk tolerance and asset allocation with mathematics
What's CER from CRRA utility function? I have seen CE of CRRA but not CER of CRRA.langlands wrote: ↑Fri Mar 06, 2020 5:19 pm Edit: I want to emphasize that the derivations for CER for CARA and CRRA are usually done under differing return distributions, simply because that's how the math comes out nicely. For CARA, the assumption is a normal distribution of outcomes while for CRRA, the assumption is lognormal distribution.
Re: Risk tolerance and asset allocation with mathematics
Whoops, by CER, I meant certainty equivalent (CE). I just kept seeing the abbreviation CER used and guess I typed that out of habit. I'm not even sure what certainty equivalent return (CER) means. Is there any substantial difference?acegolfer wrote: ↑Fri Mar 06, 2020 6:54 pmWhat's CER from CRRA utility function? I have seen CE of CRRA but not CER of CRRA.langlands wrote: ↑Fri Mar 06, 2020 5:19 pm Edit: I want to emphasize that the derivations for CER for CARA and CRRA are usually done under differing return distributions, simply because that's how the math comes out nicely. For CARA, the assumption is a normal distribution of outcomes while for CRRA, the assumption is lognormal distribution.
Re: Risk tolerance and asset allocation with mathematics
OK, FWIW then:Uncorrelated wrote: ↑Fri Mar 06, 2020 1:45 pmSorry for the delay. I don't have any particular definition for risk aversion and risk tolerance. The way I define those words is very generic, a higher risk tolerance means you're more likely to take risk. And a higher risk aversion means you're less likely to take risk. I use both interchangeably, depending on which sounds better.Fallible wrote: ↑Fri Mar 06, 2020 1:38 pmOP, I may be misreading you, but in your original post and here, I see risk tolerance and risk aversion used as if they mean the same thing. However, in definitions I'm familiar with, they don't mean the same. I asked earlier how you were defining the two, so can you say how?Uncorrelated wrote: ↑Fri Mar 06, 2020 1:15 pmYou're right, I messed up. I meant to say I'm more risk tolerant than the average investor.bobcat2 wrote: ↑Fri Mar 06, 2020 11:32 amThis is inconsistent. If the OP is more risk averse than the average investor's γ=3, then his γ would be greater than 3 and he would hold a smaller proportion of stocks in his portfolio.Uncorrelated wrote: ↑Wed Mar 04, 2020 5:28 pm
We can plug in some numbers to get a feel for the correct value of γ. I will use a risk premium of 5% and standard deviation of 16%.
γ=5 corresponds to an asset allocation of 40% stocks, 60% bonds
γ=4 corresponds to an asset allocation of 50% stocks, 50% bonds
γ=3 corresponds to an asset allocation of 65% stocks, 35% bonds
γ=2 corresponds to an asset allocation of 100% stocks
γ=1 corresponds to an asset allocation of 200% stocks
γ=0 corresponds to an asset allocation of infinitely many stocks.
I think a value of 3 is a good starting point for most investors. I personally think I'm a bit more risk averse than the average investor, and a lot less likely to panic sell, my risk aversion is between 2 and 3.
BobK
Risk aversion: “The common preference that people generally show for a sure thing over a favorable gamble of equal or slightly higher expected value.” Daniel Kahneman.
Risk tolerance: “Having the fortitude and discipline to stick with a predetermined investment strategy when the going gets rough,” also known as the “stomach acid test.”
~Larry Swedroe
"Yes, investing is simple. But it is not easy, for it requires discipline, patience, steadfastness, and that most uncommon of all gifts, common sense." ~Jack Bogle
Re: Risk tolerance and asset allocation with mathematics
Agree. And the same behavior (often overconfidence) that can muck up portfolios can also muck up otherwise good models (remembering LTCM in '98 and the '08 financial crisis).watchnerd wrote: ↑Fri Mar 06, 2020 3:10 pm+1BillyK wrote: ↑Fri Mar 06, 2020 2:54 pm Larry Swedroe's stomachacid test and the ability to sleep well at night tests makes the most sense to me for determining risk tolerance. You can try to mathematically quantify it all you want, but when it comes right down to it as an investor, how much of a loss can you stomach while still sleeping well at night before panic selling during a bear market is the question you have to honestly answer yourself.
Human behavioral issues are a major issue in personal finance.
If people can't be stopped from mucking up their portfolios, it doesn't really matter how good the model is.
"Yes, investing is simple. But it is not easy, for it requires discipline, patience, steadfastness, and that most uncommon of all gifts, common sense." ~Jack Bogle
Re: Risk tolerance and asset allocation with mathematics
Thank you so much for writing this! I was following your posts (and learned a lot from them!) in various threads over the last few months, great to have it in a single place. As others suggested, this should be a wiki page.Uncorrelated wrote: ↑Wed Mar 04, 2020 5:28 pm Hello Boglehads. I think the mathematical knowledge here is disappointing, so I'm writing a short series on mathematical fundamentals for asset allocation. I'm starting off with utility theory and risk aversion.
Two questions:
1. Could you start another thread on small and value factors? There are many factor sceptics on this board (and I count myself as one of them, mostly because of their subdued performance after they've been discovered). But I also tend to agree with everything else you write, which means there's something I (and others) could learn.
2. I really (really!) want to play with your MV model. I saw you mentioned it's not ready to be shared/reviewed yet, but if you need alpha testers/reviewers I would be extremely interested (my background is in programming and maths though, not finance/economics). I'm sure you can find a dozen of others on this board too.
Thank you again!
Re: Risk tolerance and asset allocation with mathematics
Thanks you!305pelusa wrote: ↑Fri Mar 06, 2020 2:34 pmMy cliff notes:
 Don’t focus on the most efficient portfolio. Just whichever one produces the best risk and return trade off for your circumstances (maximizes utility).
 Don’t mentally account your investments (I.e 5% in Hedgefundie Excellent Adventure). What matters is the entire portfolio.
 Target Volatility doesn’t make much sense for most people.
 If you want to time the market, small changes in allocation (say from 60% to 40%) tend to be superior than full on “all stocks or no stocks”. To fully exit the market, you must believe stocks have a negative expected return. Since it’s really hard to know if that’s the case, it’s unlikely you should ever fully exit the market.
 For those of us in accumulation, you would decrease your lifetime risk, while improving lifetime returns, by following a glide path.
I’m of the thought that if this topic isn’t too exciting, just a good old Vanguard Target Date Fund, ideally with a retirement date well after your actual retirement, is good enough.
If you want to get much closer to the true, optimal glide path, then you have to do a little more work as I’ve done in this thread:
viewtopic.php?f=10&t=274390
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Re: Risk tolerance and asset allocation with mathematics
Uncorrelated, serious question. So I see equations that require you to plug in numerical measures of risk aversion. In order to use them, you need to have some way to measure risk aversion quantitatively.
How do you do that? The formulas are useless without a fairly precise measurement instrument. What is that instrument? I don't think anybody in their right mind believes that brokeragestyle questionnaires...
There's a kind of circularity. You can try to deduce someone's risk aversion by observing the asset allocation they choseseeing what they did do instead of asking what they would do. For example, we see they did not opt out of their employer's default in their 401(k) plan, which was a 2060 target date fund with a current asset allocation of 90/10, therefore we observe that they have a high risk tolerance."
Both asking hypothetical questions and observing actual behavior suffer from the same problem: "Whatever is, is right."
Determining asset allocation by formula is useless if it simply tells you that whatever choices you've made are right. In order to be useful, the formula needs to be able to tell people that their asset allocation needs to be higher or lower than it is. To put it another way, there needs to be a way, based on data about how you answer questions and how you've chosen assets in the past, that accurately predicts how you are likely to feel and what you are likely to do in the future, for example during crises.
Absent that, formulas with risk tolerance in them are just a way of saying "If we had some ham, we could have ham and eggs, if we had some eggs."
How do you do that? The formulas are useless without a fairly precise measurement instrument. What is that instrument? I don't think anybody in their right mind believes that brokeragestyle questionnaires...
can do more than protect the broker from complaints.Generally, I prefer investments with little or no fluctuation in value, and I'm willing to accept the lower return associated with these investments. a) Strongly disagree b) Disagree c) Somewhat agree d) Agree e) Strongly agree
There's a kind of circularity. You can try to deduce someone's risk aversion by observing the asset allocation they choseseeing what they did do instead of asking what they would do. For example, we see they did not opt out of their employer's default in their 401(k) plan, which was a 2060 target date fund with a current asset allocation of 90/10, therefore we observe that they have a high risk tolerance."
Both asking hypothetical questions and observing actual behavior suffer from the same problem: "Whatever is, is right."
Determining asset allocation by formula is useless if it simply tells you that whatever choices you've made are right. In order to be useful, the formula needs to be able to tell people that their asset allocation needs to be higher or lower than it is. To put it another way, there needs to be a way, based on data about how you answer questions and how you've chosen assets in the past, that accurately predicts how you are likely to feel and what you are likely to do in the future, for example during crises.
Absent that, formulas with risk tolerance in them are just a way of saying "If we had some ham, we could have ham and eggs, if we had some eggs."
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.
 Steve Reading
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Re: Risk tolerance and asset allocation with mathematics
I’m not Uncorrolated but the way to determine the risk aversion coefficient is by answering how much % of your wealth you’d be willing to lose to have an even chance to double it.nisiprius wrote: ↑Sat Mar 07, 2020 8:46 am Uncorrelated, serious question. So I see equations that require you to plug in numerical measures of risk aversion. In order to use them, you need to have some way to measure risk aversion quantitatively.
How do you do that? The formulas are useless without a fairly precise measurement instrument. What is that instrument? I don't think anybody in their right mind believes that brokeragestyle questionnaires...can do more than protect the broker from complaints.Generally, I prefer investments with little or no fluctuation in value, and I'm willing to accept the lower return associated with these investments. a) Strongly disagree b) Disagree c) Somewhat agree d) Agree e) Strongly agree
There's a kind of circularity. You can try to deduce someone's risk aversion by observing the asset allocation they choseseeing what they did do instead of asking what they would do. For example, we see they did not opt out of their employer's default in their 401(k) plan, which was a 2060 target date fund with a current asset allocation of 90/10, therefore we observe that they have a high risk tolerance."
Both asking hypothetical questions and observing actual behavior suffer from the same problem: "Whatever is, is right."
Determining asset allocation by formula is useless if it simply tells you that whatever choices you've made are right. In order to be useful, the formula needs to be able to tell people that their asset allocation needs to be higher or lower than it is. To put it another way, there needs to be a way, based on data about how you answer questions and how you've chosen assets in the past, that accurately predicts how you are likely to feel and what you are likely to do in the future, for example during crises.
Absent that, formulas with risk tolerance in them are just a way of saying "If we had some ham, we could have ham and eggs, if we had some eggs."
The writers of Lifecycle Investing have made a little calculator that takes you to the scenario here (“Calculate a rough estimate of your RRA”):
http://www.lifecycleinvesting.net/resources.html
"... so high a present discounted value of wealth, it is only prudent for him to put more into common stocks compared to his present tangible wealth, borrowing if necessary"  Paul Samuelson
 Steve Reading
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Re: Risk tolerance and asset allocation with mathematics
They never made any sense to me because they have no mention of returns. I might be willing to tolerate a 50% haircut from equities (which I know have a generally high expected rate of return and hence might work out long term) but I’m not going to tolerate a 50% haircut from commodities, whose returns are much smaller. I could tolerate a 50% drop so that might mean a 40%+ allocation to commodities based on pure risk tolerance, but I DEFINITELY wouldn’t want that.BillyK wrote: ↑Fri Mar 06, 2020 2:54 pm Larry Swedroe's stomachacid test and the ability to sleep well at night tests makes the most sense to me for determining risk tolerance. You can try to mathematically quantify it all you want, but when it comes right down to it as an investor, how much of a loss can you stomach while still sleeping well at night before panic selling during a bear market is the question you have to honestly answer yourself.
Allocating capital isn’t just about downside risk. It must also be based on upside potential. Instead of asking what’s your risk tolerance, and your need and your willingness, etc, I think it’s far simpler to encapsulate it all in the Coefficient of Relative Risk Aversion
"... so high a present discounted value of wealth, it is only prudent for him to put more into common stocks compared to his present tangible wealth, borrowing if necessary"  Paul Samuelson
 Steve Reading
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Re: Risk tolerance and asset allocation with mathematics
Dude can you please stop just deleting your posts? It is a little bothersome to take the time to type a post to respond to you (which I just did to respond to your follow up questions), only to find you deleted that same post (how do you do it btw, are you a Mod?). It’s like the third time it’s happened to me with you 0_o
"... so high a present discounted value of wealth, it is only prudent for him to put more into common stocks compared to his present tangible wealth, borrowing if necessary"  Paul Samuelson
 Uncorrelated
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Re: Risk tolerance and asset allocation with mathematics
The unfortunate answer is that I don't know. My approach is to envision the desired asset allocation in retirement, and then abuse lifecycle investing to calculate the presentday risk tolerance. However this is somewhat circular. I personally think it's very important to be consistent with some coefficient of relative risk aversion, even if you can't estimate yours perfectly.
https://www.aacalc.com/docs/relative_risk_aversion has some research notes notes on this topic.
Re: Risk tolerance and asset allocation with mathematics
OP I tried to follow your post but most of it was far over my head. (I was a science major in college and took some calculus and statistics.) If your intent (and perhaps it was not your primary intent, which is fine too) was to help forum members make decisions with respect to their portfolio, maybe it would have been helpful if at some point you stated, perhaps with some practical examples, how what you are discussing might be applied.
But thank you for sharing your obviously carefully considered thoughts. Certainly, more interesting than another iteration of "domestic vs. international". (My apologies if that is the planned topic of a future post.)
But thank you for sharing your obviously carefully considered thoughts. Certainly, more interesting than another iteration of "domestic vs. international". (My apologies if that is the planned topic of a future post.)
Re: Risk tolerance and asset allocation with mathematics
I also don't find it useful to directly answer the "what should my allocation of stocks/bonds" be  mostly because of the bootstrap issue. The survey based methods for determining risk aversion feel more difficult for me to intuit what I consider a "real" answer.nisiprius wrote: ↑Sat Mar 07, 2020 8:46 am (...)
There's a kind of circularity. You can try to deduce someone's risk aversion by observing the asset allocation they choseseeing what they did do instead of asking what they would do. For example, we see they did not opt out of their employer's default in their 401(k) plan, which was a 2060 target date fund with a current asset allocation of 90/10, therefore we observe that they have a high risk tolerance."
Both asking hypothetical questions and observing actual behavior suffer from the same problem: "Whatever is, is right."
Determining asset allocation by formula is useless if it simply tells you that whatever choices you've made are right. In order to be useful, the formula needs to be able to tell people that their asset allocation needs to be higher or lower than it is.
(...)
But (at least for me), they key is that's not the only question you can ask based on a risk aversion. Using the value  or a range of values  one can then examine proposed changes in portfolio composition. So it's not so much saying "this is my perfect, ideal risk aversion"  but rather, "keeping my risk aversion constant, what is the change in utility I get under different portfolio constructions" (Uncorrelated's market timing example demonstrated doing this across a different type of change as well).
So, again, form my point of view it's really a question of "what question are you using this to answer", and holding the risk aversion factor constant across multiple scenarios leads, I believe, to a much more robust comparison.
Re: Risk tolerance and asset allocation with mathematics
CE is the X that satisfy the equation U(X) = E ( U(W) ), where W is a random variable consumption.
OTOH, CER is a return rather than consumption (input variable of utility function). The 2 are related but not identical. It requires a few more assumptions to derive CER.

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Re: Risk tolerance and asset allocation with mathematics
Thank you for putting in the time and effort to write this. It's very informative.
If you were to suggest that to someone riding the 100% stocks train, they would probably respond with some combination of confusion and ridicule.
What am I misunderstanding about the above quote? Or what are the 100/0 folks misunderstanding? Or is that example oversimplifying something important?
If the answer to all three of those questions is "nothing", then it seems like this utility function is out of whack with what people actually want from their investments.
This would seem to mean that someone with a high enough risk tolerance to comfortably hold 100% stocks for the long term should, given the option, instead put everything into CDs at 3%.Uncorrelated wrote: ↑Wed Mar 04, 2020 5:28 pm We can plug in some numbers to get a feel for the correct value of γ. I will use a risk premium of 5% and standard deviation of 16%.
<...>
γ=2 corresponds to an asset allocation of 100% stocks
<...>
Suppose that we have an investor with γ=2 and a 100% stock allocation. The utility of this investor is E(X)  1/2 * γ * Var(X) = .0244. The CER is 2.44%. This means that if the investor can choose between a 100% stock allocation or an investment with a guaranteed return of 2.44%, both investments are equally good.
If you were to suggest that to someone riding the 100% stocks train, they would probably respond with some combination of confusion and ridicule.
What am I misunderstanding about the above quote? Or what are the 100/0 folks misunderstanding? Or is that example oversimplifying something important?
If the answer to all three of those questions is "nothing", then it seems like this utility function is out of whack with what people actually want from their investments.
Re: Risk tolerance and asset allocation with mathematics
Actually, if someone showed me that graph and I didn't know the phenomena I'd say "looks normal or lognormal, but with a very small sample size so not very clear yet".Thesaints wrote: ↑Wed Mar 04, 2020 6:02 pm How normal is this distribution ?
https://static.seekingalpha.com/uploads ... urns1.jpg
Re: Risk tolerance and asset allocation with mathematics
OP was inconsistent with the numbers. He assumed "5%" as the risk premium but then used 5% as E(X).luckyducky99 wrote: ↑Sat Mar 07, 2020 12:27 pm Thank you for putting in the time and effort to write this. It's very informative.
This would seem to mean that someone with a high enough risk tolerance to comfortably hold 100% stocks for the long term should, given the option, instead put everything into CDs at 3%.Uncorrelated wrote: ↑Wed Mar 04, 2020 5:28 pm We can plug in some numbers to get a feel for the correct value of γ. I will use a risk premium of 5% and standard deviation of 16%.
<...>
γ=2 corresponds to an asset allocation of 100% stocks
<...>
Suppose that we have an investor with γ=2 and a 100% stock allocation. The utility of this investor is E(X)  1/2 * γ * Var(X) = .0244. The CER is 2.44%. This means that if the investor can choose between a 100% stock allocation or an investment with a guaranteed return of 2.44%, both investments are equally good.
If you were to suggest that to someone riding the 100% stocks train, they would probably respond with some combination of confusion and ridicule.
What am I misunderstanding about the above quote? Or what are the 100/0 folks misunderstanding? Or is that example oversimplifying something important?
If the answer to all three of those questions is "nothing", then it seems like this utility function is out of whack with what people actually want from their investments.
 Uncorrelated
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Re: Risk tolerance and asset allocation with mathematics
A CD of 3% doesn't have a risk premium of 3%. With the (somewhat arbitrary) return assumptions in this post, a CD has a risk premium of 0%.luckyducky99 wrote: ↑Sat Mar 07, 2020 12:27 pm Thank you for putting in the time and effort to write this. It's very informative.
This would seem to mean that someone with a high enough risk tolerance to comfortably hold 100% stocks for the long term should, given the option, instead put everything into CDs at 3%.Uncorrelated wrote: ↑Wed Mar 04, 2020 5:28 pm We can plug in some numbers to get a feel for the correct value of γ. I will use a risk premium of 5% and standard deviation of 16%.
<...>
γ=2 corresponds to an asset allocation of 100% stocks
<...>
Suppose that we have an investor with γ=2 and a 100% stock allocation. The utility of this investor is E(X)  1/2 * γ * Var(X) = .0244. The CER is 2.44%. This means that if the investor can choose between a 100% stock allocation or an investment with a guaranteed return of 2.44%, both investments are equally good.
If you were to suggest that to someone riding the 100% stocks train, they would probably respond with some combination of confusion and ridicule.
What am I misunderstanding about the above quote? Or what are the 100/0 folks misunderstanding? Or is that example oversimplifying something important?
If the answer to all three of those questions is "nothing", then it seems like this utility function is out of whack with what people actually want from their investments.
I'll modify the text to explain a 100% stock allocation is equivalent with a guaranteed return of 2.44% above the riskfree rate.
 nisiprius
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 Location: The terrestrial, globular, planetary hunk of matter, flattened at the poles, is my abode.O. Henry
Re: Risk tolerance and asset allocation with mathematics
I'm sorry, apologies. When I reread my posting on line it felt it was argumentative, and I thought it would be better just to let your posting stand. I'm not a mod, anybody can delete their own posts as long as nobody has replied to them yet.305pelusa wrote: ↑Sat Mar 07, 2020 10:00 amDude can you please stop just deleting your posts? It is a little bothersome to take the time to type a post to respond to you (which I just did to respond to your follow up questions), only to find you deleted that same post (how do you do it btw, are you a Mod?). It’s like the third time it’s happened to me with you 0_o
Can we find out whether a man is above average in driving ability by the questionnaire methodby asking him "are you above average in driving ability?" According to an AAA survey, 8 out of 10 men answered "yes." (OK, that's not mathematically impossible, but...)
Asking people their driving ability is not a reliable way to measure their driving ability.
My point, which you seemed to miss, is "how does one determine one's risk tolerance accurately?" Your link leads to a spreadsheet which asks "Maximum Percentage of Current Income You Would be Willing to Put at Risk?" But can you determine someone's risk tolerance accurately through the "questionnaire" method? Do you actually know what percentage of your current income you are willing to put at risk?
So in fact, the spreadsheet is more like a definition of risk tolerance than a reliable method for measuring risk tolerance. Or perhaps it's like a way of converting a measurement from one system of units to another, rather than a way to make a measurement. The conversion could be very precise, but the answer would not be.
The spreadsheet itself actually says "Warning: As discussed in chapter 7 of our book, it is extremely difficult to derive an unbiased and robust estimate of riskaversion from a single question."
Last edited by nisiprius on Sat Mar 07, 2020 2:48 pm, edited 1 time in total.
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.
 Steve Reading
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Re: Risk tolerance and asset allocation with mathematics
I understand your concerns and I had addressed them in my reply to your deleted post. I'm sorry but I don't feel like rewriting it.nisiprius wrote: ↑Sat Mar 07, 2020 2:18 pmI'm sorry, apologies. When I reread my posting on line it felt it was argumentative, and I thought it would be better just to let your posting stand. I'm not a mod, anybody can delete their own posts as long as nobody has replied to them yet.305pelusa wrote: ↑Sat Mar 07, 2020 10:00 amDude can you please stop just deleting your posts? It is a little bothersome to take the time to type a post to respond to you (which I just did to respond to your follow up questions), only to find you deleted that same post (how do you do it btw, are you a Mod?). It’s like the third time it’s happened to me with you 0_o
Can we find out whether a man is above average in driving ability by the questionnaire methodby asking him "are you above average in driving ability?" According to an AAA survey, 8 out of 10 men answered "yes." (OK, that's not mathematically impossible, but...)
Asking people their driving ability is not a reliable way to measure their driving ability.
My point, which you seemed to miss, is "how do you determine one's risk tolerance accurately?" Your link leads to a spreadsheet which asks "Maximum Percentage of Current Income You Would be Willing to Put at Risk?" But can you determine someone's risk tolerance accurately through the "questionnaire" method? Do you actually know what percentage of your current income you are willing to put at risk?
So in fact, the spreadsheet is more like a definition of risk tolerance than a reliable method for measuring risk tolerance. Or perhaps it's like a way of converting a measurement from one system of units to another, rather than a way to make a measurement. The conversion could be very precise, but the answer would not be.
The spreadsheet itself actually says "Warning: As discussed in chapter 7 of our book, it is extremely difficult to derive an unbiased and robust estimate of riskaversion from a single question."
For future reference, I don't find you argumentative in any way. I think most in this forum agree. Don't worry about your posts seeming like they could be taken poorly. I know you mean well and I would personally never take them that way.
"... so high a present discounted value of wealth, it is only prudent for him to put more into common stocks compared to his present tangible wealth, borrowing if necessary"  Paul Samuelson

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Re: Risk tolerance and asset allocation with mathematics
Thanks  the "above the riskfree rate" part wasn't obvious to me, though it might have been to more math savvy readers.Uncorrelated wrote: ↑Sat Mar 07, 2020 1:29 pmA CD of 3% doesn't have a risk premium of 3%. With the (somewhat arbitrary) return assumptions in this post, a CD has a risk premium of 0%.luckyducky99 wrote: ↑Sat Mar 07, 2020 12:27 pm Thank you for putting in the time and effort to write this. It's very informative.
This would seem to mean that someone with a high enough risk tolerance to comfortably hold 100% stocks for the long term should, given the option, instead put everything into CDs at 3%.Uncorrelated wrote: ↑Wed Mar 04, 2020 5:28 pm We can plug in some numbers to get a feel for the correct value of γ. I will use a risk premium of 5% and standard deviation of 16%.
<...>
γ=2 corresponds to an asset allocation of 100% stocks
<...>
Suppose that we have an investor with γ=2 and a 100% stock allocation. The utility of this investor is E(X)  1/2 * γ * Var(X) = .0244. The CER is 2.44%. This means that if the investor can choose between a 100% stock allocation or an investment with a guaranteed return of 2.44%, both investments are equally good.
If you were to suggest that to someone riding the 100% stocks train, they would probably respond with some combination of confusion and ridicule.
What am I misunderstanding about the above quote? Or what are the 100/0 folks misunderstanding? Or is that example oversimplifying something important?
If the answer to all three of those questions is "nothing", then it seems like this utility function is out of whack with what people actually want from their investments.
I'll modify the text to explain a 100% stock allocation is equivalent with a guaranteed return of 2.44% above the riskfree rate.
Re: Risk tolerance and asset allocation with mathematics
OP, one of the most important goals of writers is to know who they are writing for, their target audience. One question I have is who you identify as your target audience; another question is how successful you think you would be with that target based on the variety of responses you've seen here.
Also, in using the terms risk aversion and risk tolerance interchangebly, even though they are defined differently, wouldn't this mean you are referring only to investors with low risk tolerance, even though investors can also have high risk tolerance or somewhere in between?
Also, in using the terms risk aversion and risk tolerance interchangebly, even though they are defined differently, wouldn't this mean you are referring only to investors with low risk tolerance, even though investors can also have high risk tolerance or somewhere in between?
"Yes, investing is simple. But it is not easy, for it requires discipline, patience, steadfastness, and that most uncommon of all gifts, common sense." ~Jack Bogle
Re: Risk tolerance and asset allocation with mathematics
Determining the relative risk aversion coefficient to use to determine an asset allocation may at first appear to require circular reasoning, but doing so has real value.
First, it translates a decision about acceptable portfolio losses into something far more meaningful, uncertainty in retirement consumption.
Second, it makes the asset allocation decision self consistent over the lifecycle. Without the risk aversion framework you are left guessing what you should do, not once, but at every point in the lifecycle.
It is still a difficult decision to make. For the financial calculator I am currently working on I attempt to help the user decide by showing them the consequences of the RRA decision: for RRA=6 you can get a mean consumption of $50,000 with a standard deviation of $10,000, and this is what the distribution of outcomes looks like; or for RRA=1.5 you can get a mean consumption of $70,000 with a standard deviation of $30,000, and this is what the distribution of outcomes looks like.
Deriving an asset allocation by looking at the long term retirement consumption consequences seems far more meaningful than deriving an asset allocation from the short term psychological consequences of the commonly asked "how would you feel after a 50% drop in your investment portfolio value".
First, it translates a decision about acceptable portfolio losses into something far more meaningful, uncertainty in retirement consumption.
Second, it makes the asset allocation decision self consistent over the lifecycle. Without the risk aversion framework you are left guessing what you should do, not once, but at every point in the lifecycle.
It is still a difficult decision to make. For the financial calculator I am currently working on I attempt to help the user decide by showing them the consequences of the RRA decision: for RRA=6 you can get a mean consumption of $50,000 with a standard deviation of $10,000, and this is what the distribution of outcomes looks like; or for RRA=1.5 you can get a mean consumption of $70,000 with a standard deviation of $30,000, and this is what the distribution of outcomes looks like.
Deriving an asset allocation by looking at the long term retirement consumption consequences seems far more meaningful than deriving an asset allocation from the short term psychological consequences of the commonly asked "how would you feel after a 50% drop in your investment portfolio value".
 Uncorrelated
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Re: Risk tolerance and asset allocation with mathematics
I recognize the error, thanks for paying attention. I need to take some time to familiarize myself with the material, hopefully I'll be able to update the post in the coming week.acegolfer wrote: ↑Fri Mar 06, 2020 5:33 pmCARA and CRRA are not almost identical.Uncorrelated wrote: ↑Fri Mar 06, 2020 4:26 pm As far as I understand, the expected utility for CARA and CRRA are almost identical. Your first source only displays the derivation for CARA utility and not CRRA. The second source appears to specify µ as the geometric growth, but I use the arithmetic growth. I'm unsure if that explains the differences between the results.
My sources are http://citeseerx.ist.psu.edu/viewdoc/do ... 1&type=pdf (chapter 5) and https://www.gordoni.com/lifetime_portfo ... ection.pdf (page 5), although neither sources give an explicit derivation.
It's entirely possible I'm making an error here. Despite all my talk about math, this is pretty far outside my expertise.
http://citeseerx.ist.psu.edu/viewdoc/do ... 1&type=pdf has typo in page 31.
"γ is the investor’s coefficient of relative risk aversion"
It should be "γ is the investor’s coefficient of absolute risk aversion"
https://www.tau.ac.il/~spiegel/teaching ... riance.pdf has a detailed derivation of CER from CARA function. I learned this when I was taking a masters level economics course in 90s. You only need to know exponential and normal distribution pdf to understand the proof.
 Uncorrelated
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Re: Risk tolerance and asset allocation with mathematics
I also have a programming background, math/finance/economics is just my hobby.toisvu wrote: ↑Sat Mar 07, 2020 1:30 amThank you so much for writing this! I was following your posts (and learned a lot from them!) in various threads over the last few months, great to have it in a single place. As others suggested, this should be a wiki page.Uncorrelated wrote: ↑Wed Mar 04, 2020 5:28 pm Hello Boglehads. I think the mathematical knowledge here is disappointing, so I'm writing a short series on mathematical fundamentals for asset allocation. I'm starting off with utility theory and risk aversion.
Two questions:
1. Could you start another thread on small and value factors? There are many factor sceptics on this board (and I count myself as one of them, mostly because of their subdued performance after they've been discovered). But I also tend to agree with everything else you write, which means there's something I (and others) could learn.
2. I really (really!) want to play with your MV model. I saw you mentioned it's not ready to be shared/reviewed yet, but if you need alpha testers/reviewers I would be extremely interested (my background is in programming and maths though, not finance/economics). I'm sure you can find a dozen of others on this board too.
Thank you again!
If you want to learn more about factors, I recommend viewing Ben Felix' youtube channel. He explains it much better than I can. Half of my posts on factors are just citations from his youtube channel.
https://www.youtube.com/channel/UCDXTQ8 ... Z2vkp7QxA
I have plans for another entry in this series where I discuss meanvariance optimization in greater detail, with source code. I'm very unhappy about the bond assumptions currently in the model, that needs to be fixed first before I'm confident enough to publish it.
Re: Risk tolerance and asset allocation with mathematics
Thank you, appreciate the reply!
Re: Risk tolerance and asset allocation with mathematics
Those are not the only options. There is a whole continuum of choices with various amounts of intuitions and math. I suspect that it is not even a onedimensional space. You can try to make a plan that is less reliant on assumptions and you can try to make a model that is more robust to uncertainties and unknowns. I am a modeler. My background is not in economics; I am an astrophysicist. But I spend my day making models. I am keenly aware of the dangers of trusting complex models based on faulty assumptions simply because those assumptions make the math tractable. You are advocating the use of a complex model for which none of us has any idea how to feed correct inputs. There is a saying that you have surely heard: "garbage in, garbage out".Uncorrelated wrote: ↑Thu Mar 05, 2020 5:20 pm It's impossible to avoid modeling errors. Either you choose to make your model explicit, with math, utility functions and distributions, or you can choose to do everything by intuition.
Let me follow that thought:
I am not persuaded that the model you have presented here meets that bar. It is not hard to imagine that a simple rule of thumb (like a 60/40 stock/bonds) might be more robust than feeding wrong data to a complex model that may be dangerously susceptible to overfitting. Just as an example.Uncorrelated wrote: ↑Thu Mar 05, 2020 5:20 pm Models will never be perfect, but they don't have to be. They only need to be better than the intuitive approach, and that is not a very high bar.
Re: Risk tolerance and asset allocation with mathematics
THANK YOU. Now, here's a question: How about using Target Volatility as a market timing signal? Let me show you two problems I have with the original post. In the section about TV, Uncorrelated says:305pelusa wrote: ↑Fri Mar 06, 2020 2:34 pmMy cliff notes:
 Don’t focus on the most efficient portfolio. Just whichever one produces the best risk and return trade off for your circumstances (maximizes utility).
 Don’t mentally account your investments (I.e 5% in Hedgefundie Excellent Adventure). What matters is the entire portfolio.
 Target Volatility doesn’t make much sense for most people.
 If you want to time the market, small changes in allocation (say from 60% to 40%) tend to be superior than full on “all stocks or no stocks”. To fully exit the market, you must believe stocks have a negative expected return. Since it’s really hard to know if that’s the case, it’s unlikely you should ever fully exit the market.
 For those of us in accumulation, you would decrease your lifetime risk, while improving lifetime returns, by following a glide path.
"We assume that return is linear to volatility, r = D * σ, with D being a constant greater than zero.
... < some arithmetic > ...
What have we learned here? If you are using target volatility and have a CRRA utility, you must believe that the sharpe ratio is a constant, and that return is proportional to volatility.
"
The first problem is that the bold text is not true. The idea that return is proportional to volatility was an assumption on the part of Uncorrelated; he can't then present that as a conclusion. The second problem is that I have never heard a proponent of TV argue that returns are proportional to volatility. At least, not in the sense that periods of high volatility are periods of high returns. Finally, Uncorrelated proceeds to disprove his own assumption right in the next sentence:
"This is at odds with the research literature, return is not at all or weakly negatively correlated with volatility."
You see what I mean? Uncorrelated inserted an assumption that I haven't seen any TV proponent make, then claimed that that assumption was somehow an implication of TV, and then argued that that assumption, that he inserted, was wrong.
Let's try running through Uncorrelated's math again, but now wit the (better) assumption that returns are weakly negatively correlated with volatility:
Stock Allocation = C / σ = (1 / γ) * r / σ^2
=> C = (1 / γ) * r / σ
Ok. So we want to make r be negatively correlated with volatility. So let r = r0  a * σ where a is a positive number.
=> C = (1 / γ) * (r0  a * σ) / σ
=> C = (1 / γ) * (r0 / σ  a)
If we need C to be constant, the above result implies that a >> r0/σ so stock returns are strongly negatively correlated with volatility (rather than weakly negatively correlated). If r0/σ >> a then C is not a constant at all. Both of these assumptions can be criticized, but they are more palatable than the ones that Uncorrelated claimed. And as I've noted elsewhere, this is still all based on Uncorrelated's assumption of CCRA utility, which is known to be wrong.
Re: Risk tolerance and asset allocation with mathematics
Will that financial calculator be available online? I would love to see it. Could you perhaps PM me when it's online so I can try it out?0gordoni2 wrote: ↑Sat Mar 07, 2020 7:01 pm Determining the relative risk aversion coefficient to use to determine an asset allocation may at first appear to require circular reasoning, but doing so has real value.
First, it translates a decision about acceptable portfolio losses into something far more meaningful, uncertainty in retirement consumption.
Second, it makes the asset allocation decision self consistent over the lifecycle. Without the risk aversion framework you are left guessing what you should do, not once, but at every point in the lifecycle.
It is still a difficult decision to make. For the financial calculator I am currently working on I attempt to help the user decide by showing them the consequences of the RRA decision: for RRA=6 you can get a mean consumption of $50,000 with a standard deviation of $10,000, and this is what the distribution of outcomes looks like; or for RRA=1.5 you can get a mean consumption of $70,000 with a standard deviation of $30,000, and this is what the distribution of outcomes looks like.
Deriving an asset allocation by looking at the long term retirement consumption consequences seems far more meaningful than deriving an asset allocation from the short term psychological consequences of the commonly asked "how would you feel after a 50% drop in your investment portfolio value".
 Uncorrelated
 Posts: 1232
 Joined: Sun Oct 13, 2019 3:16 pm
Re: Risk tolerance and asset allocation with mathematics
I don't think it's likely that the model, as presented, is too complex. It makes very few and mathematically nice assumptions. The normal distribution assumption is not necessarily to get it to work although it is convenient. IMO the only really interesting part of the model is that utility is proportional to variance. as far as I'm aware of that is true for a wide range of utility functions. Not just CRRA.danielc wrote: ↑Mon Mar 09, 2020 1:22 amThose are not the only options. There is a whole continuum of choices with various amounts of intuitions and math. I suspect that it is not even a onedimensional space. You can try to make a plan that is less reliant on assumptions and you can try to make a model that is more robust to uncertainties and unknowns. I am a modeler. My background is not in economics; I am an astrophysicist. But I spend my day making models. I am keenly aware of the dangers of trusting complex models based on faulty assumptions simply because those assumptions make the math tractable. You are advocating the use of a complex model for which none of us has any idea how to feed correct inputs. There is a saying that you have surely heard: "garbage in, garbage out".Uncorrelated wrote: ↑Thu Mar 05, 2020 5:20 pm It's impossible to avoid modeling errors. Either you choose to make your model explicit, with math, utility functions and distributions, or you can choose to do everything by intuition.
Let me follow that thought:
I am not persuaded that the model you have presented here meets that bar. It is not hard to imagine that a simple rule of thumb (like a 60/40 stock/bonds) might be more robust than feeding wrong data to a complex model that may be dangerously susceptible to overfitting. Just as an example.Uncorrelated wrote: ↑Thu Mar 05, 2020 5:20 pm Models will never be perfect, but they don't have to be. They only need to be better than the intuitive approach, and that is not a very high bar.
Of course the results of the model are dependent on how accurate you are able to forecast the equity risk premium, but that is also a problem for the 60/40 portfolio.
Your formula appears to be ill formed because it results in C < 0, which implies a negative allocation to the risky asset. The assumptions r = r0  a * σ and a >> r0/σ result in a negative expected return.danielc wrote: ↑Mon Mar 09, 2020 2:17 am If we need C to be constant, the above result implies that a >> r0/σ so stock returns are strongly negatively correlated with volatility (rather than weakly negatively correlated). If r0/σ >> a then C is not a constant at all. Both of these assumptions can be criticized, but they are more palatable than the ones that Uncorrelated claimed. And as I've noted elsewhere, this is still all based on Uncorrelated's assumption of CCRA utility, which is known to be wrong.
What you described there is not target volatility. Target volatility sets the strategy such that the expected volatility is constant. This can only occur if the allocation to the risky asset is proportional to 1/forecasted_volatility. I recognize that there are other approaches, such as making the allocation to the risky asset proportional to 1/(some quadratic function of σ) that are consistent with literature's conclusion that return is weakly negatively correlated with volatility. But those other approaches are not named target volatility.
Making the assumption that TV implicitly assumes that return is proportional to volatility is the only way I can get the math to be consistent.
 Steve Reading
 Posts: 2674
 Joined: Fri Nov 16, 2018 10:20 pm
Re: Risk tolerance and asset allocation with mathematics
What that math shows is how returns must change as a function of volatility for Target Volatility to produce allocations that are consistent with CRRA. And it all makes perfect sense. Remember the CRRA formula allocates based on returns and volatility. So for a strategy that only uses volatility to allocate, then said volatility must somehow carry all the information about returns that you need. The only way that can happen is if you can convert volatility to return via a linear formula, which happens to be Sharpe.danielc wrote: ↑Mon Mar 09, 2020 2:17 amTHANK YOU. Now, here's a question: How about using Target Volatility as a market timing signal? Let me show you two problems I have with the original post. In the section about TV, Uncorrelated says:305pelusa wrote: ↑Fri Mar 06, 2020 2:34 pmMy cliff notes:
 Don’t focus on the most efficient portfolio. Just whichever one produces the best risk and return trade off for your circumstances (maximizes utility).
 Don’t mentally account your investments (I.e 5% in Hedgefundie Excellent Adventure). What matters is the entire portfolio.
 Target Volatility doesn’t make much sense for most people.
 If you want to time the market, small changes in allocation (say from 60% to 40%) tend to be superior than full on “all stocks or no stocks”. To fully exit the market, you must believe stocks have a negative expected return. Since it’s really hard to know if that’s the case, it’s unlikely you should ever fully exit the market.
 For those of us in accumulation, you would decrease your lifetime risk, while improving lifetime returns, by following a glide path.
"We assume that return is linear to volatility, r = D * σ, with D being a constant greater than zero.
... < some arithmetic > ...
What have we learned here? If you are using target volatility and have a CRRA utility, you must believe that the sharpe ratio is a constant, and that return is proportional to volatility.
"
The first problem is that the bold text is not true. The idea that return is proportional to volatility was an assumption on the part of Uncorrelated; he can't then present that as a conclusion. The second problem is that I have never heard a proponent of TV argue that returns are proportional to volatility. At least, not in the sense that periods of high volatility are periods of high returns. Finally, Uncorrelated proceeds to disprove his own assumption right in the next sentence:
"This is at odds with the research literature, return is not at all or weakly negatively correlated with volatility."
You see what I mean? Uncorrelated inserted an assumption that I haven't seen any TV proponent make, then claimed that that assumption was somehow an implication of TV, and then argued that that assumption, that he inserted, was wrong.
The lesson is that: if volatility is weakly or even negatively correlated with returns, then using only volatility will not lead to allocations consistent with CRRA.
Now you can argue it's not supposed to mimic allocations consistent with CRRA. Otherwise, you would just use the CRRA formula. If you judge a duck (Target Volatility) or a shark (Market timing fully in and out) by how fast they can run, then a cheetah (the CRRA allocation formula) will always be ideal. So you can argue how relevant Uncorrelated's formulations are, but that they're correct, they're correct.
"... so high a present discounted value of wealth, it is only prudent for him to put more into common stocks compared to his present tangible wealth, borrowing if necessary"  Paul Samuelson
Re: Risk tolerance and asset allocation with mathematics
I'm not sure about that. I don't understand it at all, and I'm mathematically inclined. To be clear, the problem is not (for example) that I don't know what a logarithm is; the problem is that I don't know why I should expect it to model my investment goals.Uncorrelated wrote: ↑Mon Mar 09, 2020 4:49 am I don't think it's likely that the model, as presented, is too complex.
It's definitely ill formed if a >> r0/σ. But that's not my fault; that's what your utility function gives if we combine it TV and the idea that stocks returns are negatively correlated with volatility. That just means that at least one of the assumptions in the preceding two sentences is very likely false. I never advocated a >> r0/σ or CCRA utility.Uncorrelated wrote: ↑Mon Mar 09, 2020 4:49 amYour formula appears to be ill formed because it results in C < 0, which implies a negative allocation to the risky asset. The assumptions r = r0  a * σ and a >> r0/σ result in a negative expected return.danielc wrote: ↑Mon Mar 09, 2020 2:17 am If we need C to be constant, the above result implies that a >> r0/σ so stock returns are strongly negatively correlated with volatility (rather than weakly negatively correlated). If r0/σ >> a then C is not a constant at all. Both of these assumptions can be criticized, but they are more palatable than the ones that Uncorrelated claimed. And as I've noted elsewhere, this is still all based on Uncorrelated's assumption of CCRA utility, which is known to be wrong.
What? No. How? We must be talking past each other. How could it ever make sense to set your stock allocation to C/σ where C is a constant if you thought that σ is constant too? The whole point of TV is that stock volatility varies in time, and you are supposed to reduce your stock allocation when stocks are volatile.Uncorrelated wrote: ↑Mon Mar 09, 2020 4:49 am What you described there is not target volatility. Target volatility sets the strategy such that the expected volatility is constant.
One way to make your statement true is to say that "volatility" refers to the volatility of the entire portfolio rather than just stocks, and then inject the assumption that your only investments are stocks and cash. Nobody in this forum (as far as I can tell) who uses TV chooses cash as their nonstock asset.
But you already know that that assumption is counterfactual. Whether your "TV" portfolio is UPRO+TMF or plain old stocks+cash, nobody that I know has suggested that returns are proportional to any kind of volatility. In fact, the only assertion that I have seen about returns is that they are unpredictable. So the most accurate way to describe what people are assuming is that the expected return is constant.Uncorrelated wrote: ↑Mon Mar 09, 2020 4:49 am Making the assumption that TV implicitly assumes that return is proportional to volatility is the only way I can get the math to be consistent.
=> C = r / ( σ γ )
So either C is not constant, or γ is not constant, or CCRA is not the right utility, or something else. I believe you when you say that you have not found any way to make the math consistent other than assuming that returns are proportional to volatility. But that is an argument from lack of imagination. As far as I know, nobody thinks that volatility forecasts excess stock returns. The problem must be elsewhere.
Re: Risk tolerance and asset allocation with mathematics
This might sound pedantic on my part, but your last sentence does not follow. Let's grant that "volatility must somehow carry all the information about returns that you need"  It does not follow that the relation must be linear. There is an infinite number of functions of one variable. A trivial example is to assume that expected returns from stocks, bonds, and cash are all constant. That assumption seems more in line with the limited comments that I have read about returns in the context of TV: I've seen people argue that there's no way to forecast returns, but you can volatility is somewhat predictable. Inability to forecast returns means that expected returns are constant.305pelusa wrote: ↑Mon Mar 09, 2020 7:45 am What that math shows is how returns must change as a function of volatility for Target Volatility to produce allocations that are consistent with CRRA. And it all makes perfect sense. Remember the CRRA formula allocates based on returns and volatility. So for a strategy that only uses volatility to allocate, then said volatility must somehow carry all the information about returns that you need. The only way that can happen is if you can convert volatility to return via a linear formula, which happens to be Sharpe.
I'm not sure in which sense you call them correct. If they don't model reality, what exactly is correct about them?305pelusa wrote: ↑Mon Mar 09, 2020 7:45 am Now you can argue it's not supposed to mimic allocations consistent with CRRA. Otherwise, you would just use the CRRA formula. If you judge a duck (Target Volatility) or a shark (Market timing fully in and out) by how fast they can run, then a cheetah (the CRRA allocation formula) will always be ideal. So you can argue how relevant Uncorrelated's formulations are, but that they're correct, they're correct.
Here is what I think I see:
The fundamental premise of TV is that volatility varies with time, and is autocorrelated so that the current volatility is a predictor of near future volatility, but returns are not predictable beyond a simple constant expected value. Uncorrelated has not tried to model any of this. Instead, (s)he modeled TV by assuming the exact opposite of the premises of TV, and found that TV is inconsistent with a dubious utility function (CRRA). It seems to me that, at best, Uncorrelated discovered that if all the premises of TV are completely wrong then TV is unjustified. Did we need any math to discover that? Importantly, Uncorrelated's chief conclusion about TV, which you eloquently summarized,
"Target Volatility doesn’t make much sense for most people."
is completely unfounded because Uncorrelated's model of TV violates all the premises of TV, while also injecting a new unfounded premise (CRRA).
 Steve Reading
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Re: Risk tolerance and asset allocation with mathematics
Of course there are. But only one will make Target Volatility allocate identically to how CRRA allocates. And that's a linear relationship.danielc wrote: ↑Tue Mar 10, 2020 3:05 am This might sound pedantic on my part, but your last sentence does not follow. Let's grant that "volatility must somehow carry all the information about returns that you need"  It does not follow that the relation must be linear. There is an infinite number of functions of one variable.
In this case, Target Volatility does not allocate identically to CRRA. If returns were constant and SD doubled, then Target Volatility would allocate half as much to stock while CRRA would allocate a quarter as much to stocks (since it goes by the square inverse of volatility).
Again, you might not say it's applicable or relevant, but it is true. If I tell you that "If the market were randomly distributed, allocation does not depend on time horizon"". That is a true and correct statement. You might not believe it applies well to reality because market movements aren't Gaussian. This latter criticism is reasonable but not the former.danielc wrote: ↑Tue Mar 10, 2020 3:05 amI'm not sure in which sense you call them correct. If they don't model reality, what exactly is correct about them?305pelusa wrote: ↑Mon Mar 09, 2020 7:45 am Now you can argue it's not supposed to mimic allocations consistent with CRRA. Otherwise, you would just use the CRRA formula. If you judge a duck (Target Volatility) or a shark (Market timing fully in and out) by how fast they can run, then a cheetah (the CRRA allocation formula) will always be ideal. So you can argue how relevant Uncorrelated's formulations are, but that they're correct, they're correct.
Daniel based on some of your questions, I'm not sure you actually followed the math he did. He didn't model anything about TV and did not assume anything about it. He simply looked at the allocation rule of TV (NO assumptions), and then equated it to the allocation rule of CRRA and asked "what would it take for these two to allocate in similar ways?". What it takes is something that isn't empirically shown (a linear relationship between volatility and return). Hence, if you do have CRRA, you shouldn't use TV.danielc wrote: ↑Tue Mar 10, 2020 3:05 am
The fundamental premise of TV is that volatility varies with time, and is autocorrelated so that the current volatility is a predictor of near future volatility, but returns are not predictable beyond a simple constant expected value. Uncorrelated has not tried to model any of this. Instead, (s)he modeled TV by assuming the exact opposite of the premises of TV, and found that TV is inconsistent with a dubious utility function (CRRA).
To the extent you have some other utility function, then use an allocation consistent with that other utility function.
"... so high a present discounted value of wealth, it is only prudent for him to put more into common stocks compared to his present tangible wealth, borrowing if necessary"  Paul Samuelson
 Uncorrelated
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Re: Risk tolerance and asset allocation with mathematics
Forget about bonds for a moment, use a simple investment universe where the only available instruments are stocks and a riskfree asset. If we consider multiple investments, we must use meanvariance optimization (which TV doesn't do, another good argument to avoid TV).danielc wrote: ↑Tue Mar 10, 2020 2:57 amt
What? No. How? We must be talking past each other. How could it ever make sense to set your stock allocation to C/σ where C is a constant if you thought that σ is constant too? The whole point of TV is that stock volatility varies in time, and you are supposed to reduce your stock allocation when stocks are volatile.
One way to make your statement true is to say that "volatility" refers to the volatility of the entire portfolio rather than just stocks, and then inject the assumption that your only investments are stocks and cash. Nobody in this forum (as far as I can tell) who uses TV chooses cash as their nonstock asset.
TV uses a stock allocation that is proportion to 1/σ. In my calculations I assumed C/σ, where C must be a constant. If we assume CRRA or CARA, we find C = (1/γ) * r/σ, the only way C can be a constant is if r/σ is a constant, which implies that the sharpe ratio is a constant. Which implies that return is linear with σ. Nowhere I'm making the assumption that σ is constant.
I prefer to infer facts from math, hard data or arguments, not by listening to the suggestions of others. If people are assuming that return is unpredictable, perhaps they shouldn't be using use a strategy that makes hidden assumptions about return.But you already know that that assumption is counterfactual. Whether your "TV" portfolio is UPRO+TMF or plain old stocks+cash, nobody that I know has suggested that returns are proportional to any kind of volatility. In fact, the only assertion that I have seen about returns is that they are unpredictable. So the most accurate way to describe what people are assuming is that the expected return is constant.Uncorrelated wrote: ↑Mon Mar 09, 2020 4:49 am Making the assumption that TV implicitly assumes that return is proportional to volatility is the only way I can get the math to be consistent.
Here is a very simple example that shows that TV is irrational under the assumption that return is unrelated to volatility, without making any assumptions on the utility function. Assume that the equity risk premium is 6% annually, normally distributed, target volatility is 10%. You happen to know that the average volatility is around 50%.
In the first period, the forecasted volatility is 100% and therefore the stock allocation is 10%, expected return 0.6%.
In the next period, the forecasted volatiltiy is 15%, the stock allocation is 2/3 and expected return is 4%.
Don't you think it makes much more sense to not invest anything in the first period, and invest more than 2/3 in the second period? This would result in more return for less risk.
The problem is that target volatility is wrong. If you believe that return is unrelated to risk, then one should use target variance and not target volatility. With target variance, the allocation to the risky asset is proportional to C/σ^{2}, then C = r/γ (a constant), which is consistent with the assumption of a constant or completely unpredictable risk premium.=> C = r / ( σ γ )
So either C is not constant, or γ is not constant, or CCRA is not the right utility, or something else. I believe you when you say that you have not found any way to make the math consistent other than assuming that returns are proportional to volatility. But that is an argument from lack of imagination. As far as I know, nobody thinks that volatility forecasts excess stock returns. The problem must be elsewhere.
As far as I'm aware of, the optimal allocation to the risky asset is some form of C/σ^{2} for all utility functions mentioned in https://en.wikipedia.org/wiki/Risk_aversion. If you believe that CRRA is not the right utility function, I would be interested to know which utility function results in an asset allocation proportional to 1/σ.

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Re: Risk tolerance and asset allocation with mathematics
There are some mathematicians on Bogleheads with a deeper understanding of mathematics than the authors of some peerreviewed mathematical finance papers.Uncorrelated wrote: Hello Boglehads. I think the mathematical knowledge here is disappointing, so I'm writing a short series on mathematical fundamentals for asset allocation.
In the 33 years I’ve been following capital markets as an observer and as an investor, the investing landscape has been littered with the corpses of failed mathematical models. An astute mathematician doing postmortem analyses would note a commonality: the incorrectness or failure of one or more assumptions needed to complete the model and/or make the math tractable.
Merton has noted that the time series of future returns is not a stationary process. When viewed as a Markov process, the distribution of future returns is both unknown and changing. Analyzing riskadjusted returns quantitatively is certainly useful, but mathematical sophistication and rigor should never be construed as compensating for simplifying assumptions or assumptions about that which is unknown or poorly understood.
I’m reminded of the failure in the design of the original mirror of the Hubble telescope, where engineers were eager to use very sophisticated, cutting edge mirror production techniques that could produce a mirror that conformed to its specification by a much tighter tolerance. But they were so distracted by that possibility, that insufficient due diligence was paid to the calculation of the specification to which the mirror should conform, and a simple arithmetic error was made. This resulted in the use of a very sophisticated process to produce a mirror that was two orders of magnitude more precise than necessary in conforming to the incorrect specification.
A rigorous mathematical model of riskadjusted return and the utility that is needed and delivered certainly informs the process of choosing an asset allocation, may be used to derive an allocation, and may be useful as a sanity check on one’s asset allocation. I nonetheless have not seen any evidence that it will be more accurate (in whatever measure one prefers) than one derived using basic calculations combined with qualitative reasoning. And there are technical reasons why this is very difficult at best to test.
Risk is not a guarantor of return.