## Risk tolerance and asset allocation with mathematics

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Uncorrelated
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### Risk tolerance and asset allocation with mathematics

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.

This has many different applications. It can be used to determine the correct mixture of stocks and bonds for your personal risk tolerance, and to calculate the optimal glidepath for accumulation. It can also be used to calculate the correct proportion of stocks and bonds in a market timing strategy where the return assumptions change continuously.

In this post I will be writing about the CRRA utility, some math details, the relation between risk tolerance and the kelly criterion, assumptions of target volatility, multi-asset optimization, effect of time horizon on asset allocation, and uninformed market timing.

Introduction to utility functions

A fundamental concept in asset allocation is that of risk aversion. Risk aversion models the behavior of humans with respect to uncertainty. Most investors are risk averse, that means that they would rather accept a certain payment instead of taking a gamble with the same expected return. Your risk aversion is encoded mathematically in what's called an utility function. If you lose money, you're unhappy and the utility function responds by outputting a small value. If you make money, you're supposed to be happy and the utility function gives a large value. Most investors are risk averse, that means that losing \$10 hurts more than twice as much as losing \$5.

The goal of every investor is to maximize their utility function. Obviously it's extremely difficult to determine your exact utility function, but there are a few functions that are common in literature. One of those is the constant relative risk aversion (CRRA). Under this form of risk aversion, your decisions depend only on the bet size relative to your total net worth, not on the absolute dollar amount. That means that a billionaire that loses 5% of his net worth is equally unhappy as a middle class worker that loses 5% of his net worth, if both have the same risk aversion.

Another form of risk aversion is constant absolute risk aversion (CARA). This is the opposite of CRRA, the decision depends only on absolute dollar amounts. I will not be discussing this type of risk aversion further.

Math stuff

The CRRA utility is shown below, c stands for consumption.

Throughout the remainder of this post I will use the symbol γ (gamma) to indicate the coefficient of relative risk aversion. A γ=0 indicates a risk neutral investor. The value of the parameter γ depends on your personal risk tolerance.

This utility function is quite abstract. But if we make the assumption that returns are normally distributed, we fund two extremely useful and simple results. The first result is the expression for the investor's utility, which can be expressed as:

U(X) = E(X) - 1/2 * γ * Var(X)

X indicates the random variable for your investment. E indicates the expectancy, or the average return (not the geometric return) of the investment. Var indicates the variance (standard deviation squared). The investor's utility decreases linearly with the variance (not the standard deviation).

The second result is the expression we get when maximizing the investor's utility. The following expression gives the optimal proportion allocated to the risky asset (stocks).

proportion of risky asset = (1 / γ) * E(X - Rf) / Var(X - Rf)

Rf indicates the risk free rate. It is important to realize that the optimal proportion does not depend on the absolute return, but on the risk premium compared to a risk free benchmark.

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 tolerant than the average investor, and a lot less likely to panic sell, my risk aversion is between 2 and 3.

Certainty equivalent return

Also related to the concept of utility is certainty equivalent return (CER). The CER is the inverse of the utility function. Conveniently, the inverse utility of a normal random variable simplifies to U-1(u) = N(u, 0) = an investment with a constant return of u

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% + risk free rate, both investments are equally good.

The certainty equivalent return is extremely useful for comparing investment decisions. If we have two strategies A and B, and A has a higher return and more risk than B, we don't have enough information to decide which is better. The better investment depends on the personal utility function of the investor. But if we say that A has a higher certainty equivalent return than B for an investor with γ=3, we can immediately conclude that it is better.

Arithmetic, geometric returns, kelly criterion, and sharpe ratio

How does utility relate to commonly used metrics, such as geometric returns and sharpe ratio?

If we use γ=0, we find that the investor's expected utility is simply E(X), the investment's expected return.

If we use γ=1, we find that the investor's expected utility is E(X) - 1/2 * Var(X), which is equal to the geometric return. (under the assumption of normally distributed returns).

How about the kelly criterion? The kelly criterion is an optimization method that attempts to maximize the geometric growth rate. This is equivalent to maximizing the log wealth. Which is equivalent to maximizing the CRRA utility function with γ=1. We can say that the kelly criterion is useful if and only if γ=1. Have you ever wondered why you should use the kelly criterion? Now you know.

If you're a frequent visitor on this forum, you may have heard that there is no reason to go past the kelly criterion. But that is false. For investors that have γ<1 (very risk tolerant), it is rational to take more risk than the kelly criterion.

What about the sharpe ratio? The max sharpe ratio portfolio does not correspond with any specific γ. Therefore, for the investor with a CRRA utility, optimizing for the maximum sharpe ratio is completely pointless.

The assumptions of target volatility strategies

Target volatility is a strategy that attempts to forecast future volatility. The strategy then adjusts the asset allocation such that the volatility is constant throughout time. If we have an investor with a CRRA, which assumptions do we need to make to rationalize this strategy? We denote the target volatility investor's allocation to stocks with the following formula:

C / σ

Where C is a constant related to the investor's personal risk tolerance. The question we are interested in is: when does the target volatility investor agree with an investor with CCRA utility? Recall that the CRRA investor's optimal stock allocation is:

(1 / γ) * r / σ2

With r being the return, and σ2 the variance. We assume that return is linear to volatility, r = D * σ, with D being a constant greater than zero. We now have:

= (1 / γ) * (D * σ) / σ2
= (1 / γ) * D / σ

Therefore C = (1 / γ) * D, with D being equal to r / σ, which is the sharpe ratio (a constant).

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. This is at odds with the research literature, return is not at all or weakly negatively correlated with volatility. for example see this blog.

Mean-variance optimization.

The equations above are capable of finding the optimal allocation to a single asset. It is possible to use mean-variance optimization to find the optimal proportion of multiple assets. Two such examples are shown below (UPRO is a 3x leveraged S&P500 fund, and TMF is a 3x leveraged 20 year treasury fund):

The top panel shows the efficient frontier and utility. The bottom panel shows the proportion of different funds.

I advice against using these specific images to determine your asset allocation, as these asset allocations have not been peer reviewed. Nevertheless, this should give some indication how your asset allocation changes as your risk tolerance increases or decreases. It also shows that specific proportions of funds are only optimal for one specific risk tolerance. This is especially concerning for portfolio's such as Dalio's all-seasons portfolio, or HEDGEFUNDIE's 55/45 strategy. I always find myself wondering: for which investor, which assumptions and which risk tolerance is this strategy appropriate? It turns out nobody knows.

A limitation of mean variance optimization is that returns must be normally distributed and independent throughout time (no mean reversion). It is possible to use other optimization techniques to calculate the optimal allocations for non-normally distributed asset returns, but this is a significantly more complicated. Usually, a normal distribution results in an asset allocation that is good enough. See also https://www.aacalc.com/docs/ef_non_normal.

By tweaking the inputs to mean variance optimization, it's possible to create many different asset allocations strategies. For example, if you assume all assets have the same return and no correlation, it becomes equivalent with risk parity. The advantage of this method is that it forces you to make your assumptions explicit. Rather than running a backtest with some specific portfolio (e.g. all-seasons) we can say: all-seasons is optimal under assumptions X Y and Z. This (potentially) results in much more robust asset allocations because the individual assumptions can be tested. My favorite example is that the statistical significance of the value factor is greater than that of gold. Therefore, it would appear to be irrational to have a portfolio with gold that does not contains the value factor. See also viewtopic.php?f=10&t=277661 for some inspiration for different assumptions.

The effect of time horizon on investment strategy.

What happens if we try to optimize the utility over time, instead of on a fixed time horizon? This problem is known as Merton's portfolio problem. This is essentially a continuous time version of modern portfolio theory. Conveniently, if consumption is optimally chosen the asset allocation does not depend on the time horizon!

Various improvements to Merton's portfolio problem have been proposed. If human capital is taken into account, then the strategy turns into lifecycle investing. In the case of lifecycle investing, a constant amount of lifetime wealth is invested into stocks instead of a proportion of current wealth. If a mean reverting market is considered, or stochastic lifespan is taken into account, the solution also takes slightly different forms. Nevertheless, a simple application with lifecycle investing is good enough for most investors.

The utility loss of uninformed market timing.

The optimal proportion of stocks for an investor with γ=4 is approximately 50%. This investor has a CER (certainty equivalent return) of 1.22%.

Now suppose an investor that is market timing. 50% of the time, the asset allocation is 100% stocks, and 50% of the time the asset allocation is 100% bonds. We assume that the equity risk premium is constant at 5%. You would think that the utility of the market timing investor is similar to that of the buy and hold investor because the average allocation is the same. The CER for the market timing investor is:

= P(asset allocation stocks) * U(100% stocks) + P(asset allocation bonds) * U(100% bonds)
= .5 * U(100% stocks) + .5 * U(100% bonds)
= .5 * -0.12% + .5 * 0
= -.06%

Which is worse than just investing everything in a savings account. If the stocks outperform by 2.56% in periods where the market timer holds stocks, then both strategies are equivalent, that's quite a high bar to reach. This shows that you should never try to time the market by going 100% in and out of stocks.

We can also reverse-engineer the market timer's decisions. If the market timer with γ=4 is in bonds, he must believe that the equity risk premium is zero or negative. And if the market timer is in 100% stocks, he must believe that the equity risk premium is 10% or greater.

Can the market timer do better? Instead of going 100% in and out of stocks, suppose that the market timer estimates that the equity risk premium is 4% in bad times and 6% in good time. This corresponds to an asset allocation of approximately 40% stocks and 60% stocks, respectively. If the market timer is completely wrong and the underlying equity risk premium is actually a constant 5%, the utility is:

= P(erp forecast = 4%) * U(40% stocks) + P(erp forecast = 6%) * U(60% stocks)
= .5 * U(40% stocks) + .5 * U(60% stocks)
= .5 * 1.18% + .5 * 1.16%
= 1.17%

Which is very close to the buy & hold investor's CER of 1.22%. Even if the market timer is completely wrong, the damage is limited. If the marker timer forecasts the equity risk premium perfectly, the CER increases to 1.27%.
Last edited by Uncorrelated on Sat Mar 07, 2020 1:30 pm, edited 3 times in total.
Thesaints
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### Re: Risk tolerance and asset allocation with mathematics

Uncorrelated wrote: Wed Mar 04, 2020 5:28 pm But if we make the assumption that returns are normally distributed,
Quick! Erase the post before Taleb reads it !
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### Re: Risk tolerance and asset allocation with mathematics

Hope this made you feel less disappointed.
"I think it's much more interesting to live not knowing than to have answers which might be wrong." - Richard Feynman
NotTooDeepLearning
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### Re: Risk tolerance and asset allocation with mathematics

Great post!
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### Re: Risk tolerance and asset allocation with mathematics

Should be a Wiki article.
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.
Thesaints
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### Re: Risk tolerance and asset allocation with mathematics

Fallible
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### Re: Risk tolerance and asset allocation with mathematics

Uncorrelated wrote: Wed Mar 04, 2020 5:28 pm Hello Boglehads. I think the mathematical knowledge here is disappointing...
Whoa, I am not a math person, but from what I have seen on this forum for many years, the math knowledge is excellent, even outstanding. Do you mean it's not being applied correctly?
"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
danielc
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### Re: Risk tolerance and asset allocation with mathematics

Uncorrelated wrote: Wed Mar 04, 2020 5:28 pm The goal of every investor is to maximize their utility function. Obviously it's extremely difficult to determine your exact utility function, but there are a few functions that are common in literature. One of those is the constant relative risk aversion (CRRA). Under this form of risk aversion, your decisions depend only on the bet size relative to your total net worth, not on the absolute dollar amount. That means that a billionaire that loses 5% of his net worth is equally unhappy as a middle class worker that loses 5% of his net worth, if both have the same risk aversion.
That seems completely disconnected from the way I perceive risk. If I only have enough money to make ends meet, a loss of 5% means that either I don't pay the utility bill, or I don't pay rent. If I have a billion dollars, and I lose 5%, I still have more money than I could ever spend.
Thesaints
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### Re: Risk tolerance and asset allocation with mathematics

"if both have the same risk aversion" ...and ignore the marginal utility of money.
typical.investor
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### Re: Risk tolerance and asset allocation with mathematics

Uncorrelated wrote: Wed Mar 04, 2020 5:28 pm But if we make the assumption that returns are normally distributed, we fund two extremely useful and simple results. The first result is the expression for the investor's utility ...

The second result is the expression we get when maximizing the investor's utility.
And what if we more realistically assume that returns will have fat tails?

Fama did half his Phd dissertation on the topic and states:
Distributions of daily and monthly stock returns are rather symmetric about their means, but the tails are fatter (i.e., there are more outliers) than would be expected with normal distributions...The message for investors is: expect extreme returns, negative as well as positive.
https://famafrench.dimensional.com/ques ... buted.aspx

To put it in further context, Research Affiliates examined actual factor returns and compared them to what would be expected in a normal distribution. They found:
The worst month for 11 of the 14 factors should have occurred less than once in the past 2,000 years. The worst month for 9 of the factors should have occurred less than once during the time modern humans have roamed the earth, and the worst month for 3 of the factors should have occurred less than once since the universe was created! So much for a normal distribution!
The 14 factors they examined were firstly value, size, operating profitability, investment, momentum, and low beta.
The second group includes another 8 popular factors: idiosyncratic volatility, short-term reversals, illiquidity, accruals, cash flow to price, earnings to price, long-term reversals, and net share issues.
Thesaints
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### Re: Risk tolerance and asset allocation with mathematics

Everybody knows the distribution is a power law and then some.
MarkRoulo
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### Re: Risk tolerance and asset allocation with mathematics

Fallible wrote: Wed Mar 04, 2020 7:14 pm
Uncorrelated wrote: Wed Mar 04, 2020 5:28 pm Hello Boglehads. I think the mathematical knowledge here is disappointing...
Whoa, I am not a math person, but from what I have seen on this forum for many years, the math knowledge is excellent, even outstanding. Do you mean it's not being applied correctly?
The math is often applied incorrectly.

Two quick examples:
• We like to talk about the 4% rule (even those who are quite clear that it is NOT a draw down strategy) and the historical data that went into the conclusion. But ... we have only THREE non-overlapping (and thus *maybe* independent) periods of data from 1926 until today. The periods *ARE* probably correlated, so we likely have less than three independent pieces or data.

So ... "in the last 90 years 4% has been sufficiently cautious to succeed" is the about the same as "with three coin flips we've come up heads each time."

People *want* more certainty, but it isn't clear that that certainty is available.

Extending the data set to ex-USA helps, but only a little bit (foreign markets do have a correlation with the US), and ... doing so means that the 4% rule doesn't work anymore, either.
• We (and lots of finance professionals and academics) like to calculate the standard deviation of returns.

The *samples* of returns have a standard deviation (by definition). It is *FAR* from clear that the underlying distribution does.

There are known distributions that don't have a standard deviation, so this isn't entirely theoretical. And the stock market has "fat tails" enough that there is a lot of empirical evidence that the returns aren't normal/gaussian, either. Nevertheless, we continue on with Sharpe ratios and such ... partially because we don't have anything better. But not having a better model doesn't make the clearly broken one any better. Again, we just may not be able to calculate the risk because there it too much uncertainty in the real world.
I don't have any obvious "fix" other than the non-quantified suggestion to expect less certainty than one is comfortable with.
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### Re: Risk tolerance and asset allocation with mathematics

Uncorrelated wrote: Wed Mar 04, 2020 5:28 pm Hello Boglehads.
Dude awesome post. Made a ton of sense, clean math and good explanations. You keep these coming please!
Uncorrelated wrote: Wed Mar 04, 2020 5:28 pm
Various improvements to Merton's portfolio problem have been proposed. If human capital is taken into account, then the strategy turns into lifecycle investing. In the case of lifecycle investing, a constant amount of lifetime wealth is invested into stocks instead of a proportion of current wealth. If a mean reverting market is considered, or stochastic lifespan is taken into account, the solution also takes slightly different forms. Nevertheless, a simple application with lifecycle investing is good enough for most investors.
Ah thanks for the shout-out. Here's to hoping more BHs try to utilize some of these concepts
"... 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
Thesaints
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### Re: Risk tolerance and asset allocation with mathematics

MarkRoulo wrote: Wed Mar 04, 2020 8:41 pm The math is often applied incorrectly.

Two quick examples:
• We like to talk about the 4% rule (even those who are quite clear that it is NOT a draw down strategy) and the historical data that went into the conclusion. But ... we have only THREE non-overlapping (and thus *maybe* independent) periods of data from 1926 until today. The periods *ARE* probably correlated, so we likely have less than three independent pieces or data.

So ... "in the last 90 years 4% has been sufficiently cautious to succeed" is the about the same as "with three coin flips we've come up heads each time."

People *want* more certainty, but it isn't clear that that certainty is available.

Extending the data set to ex-USA helps, but only a little bit (foreign markets do have a correlation with the US), and ... doing so means that the 4% rule doesn't work anymore, either.
• We (and lots of finance professionals and academics) like to calculate the standard deviation of returns.

The *samples* of returns have a standard deviation (by definition). It is *FAR* from clear that the underlying distribution does.

There are known distributions that don't have a standard deviation, so this isn't entirely theoretical. And the stock market has "fat tails" enough that there is a lot of empirical evidence that the returns aren't normal/gaussian, either. Nevertheless, we continue on with Sharpe ratios and such ... partially because we don't have anything better. But not having a better model doesn't make the clearly broken one any better. Again, we just may not be able to calculate the risk because there it too much uncertainty in the real world.
I don't have any obvious "fix" other than the non-quantified suggestion to expect less certainty than one is comfortable with.
Absolutely!

A first step should be trying to understand the "physics" behind the 4% rule being successful in the past, rather than blindly trusting back testing, which in this case is almost equivalent to trusting our lucky star.
Let's begin with observing that the 60/40 portfolio had an average return above 4% in all those periods and analyze what happens when it does not, as it will probably be the case in the future.
also, try to understand the effect of market volatility: the higher it is, the larger are the chances of running out of money.
Last edited by Thesaints on Wed Mar 04, 2020 9:39 pm, edited 1 time in total.
Hydromod
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### Re: Risk tolerance and asset allocation with mathematics

Very nice. I like a clean presentation like this.

One question. You said you were more risk adverse and less likely to sell than the typical investor, with an implied weight between 65 and 100 percent equities. I presume that this is consistent with an early investor? It seems like late investors around here would find that to be quite risky.
MoneyMarathon
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### Re: Risk tolerance and asset allocation with mathematics

Uncorrelated wrote: Wed Mar 04, 2020 5:28 pm I think the mathematical knowledge here is disappointing
Sir, this is a forum. If something is imprecise or even technically wrong, it's often just some combination of the person not putting in that kind of effort & not wanting to confuse people with irrelevant jargon (which is technically wrong itself, about reality... which is not a spherical cow).
rkhusky
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### Re: Risk tolerance and asset allocation with mathematics

Frankly, I don't think that standard statistics (means and std's) can be applied to stock investing except in a very broad-brush sense. Something like game theory or one-shot learning is more applicable.
beehivehave
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### Re: Risk tolerance and asset allocation with mathematics

danielc wrote: Wed Mar 04, 2020 8:04 pm
Uncorrelated wrote: Wed Mar 04, 2020 5:28 pm The goal of every investor is to maximize their utility function. Obviously it's extremely difficult to determine your exact utility function, but there are a few functions that are common in literature. One of those is the constant relative risk aversion (CRRA). Under this form of risk aversion, your decisions depend only on the bet size relative to your total net worth, not on the absolute dollar amount. That means that a billionaire that loses 5% of his net worth is equally unhappy as a middle class worker that loses 5% of his net worth, if both have the same risk aversion.
That seems completely disconnected from the way I perceive risk. If I only have enough money to make ends meet, a loss of 5% means that either I don't pay the utility bill, or I don't pay rent. If I have a billion dollars, and I lose 5%, I still have more money than I could ever spend.
Exactly right. Thanks for the post that empowers me to not have to plow through all the math.
As I understand it, the post limits the possibilities to just two choices: Absolute and constant, regardless of wealth and need.
jimbomahoney
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### Re: Risk tolerance and asset allocation with mathematics

Thanks for this Uncorrelated.

Very nice work!

PS - I've updated my BTM strategy code to use the risk-free premium instead of the absolute (arithmetic) return.

It makes a teeny tiny improvement, so thanks!
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### Re: Risk tolerance and asset allocation with mathematics

rkhusky wrote: Wed Mar 04, 2020 9:44 pm Frankly, I don't think that standard statistics (means and std's) can be applied to stock investing except in a very broad-brush sense. Something like game theory or one-shot learning is more applicable.
Behavioral economics.

Hot and cold cognition.
70% Global Market Weight Equities | 15% Long Treasuries 15% short TIPS & cash || RSU + ESPP
siamond
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### Re: Risk tolerance and asset allocation with mathematics

Hats off to the OP for the effort put in this post and the quality of the outcome. You are very knowledgeable and you are a terrific writer.

We could debate the practical application of those mathematical constructs until we're blue on the face. Personally, I have yet to see a utility function that even remotely matches anything significant in MY real life. For sure, stocks are NOT normally distributed. I think most efficient frontier charts are totally off by assuming risk is a short-term concept and unidimensional. Etc.

But this doesn't take away from this awesome post. It is one thing to describe the mechanics of a given set of mathematical constructs, it a completely different thing to debate its possible applicationS to real-life, notably in the context of very diverse personal situations and personalities.

It's like valuations and constructs like P/E and CAPE. It is VERY useful to define them in a precise manner (e.g. via a wiki page) and it is definitely interesting to see how SOME people make use of them. It drives me nuts when people react to discussions about corresponding mechanics by getting all judgmental (if not downright integrist) about market timing, associating a construct with ONE possible application, then throwing everything by the window.

My advice to the OP is the following. Take your post (and follow-up ideas) and see what part of it is truly descriptive and non-controversial (i.e. the mechanics) and could be used to improve one wiki page or another (or to create another if you feel something is missing). Then consider contributing to the Bogleheads blog to describe possible applications of such constructs (then it's ok to be more opinionated). And keep this forum for what it is good at, a fleeting discussion expressing opinions on such constructs and/or use cases. But do make the extra effort to ensure that your excellent work is captured by more lasting vehicles.
acegolfer
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### Re: Risk tolerance and asset allocation with mathematics

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
hoops777
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### Re: Risk tolerance and asset allocation with mathematics

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
My take is that the people who are able to understand your post are smart enough to not need it
K.I.S.S........so easy to say so difficult to do.
firebirdparts
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### Re: Risk tolerance and asset allocation with mathematics

Well done, thanks! That is definitely an important topic.
A fool and your money are soon partners
Topic Author
Uncorrelated
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### Re: Risk tolerance and asset allocation with mathematics

Fallible wrote: Wed Mar 04, 2020 7:14 pm
Uncorrelated wrote: Wed Mar 04, 2020 5:28 pm Hello Boglehads. I think the mathematical knowledge here is disappointing...
Whoa, I am not a math person, but from what I have seen on this forum for many years, the math knowledge is excellent, even outstanding. Do you mean it's not being applied correctly?
Maybe that was a bit harsh. In the topics about market timing, volatility targeting, gold, international investing, it appears that the vast majority of posters has no clue what they are doing. In the case of volatility targeting, it seems I'm the only one that understands the underlying assumptions are horrendous. In the case of market timing, it seems that the overlap between market timers and people that know risk tolerance exists is zero.

Of course there are a lot of people on bogleheads that do know what they are doing. I've certainly learned a lot from them.

danielc wrote: Wed Mar 04, 2020 8:04 pm
Uncorrelated wrote: Wed Mar 04, 2020 5:28 pm The goal of every investor is to maximize their utility function. Obviously it's extremely difficult to determine your exact utility function, but there are a few functions that are common in literature. One of those is the constant relative risk aversion (CRRA). Under this form of risk aversion, your decisions depend only on the bet size relative to your total net worth, not on the absolute dollar amount. That means that a billionaire that loses 5% of his net worth is equally unhappy as a middle class worker that loses 5% of his net worth, if both have the same risk aversion.
That seems completely disconnected from the way I perceive risk. If I only have enough money to make ends meet, a loss of 5% means that either I don't pay the utility bill, or I don't pay rent. If I have a billion dollars, and I lose 5%, I still have more money than I could ever spend.

I can see that, yeah. There are two ways to resolve this problem, we can say that the middle class worker and the billionaire are very unlikely to have the same risk tolerance. Some research indicates that risk tolerance increases with wealth.

The second possible approach is to use an utility function where the risk aversion changes based on your total net wealth, such as HARA (hyperbolic absolute risk aversion). But this makes parameter estimation significantly more difficult.

Thing is, if you make a very small amount of assumptions about your utility function (more wealth is always better than less wealth. Marginal utility of wealth), there are only a few sensible utility functions.
Thesaints wrote: Wed Mar 04, 2020 8:07 pm "if both have the same risk aversion" ...and ignore the marginal utility of money.
A CRRA utility includes marginal utility of wealth. If the middle class worker gains \$1m the utility function increases by a very large amount, but the billionaire's utility function barely responds.

A CRRA has the property that the difference in utility between relative wealth changes is constant. That means that questions such as: "would you be willing to bet X% of your net worth in exchange for Y% gain?" are independent of the net wealth.
Topic Author
Uncorrelated
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### Re: Risk tolerance and asset allocation with mathematics

typical.investor wrote: Wed Mar 04, 2020 8:17 pm
Uncorrelated wrote: Wed Mar 04, 2020 5:28 pm But if we make the assumption that returns are normally distributed, we fund two extremely useful and simple results. The first result is the expression for the investor's utility ...

The second result is the expression we get when maximizing the investor's utility.
And what if we more realistically assume that returns will have fat tails?

Fama did half his Phd dissertation on the topic and states:
Distributions of daily and monthly stock returns are rather symmetric about their means, but the tails are fatter (i.e., there are more outliers) than would be expected with normal distributions...The message for investors is: expect extreme returns, negative as well as positive.
https://famafrench.dimensional.com/ques ... buted.aspx

To put it in further context, Research Affiliates examined actual factor returns and compared them to what would be expected in a normal distribution. They found:
The worst month for 11 of the 14 factors should have occurred less than once in the past 2,000 years. The worst month for 9 of the factors should have occurred less than once during the time modern humans have roamed the earth, and the worst month for 3 of the factors should have occurred less than once since the universe was created! So much for a normal distribution!
The 14 factors they examined were firstly value, size, operating profitability, investment, momentum, and low beta.
The second group includes another 8 popular factors: idiosyncratic volatility, short-term reversals, illiquidity, accruals, cash flow to price, earnings to price, long-term reversals, and net share issues.

Thanks for the insight. Gordon Irlam investigated the difference between the optimal retirement asset allocation for both normal and actual stock returns, and found a CER difference of ~ 0.1%. I think he used annual returns for stocks, bonds and SCV. https://www.aacalc.com/docs/ef_non_normal

For daily returns, a normal distribution is definitely inadequate because daily stock returns are not normal. For monthly and annual returns, I'm less convinced that it matters. The real question isn't whether returns are normally distributed or not, but if a normal distribution is an acceptable compromise between accuracy and simplicity. This should be judged on a case-by-case basis.

It's worth noting that the CRRA itself doesn't depend on any particular distribution. you could calculate the utility by running a backtest, chopping up the backtest in 1-year blocks and then calculating the average utility over all blocks. But that is significantly more complex and error-prone than the normal distribution assumption.
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### Re: Risk tolerance and asset allocation with mathematics

danielc wrote: Wed Mar 04, 2020 8:04 pm
Uncorrelated wrote: Wed Mar 04, 2020 5:28 pm The goal of every investor is to maximize their utility function. Obviously it's extremely difficult to determine your exact utility function, but there are a few functions that are common in literature. One of those is the constant relative risk aversion (CRRA). Under this form of risk aversion, your decisions depend only on the bet size relative to your total net worth, not on the absolute dollar amount. That means that a billionaire that loses 5% of his net worth is equally unhappy as a middle class worker that loses 5% of his net worth, if both have the same risk aversion.
That seems completely disconnected from the way I perceive risk. If I only have enough money to make ends meet, a loss of 5% means that either I don't pay the utility bill, or I don't pay rent. If I have a billion dollars, and I lose 5%, I still have more money than I could ever spend.
Well but risk aversion isn’t just about how much you’re willing to lose. It’s about the trade off of loss vs gain. It’s true that if you can barely make ends meet, a 5% loss will be more painful than for a billionaire, but a +20% gain will also be far more useful to you than it will be for a billionaire.
"... 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

305pelusa wrote: Thu Mar 05, 2020 3:26 pm
danielc wrote: Wed Mar 04, 2020 8:04 pm That seems completely disconnected from the way I perceive risk. If I only have enough money to make ends meet, a loss of 5% means that either I don't pay the utility bill, or I don't pay rent. If I have a billion dollars, and I lose 5%, I still have more money than I could ever spend.
Well but risk aversion isn’t just about how much you’re willing to lose. It’s about the trade off of loss vs gain. It’s true that if you can barely make ends meet, a 5% loss will be more painful than for a billionaire, but a +20% gain will also be far more useful to you than it will be for a billionaire.
Although it is true that a +20% gain will be more useful to me if I am poor, that would not nearly offset the fact that a 5% loss is unacceptable. The "constant trade off" model definitely does not match my risk perception. I suspect that I would be most willing to accept a -5%/+20% loss/gain trade off if I have a "middle of the road" wealth, in which a 5% loss won't ruin me but a 20% gain would usefully impact my life. So... if we measure risk in terms of the loss/gain tradeoff you are willing to take, then perhaps a plot of my risk perception might look like a bump (e.g. like an inverted parabola or similar).

In any case, I am very skeptical of drawing conclusions based on a model of risk and a model of returns that we KNOW does not match reality.
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### Re: Risk tolerance and asset allocation with mathematics

Uncorrelated wrote: Thu Mar 05, 2020 2:52 pm
danielc wrote: Wed Mar 04, 2020 8:04 pm
Uncorrelated wrote: Wed Mar 04, 2020 5:28 pm The goal of every investor is to maximize their utility function. Obviously it's extremely difficult to determine your exact utility function, but there are a few functions that are common in literature. One of those is the constant relative risk aversion (CRRA). Under this form of risk aversion, your decisions depend only on the bet size relative to your total net worth, not on the absolute dollar amount. That means that a billionaire that loses 5% of his net worth is equally unhappy as a middle class worker that loses 5% of his net worth, if both have the same risk aversion.
That seems completely disconnected from the way I perceive risk. If I only have enough money to make ends meet, a loss of 5% means that either I don't pay the utility bill, or I don't pay rent. If I have a billion dollars, and I lose 5%, I still have more money than I could ever spend.

I can see that, yeah. There are two ways to resolve this problem, we can say that the middle class worker and the billionaire are very unlikely to have the same risk tolerance. Some research indicates that risk tolerance increases with wealth.

The second possible approach is to use an utility function where the risk aversion changes based on your total net wealth, such as HARA (hyperbolic absolute risk aversion). But this makes parameter estimation significantly more difficult.

Thing is, if you make a very small amount of assumptions about your utility function (more wealth is always better than less wealth. Marginal utility of wealth), there are only a few sensible utility functions.

Yeah. So that's the thing. How am I supposed to take all the math in your post and translate it into an investment strategy that makes sense for me? The model of risk, the model of return, and the utility function in your example are all known to be wrong, and some are possibly unknowable. What am I to do then? As far as I can tell, for all the severe problems with choosing an asset allocation based on a fuzzy rule of thumb, it seems to me like that's all we have.
azanon
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### Re: Risk tolerance and asset allocation with mathematics

Uncorrelated wrote: Wed Mar 04, 2020 5:28 pmI advice against using these specific images to determine your asset allocation, as these asset allocations have not been peer reviewed. Nevertheless, this should give some indication how your asset allocation changes as your risk tolerance increases or decreases. It also shows that specific proportions of funds are only optimal for one specific risk tolerance. This is especially concerning for portfolio's such as Dalio's all-seasons portfolio, or HEDGEFUNDIE's 55/45 strategy. I always find myself wondering: for which investor, which assumptions and which risk tolerance is this strategy appropriate? It turns out nobody knows.
Speaking specifically for the Dalio All-weather (DAW) portfolio (Bridgewater version), the strategy (at least in theory) is for someone that wants to be emotionally indifferent to any investing climate. So you're not going to be able to put that particular portfolio in this mathematical box that you'd like to put it in here, such as it falling somewhere on a risk tolerance scale.

In layman's terms, DAW is for, if you want to completely opt out of trying to determine what your risk tolerance is, since the entire purpose of the portfolio design is only considering the following facts: 1. Consideration of investments that have a higher return than cash and 2. Trying to determine the approximate percentages that each viable asset class should be so that have so that the portfolio's estimated risk in 4 theoretical economic scenarios that we could be in at any given moment, is approximated to be the same. Now does it accomplish that objective 100% successfully? Probably not. But that's what it's attempting to do.

So most portfolios have bias to a certain economic climate, with that bias usually being to times of economic prosperity. At least in theory, the DAW portfolio holder could not care less what is happening at any given moment, such as Coronavirus now.

In short, it's not applicable to your work here. Hope that helps.*

* I acknowledged that the Tony Robbins All seasons portfolio is flawed, and went into great detail about that here: viewtopic.php?t=206028 , so yeah maybe the Robbins (non-risk parity) portfolio can be evaluated here. But variants of the Bridgewater All-Weather portfolio are designed/attempting to do what I discussed above, so therefore looking at them from the lens of "risk tolerance" is completely missing the point and primary objective of the portfolio.

> I'll add that the same comments above also apply to the Permanent Portfolio (PP), for all intents and purposes. As with the Bridgewater All Weather portfolio, one is correctly selecting the PP if they are trying to minimize bias to any particular investing climate. It is merely an observation, that the portfolio exhibits low volatility and, therefore, might be even more attractive to someone with a low risk tolerance. In other words, you could have a high risk tolerance, yet still rationalize that it makes more sense to select a PP despite your risk tolerance. Just because you can tolerate losing a lot of money in a short period of time, doesn't automatically make doing so a good idea.
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### Re: Risk tolerance and asset allocation with mathematics

danielc wrote: Thu Mar 05, 2020 4:10 pm
305pelusa wrote: Thu Mar 05, 2020 3:26 pm
danielc wrote: Wed Mar 04, 2020 8:04 pm That seems completely disconnected from the way I perceive risk. If I only have enough money to make ends meet, a loss of 5% means that either I don't pay the utility bill, or I don't pay rent. If I have a billion dollars, and I lose 5%, I still have more money than I could ever spend.
Well but risk aversion isn’t just about how much you’re willing to lose. It’s about the trade off of loss vs gain. It’s true that if you can barely make ends meet, a 5% loss will be more painful than for a billionaire, but a +20% gain will also be far more useful to you than it will be for a billionaire.
Although it is true that a +20% gain will be more useful to me if I am poor, that would not nearly offset the fact that a 5% loss is unacceptable. The "constant trade off" model definitely does not match my risk perception. I suspect that I would be most willing to accept a -5%/+20% loss/gain trade off if I have a "middle of the road" wealth, in which a 5% loss won't ruin me but a 20% gain would usefully impact my life. So... if we measure risk in terms of the loss/gain tradeoff you are willing to take, then perhaps a plot of my risk perception might look like a bump (e.g. like an inverted parabola or similar).

In any case, I am very skeptical of drawing conclusions based on a model of risk and a model of returns that we KNOW does not match reality.
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. If you choose the latter, it doesn't mean that you don't have an utility function. It just means that your utility function is stuck inside your head, it might be subject to behavioral biases, misconceptions, recency bias, survivorship bias, anchoring.

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.

Recently there was a discussion about lifecycle investing, I claimed that your future income is similar to bonds, and therefore the asset allocation should be adjusted to take this into account. A boglehead rejected the entire idea, saying that future income is too uncertain and that he wasn't willing to make assumptions based on future income. My opinion: saying that there is no future income is also an explicit modeling assumption. Having a best-effort estimate on future income results in a better asset allocation than having none at all.
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### Re: Risk tolerance and asset allocation with mathematics

First off, just want to echo everyone and say thanks for the great post!
Uncorrelated wrote: Thu Mar 05, 2020 2:52 pm
danielc wrote: Wed Mar 04, 2020 8:04 pm
Uncorrelated wrote: Wed Mar 04, 2020 5:28 pm The goal of every investor is to maximize their utility function. Obviously it's extremely difficult to determine your exact utility function, but there are a few functions that are common in literature. One of those is the constant relative risk aversion (CRRA). Under this form of risk aversion, your decisions depend only on the bet size relative to your total net worth, not on the absolute dollar amount. That means that a billionaire that loses 5% of his net worth is equally unhappy as a middle class worker that loses 5% of his net worth, if both have the same risk aversion.
That seems completely disconnected from the way I perceive risk. If I only have enough money to make ends meet, a loss of 5% means that either I don't pay the utility bill, or I don't pay rent. If I have a billion dollars, and I lose 5%, I still have more money than I could ever spend.
I can see that, yeah. There are two ways to resolve this problem, we can say that the middle class worker and the billionaire are very unlikely to have the same risk tolerance. Some research indicates that risk tolerance increases with wealth.

The second possible approach is to use an utility function where the risk aversion changes based on your total net wealth, such as HARA (hyperbolic absolute risk aversion). But this makes parameter estimation significantly more difficult.
To baseline - I always assumed we were referring to discretionary, or "excess income" - i.e. take out cost of living + budgeted entertainment, etc. - and the excess each time period is really what I care about. (This was the value I used for the lifecycle future income to calculate the present discounted value)

So a 5% drop isn't going to impact my ability to pay bills, rent, or really compromise by standard of living. Looking at it like this the CRRA model seems to be much more reasonable.

If we are investing funds that, when reduced in value during a market downturn, could impact ones ability to pay bills it seems to me like the actual problem is investing money that shouldn't be invested? - and the CRRA exemplar is no longer intuitive because we are sort of crossing the beams (investing or obligating already earmarked funds)?
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### Re: Risk tolerance and asset allocation with mathematics

danielc wrote: Thu Mar 05, 2020 4:10 pm
305pelusa wrote: Thu Mar 05, 2020 3:26 pm
danielc wrote: Wed Mar 04, 2020 8:04 pm That seems completely disconnected from the way I perceive risk. If I only have enough money to make ends meet, a loss of 5% means that either I don't pay the utility bill, or I don't pay rent. If I have a billion dollars, and I lose 5%, I still have more money than I could ever spend.
Well but risk aversion isn’t just about how much you’re willing to lose. It’s about the trade off of loss vs gain. It’s true that if you can barely make ends meet, a 5% loss will be more painful than for a billionaire, but a +20% gain will also be far more useful to you than it will be for a billionaire.
Although it is true that a +20% gain will be more useful to me if I am poor, that would not nearly offset the fact that a 5% loss is unacceptable. The "constant trade off" model definitely does not match my risk perception. I suspect that I would be most willing to accept a -5%/+20% loss/gain trade off if I have a "middle of the road" wealth, in which a 5% loss won't ruin me but a 20% gain would usefully impact my life. So... if we measure risk in terms of the loss/gain tradeoff you are willing to take, then perhaps a plot of my risk perception might look like a bump (e.g. like an inverted parabola or similar).

In any case, I am very skeptical of drawing conclusions based on a model of risk and a model of returns that we KNOW does not match reality.
I mean, then maybe you shift your RA coefficient depending on your wealth. Or just use one as an estimate and then tweak a bit based on what makes sense. This is IMO superior to not doing it at all.

I’m personally the opposite; I’m very skeptical of any allocation that doesn’t take both return and risk. Many choose their allocation to stocks based on max drawdown (“if you can tolerate X loss, have Y in stocks”). How does that make any sense? I might be very willing to tolerate some loss given a high enough expected return. Target volatility is the same way.

And don’t get me started on “Age in bonds” or other useless rules of thumbs that don’t have anything to do with risk OR return.
"... 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

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 probably wouldn't have phrased it like that but I also often find myself disappointed that so few people engage with topics I find intellectually interesting. But then I remind myself that, even for the overwhelming majority of Bogleheads, money is simply something instrumental. Instrumental in the sense of "a means for pursuing an ends". In the same way that most people aren't interested in the nitty gritty of how their car works even though they might use it for 2+ hours a day. So it is totally understandable that not everyone has the same hobbies and interests.

In any case, I agree with others that you should consider adding this to the wiki. Adding to the wiki is a bit of extra work but it would be nice to have an official Bogleheads wiki entry on some of these topics.
Last edited by AlohaJoe on Thu Mar 05, 2020 7:52 pm, edited 1 time in total.
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### Re: Risk tolerance and asset allocation with mathematics

AlohaJoe wrote: Thu Mar 05, 2020 7:16 pm
I probably wouldn't have phrased it like that but I also often find myself disappointed that so few people engage with topics I find intellectually interesting. But then I remind myself that, even for the overwhelming majority of Bogleheads, money is simply something instrumental. Instrumental in the sense of "a means for pursuing an ends". In the same way that most people aren't interested in the nitty gritty of how their car works even though they might use it for 2+ hours a day. So it is totally understandable that not every has the same hobbies and interests.

In any case, I agree with others that you should consider adding this to the wiki. Adding to the wiki is a bit of extra work but it would be nice to have an official Bogleheads wiki entry on some of these topics.
I get what you're saying.

I can be fun mental stimulation.

In reality, though, I only have so many sharp brain cycles in a day and there is an ROI tradeoff on re-arranging ways to get returns from public markets (dominated as those returns are by beta), vs spending the same mental energy on other investments, entrepreneurial activities, or my career.
70% Global Market Weight Equities | 15% Long Treasuries 15% short TIPS & cash || RSU + ESPP
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### Re: Risk tolerance and asset allocation with mathematics

watchnerd wrote: Thu Mar 05, 2020 7:24 pm In reality, though, I only have so many sharp brain cycles in a day and there is an ROI tradeoff on re-arranging ways to get returns from public markets (dominated as those returns are by beta), vs spending the same mental energy on other investments, entrepreneurial activities, or my career.
Sure, definitely. It is pretty clear that savings rate, spending rate, and market beta probably explain 80-90% of saving & retirement outcomes. Even dramatic changes to asset allocation are surprisingly pointless in an ex ante way. And you don't need to spend 10 years reading Bogleheads to get to that point. I mean, even among the posters with thousands of posts or who are really into the research or whatever, how often does it really translate it any portfolio changes? If I look back over all the portfolio changes I've made in the 20+ years since I first discovered A Random Walk Down Wall Street and William Bernstein's now-decrepit blog....how many things would I actually have done differently?

Just 2, I think.

1) Long ago Bernstein recommended splitting Europe and Asia-Pacific into separate holdings (partly because few good international funds existed back then and partly to get rebalancing bonuses) and I followed that for a number of years before selling them (easy, since Pacific had no capital gains after a decade....) and just getting a single international holding.

2) If I didn't have large embedded capital gains in my REIT holdings, I'd probably sell them off. While I don't think the recent research (like Jared Kizer's) is conclusive it raises enough doubts that I probably wouldn't get REITs if I did it all over again.
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### Re: Risk tolerance and asset allocation with mathematics

AlohaJoe wrote: Thu Mar 05, 2020 8:02 pm
2) If I didn't have large embedded capital gains in my REIT holdings, I'd probably sell them off. While I don't think the recent research (like Jared Kizer's) is conclusive it raises enough doubts that I probably wouldn't get REITs if I did it all over again.
If rates keep going down, I'm afraid I'll be in the same boat with long Treasuries.
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### Re: Risk tolerance and asset allocation with mathematics

Would have been a much better post without the unnecessary pompous intro. “I think I can bring some additional math insights to the forum, so here’s the first of a series of posts!” Et voila.
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### Re: Risk tolerance and asset allocation with mathematics

Uncorrelated wrote: Thu Mar 05, 2020 2:52 pm
Fallible wrote: Wed Mar 04, 2020 7:14 pm
Uncorrelated wrote: Wed Mar 04, 2020 5:28 pm Hello Boglehads. I think the mathematical knowledge here is disappointing...
Whoa, I am not a math person, but from what I have seen on this forum for many years, the math knowledge is excellent, even outstanding. Do you mean it's not being applied correctly?
Maybe that was a bit harsh. In the topics about market timing, volatility targeting, gold, international investing, it appears that the vast majority of posters has no clue what they are doing. In the case of volatility targeting, it seems I'm the only one that understands the underlying assumptions are horrendous. In the case of market timing, it seems that the overlap between market timers and people that know risk tolerance exists is zero.

Of course there are a lot of people on bogleheads that do know what they are doing. I've certainly learned a lot from them.
I agree there are plenty of folks on the forum who know what they're doing and I don't just mean the the pros we are lucky to have such as Ferri, Swedroe, Bernstein, etc.

Anyway, some questions:

_How are you defining risk tolerance and risk aversion? It appears you are using them interchangeably, but I may be misreading.

_In on of your posts you said this below about modeling errors (boldface mine):

"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. If you choose the latter, it doesn't mean that you don't have an utility function. It just means that your utility function is stuck inside your head, it might be subject to behavioral biases, misconceptions, recency bias, survivorship bias, anchoring."

_EDIT to update questions (and better reflect my confusion over the above): How would the unused utility function be subject to behavioral biases, misconceptions, etc.? Can intuition, in particular "expert intuition," ever be excluded from our thinking and would we even want it to be?
Last edited by Fallible on Fri Mar 06, 2020 10:20 am, edited 1 time in total.
"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
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Uncorrelated
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### Re: Risk tolerance and asset allocation with mathematics

Hydromod wrote: Wed Mar 04, 2020 9:21 pm Very nice. I like a clean presentation like this.

One question. You said you were more risk adverse and less likely to sell than the typical investor, with an implied weight between 65 and 100 percent equities. I presume that this is consistent with an early investor? It seems like late investors around here would find that to be quite risky.
My current asset allocation is around 90% equities, I plan to hold this asset allocation for a very, very long time. Possibly until the day I die. You're right that this makes me significantly more risk tolerant than most investors. I try to avoid talking about my personal situation because my circumstances are quite unique, but glidepaths (decreasing risk over time) don't make any sense for my personal circumstances.

siamond wrote: Thu Mar 05, 2020 11:31 am Hats off to the OP for the effort put in this post and the quality of the outcome. You are very knowledgeable and you are a terrific writer.

We could debate the practical application of those mathematical constructs until we're blue on the face. Personally, I have yet to see a utility function that even remotely matches anything significant in MY real life. For sure, stocks are NOT normally distributed. I think most efficient frontier charts are totally off by assuming risk is a short-term concept and unidimensional. Etc.

...
Certainly. The most important thing is to have a model, be aware what your assumptions are, and try to make a decision in a way that minimizes the chance of behavioral issues. The methods I described here are only a small selection of possible methods.

Why do you think that efficient frontier charts are totally off by assuming that risk is a short term concept and is unidimensional? Unless you take an active position for predicting stock returns (time-varying return assumptions), the efficient frontier does not seem to depend on the time horizon or just barely. If risk is multi dimensional and can't be measured by a probability distribution alone, then what are those other risks? I recognize that the normal distribution has issues, but I seem to have have difficulty rationalizing the other dimensions of risk.
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### Re: Risk tolerance and asset allocation with mathematics

Uncorrelated wrote: Fri Mar 06, 2020 7:53 amWhy do you think that efficient frontier charts are totally off by assuming that risk is a short term concept and is unidimensional? Unless you take an active position for predicting stock returns (time-varying return assumptions), the efficient frontier does not seem to depend on the time horizon or just barely. If risk is multi dimensional and can't be measured by a probability distribution alone, then what are those other risks? I recognize that the normal distribution has issues, but I seem to have have difficulty rationalizing the other dimensions of risk.
Let me suggest that you take a good look at the 'Deep Risk' booklet from W. Bernstein (easy to find on Amazon for a few bucks). This was instrumental to help me move away from the standard-deviation, Sharpe/Sortino ratios and all this non-sense to longer-term risk considerations which are much more the reality of investors, whether they are accumulators or retirees. It also makes the key consideration that risk(s) & return are NOT orthogonal considerations (the two-dimensional fabric of an efficient frontier chart!) when thinking long-term, as risk and -cumulative- return become kind of the same thing by then. The booklet isn't perfect, there is probably too much emphasis on things beyond our control and not enough emphasis on sustained purchasing power over the long-term, but still, this is an eye-opening read allowing to move away from financial theory to real-life thinking. I also wrote a blog article a little while ago explaining my practical perspective on risk & reward for retirees (I am an early retiree with an hopefully long time horizon, I've been reflecting on such topic for a while now).

But again, any financial theory is good to document (e.g. in a wiki page) and any use case/application is good to discuss (e.g. blog article, forum post, etc), irrespective of individual opinions. This was really my main feedback to you, you clearly like to explain things and work hard at doing so, this is cool, I would just suggest you try to do it using more lasting vehicles than a forum post.

PS. and yes, glide paths are pretty much non-sensical imho, totally agree with you here. Fixed AA for life, it is. That is, in my case.
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### Re: Risk tolerance and asset allocation with mathematics

siamond wrote: Fri Mar 06, 2020 8:48 am PS. and yes, glide paths are pretty much non-sensical imho, totally agree with you here. Fixed AA for life, it is. That is, in my case.
Lmao dude what? Uncorrolated is saying the opposite; a glide path is almost certainly superior to a fixed AA for those accumulating except in very rare circumstances. Which Uncorrolated actually fits so he uses a fixed AA; but doesn’t bring it up much to avoid confusing people. Which is ironic because it did just that by the looks of it.
"... 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

305pelusa wrote: Fri Mar 06, 2020 9:08 amUncorrolated is saying the opposite; a glide path is almost certainly superior to a fixed AA for those accumulating except in very rare circumstances. Which Uncorrolated actually fits so he uses a fixed AA; but doesn’t bring it up much to avoid confusing people. Which is ironic because it did just that by the looks of it.
Hm, yeah, you're right, I just reacted to his last post and forgot the OP, as the reasoning is based on assumptions I just don't agree with... My bad. Stupid flu, can't think straight. Thanks for correcting me. Anyhoo, those things are highly personal, there is no 'right' and 'wrong' way...
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### Re: Risk tolerance and asset allocation with mathematics

siamond wrote: Fri Mar 06, 2020 9:26 am
305pelusa wrote: Fri Mar 06, 2020 9:08 amUncorrolated is saying the opposite; a glide path is almost certainly superior to a fixed AA for those accumulating except in very rare circumstances. Which Uncorrolated actually fits so he uses a fixed AA; but doesn’t bring it up much to avoid confusing people. Which is ironic because it did just that by the looks of it.
Hm, yeah, you're right, I just reacted to his last post and forgot the OP, as the reasoning is based on assumptions I just don't agree with... My bad. Stupid flu, can't think straight. Thanks for correcting me. Anyhoo, those things are highly personal, there is no 'right' and 'wrong' way...
Boy you get better with that, we need you in this forum for real
"... 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

siamond wrote: Fri Mar 06, 2020 8:48 am
Uncorrelated wrote: Fri Mar 06, 2020 7:53 amWhy do you think that efficient frontier charts are totally off by assuming that risk is a short term concept and is unidimensional? Unless you take an active position for predicting stock returns (time-varying return assumptions), the efficient frontier does not seem to depend on the time horizon or just barely. If risk is multi dimensional and can't be measured by a probability distribution alone, then what are those other risks? I recognize that the normal distribution has issues, but I seem to have have difficulty rationalizing the other dimensions of risk.
Let me suggest that you take a good look at the 'Deep Risk' booklet from W. Bernstein (easy to find on Amazon for a few bucks). This was instrumental to help me move away from the standard-deviation, Sharpe/Sortino ratios and all this non-sense to longer-term risk considerations which are much more the reality of investors, whether they are accumulators or retirees. It also makes the key consideration that risk(s) & return are NOT orthogonal considerations (the two-dimensional fabric of an efficient frontier chart!) when thinking long-term, as risk and -cumulative- return become kind of the same thing by then. The booklet isn't perfect, there is probably too much emphasis on things beyond our control and not enough emphasis on sustained purchasing power over the long-term, but still, this is an eye-opening read allowing to move away from financial theory to real-life thinking. I also wrote a blog article a little while ago explaining my practical perspective on risk & reward for retirees (I am an early retiree with an hopefully long time horizon, I've been reflecting on such topic for a while now).

But again, any financial theory is good to document (e.g. in a wiki page) and any use case/application is good to discuss (e.g. blog article, forum post, etc), irrespective of individual opinions. This was really my main feedback to you, you clearly like to explain things and work hard at doing so, this is cool, I would just suggest you try to do it using more lasting vehicles than a forum post.

PS. and yes, glide paths are pretty much non-sensical imho, totally agree with you here. Fixed AA for life, it is. That is, in my case.
It appears that we disagree on the definition of risk. You define risk (with your personal circumstances) as the probability not meeting your financial goals. For your situation, bonds are high-risk because an allocation to stocks increases the chance of a successful retirement. But when I talk about risk, I mean the probability distribution of outcomes over a certain timescale. The overall overall happiness I receive from not preemptively running out of money during retirement is what I call utility. This lies at the heart of Merton's portfolio problem: the goal of every investor is to maximize lifetime utility.

Merton considers the simple case, where there are no additions and withdrawals from the portfolio. In this case, the optimal allocation is constant. With lifecycle investing, the accumulator periodically makes additions to the portfolio from his salary, and the optimal allocation is no longer constant. If you are a retiree and you are withdrawing money from your portfolio, then the optimal allocation is no longer a constant.

It's really a two step process. The first step is to gather your assumptions and calculate the efficient frontier. The mathematical methods I gave here are well-suited for this task, you can even include active (market timing) views in the allocation. The second step is investigate your goals and determine the portfolio, among all portfolio's on the frontier, that maximizes the chance of hitting your goals. The mathematical methods I gave here are not suitable for this application (except in the extremely limited case of no additions/withdrawals to the portfolio).

I'm going to refer to Gordon Irlam again, in the paper Floor and Upside Investing in Retirement without Annuities, he investigates the optimal asset allocation for a hypothetical retiree. He defines a high risk aversion γ=4 for 'essential spending', and a low risk aversion γ=1 for 'extra spending'. After a bunch of complicated math, Gordon finds that the optimal allocation in a portfolio with stocks and bonds is as:

If we assume normal returns, then every candidate portfolio is on the efficient frontier.

I suspect we agree on most things, but are just hung up about the terminology.
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### Re: Risk tolerance and asset allocation with mathematics

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.
This 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.

BobK
In finance risk is defined as uncertainty that is consequential (nontrivial). | The two main methods of dealing with financial risk are the matching of assets to goals & diversifying.
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### Re: Risk tolerance and asset allocation with mathematics

Wow! Don't think I'll go back to school to understand this one and will stick with a simplistic approach. Does show the importance of being comfortable with our choices and that there are a number of ways to do that, though.

Tim
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### Re: Risk tolerance and asset allocation with mathematics

bobcat2 wrote: Fri Mar 06, 2020 11:32 am
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.
This 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.

BobK
You're right, I messed up. I meant to say I'm more risk tolerant than the average investor.
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### Re: Risk tolerance and asset allocation with mathematics

Uncorrelated wrote: Fri Mar 06, 2020 1:15 pm
bobcat2 wrote: Fri Mar 06, 2020 11:32 am
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.
This 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.
BobK
You're right, I messed up. I meant to say I'm more risk tolerant than the average investor.
OP, 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?
"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
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Uncorrelated
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### Re: Risk tolerance and asset allocation with mathematics

Fallible wrote: Fri Mar 06, 2020 1:38 pm
Uncorrelated wrote: Fri Mar 06, 2020 1:15 pm
bobcat2 wrote: Fri Mar 06, 2020 11:32 am
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.
This 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.
BobK
You're right, I messed up. I meant to say I'm more risk tolerant than the average investor.
OP, 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?
Sorry 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.