How do you price risk?

[Boofinator]

Dear Mustachians,

What model do you use to price risk when comparing returns from different investments? For example, let's say stocks have an expected yield of 7% based on historical observation, at what expected yield would you choose bonds or fixed income over stocks?

I'm not a finance major, so interested in hearing any conceptual or mathematical models you might use or be familiar with (though let's leave feelings to the degree we can out of it).

B.

[Louisville]

Risk is usually expressed as R-squared, a measure of volatility. How one uses historical data to arrive at R-squared for a particular asses class, I don't know.
Generally speaking, over longer terms, more risk/volatility means greater returns.

[DS]

There is Finance and also Risk Management, two entire areas of study to really go deep into this. There is no feelings category for measuring risk. That could be discussed in the realm of Behavioral Economics, another entire area of study.

Either way, I highly doubt anyone on this forum is actively making these measurements. Not really necessary in the context of MMM, Bogleheads, JLCollins approach.

In simple terms, you would choose one over the other when the present value of future cash flows from one is greater than the other. All of this requires assumptions of future returns, which can be measured using past data, economic analysis, and throwing a dart at a wall.

The index investing approach? Choose an allocation of stocks and bonds, then buy the whole market and spend the time you saved enjoying your life.




Here is some basic info that might help with the various topics:

https://www.investopedia.com/terms/r/riskmeasures.asp

https://www.investopedia.com/ask/answers/041415/what-are-some-common-measures-risk-used-risk-management.asp

https://en.wikipedia.org/wiki/Behavioral_economics

https://hbr.org/2017/10/the-rise-of-behavioral-economics-and-its-influence-on-organizations

https://www.accountingcoach.com/blog/npv-net-present-value

https://www.investopedia.com/exam-guide/cfa-level-1/quantitative-methods/discounted-cash-flow-npv-irr.asp

[MDM]

...any conceptual or mathematical models you might use or be familiar with (though let's leave feelings to the degree we can out of it).

Feelings (in the form of risk aversion) are at the heart of this.

[DS]

...any conceptual or mathematical models you might use or be familiar with (though let's leave feelings to the degree we can out of it).

Feelings (in the form of risk aversion) are at the heart of this.

Feelings can determine risk tolerance of a human, but are not a component in measuring risk of an investment.

[MDM]

...any conceptual or mathematical models you might use or be familiar with (though let's leave feelings to the degree we can out of it).

Feelings (in the form of risk aversion) are at the heart of this.

Feelings can determine risk tolerance of a human, but are not a component in measuring risk of an investment.

Yes, but that is not the OP's question.  The question here is how one "price(s) risk when comparing returns from different investments?"

[DS]

...any conceptual or mathematical models you might use or be familiar with (though let's leave feelings to the degree we can out of it).

Feelings (in the form of risk aversion) are at the heart of this.

Feelings can determine risk tolerance of a human, but are not a component in measuring risk of an investment.

Yes, but that is not the OP's question.  The question here is how one "price(s) risk when comparing returns from different investments?"

So it's more of an Investment Order question? I misinterpreted then due to the terminology.

[MDM]

...any conceptual or mathematical models you might use or be familiar with (though let's leave feelings to the degree we can out of it).

Feelings (in the form of risk aversion) are at the heart of this.

Feelings can determine risk tolerance of a human, but are not a component in measuring risk of an investment.

Yes, but that is not the OP's question.  The question here is how one "price(s) risk when comparing returns from different investments?"

So it's more of an Investment Order question?

Pretty much.

[Boofinator]

There is Finance and also Risk Management, two entire areas of study to really go deep into this. There is no feelings category for measuring risk. That could be discussed in the realm of Behavioral Economics, another entire area of study.

Either way, I highly doubt anyone on this forum is actively making these measurements. Not really necessary in the context of MMM, Bogleheads, JLCollins approach.

In simple terms, you would choose one over the other when the present value of future cash flows from one is greater than the other. All of this requires assumptions of future returns, which can be measured using past data, economic analysis, and throwing a dart at a wall.

The index investing approach? Choose an allocation of stocks and bonds, then buy the whole market and spend the time you saved enjoying your life.




Here is some basic info that might help with the various topics:

https://www.investopedia.com/terms/r/riskmeasures.asp

https://www.investopedia.com/ask/answers/041415/what-are-some-common-measures-risk-used-risk-management.asp

https://en.wikipedia.org/wiki/Behavioral_economics

https://hbr.org/2017/10/the-rise-of-behavioral-economics-and-its-influence-on-organizations

https://www.accountingcoach.com/blog/npv-net-present-value

https://www.investopedia.com/exam-guide/cfa-level-1/quantitative-methods/discounted-cash-flow-npv-irr.asp

I appreciate the links.

Let's use your approach to rephrase this question: How would you choose an appropriate asset allocation to account for risk?

[maizefolk]

@brooklynguy has had some very good discussions of the different concepts that people may mean when they use the work risk. I'm going to mangle it trying to reproduce from memory, but at a bare minimum, risk can be volatility and uncertainty (the price moves around a lot and it's hard to predict exactly how much money you'll have in 30 years) which is a different meaning of risk than looking at the probability you'll have less than X dollars after 30 years.

My own analogy about why it is important to define what you mean by risk:

If you need $10,000, and have the choice of either betting on a dice roll where you receive $10,000 is you roll a 6, or $9,000 if you roll 1-5, or a separate roll were you receive $10,000 if you roll a one, $20,000 if you roll a 2, $30,000 if you roll a 3 and so on up to 6, the second scenario has a lot more volatility and a much higher standard deviation of returns, but the first scenario is a lot "riskier."

[Boofinator]

Here's my conceptual approach. I wrote another post that was way too long with too many details, so I started fresh with minimal details to get the point across and avoid the tl;dr's.

First, the marginal utility value of money is an indispensable concept to understanding risk, in that each dollar we obtain is worth less than or equal to the last dollar that was obtained.  (https://www.mrmoneymustache.com/2012/02/09/brave-new-life/) A monotonically decreasing function is obviously a necessity to show the additional value added for each dollar. Let's use the inverse function (which is the pdf to the logarithm cdf). (Of course the actual function is a nonlinear stepwise function, but 1/x serves as an approximation.)

Stock market returns have been shown to have a decent relationship with the Laplacian pdf, so I'll use that to model returns.

To obtain the risk-adjusted return, one should multiply the Laplacian market returns by the utility value function. Since the utility function is monotonically decreasing, it is clear the result must skew left, thereby reducing the average "risk-adjusted" return.

As should be self-evident, one can reduce the risk in one of two ways: 1) Have a large asset-to-spending ratio, thereby reaching the essentially flat part of the utility value curve. 2) Have a long time horizon, which reduces the standard error in the Laplacian and results in multiplying by a fairly flat stretch of the utility curve. So the time when risk plays a role is during short time horizons with low assets.

I ran the numbers in my model, and came out with the result that stocks averaging 7% historical returns with 5% standard deviation would have a 5.4% "risk-adjusted" return over a one-year time period when the utility value of money was very high. Bonds with up to a 2% standard deviation had about a 0.1% difference between returns and risk-adjusted returns, so if I expected 7% real returns in the stock market and absolutely needed the money in a year (otherwise I would need to take on high-cost debt), I would invest in bonds rather than stock if the yield was 5.5%.

[Boofinator]

@brooklynguy has had some very good discussions of the different concepts that people may mean when they use the work risk. I'm going to mangle it trying to reproduce from memory, but at a bare minimum, risk can be volatility and uncertainty (the price moves around a lot and it's hard to predict exactly how much money you'll have in 30 years) which is a different meaning of risk than looking at the probability you'll have less than X dollars after 30 years.

My own analogy about why it is important to define what you mean by risk:

If you need $10,000, and have the choice of either betting on a dice roll where you receive $10,000 is you roll a 6, or $9,000 if you roll 1-5, or a separate roll were you receive $10,000 if you roll a one, $20,000 if you roll a 2, $30,000 if you roll a 3 and so on up to 6, the second scenario has a lot more volatility and a much higher standard deviation of returns, but the first scenario is a lot "riskier."

Though I personally have no desire to gamble (unless I owned the casino), it presents many good analogies for discussing risk. Consider the roulette wheel. What's a better bet, a number or a color? Both have the same expected return, so we can't use that to make a determination. Let's instead maximize our playing time (which is why people bet?). We have $4, and want it to last at least 5 rolls at $1 a roll. Number or color?

The probability of lasting at least 5 rolls choosing a number is about 10%. The probability of lasting at least 5 rolls choosing a color is 95%.

This shows the risk present when you have a high utility value for money (due to the low bankroll).

[ILikeDividends]

Though I personally have no desire to gamble (unless I owned the casino), it presents many good analogies for discussing risk. Consider the roulette wheel. What's a better bet, a number or a color?

European roulette is slightly friendlier to the player than American style roulette, so let's use European roulette for an example.

A bet on red or black pays even money; $1 payout for every $1 bet.  But the odds of winning are only 48.6%; not 50/50.  If the payout for this bet were statistically fair, it would pay ~$1.056 for every winning $1 bet. So you are short-changed by about 5.6% every time you win.

A bet on a single number pays 35 to 1; a $35 payout for every $1 bet.  But the odds of winning are only 2.7%.  If the payout for this bet were statistically fair, it would pay $36 for every winning $1 bet. So you are short-changed by about 2.9% every time you win.

In both cases, the house's edge is the difference between what you're paid when you win versus what you would have been paid if the payouts were  statistically fair. On the surface, it might seem the single number bet is paid better odds than the color bet.  But that is deceptive because it doesn't account for the different win/loss ratios of the two bets.

With both bets, a series of 37 $1 bets is needed, assuming statistically perfect outcomes of the roulette wheel (hitting each number once and only once), in order to evaluate the two types of bets.  With the color bet, your initial $37 bankroll would become $36 (winning 18 times, and losing 19 times).  With the single number bet, your $37 bankroll would also become $36 (losing 36 times and winning 1 time).

So the house's cut, about $1 for every $37 bet across 37 games, is about equal for both bets over a series of bets.  But both bets are statistically destined to lose more money the more times you make that bet, because your bankroll is consistently shrinking at a fairly constant rate over time. The single number bet is highly volatile as compared to the color bet, but that's about the only difference.

Reference: https://www.roulettesites.org/rules/odds/

We have $4, and want it to last at least 5 rolls at $1 a roll. Number or color?

That's the wrong approach to risk mitigation in terms of roulette.  Ideally, you would place your entire betting stash on red or black, spin the wheel once, and whether you win or lose, you walk out of the casino, and never return.

The house's statistical cut doesn't change from one spin to the next.  The more times you expose fractions of your bankroll to that game, the more likely you are to realize a negatively compounding loss that grows with the number of times you play.

But casino patrons do not go to casinos with the intent of mitigating risk.  Where's the fun in that?  ;)

Edited to correct some miscalculations of the house's cut.

[maizefolk]

Though I personally have no desire to gamble (unless I owned the casino), it presents many good analogies for discussing risk. Consider the roulette wheel. What's a better bet, a number or a color? Both have the same expected return, so we can't use that to make a determination.

It depends on how much money you need to be happy.

[Boofinator]

Though I personally have no desire to gamble (unless I owned the casino), it presents many good analogies for discussing risk. Consider the roulette wheel. What's a better bet, a number or a color?

European roulette is slightly friendlier to the player then American style roulette, so let's use European roulette for an example.

A bet on red or black pays even money; $1 payout for every $1 bet.  But the odds of winning are only 48.6%; not 50/50.  If the payout for this bet were statistically fair, it would pay ~$1.056 for every winning $1 bet. So you are short-changed by about 5.6% every time you win.

A bet on a single number pays 35 to 1; a $35 payout for every $1 bet.  But the odds of winning are only 2.7%.  If the payout for this bet were statistically fair, it would pay $36 for every winning $1 bet. So you are short-changed by about 2.9% every time you win.

In both cases, the house's edge is the difference between what you're paid when you win versus what you would have been paid if the payouts were  statistically fair. On the surface, it might seem the single number bet is paid better odds than the color bet.  But that is deceptive because it doesn't account for the different win/loss ratios of the two bets.

With both bets, a series of 37 $1 bets is needed, assuming statistically perfect outcomes of the roulette wheel, in order to evaluate the two types of bets.  With the color bet, your initial $37 bankroll would become $36 (winning 18 times, and losing 19 times).  With the single number bet, your $37 bankroll would also become $36 (losing 36 times and winning 1 time).

So the house's cut, about $1 for every $37 bet across 37 games, is about equal for both bets over a series bets.  But both bets are statistically destined to lose more the more times you make that bet, because your bankroll is consistently shrinking at a fairly constant rate over time. The single number bet is highly volatile as compared to the color bet, but that's about the only difference.

Reference: https://www.roulettesites.org/rules/odds/

We have $4, and want it to last at least 5 rolls at $1 a roll. Number or color?

That's the wrong approach to risk mitigation in terms of gambling.  Ideally, you would place your entire betting stash on red or black, spin the wheel once, and whether you win or lose, you walk out of the casino, and never return.

The house's statistical cut doesn't change from one spin to the next.  The more times you expose fractions of your bankroll to that game, the more likely you are to realize a negatively compounding loss that grows with the number of times you play.

But casino goers do not go to casinos with the intent of mitigating risk.  Where's the fun in that?  ;)

Edited to correct some miscalculations of the house's cut.

That's the wrong approach to risk mitigation in terms of gambling.  Ideally, you would place your entire betting stash on red or black, spin the wheel once, and whether you win or lose, you walk out of the casino, and never return.

Now, that's risk mitigation.

The house's statistical cut doesn't change from one spin to the next.  The more times you expose fractions of your bankroll to that game, the more likely you are to realize a negatively compounding loss that grows with the number of times you play.

In roulette (as in risky investments) you don't get expected returns on a small number of spins, you get actual returns. So in roulette a color and number have the same statistical payout, but on small bankrolls you need to minimize risk (if you want to keep playing). And calculating the probability of losing four times in a row is trivial, as compared to calculating investment risks.

This is supposed to illustrate that on short time horizons and with a limited bankroll, risky assets (such as stocks) might cause you to lose your bankroll before less risky assets.

[ILikeDividends]

This is supposed to illustrate that on short time horizons and with a limited bankroll, risky assets (such as stocks) might cause you to lose your bankroll before less risky assets.

No arguments there.  I was extending the time horizon to illustrate a nuanced point.

With roulette, time (the number of times you spin) is against you.  The shorter the better.

With equities, time (literally, time in the market) is on your side.  The longer the better.

In that regard, with roulette, color bets versus number bets have very different risk profiles if you have a very short time horizon; say 5 spins of the wheel.  Color bets could be likened to bond investments where you stand something close to a 50/50 chance of breaking even over the very short term.  With a single number bet (I'm using it as a proxy for equities), you have an extremely low chance of leaving the casino with any chips at all after only 5 spins.  The two bets are only comparable (meaning equally likely outcomes) over a series of games played in 37 game increments.

With only 5 spins of the wheel, the number bet is bucking pretty severe odds of walking out of the casino with any money left at all.  If you beat those odds, on the other hand, you of course could walk out with substantially more money than you came in with.

So mathematically, the odds of the two bets are about the same; which you can't really say about stocks vs bonds.  But the volatility of the two bets are at opposite ends of the spectrum; which I think goes more to your point, i.e., the shorter your time horizon, the less tolerance for volatility you can bear.  This is true both in roulette and in financial markets.  This makes the single number bet much more risky, in terms of realizing a total loss, than the color bet, in the very short term.

In financial markets you can mitigate the volatility by extending the time horizon if you are properly diversified.  In roulette, extending the time horizon mitigates volatility too, but then you suffer a locked-in negative rate of return that becomes increasingly more probable as time goes by, by doing so, regardless of which bet you select.  Because of differences in volatility, the two bets are only equivalently bad, i.e., "risky," over the long term, but not over the short term.

If you only get 5 spins and you are risk-averse, you bet the colors.  If you want to go for broke and swing for the fences, you bet the numbers.  You could win or lose everything either way, but the odds of losing everything are very different in a series of games lasting fewer than 37 spins of the wheel.

If you want to enjoy your free drinks and kill a bunch of time, you can mix and match your roulette bets however you please.  The longer you play, the less your choice of bets will matter, and the less likely you are to realize that those drinks weren't really free after all; not until after you sober up, that is.

Edit to add: And yes, I stand corrected.  The best way to mitigate risk in roulette is to go see a movie, instead of playing roulette.  ;)