Interest rates low: leverage up?

Have a question about your personal investments? No matter how simple or complex, you can ask it here.
User avatar
ThereAreNoGurus
Posts: 331
Joined: Fri Jan 24, 2014 11:41 pm

Re: Interest rates low: leverage up?

Post by ThereAreNoGurus » Thu Aug 15, 2019 2:10 am

rascott wrote:
Tue Aug 13, 2019 12:30 pm
I'm not opposed to leverage....the ideal leverage on the SP500 is probably around 1.5x for the long term. If you actually could manage to hold that for 30+ years you would most very likely end up with a lot more money. However it would be a wild ride and there is the tail risk of a Great Depression type market where you would be wiped out (or close). And even a "regular" bear market would take you down 75%....that would take serious stones.

However nobody (or very few crazy souls) would be looking at holding a leveraged portfoilo all the way into and during retirement. So what's your real time frame?

A trend following/momentum method like mentioned in the white paper I linked might be a good compromise. When in an up-trending market push the leverage button on....when not just go back to 100% equity. This has added a couple points to the SP500 over the last 4 decades as well, without the major drawdowns of leveraged buy and hold. I'm only looking at going 1.2-1.3x.
That was a very interesting paper you posted up-thread.... thanks. Out of curiosity, how did you run across it?
Last edited by ThereAreNoGurus on Sat Aug 17, 2019 8:44 pm, edited 1 time in total.
Trade the news and you will lose.

User avatar
nisiprius
Advisory Board
Posts: 39459
Joined: Thu Jul 26, 2007 9:33 am
Location: The terrestrial, globular, planetary hunk of matter, flattened at the poles, is my abode.--O. Henry

Re: Interest rates low: leverage up?

Post by nisiprius » Thu Aug 15, 2019 7:30 am

140% leverage, Kelly criterion. It's coming back to me now. I'm going to take this only as far as I can while having confidence that I'm explaining it correctly. Warning, other Kelly-cognoscenti may come in and discover that I've made glaring blunders--but I think this is a responsible presentation.

The canonical illustration is a situation in which someone offers to bet with you, as long as you like, on a series of coin tosses of a fair coin, on the condition that if you win, he pays you an amount equal to your bet--your money doubles--but if you lose, you have to pay him half of the bet--your money halves.

Notice: this is not "double-or-nothing," this is "double-or-halve," and half is more than nothing.

Now the first thing that can be said is that this is not a fair bet, it is extremely favorable to you. In real life, in a zero-sum or slightly-negative-sum situation--like horse racing with a parimutuel system--you can only have this situation if you have superior information to the other bettors you are competing with. It is, however, the case in the stock market, because the stock market has an expected positive return (I hope).

The question that the Kelly criterion answers is, "what is the best betting strategy?"

Except that is far too vague, so we make it definite: "how should I size my bets to as to create the highest expected rate of compound growth in my wealth?"

I can't say this too strongly: this situation--the double-or-halving bet--is quite puzzling, even mind-boggling, and do not assume you "get" it if you haven't thought about it for many minutes. There is a paradoxical quality to it all, sort of a gentle version of the St. Petersburg paradox, and it is not easy to sort through it.

The issue is that, on the one hand, with, say, a $100 bet, you have a 50% chance of ending up with $50 and 50% of ending up with $200, so your mathematical expectation is $125. On the average, you will increase your wealth by 25% of your bet every time you play.

So one's naïve first thought--and this corresponds more or less to the way MPT as usually presented, efficient frontier and capital markets line--is that since the bet is hugely favorable to you, you should bet as much as possible. For example, parlay everything, bet everything you have every time. Furthermore, if you can borrow money, the naïve thinking is you should exploit this incredible opportunity by using leverage.

On the other hand, you get puzzled when you consider the statistically-most-likely outcome, equal number of heads and tails. The most likely outcome, the one that happens most of the time, is that if you bet a hundred times, betting all your money every time, your money is doubled fifty times and halved fifty times--which puts you back exactly where you were before.

A way to see this is to assume 2 bets and a result of one win, then one loss. You start with $100. After the win, you have $200. After the loss, you are back to $100. The same thing happens if you have a long sequence with equal numbers of heads and tails, and the order doesn't matter.

This does not mean that your mathematical expectation for a long series of bets is zero gain. Zero gain is merely the most likely outcome, and the actual distribution of outcomes is weird, because it is highly skew. If you are always betting all you have, your wealth cannot go below zero, but it can go incredibly high. The most likely outcome is you end up with the same amount you started with, but if you look at the less-likely cases, you win more in the cases where you win than you lose in the cases where you lose, and the wins outweigh losses.

The weird thing is if you only bet half of your money every time, it is no longer true that the most likely outcome is zero gain. This changes the situation to one where, if you have $100, you bet $50, you have equal chances of ending up with $25 or $100 on the bet itself. So, you have equal chances of ending up with wealth of $75 or $150 after the bet. So, you have equal chances of changing your wealth by +50% or -25% and, again, looking at the 2-bet case, a win takes you up to $150. On your second bet, you bet $75 and lose, so you end up with $37.50 on the bet, so your total wealth is now $112.50. You are not back to zero. The most likely outcome has been changed from zero gain to a gain of 12.5%.

In other words, in the most likely scenario--not in the average of all scenarios, not in mathematical expectation, but in the most likely scenario--equal numbers of heads and tails--betting only half your wealth every time will outperform betting it all every time.

So in one sense betting half each time is "better" than betting it all each time.

And if you do the same math, using leverage is even "worse" than betting it all each time. If you use 100% leverage, then if you have $100, you borrow $100, you bet $200, you have half a chance each of the result after the bet being $100 or $400, you pay back the bet, overall you have a 50% chance of ending up with $0 or $300, your mathematical expectation is $150, so your expectation is that you make $50 on the bet instead of the $25 you would make if you didn't use leverage. Except. If we again consider the most-likely sequence--equal numbers of wins and losses--no matter how many wins you have, the first loss you have results in your ruin, you're broke. Your wealth goes to zero and stays there.

So, again, we have the pseudo-paradox, that using leverage makes the mathematical expectation better, but makes the most likely outcome worse, much much worse.

Now, the Kelly analysis asks the question "how do we maximize our CAGR, our compound average growth rate?" On the one hand this sounds like exactly like what we want to do. On the other hand, we've smuggled in a logarithm. By choosing to maximize compound average growth rate, we are choosing to maximize the logarithm of our total wealth, not our total wealth itself. In the Kelly framework, doubling your money and halving your money are balanced outcomes. One of them adds 0.301 to your log(wealth), the other subtracts 0.301.

The use of the logarithm matters hugely in the analysis. With the double-or-halve bet and the bet-it-all strategy, If you look at the average of all outcomes--not just those with equal numbers of heads and tails--ordinary high-school-probability says that betting it all each time is preferable to only betting half... and that betting with leverage is more preferable yet. You have a balance of small probabilities of huge wins or losses, and the wins outweigh the losses. However, when you calculate the average logarithm of the outcomes, then betting half gives you a higher average logarithm than betting it all, so it becomes preferable to betting it all... while using leverage becomes even worse than betting it all.

So, this logarithm business. If you call it "logarithm of wealth" it sounds artificial, arbitrary, and weird. If you call it "maximizing the compound average growth rate," it sounds exactly what we all do when comparing investments and strategies.

(By the way, I'm not sure how the Kelly theorists deal with the possibility of actual ruin, losing everything--or, with leverage, more than everything-- since the logarithm of zero is minus infinity or "not a number," "NaN.")

Worse yet, in terms of Kelly criterion discussions, many people feel that the psychological value of wealth is logarithmic--a 10% raise feels just as good whether your salary is being raised from $50,000 to $55,000 or from $100,000 to $110,000, even though the number of dollars is different. The consideration of "whether it makes sense to take the logarithm" gets inextrictably tangled with the question of whether or not any "utility function" is involved, and, if so, whether a logarithmic utility function is "right."

Now, finally, there are some questions that must be asked before trying to use the Kelly criterion in real life. It appears that to calculate the optimum bet using the Kelly criterion, you need to have information that you almost never have in real life. In our examples above, we had a permanent agreement with a betting partner to accept a potentially endless series of bets, and we assumed that we had a fair coin.

If we are, let's say, playing roulette, a bet on #22 pays off at 35:1, i.e. our bet gets multiplied by 36. If the wheel is fair, #22 should come up 1/38th of the time. However, if we know (somehow!) that on this particular wheel, #22 actually comes up more than 1/36th of the time and we know just what that percentage is, then we have a betting situation in our favor, with known odds, and the possibility of an almost unlimited series of bets. The Kelly criterion would be relevant here in telling us how to bet.

But even this situation is unrealistically contrived. You do not have anything resembling comparable information about stock market... bets. Nevertheless, numbers like 140% and 122% have been bruited about as appropriate for the stock market. There is a bunch of handwaving about the use of "partial Kelly," which is an idea that you should not actually go to the maximum suggested by the Kelly criterion because of uncertainty about what the odds actually are, and so you should cut it down by some tasteful or prudent or guessed amount dictated by wisdom and experience.

It is sometimes asserted that some hugely successful investors actually make use of the Kelly criterion in their investing, perhaps even that it is the secret sauce that has made them successful. A second set of questions that must be asked is: is it really true that they use it? and, even if they do, is it really an element in their success?
Last edited by nisiprius on Thu Aug 15, 2019 7:53 am, edited 3 times in total.
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.

User avatar
welderwannabe
Posts: 1113
Joined: Fri Jun 16, 2017 8:32 am

Re: Interest rates low: leverage up?

Post by welderwannabe » Thu Aug 15, 2019 7:48 am

whodidntante wrote:
Tue Aug 13, 2019 11:14 am
I think you missed the best part of the party, friend. All the cool people already left and the host is starting to look at her watch and wonder if anyone will help clean up this mess. Most bogleheads insist they are immune to market timing, and robotically buy no matter what, and I believe them all. But I think most of us can feel a chill in the air, even if we don't want to wear a coat yet. I feel a chill in the air. Maybe it's a pleasant and lengthy and uneventful autumn and maybe winter is coming. We will see.

I already set a course to max all tax advantaged accounts by the end of the year, buying equities with every penny of it.
I am now shedding my most expensive debt, which is or is the same as deleveraging. This reduces my overall risk and is altogether fitting for my situation. If you feel differently and want to increase leverage, be my guest. I'm happy to give some ideas on what I would do, if I wanted to take more risk.
Great post. :beer
I am not an investment professional, but I did stay at a Holiday Inn Express last night.

ryman554
Posts: 1238
Joined: Sun Jan 12, 2014 9:44 pm

Re: Interest rates low: leverage up?

Post by ryman554 » Thu Aug 15, 2019 9:25 am

MotoTrojan wrote:
Wed Aug 14, 2019 6:01 pm
ryman554 wrote:
Wed Aug 14, 2019 5:42 pm
I do know that chart, but also thought I saw something similar with leverage showing about a 140% sweet spot. I can't be certain.

Still,.I have wondered what "risk adjusted return" really means...

Clearly 100/0 will give you a larger expected return than 80/20 (or 0/100)! Over the long run.

So, how does one adjust for risk? Is it defined as portfolio volatility? And then how does one adjust for withdrawal (I guess) when it's down? Or is risk defined in another way?

Thanks for the great discussion!
There are different ways but the most widely accepted is sharpe ratio which is the ratio of return/volatility. Generally when you leverage things your volatility will increase more than your return, thus reducing your sharpe ratio. This is why people will often take portfolios with very high sharpe ratios (but low returns) and then leverage them, such as hedgefundies risk-parity portfolio.
You see, this is what bugs me.

Why is volatility related to risk? Well, it is *risk*, as in you ask yourself what is the chance my portfolio will be up or down on a given day, but most people aren't like that.

They want to know what their portfolio will be *over the long haul* -> and if volatility is measured in terms of days/weeks, it is not particularly useful for timeframes of years/decades. On such a timescale, does not the expected value have more "predictive power"? Said another way, if you look at the distribution of returns for equities over 15 year periods, it is very difficult to find one that has a negative CAGR. But the CAGR is usually *much* better than bonds/cash/etc. I haven't calculated a long-term variability, but that's probably more important in the accumulation stage.

While I fully subscribe to "low volatility" for funds which have a distinct purpose (such as eating tonight, or a housing budget) out to ca 2-5 years or so, I can't bring myself to accept that volatility is in any way risky for a long-term horizon. Why, then, do folks think sharpe is something worth looking into? (Note: I honestly want to know where my logic is flawed.... Too many smart folks seem to think otherwise)

User avatar
Cyclesafe
Posts: 972
Joined: Wed Dec 31, 2014 1:03 pm

Re: Interest rates low: leverage up?

Post by Cyclesafe » Thu Aug 15, 2019 9:53 am

ryman554 wrote:
Thu Aug 15, 2019 9:25 am
MotoTrojan wrote:
Wed Aug 14, 2019 6:01 pm
ryman554 wrote:
Wed Aug 14, 2019 5:42 pm
I do know that chart, but also thought I saw something similar with leverage showing about a 140% sweet spot. I can't be certain.

Still,.I have wondered what "risk adjusted return" really means...

Clearly 100/0 will give you a larger expected return than 80/20 (or 0/100)! Over the long run.

So, how does one adjust for risk? Is it defined as portfolio volatility? And then how does one adjust for withdrawal (I guess) when it's down? Or is risk defined in another way?

Thanks for the great discussion!
There are different ways but the most widely accepted is sharpe ratio which is the ratio of return/volatility. Generally when you leverage things your volatility will increase more than your return, thus reducing your sharpe ratio. This is why people will often take portfolios with very high sharpe ratios (but low returns) and then leverage them, such as hedgefundies risk-parity portfolio.
You see, this is what bugs me.

Why is volatility related to risk? Well, it is *risk*, as in you ask yourself what is the chance my portfolio will be up or down on a given day, but most people aren't like that.

They want to know what their portfolio will be *over the long haul* -> and if volatility is measured in terms of days/weeks, it is not particularly useful for timeframes of years/decades. On such a timescale, does not the expected value have more "predictive power"? Said another way, if you look at the distribution of returns for equities over 15 year periods, it is very difficult to find one that has a negative CAGR. But the CAGR is usually *much* better than bonds/cash/etc. I haven't calculated a long-term variability, but that's probably more important in the accumulation stage.

While I fully subscribe to "low volatility" for funds which have a distinct purpose (such as eating tonight, or a housing budget) out to ca 2-5 years or so, I can't bring myself to accept that volatility is in any way risky for a long-term horizon. Why, then, do folks think sharpe is something worth looking into? (Note: I honestly want to know where my logic is flawed.... Too many smart folks seem to think otherwise)
IMHO, the Sharpe Ratio or any other relative measure of volatility is merely a proxy for risk, and only tangentially related to what the layperson considers risk to be; the chance that future obligations may not be met. Better for the long-term-investor to think instead about fixed cost coverage over time rather than equity volatility. There, I said it.
"Plans are useless; planning is indispensable.” - Dwight Eisenhower

alex_686
Posts: 5097
Joined: Mon Feb 09, 2015 2:39 pm

Re: Interest rates low: leverage up?

Post by alex_686 » Thu Aug 15, 2019 10:00 am

ryman554 wrote:
Thu Aug 15, 2019 9:25 am
MotoTrojan wrote:
Wed Aug 14, 2019 6:01 pm
ryman554 wrote:
Wed Aug 14, 2019 5:42 pm
I do know that chart, but also thought I saw something similar with leverage showing about a 140% sweet spot. I can't be certain.

Still,.I have wondered what "risk adjusted return" really means...

Clearly 100/0 will give you a larger expected return than 80/20 (or 0/100)! Over the long run.

So, how does one adjust for risk? Is it defined as portfolio volatility? And then how does one adjust for withdrawal (I guess) when it's down? Or is risk defined in another way?

Thanks for the great discussion!
There are different ways but the most widely accepted is sharpe ratio which is the ratio of return/volatility. Generally when you leverage things your volatility will increase more than your return, thus reducing your sharpe ratio. This is why people will often take portfolios with very high sharpe ratios (but low returns) and then leverage them, such as hedgefundies risk-parity portfolio.
You see, this is what bugs me.

Why is volatility related to risk? Well, it is *risk*, as in you ask yourself what is the chance my portfolio will be up or down on a given day, but most people aren't like that.

They want to know what their portfolio will be *over the long haul* -> and if volatility is measured in terms of days/weeks, it is not particularly useful for timeframes of years/decades. On such a timescale, does not the expected value have more "predictive power"? Said another way, if you look at the distribution of returns for equities over 15 year periods, it is very difficult to find one that has a negative CAGR. But the is CAGR usually *much* better than bonds/cash/etc. I haven't calculated a long-term variability, but that's probably more important in the accumulation stage.

While I fully subscribe to "low volatility" for funds which have a distinct purpose (such as eating tonight, or a housing budget) out to ca 2-5 years or so, I can't bring myself to accept that volatility is in any way risky for a long-term horizon. Why, then, do folks think sharpe is something worth looking into? (Note: I honestly want to know where my logic is flawed.... Too many smart folks seem to think otherwise)
So, on the glass is half full...

Volatility is very easy to calculate. You only need a few data points. It works across a wide array of asset classes. There is a big tool kit that support volatility calculations. The limitations and defects are well known, and can be detected using above tool kits.

Volatility scales beautifully over time. In theory I can take a days worth if data and scale it up to 30 years. From a practical standpoint it tends to be months to years.

Next, how do you compare apples to oranges? The Sharpe ratio puts bonds and stocks on the same scale so they can be measured.

CAGR Is fine for what it does. However it is for a fixed historical time frame, backwards looking, and does not scale. Critically it does not have a risk dimension.

On the other hand I can take historic and futures data and spin that around to make forward projections over multiple time frames.

alex_686
Posts: 5097
Joined: Mon Feb 09, 2015 2:39 pm

Re: Interest rates low: leverage up?

Post by alex_686 » Thu Aug 15, 2019 10:08 am

Cyclesafe wrote:
Thu Aug 15, 2019 9:53 am
IMHO, the Sharpe Ratio or any other relative measure of volatility is merely a proxy for risk, and only tangentially related to what the layperson considers risk to be; the chance that future obligations may not be met. Better for the long-term-investor to think instead about fixed cost coverage over time rather than equity volatility. There, I said it.
While this may be true, what is your point? How do you model this? How do you translate your unique situation into a generalized model to share?

Yes, Sharpe is a proxy. A nice simple generic proxy, which is both its strength and weakness.

aristotelian
Posts: 6473
Joined: Wed Jan 11, 2017 8:05 pm

Re: Interest rates low: leverage up?

Post by aristotelian » Thu Aug 15, 2019 10:13 am

What do low interest rates have to do with making UPRO more favorable? You are not actually borrowing money to invest in UPRO. Is your assumption that UPROs expenses will go down?

MotoTrojan
Posts: 6877
Joined: Wed Feb 01, 2017 8:39 pm

Re: Interest rates low: leverage up?

Post by MotoTrojan » Thu Aug 15, 2019 12:29 pm

ryman554 wrote:
Thu Aug 15, 2019 9:25 am
MotoTrojan wrote:
Wed Aug 14, 2019 6:01 pm
ryman554 wrote:
Wed Aug 14, 2019 5:42 pm
I do know that chart, but also thought I saw something similar with leverage showing about a 140% sweet spot. I can't be certain.

Still,.I have wondered what "risk adjusted return" really means...

Clearly 100/0 will give you a larger expected return than 80/20 (or 0/100)! Over the long run.

So, how does one adjust for risk? Is it defined as portfolio volatility? And then how does one adjust for withdrawal (I guess) when it's down? Or is risk defined in another way?

Thanks for the great discussion!
There are different ways but the most widely accepted is sharpe ratio which is the ratio of return/volatility. Generally when you leverage things your volatility will increase more than your return, thus reducing your sharpe ratio. This is why people will often take portfolios with very high sharpe ratios (but low returns) and then leverage them, such as hedgefundies risk-parity portfolio.
You see, this is what bugs me.

Why is volatility related to risk? Well, it is *risk*, as in you ask yourself what is the chance my portfolio will be up or down on a given day, but most people aren't like that.

They want to know what their portfolio will be *over the long haul* -> and if volatility is measured in terms of days/weeks, it is not particularly useful for timeframes of years/decades. On such a timescale, does not the expected value have more "predictive power"? Said another way, if you look at the distribution of returns for equities over 15 year periods, it is very difficult to find one that has a negative CAGR. But the CAGR is usually *much* better than bonds/cash/etc. I haven't calculated a long-term variability, but that's probably more important in the accumulation stage.

While I fully subscribe to "low volatility" for funds which have a distinct purpose (such as eating tonight, or a housing budget) out to ca 2-5 years or so, I can't bring myself to accept that volatility is in any way risky for a long-term horizon. Why, then, do folks think sharpe is something worth looking into? (Note: I honestly want to know where my logic is flawed.... Too many smart folks seem to think otherwise)
Volatility decay kills leveraged portfolios so in that sense, comparing the sharpe ratio of various unleveraged portfolios to help determine which are worth considering for leverage can be quite useful.

pkcrafter
Posts: 13655
Joined: Sun Mar 04, 2007 12:19 pm
Location: CA
Contact:

Re: Interest rates low: leverage up?

Post by pkcrafter » Fri Aug 16, 2019 11:35 am

Nisiprius, thank you for the detailed replies on leverage. It doesn't sound smart to me and certainly no one who calls themselves a Boglehead would ever consider using it.

As for volatility and risk, the definition of risk I like is this: Not having the money for something important when you need it.

This is also worth thinking about - “Risk comes from not knowing what you’re doing.” — Warren Buffett

Certainly investing in things you don't fully understand, leveraging or instance, falls into that definition. This is one of the big reasons why Boglehead investing is the best choice for the majority of investors is, it's easy to understand and implement, and due to low cost and consistency it outperforms the great majority of active strategies. Ask yourself why leveraging isn't embraced by all active investors.

Bogleheads, do not waste time or money, even small play amounts, on lowering your investing returns. The advantage of the Boglehead strategy is still a fine line.

Paul
When times are good, investors tend to forget about risk and focus on opportunity. When times are bad, investors tend to forget about opportunity and focus on risk.

HEDGEFUNDIE
Posts: 3656
Joined: Sun Oct 22, 2017 2:06 pm

Re: Interest rates low: leverage up?

Post by HEDGEFUNDIE » Fri Aug 16, 2019 12:40 pm

aristotelian wrote:
Thu Aug 15, 2019 10:13 am
What do low interest rates have to do with making UPRO more favorable? You are not actually borrowing money to invest in UPRO. Is your assumption that UPROs expenses will go down?
You are not borrowing money but UPRO is borrowing money. So UPRO’s costs go down and returns should go up.

Also low interest rates should correlate with stock bull markets. Which drives up returns even more.

aristotelian
Posts: 6473
Joined: Wed Jan 11, 2017 8:05 pm

Re: Interest rates low: leverage up?

Post by aristotelian » Fri Aug 16, 2019 1:08 pm

HEDGEFUNDIE wrote:
Fri Aug 16, 2019 12:40 pm
aristotelian wrote:
Thu Aug 15, 2019 10:13 am
What do low interest rates have to do with making UPRO more favorable? You are not actually borrowing money to invest in UPRO. Is your assumption that UPROs expenses will go down?
You are not borrowing money but UPRO is borrowing money. So UPRO’s costs go down and returns should go up.

Also low interest rates should correlate with stock bull markets. Which drives up returns even more.
Market timing seems like a terrible justification for using leverage. There are lots of reasons to think a recession may be coming regardless of what the Fed does right now.

fujiters
Posts: 283
Joined: Tue Mar 06, 2018 2:17 pm

Re: Interest rates low: leverage up?

Post by fujiters » Fri Aug 16, 2019 4:30 pm

nisiprius wrote:
Thu Aug 15, 2019 7:30 am
140% leverage, Kelly criterion. It's coming back to me now. I'm going to take this only as far as I can while having confidence that I'm explaining it correctly. Warning, other Kelly-cognoscenti may come in and discover that I've made glaring blunders--but I think this is a responsible presentation.

The canonical illustration is a situation in which someone offers to bet with you, as long as you like, on a series of coin tosses of a fair coin, on the condition that if you win, he pays you an amount equal to your bet--your money doubles--but if you lose, you have to pay him half of the bet--your money halves.

Notice: this is not "double-or-nothing," this is "double-or-halve," and half is more than nothing.

Now the first thing that can be said is that this is not a fair bet, it is extremely favorable to you. In real life, in a zero-sum or slightly-negative-sum situation--like horse racing with a parimutuel system--you can only have this situation if you have superior information to the other bettors you are competing with. It is, however, the case in the stock market, because the stock market has an expected positive return (I hope).

The question that the Kelly criterion answers is, "what is the best betting strategy?"

Except that is far too vague, so we make it definite: "how should I size my bets to as to create the highest expected rate of compound growth in my wealth?"

I can't say this too strongly: this situation--the double-or-halving bet--is quite puzzling, even mind-boggling, and do not assume you "get" it if you haven't thought about it for many minutes. There is a paradoxical quality to it all, sort of a gentle version of the St. Petersburg paradox, and it is not easy to sort through it.

The issue is that, on the one hand, with, say, a $100 bet, you have a 50% chance of ending up with $50 and 50% of ending up with $200, so your mathematical expectation is $125. On the average, you will increase your wealth by 25% of your bet every time you play.

So one's naïve first thought--and this corresponds more or less to the way MPT as usually presented, efficient frontier and capital markets line--is that since the bet is hugely favorable to you, you should bet as much as possible. For example, parlay everything, bet everything you have every time. Furthermore, if you can borrow money, the naïve thinking is you should exploit this incredible opportunity by using leverage.

On the other hand, you get puzzled when you consider the statistically-most-likely outcome, equal number of heads and tails. The most likely outcome, the one that happens most of the time, is that if you bet a hundred times, betting all your money every time, your money is doubled fifty times and halved fifty times--which puts you back exactly where you were before.

A way to see this is to assume 2 bets and a result of one win, then one loss. You start with $100. After the win, you have $200. After the loss, you are back to $100. The same thing happens if you have a long sequence with equal numbers of heads and tails, and the order doesn't matter.

This does not mean that your mathematical expectation for a long series of bets is zero gain. Zero gain is merely the most likely outcome, and the actual distribution of outcomes is weird, because it is highly skew. If you are always betting all you have, your wealth cannot go below zero, but it can go incredibly high. The most likely outcome is you end up with the same amount you started with, but if you look at the less-likely cases, you win more in the cases where you win than you lose in the cases where you lose, and the wins outweigh losses.

The weird thing is if you only bet half of your money every time, it is no longer true that the most likely outcome is zero gain. This changes the situation to one where, if you have $100, you bet $50, you have equal chances of ending up with $25 or $100 on the bet itself. So, you have equal chances of ending up with wealth of $75 or $150 after the bet. So, you have equal chances of changing your wealth by +50% or -25% and, again, looking at the 2-bet case, a win takes you up to $150. On your second bet, you bet $75 and lose, so you end up with $37.50 on the bet, so your total wealth is now $112.50. You are not back to zero. The most likely outcome has been changed from zero gain to a gain of 12.5%.

In other words, in the most likely scenario--not in the average of all scenarios, not in mathematical expectation, but in the most likely scenario--equal numbers of heads and tails--betting only half your wealth every time will outperform betting it all every time.

So in one sense betting half each time is "better" than betting it all each time.

And if you do the same math, using leverage is even "worse" than betting it all each time. If you use 100% leverage, then if you have $100, you borrow $100, you bet $200, you have half a chance each of the result after the bet being $100 or $400, you pay back the bet, overall you have a 50% chance of ending up with $0 or $300, your mathematical expectation is $150, so your expectation is that you make $50 on the bet instead of the $25 you would make if you didn't use leverage. Except. If we again consider the most-likely sequence--equal numbers of wins and losses--no matter how many wins you have, the first loss you have results in your ruin, you're broke. Your wealth goes to zero and stays there.

So, again, we have the pseudo-paradox, that using leverage makes the mathematical expectation better, but makes the most likely outcome worse, much much worse.

Now, the Kelly analysis asks the question "how do we maximize our CAGR, our compound average growth rate?" On the one hand this sounds like exactly like what we want to do. On the other hand, we've smuggled in a logarithm. By choosing to maximize compound average growth rate, we are choosing to maximize the logarithm of our total wealth, not our total wealth itself. In the Kelly framework, doubling your money and halving your money are balanced outcomes. One of them adds 0.301 to your log(wealth), the other subtracts 0.301.

The use of the logarithm matters hugely in the analysis. With the double-or-halve bet and the bet-it-all strategy, If you look at the average of all outcomes--not just those with equal numbers of heads and tails--ordinary high-school-probability says that betting it all each time is preferable to only betting half... and that betting with leverage is more preferable yet. You have a balance of small probabilities of huge wins or losses, and the wins outweigh the losses. However, when you calculate the average logarithm of the outcomes, then betting half gives you a higher average logarithm than betting it all, so it becomes preferable to betting it all... while using leverage becomes even worse than betting it all.

So, this logarithm business. If you call it "logarithm of wealth" it sounds artificial, arbitrary, and weird. If you call it "maximizing the compound average growth rate," it sounds exactly what we all do when comparing investments and strategies.

(By the way, I'm not sure how the Kelly theorists deal with the possibility of actual ruin, losing everything--or, with leverage, more than everything-- since the logarithm of zero is minus infinity or "not a number," "NaN.")

Worse yet, in terms of Kelly criterion discussions, many people feel that the psychological value of wealth is logarithmic--a 10% raise feels just as good whether your salary is being raised from $50,000 to $55,000 or from $100,000 to $110,000, even though the number of dollars is different. The consideration of "whether it makes sense to take the logarithm" gets inextrictably tangled with the question of whether or not any "utility function" is involved, and, if so, whether a logarithmic utility function is "right."

Now, finally, there are some questions that must be asked before trying to use the Kelly criterion in real life. It appears that to calculate the optimum bet using the Kelly criterion, you need to have information that you almost never have in real life. In our examples above, we had a permanent agreement with a betting partner to accept a potentially endless series of bets, and we assumed that we had a fair coin.

If we are, let's say, playing roulette, a bet on #22 pays off at 35:1, i.e. our bet gets multiplied by 36. If the wheel is fair, #22 should come up 1/38th of the time. However, if we know (somehow!) that on this particular wheel, #22 actually comes up more than 1/36th of the time and we know just what that percentage is, then we have a betting situation in our favor, with known odds, and the possibility of an almost unlimited series of bets. The Kelly criterion would be relevant here in telling us how to bet.

But even this situation is unrealistically contrived. You do not have anything resembling comparable information about stock market... bets. Nevertheless, numbers like 140% and 122% have been bruited about as appropriate for the stock market. There is a bunch of handwaving about the use of "partial Kelly," which is an idea that you should not actually go to the maximum suggested by the Kelly criterion because of uncertainty about what the odds actually are, and so you should cut it down by some tasteful or prudent or guessed amount dictated by wisdom and experience.

It is sometimes asserted that some hugely successful investors actually make use of the Kelly criterion in their investing, perhaps even that it is the secret sauce that has made them successful. A second set of questions that must be asked is: is it really true that they use it? and, even if they do, is it really an element in their success?
👏🏻

Nisipedia should exist (with this filed under "Kelly Criterion"). I'm grateful I can read the entries on Bogleheads.org though.
“The purpose of the margin of safety is to render the forecast unnecessary.” -Benjamin Graham

rascott
Posts: 1087
Joined: Wed Apr 15, 2015 10:53 am

Re: Interest rates low: leverage up?

Post by rascott » Mon Aug 19, 2019 7:38 am

ThereAreNoGurus wrote:
Thu Aug 15, 2019 2:10 am
rascott wrote:
Tue Aug 13, 2019 12:30 pm
I'm not opposed to leverage....the ideal leverage on the SP500 is probably around 1.5x for the long term. If you actually could manage to hold that for 30+ years you would most very likely end up with a lot more money. However it would be a wild ride and there is the tail risk of a Great Depression type market where you would be wiped out (or close). And even a "regular" bear market would take you down 75%....that would take serious stones.

However nobody (or very few crazy souls) would be looking at holding a leveraged portfoilo all the way into and during retirement. So what's your real time frame?

A trend following/momentum method like mentioned in the white paper I linked might be a good compromise. When in an up-trending market push the leverage button on....when not just go back to 100% equity. This has added a couple points to the SP500 over the last 4 decades as well, without the major drawdowns of leveraged buy and hold. I'm only looking at going 1.2-1.3x.
That was a very interesting paper you posted up-thread.... thanks. Out of curiosity, how did you run across it?

Was linked to in this article:

https://seekingalpha.com/article/422616 ... since-1928

Post Reply