Help me figure out the Target Volatility formula

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Help me figure out the Target Volatility formula

Post by danielc » Fri Jan 31, 2020 5:43 am

I am trying to reproduce the Target Volatility allocations from Portfolio Visualizer. Here is an example portfolio. I tried to keep it simple: There is only one risky asset (UPRO -- a 3x daily leveraged S&P 500 ETF) and the out-of-market asset is just cash. Since cash contributes virtually zero volatility, the entire volatility of the portfolio is produced by UPRO.

I set a target volatility of 25%, and PortfolioVisualizer reports on the monthly returns of UPRO and its recommended UPRO/CASH asset allocation to hit that 25% volatility. Here's a sample:

Code: Select all

START		END		 UPRO   / CASH			UPRO RETURN
Jan 2019	Jan 2019	 28.22% / 71.78%		 23.89%
Feb 2019	Feb 2019	 45.30% / 54.70%		  9.28%
Mar 2019	Mar 2019	100.00% / -		  	  4.85%
Apr 2019	Apr 2019	 71.32% / 28.68%		 11.81%
May 2019	May 2019	100.00% / -			-18.89%
Jun 2019	Jun 2019	 58.49% / 41.51%		 21.37%
I am trying to use the UPRO RETURN column to reproduce the UPRO/CASH column, and I'm failing. I told PV to use a 1 month volatility period. So the 1-month volatilities here, annualized, would be:

Code: Select all

Period      Annualized volatility
Jan - Feb   sqrt(23.89% ,  9.28%) * sqrt(12) = 35.79%
Feb - Mar   sqrt( 9.29% ,  4.85%) * sqrt(12) = 10.85%
Mar - Apr   sqrt( 4.85% , 11.81%) * sqrt(12) = 17.05%
Apr - May   sqrt(11.81% ,-18.89%) * sqrt(12) = 75.20%
Two of these are above the 25% volatility threshold and two are below. So I would expect that a target volatility of 25% would require that after Jan+Feb the UPRO allocation should be 25 / 35.79 = 69.86%, then 100%, then 100%, and finally 25 / 75.20 = 33.24%. But these values don't look anything like the allocations that PV is suggesting. So what am I missing?

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Re: Help me figure out the Target Volatility formula

Post by Uncorrelated » Fri Jan 31, 2020 6:34 am

The PV FAQ says:


Target volatility model

The target volatility model uses dynamic asset allocation to achieve a stable level of volatility. The model manages volatility by forecasting future equity volatility based on historic realized volatility and then dynamically adjusts the market exposure to target a set level of volatility. The historic realized volatility calculation assumes that the given portfolio allocation is maintained (no adjustment for asset allocation drift). The model calculates the realized portfolio volatility (annualized daily volatility) based on daily total returns, and then either increases or decreases the equity exposure of the portfolio to maintain the target risk level. If the predicted volatility of the portfolio is above the set target level, the model shifts assets to a risk free asset, typically cash, so that the predicted volatility is in line with the target volatility. For example, if the target annual volatility is 4% and the realized volatility is 6%, then the portfolio asset weights are multiplied by 40/60 and the remaining assets are allocated to the selected risk free asset. The strategy aims to sell assets when their risk-adjusted expected return is falling (rising market volatility) and buying equities when their risk-adjusted expected return is rising (falling market volatility) to provide better risk-adjusted portfolio returns and to account for investor's risk tolerance. However, volatility targeting tends to reduce equity exposure after a sharp market drop, which means the portfolio may miss the early part of the market recovery.

References:

Diana Barro, Elio Canestrelli, Fabio Lanza - Volatility vs. Downside Risk: Optimally Protecting Against Drawdowns and Maintaining Portfolio Performance
I digged into the reference but was unable to find anything of interest.



Although I think that target volatility is a big joke, if I were to implement something like this I would use VIX as my volatility forecast. If that wasn't available I would use the method from The Impact of Volatility Targeting which uses an exponentially decaying window, and then use re-balance bands instead of monthly rebalancing to adjust my asset allocation.

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Re: Help me figure out the Target Volatility formula

Post by danielc » Fri Jan 31, 2020 6:44 am

Uncorrelated wrote:
Fri Jan 31, 2020 6:34 am
The PV FAQ says:
...
I digged into the reference but was unable to find anything of interest.
Thanks! I really think that what I did should be equivalent to what the FAQ says, but I'll come back to this when I'm less tired. Maybe there's a subtle difference that I've missed. I'll also read the paper.

Uncorrelated wrote:
Fri Jan 31, 2020 6:34 am
Although I think that target volatility is a big joke, ...
I would love to know why you think that target volatility is a big joke. I tried some backtests. In most cases it doesn't seem to accomplish anything interesting, but when you do it with UPRO, using TMF as the "safe" asset (ha ha) it gives a really impressive result. That's why I'm interested.

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Re: Help me figure out the Target Volatility formula

Post by ocrtech » Fri Jan 31, 2020 8:22 am

danielc wrote:
Fri Jan 31, 2020 5:43 am
I am trying to reproduce the Target Volatility allocations from Portfolio Visualizer. Here is an example portfolio. I tried to keep it simple: There is only one risky asset (UPRO -- a 3x daily leveraged S&P 500 ETF) and the out-of-market asset is just cash. Since cash contributes virtually zero volatility, the entire volatility of the portfolio is produced by UPRO.

I set a target volatility of 25%, and PortfolioVisualizer reports on the monthly returns of UPRO and its recommended UPRO/CASH asset allocation to hit that 25% volatility. Here's a sample:

Code: Select all

START		END		 UPRO   / CASH			UPRO RETURN
Jan 2019	Jan 2019	 28.22% / 71.78%		 23.89%
Feb 2019	Feb 2019	 45.30% / 54.70%		  9.28%
Mar 2019	Mar 2019	100.00% / -		  	  4.85%
Apr 2019	Apr 2019	 71.32% / 28.68%		 11.81%
May 2019	May 2019	100.00% / -			-18.89%
Jun 2019	Jun 2019	 58.49% / 41.51%		 21.37%
I am trying to use the UPRO RETURN column to reproduce the UPRO/CASH column, and I'm failing. I told PV to use a 1 month volatility period. So the 1-month volatilities here, annualized, would be:

Code: Select all

Period      Annualized volatility
Jan - Feb   sqrt(23.89% ,  9.28%) * sqrt(12) = 35.79%
Feb - Mar   sqrt( 9.29% ,  4.85%) * sqrt(12) = 10.85%
Mar - Apr   sqrt( 4.85% , 11.81%) * sqrt(12) = 17.05%
Apr - May   sqrt(11.81% ,-18.89%) * sqrt(12) = 75.20%
Two of these are above the 25% volatility threshold and two are below. So I would expect that a target volatility of 25% would require that after Jan+Feb the UPRO allocation should be 25 / 35.79 = 69.86%, then 100%, then 100%, and finally 25 / 75.20 = 33.24%. But these values don't look anything like the allocations that PV is suggesting. So what am I missing?
I ran the numbers using a spreadsheet for Jan and Feb assuming a targeted volatility goal of 25%. Stdev for Jan is 3.55%, annualized volatility is 56.38%, and targeted volatility is 44.34%. Stdev for Feb is 15.63%, annualized is 24.82%, and targeted volatility is 100%. These numbers correspond well with PV shows in your first set of numbers.

It's not clear to me what you are trying to get at with your second set of numbers.

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Re: Help me figure out the Target Volatility formula

Post by Hydromod » Fri Jan 31, 2020 10:08 am

I think you are somehow using returns as a surrogate for volatility. PV doesn't report volatility, so you need to externally pull down the daily returns and calculate the standard deviations yourself.

For those looking for the Excel approach (consider Google Sheets too, which directly downloads returns):

Formulas for one-month lookback when
  • column A is UPRO daily adjusted price
  • column B is TMF daily adjusted price
  • one trading month (21 days + 1 end of last month) are rows 2 through 23
  • targU is a target annualized volatility
Construct new columns C and D for daily returns

C3 = (A3/A2) - 1, C4 = (A4/A3) - 1, ...
D3 = (B3/B2) - 1, D4 = (B4/B3) - 1, ...

If working with downward-only volatility (some like this measure) replace with

C3 = IF(A3<A2,"",A3-A2)

Calculate annualized volatility

SDU = stdev(C3:C23) * sqrt(252)
SDT = stdev(D3:D23) * sqrt(252)

(the sqrt(252) is only needed to match target volatility)

rSDU = 1/SDU
rSDT = 1/SDT

Risk parity (volatility):
fracU = rSDU/(rSDU + rSDT)

Risk parity (variance):
fracU = rSDU^2/(rSDU^2 + rSDT^2)

Target volatility:
fracU = min(targU/SDU, fracUmax)

where fracUmax is the maximum UPRO fraction allowed (e.g., 0.8)

In all cases
fracT = 1 - fracU

The risk parity formulas assume that UPRO and TMF are both assigned half of the risk budget. One can also adaptively change the risk budget, perhaps based on the unemployment rate index (UEI).

Assume the fraction of risk assigned to UPRO is riskfracU.

While UEI is falling (signal for non-recession), perhaps riskfracU might be 0.7 or 0.8, and while UEI is rising and accelerating (signal for recession), perhaps riskfracU might be 0 to 0.3. Adjusting riskfracU step by step over several months may lessen whipsaw.

Risk parity (volatility):
rSDU = riskfracU * rSDU

Risk parity (variance):
rSDU = sqrt(riskfracU) * rSDU


I think these are right, folks with more experience please confirm so I don't inadvertently mislead.

Edit: Of course I was wrong right and left to start. That's what I get for doing this off the top of my head. Thanks OCRtech. I've fixed the formulas above.
Last edited by Hydromod on Fri Jan 31, 2020 12:18 pm, edited 1 time in total.

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Re: Help me figure out the Target Volatility formula

Post by Forester » Fri Jan 31, 2020 11:38 am

Target vol tests better than equity trend even with two recent deep V bear markets which would be better for trend following, it seems to do a little better especially with more volatile assets.

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Re: Help me figure out the Target Volatility formula

Post by ocrtech » Fri Jan 31, 2020 11:52 am

Hydromod wrote:
Fri Jan 31, 2020 10:08 am

NDAY = 21
SDU = stdev(C3:C23) * sqrt(252/NDAY)
SDT = stdev(D3:D23) * sqrt(252/NDAY)

(the sqrt(252/NDAY) is only needed for target volatility)

rSDU = 1/SDU
rSDT = 1/SDT

Target volatility:
fracU = min(targU, rSDU/(rSDU + rSDT))
I've never seen Target Volatility calculated that way. I use:

C3 = A3/A2-1 and so on

fracU = targU/(stdev.s(c3:c23) * sqrt(252))

fracU is the percentage of dollars you would invest in UPRO.

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Re: Help me figure out the Target Volatility formula

Post by Hydromod » Fri Jan 31, 2020 12:03 pm

ocrtech wrote:
Fri Jan 31, 2020 11:52 am
Hydromod wrote:
Fri Jan 31, 2020 10:08 am

NDAY = 21
SDU = stdev(C3:C23) * sqrt(252/NDAY)
SDT = stdev(D3:D23) * sqrt(252/NDAY)

(the sqrt(252/NDAY) is only needed for target volatility)

rSDU = 1/SDU
rSDT = 1/SDT

Target volatility:
fracU = min(targU, rSDU/(rSDU + rSDT))
I've never seen Target Volatility calculated that way. I use:

C3 = A3/A2-1 and so on

fracU = targU/(stdev.s(c3:c23) * sqrt(252))

fracU is the percentage of dollars you would invest in UPRO.
Dang it, that's what I get when I do this of the top of my head. Of course. It was looking a little weird. I'll fix in my OP.

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Re: Help me figure out the Target Volatility formula

Post by danielc » Fri Jan 31, 2020 12:56 pm

ocrtech wrote:
Fri Jan 31, 2020 8:22 am
I ran the numbers using a spreadsheet for Jan and Feb assuming a targeted volatility goal of 25%. Stdev for Jan is 3.55%, annualized volatility is 56.38%, and targeted volatility is 44.34%. Stdev for Feb is 15.63%, annualized is 24.82%, and targeted volatility is 100%. These numbers correspond well with PV shows in your first set of numbers.

It's not clear to me what you are trying to get at with your second set of numbers.
Ok. How did you get those numbers? How did you get the Stdev for just January? If you enter just one month on PV the Stdev always says N/A so I figured that they must be using the monthly returns. So you need at least two to compute a standard deviation. That's where my second set of numbers came from.

EDIT: Yup... if I use daily returns I get the same numbers as you. Than you so much!
Last edited by danielc on Fri Jan 31, 2020 1:20 pm, edited 2 times in total.

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Re: Help me figure out the Target Volatility formula

Post by danielc » Fri Jan 31, 2020 1:15 pm

Hydromod wrote:
Fri Jan 31, 2020 10:08 am
I think you are somehow using returns as a surrogate for volatility. PV doesn't report volatility, so you need to externally pull down the daily returns and calculate the standard deviations yourself.

For those looking for the Excel approach (consider Google Sheets too, which directly downloads returns):

Formulas for one-month lookback when
  • column A is UPRO daily adjusted price
  • column B is TMF daily adjusted price
  • one trading month (21 days + 1 end of last month) are rows 2 through 23
  • targU is a target annualized volatility
Construct new columns C and D for daily returns

C3 = (A3/A2) - 1, C4 = (A4/A3) - 1, ...
D3 = (B3/B2) - 1, D4 = (B4/B3) - 1, ...

If working with downward-only volatility (some like this measure) replace with

C3 = IF(A3<A2,"",A3-A2)

Calculate annualized volatility

SDU = stdev(C3:C23) * sqrt(252)
SDT = stdev(D3:D23) * sqrt(252)

(the sqrt(252) is only needed to match target volatility)
Thank you for showing it step-by-step. What I was doing wrong is that I thought I was supposed to use the volatility of monthly returns. Notice that PV doesn't report a Stdev if you ask it to plot only 1 month. So I made a wrong assumption. Switching to daily returns, I can now get very close to the numbers reported by PV.

Hydromod wrote:
Fri Jan 31, 2020 10:08 am
rSDU = 1/SDU
rSDT = 1/SDT

Risk parity (volatility):
fracU = rSDU/(rSDU + rSDT)

Risk parity (variance):
fracU = rSDU^2/(rSDU^2 + rSDT^2)

Target volatility:
fracU = min(targU/SDU, fracUmax)

where fracUmax is the maximum UPRO fraction allowed (e.g., 0.8)

In all cases
fracT = 1 - fracU

The risk parity formulas assume that UPRO and TMF are both assigned half of the risk budget...
Thanks! This is interesting for me because, as in your example, my real plan is to mix UPRO + TMF rather than UPRO + cash. So I had been wondering about how to do that. I'll run some tests with risk parity. Do you have any thoughts on the merits of risk parity using volatility vs variance?

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Re: Help me figure out the Target Volatility formula

Post by Hydromod » Fri Jan 31, 2020 2:38 pm

danielc wrote:
Fri Jan 31, 2020 1:15 pm
Thanks! This is interesting for me because, as in your example, my real plan is to mix UPRO + TMF rather than UPRO + cash. So I had been wondering about how to do that. I'll run some tests with risk parity. Do you have any thoughts on the merits of risk parity using volatility vs variance?
I did a long series of examples in Refinements to Hedgefundie's excellent approach to try and get at this and similar questions.

I found that backtesting arguably worked best with downward-only variance with a 3-month lookback, but I wouldn't go so far as to say that this was necessarily the way to go moving forward.

The variance approach normally weights UPRO lower than the volatility approach except when things are going smooth. It usually didn't make too much difference exactly what the weights were, though, because TMF had such good returns over this period. I don't see that as still being the case going forward, so one might prefer the volatility approach in the future.

I just made up an example Google Sheet with all of these options. For a 12-week lookback, over which things were going well, I get

Symmetric UPRO TMF
risk par vol 0.597 0.403
risk par var 0.688 0.312
target vol 0.823 0.177

Downward-only
risk par vol 0.501 0.499
risk par var 0.502 0.498
target vol 1.000 0.000

For 5-week lookback, in which the last week or so is prominent,

Symmetric UPRO TMF
risk par vol 0.554 0.446
risk par var 0.606 0.394
target vol 0.662 0.338

Downward-only
risk par vol 0.333 0.667
risk par var 0.199 0.801
target vol 0.868 0.132

You can see how the variance one quickly gets pessimistic.

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Re: Help me figure out the Target Volatility formula

Post by danielc » Fri Jan 31, 2020 3:41 pm

Hydromod wrote:
Fri Jan 31, 2020 2:38 pm
I did a long series of examples in Refinements to Hedgefundie's excellent approach to try and get at this and similar questions.
Thanks! Reading it now.

The risk parity approach usually suggests a lower UPRO allocations than target volatility (as seen in your examples). I am interested in the idea of targeting volatility but I think that ignoring the volatility of TMF is stupid. So I'm experimenting with a modified target volatility that takes into account the variance and covariance of both assets:

Target Variance = f^2 * VAR(UPRO) + (1-f)^2 * VAR(TMF) + 2*f*(1-f)*COV(UPRO,TMF)

Where 'f' is the allocation of UPRO. For the covariance we can look at the sample covariance or assume a historical correlation.

What I like about target volatility is that it gives you a better sense of the risk of the portfolio. Unfortunately stock returns are not Gaussian; the distribution has fat tails and 3-sigma and 4-sigma events happen a lot more often than would be true for a Gaussian. But empirically we can note that the annual Stdev of stocks is around 15%, and the dot-com bubble/bust and the global financial crisis had max drop downs of 45% and 51% respectively. So empirically a 3.5-sigma event is probably close to the worst that can happen. So you can convert your target volatility into a "target maximum drop".

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Re: Help me figure out the Target Volatility formula

Post by ocrtech » Fri Jan 31, 2020 4:35 pm

If you set a max cap of UPRO at 60%, risk parity and target volatility (at any target) have approximately the same returns. It's only when you unbound UPRO and increase the targeted volatility past 17% that you start to see excess returns from the target volatility approach.

I originally thought about going the targeted volatility route but ultimately wasn't comfortable with having to have 100% of my dollars deployed in UPRO fairly frequently. Especially since this also increased the drawdowns by another 10 percentage points.

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Re: Help me figure out the Target Volatility formula

Post by ocrtech » Fri Jan 31, 2020 4:47 pm

danielc wrote:
Fri Jan 31, 2020 12:56 pm
ocrtech wrote:
Fri Jan 31, 2020 8:22 am
I ran the numbers using a spreadsheet for Jan and Feb assuming a targeted volatility goal of 25%. Stdev for Jan is 3.55%, annualized volatility is 56.38%, and targeted volatility is 44.34%. Stdev for Feb is 15.63%, annualized is 24.82%, and targeted volatility is 100%. These numbers correspond well with PV shows in your first set of numbers.

It's not clear to me what you are trying to get at with your second set of numbers.
Ok. How did you get those numbers? How did you get the Stdev for just January? If you enter just one month on PV the Stdev always says N/A so I figured that they must be using the monthly returns. So you need at least two to compute a standard deviation. That's where my second set of numbers came from.

EDIT: Yup... if I use daily returns I get the same numbers as you. Than you so much!
You can actually back into the stddev number from the PV numbers if you want. Take the Targeted Volatility number you originally entered into PV and divide that by the UPRO pecentage listed in the third column of the PV info. This will give you the annualized volatility. Take that number and divide it by sqrt(252) to get the standard deviation for the month.

As an example, for the month of Feb:

25% / 45.30% = 55.6% annualized volatility
55.6% / stddev(252) = 3.5% stddev for the month

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Re: Help me figure out the Target Volatility formula

Post by Uncorrelated » Fri Jan 31, 2020 4:53 pm

danielc wrote:
Fri Jan 31, 2020 6:44 am
Uncorrelated wrote:
Fri Jan 31, 2020 6:34 am
Although I think that target volatility is a big joke, ...
I would love to know why you think that target volatility is a big joke. I tried some backtests. In most cases it doesn't seem to accomplish anything interesting, but when you do it with UPRO, using TMF as the "safe" asset (ha ha) it gives a really impressive result. That's why I'm interested.

I've written a short piece a few days ago about my thoughts on volatility targeting after about half a day of research:
The whole target volatility thing looks like an absolute joke.

We only have 10 year of data available about UPRO/TMF and I see people making decisions based on market timing backtests. How large would the confidence bands on that backtest be?

There is a backtest that goes 30 years back in another thread. The entire backtest looks like random noise except 2008. The strategy basically hinges on a single data point. Since the strategy was developed after 2008, it's not even an out-of-sample data point.

Multiple posters are referencing academic literature, misunderstanding critical details of the literature, and replacing t-bills with bonds with an equivalent duration of 5000 years. You can't just extrapolate the research literature to a portfolio where the safe asset consists of a risky asset.

The paper I have in front of me right now claims that a target volatility approach increases the sharpe ratio on stocks from .40 to .50, assuming no trading costs, no friction costs. Simply replacing t-bills with total bond market results in a similar increase in sharpe ratio. The literature isn't exactly clear on whether target volatility improves risk adjusted returns in a realistic setting.

There have been zero attempts to quantify the trading costs of this strategy. When I checked portfoliovisualizer, the annual turnover was around 6, this can easily result in trading costs up to 1% per year even if you know what you are doing. I've seen academics use figures up to 0.8% (multiplied by the turnover) to estimate friction costs.

If the strategy doesn't work (i.e. the signals are random noise), you lose utility because randomly switching between two asset allocations is worse than holding a constant asset allocation. That means that if the strategy is good enough to overcome friction costs, it still doesn't mean it's better than a constant asset allocation.

Target volatility attempts to keep the volatility constant. i.e. the optimal allocation to the risky asset is linear to 1/stddev. However if you solve for maximum utility under a CRRA, you will find that the optimal stock allocation is linear to 1/stddev^2 :oops:

On the Optimality of Target Volatility Strategies says:
Recent studies have focused on proving that periods of high volatility are associated with negative returns and vice versa. However, there is less evidence on the negative link between realized volatility and expected returns. Glosten, Jagannathan, and Runkle [1993] offered reviews of studies showing both positive and negative relationships between risk and return; they also provided evidence of the absence of volatility persistence of monthly volatility.
[...]
Here, the results are in line with the estimated correlation of –0.43 between realized volatility and realized returns and 0% between realized volatility and expected returns.
Researchers can't agree whether or not there is persistence in monthly volatility, and also can't agree whether the relation between volatility and expected returns is positive or negative. They don't know nothing.

The same paper also claims that if you use target volatility on an efficient portfolio, the resulting portfolio isn't necessarily efficient. This is problematic because it means conventional finance is unsuited for selecting the correct mix of asset if you use a timing model.

The only reasonable approach in face of such uncertainty is to buy low-cost index funds and lean back.
There are so many things wrong. I really don't understand why this strategy is being taken seriously by anyone. If active managers can't beat the stock market, why do you think you can with three lines of code?

Backtests are only useful as a tool to deceive people. There is absolutely zero chance that somebody who shows you a PV timing model didn't overfit the data massively.

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Re: Help me figure out the Target Volatility formula

Post by danielc » Fri Jan 31, 2020 6:07 pm

ocrtech wrote:
Fri Jan 31, 2020 4:35 pm
If you set a max cap of UPRO at 60%, risk parity and target volatility (at any target) have approximately the same returns. It's only when you unbound UPRO and increase the targeted volatility past 17% that you start to see excess returns from the target volatility approach.

I originally thought about going the targeted volatility route but ultimately wasn't comfortable with having to have 100% of my dollars deployed in UPRO fairly frequently. Especially since this also increased the drawdowns by another 10 percentage points.
I'm thinking of putting a UPRO cap at 90%. I am also thinking of including TMF in the target volatility calculation. The upshot is somewhere between risk parity and the target volatility values from PV. For example, to continue using January 2019 as an example:

* PV Target Volatility = 25% (i.e. ignoring TMF) ----> UPRO = 44.3%
* Target volatility using UPRO + TMF with correlation = -0.1 ----> UPRO = 37.5%
* Risk parity using Volatility ----> UPRO = 30.7%

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Re: Help me figure out the Target Volatility formula

Post by danielc » Fri Jan 31, 2020 6:19 pm

Uncorrelated wrote:
Fri Jan 31, 2020 4:53 pm
danielc wrote:
Fri Jan 31, 2020 6:44 am
I would love to know why you think that target volatility is a big joke. I tried some backtests. In most cases it doesn't seem to accomplish anything interesting, but when you do it with UPRO, using TMF as the "safe" asset (ha ha) it gives a really impressive result. That's why I'm interested.

I've written a short piece a few days ago about my thoughts on volatility targeting after about half a day of research:
...
There are so many things wrong. I really don't understand why this strategy is being taken seriously by anyone. If active managers can't beat the stock market, why do you think you can with three lines of code?

Backtests are only useful as a tool to deceive people. There is absolutely zero chance that somebody who shows you a PV timing model didn't overfit the data massively.
Ah. I remember that post. Thank you. I agree in particular that it's dumb to take research based on T-bills as the safe asset and just replace that with 3X daily leveraged long treasuries (!!!!!). But assuming that we fix the volatility calculation to account for the variance of both assets and their correlation, I think that there's a lot of merit to the idea of keeping the volatility constant, even it doesn't actually increase the Sharpe ratio.

Regarding the issue about there being only one data point... I think that there are three data points. You can see the target volatility doing its thing in 1987, 2000, and 2008. Still, three data points is not a lot, so we need to take this with a huge grain of salt.

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Re: Help me figure out the Target Volatility formula

Post by Hydromod » Fri Jan 31, 2020 6:24 pm

The 1987 event was so sudden that it can't be considered as a test of any of these strategies.

Whether or not a strategy handles it during backtesting is purely due to the luck of the assumed starting date.

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Re: Help me figure out the Target Volatility formula

Post by ocrtech » Fri Jan 31, 2020 7:24 pm

danielc wrote:
Fri Jan 31, 2020 6:07 pm
ocrtech wrote:
Fri Jan 31, 2020 4:35 pm
If you set a max cap of UPRO at 60%, risk parity and target volatility (at any target) have approximately the same returns. It's only when you unbound UPRO and increase the targeted volatility past 17% that you start to see excess returns from the target volatility approach.

I originally thought about going the targeted volatility route but ultimately wasn't comfortable with having to have 100% of my dollars deployed in UPRO fairly frequently. Especially since this also increased the drawdowns by another 10 percentage points.
I'm thinking of putting a UPRO cap at 90%. I am also thinking of including TMF in the target volatility calculation. The upshot is somewhere between risk parity and the target volatility values from PV. For example, to continue using January 2019 as an example:

* PV Target Volatility = 25% (i.e. ignoring TMF) ----> UPRO = 44.3%
* Target volatility using UPRO + TMF with correlation = -0.1 ----> UPRO = 37.5%
* Risk parity using Volatility ----> UPRO = 30.7%
Risk Parity incorporates the volatility of both UPRO and and TMF into dynamically determining the allocation of each. What do you believe adding the correlation of their variance will do?

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Re: Help me figure out the Target Volatility formula

Post by bogglizer » Fri Jan 31, 2020 8:11 pm

danielc wrote:
Fri Jan 31, 2020 3:41 pm
What I like about target volatility is that it gives you a better sense of the risk of the portfolio. Unfortunately stock returns are not Gaussian; the distribution has fat tails and 3-sigma and 4-sigma events happen a lot more often than would be true for a Gaussian. But empirically we can note that the annual Stdev of stocks is around 15%, and the dot-com bubble/bust and the global financial crisis had max drop downs of 45% and 51% respectively. So empirically a 3.5-sigma event is probably close to the worst that can happen. So you can convert your target volatility into a "target maximum drop".
If you add fat tails into the mix, then the variance is infinite and the entire approach is bogus.

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Re: Help me figure out the Target Volatility formula

Post by danielc » Fri Jan 31, 2020 11:18 pm

bogglizer wrote:
Fri Jan 31, 2020 8:11 pm
If you add fat tails into the mix, then the variance is infinite and the entire approach is bogus.
1) The variance of a finite sample of finite values is always defined and finite.

2) We've known for a very long time that stocks are not Gaussians, and that hasn't stopped anyone from computing standard deviations. All we can do is hope that the reader knows that computing a standard deviation does not magically turn every distribution into a Gaussian. For a non-Gaussian distribution, the Stdev is just another ad-hoc measure of width, just like MAD, FWHM, IQR, and others.

3) Not that it matters, but fat tails do not automatically require that the variance has to diverge. This may be true for some fat tail distributions, like the Cauchy distribution, but it is in no way a universal fact. For example the Laplace distribution has fatter tails than a Gaussian and its variance is perfectly well defined and finite.

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Re: Help me figure out the Target Volatility formula

Post by danielc » Fri Jan 31, 2020 11:35 pm

ocrtech wrote:
Fri Jan 31, 2020 7:24 pm
danielc wrote:
Fri Jan 31, 2020 6:07 pm
I'm thinking of putting a UPRO cap at 90%. I am also thinking of including TMF in the target volatility calculation. The upshot is somewhere between risk parity and the target volatility values from PV. For example, to continue using January 2019 as an example:

* PV Target Volatility = 25% (i.e. ignoring TMF) ----> UPRO = 44.3%
* Target volatility using UPRO + TMF with correlation = -0.1 ----> UPRO = 37.5%
* Risk parity using Volatility ----> UPRO = 30.7%
Risk Parity incorporates the volatility of both UPRO and and TMF into dynamically determining the allocation of each. What do you believe adding the correlation of their variance will do?
The problem with risk parity is that there is no fundamental reason why spreading the risk equally should be the goal. If two assets, "A" and "B", have the exact same volatility, but "A" has higher returns, I will want to weigh "A" more than "B". Risk parity would force equal weights. As for what you gain with correlations, the only reason why I would want to have any "B" at all is to the extent that "B" is not perfectly correlated with "A". If I'm lucky, maybe "B" is even negatively correlated. The less correlated (or the more negatively correlated) the more B I want to have.

In the UPRO/TMF adventure portfolio, UPRO is meant to be the main source of return, but TMF is included purely because of its negative correlation in bad times. The correlation is the only reason why TMF is included at all. If I could forecast the returns and covariance matrix accurately, I could come up with a mean variance optimum portfolio. But I can't do that. I just think UPRO will return more than TMF. So I'm considering a strategy where I prefer UPRO over TMF to the extent permitted by the additional goal of not allowing the overall volatility to rise above (say) 25%. That means that I need to estimate the volatility of the overall portfolio. To do that correctly, I need the covariance.

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Re: Help me figure out the Target Volatility formula

Post by bogglizer » Sat Feb 01, 2020 3:03 am

danielc wrote:
Fri Jan 31, 2020 11:18 pm
bogglizer wrote:
Fri Jan 31, 2020 8:11 pm
If you add fat tails into the mix, then the variance is infinite and the entire approach is bogus.
1) The variance of a finite sample of finite values is always defined and finite.

2) We've known for a very long time that stocks are not Gaussians, and that hasn't stopped anyone from computing standard deviations. All we can do is hope that the reader knows that computing a standard deviation does not magically turn every distribution into a Gaussian. For a non-Gaussian distribution, the Stdev is just another ad-hoc measure of width, just like MAD, FWHM, IQR, and others.

3) Not that it matters, but fat tails do not automatically require that the variance has to diverge. This may be true for some fat tail distributions, like the Cauchy distribution, but it is in no way a universal fact. For example the Laplace distribution has fatter tails than a Gaussian and its variance is perfectly well defined and finite.
1) The underlying distribution always has infinite variance. Variance is a flawed measure of volatility.

2) Yes, people can always go off and compute stuff. And they do. But variance goes to infinity, always, as the samples increase. So estimators of variance will do so as well.

3) All stable distributions except the Gaussian have infinite variance. If you are using a non-stable distribution to fit a financial time series, then you have bigger problems. Of course, university professors write papers on such things, since they have to produce something.

Although wikipedia is not the greatest source of knowledge, the article on stable distributions has all this.

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Re: Help me figure out the Target Volatility formula

Post by 305pelusa » Sat Feb 01, 2020 10:19 am

Uncorrelated wrote:
Fri Jan 31, 2020 4:53 pm
danielc wrote:
Fri Jan 31, 2020 6:44 am
Uncorrelated wrote:
Fri Jan 31, 2020 6:34 am
Although I think that target volatility is a big joke, ...
I would love to know why you think that target volatility is a big joke. I tried some backtests. In most cases it doesn't seem to accomplish anything interesting, but when you do it with UPRO, using TMF as the "safe" asset (ha ha) it gives a really impressive result. That's why I'm interested.

I've written a short piece a few days ago about my thoughts on volatility targeting after about half a day of research:
The whole target volatility thing looks like an absolute joke.

We only have 10 year of data available about UPRO/TMF and I see people making decisions based on market timing backtests. How large would the confidence bands on that backtest be?

There is a backtest that goes 30 years back in another thread. The entire backtest looks like random noise except 2008. The strategy basically hinges on a single data point. Since the strategy was developed after 2008, it's not even an out-of-sample data point.

Multiple posters are referencing academic literature, misunderstanding critical details of the literature, and replacing t-bills with bonds with an equivalent duration of 5000 years. You can't just extrapolate the research literature to a portfolio where the safe asset consists of a risky asset.

The paper I have in front of me right now claims that a target volatility approach increases the sharpe ratio on stocks from .40 to .50, assuming no trading costs, no friction costs. Simply replacing t-bills with total bond market results in a similar increase in sharpe ratio. The literature isn't exactly clear on whether target volatility improves risk adjusted returns in a realistic setting.

There have been zero attempts to quantify the trading costs of this strategy. When I checked portfoliovisualizer, the annual turnover was around 6, this can easily result in trading costs up to 1% per year even if you know what you are doing. I've seen academics use figures up to 0.8% (multiplied by the turnover) to estimate friction costs.

If the strategy doesn't work (i.e. the signals are random noise), you lose utility because randomly switching between two asset allocations is worse than holding a constant asset allocation. That means that if the strategy is good enough to overcome friction costs, it still doesn't mean it's better than a constant asset allocation.

Target volatility attempts to keep the volatility constant. i.e. the optimal allocation to the risky asset is linear to 1/stddev. However if you solve for maximum utility under a CRRA, you will find that the optimal stock allocation is linear to 1/stddev^2 :oops:

On the Optimality of Target Volatility Strategies says:
Recent studies have focused on proving that periods of high volatility are associated with negative returns and vice versa. However, there is less evidence on the negative link between realized volatility and expected returns. Glosten, Jagannathan, and Runkle [1993] offered reviews of studies showing both positive and negative relationships between risk and return; they also provided evidence of the absence of volatility persistence of monthly volatility.
[...]
Here, the results are in line with the estimated correlation of –0.43 between realized volatility and realized returns and 0% between realized volatility and expected returns.
Researchers can't agree whether or not there is persistence in monthly volatility, and also can't agree whether the relation between volatility and expected returns is positive or negative. They don't know nothing.

The same paper also claims that if you use target volatility on an efficient portfolio, the resulting portfolio isn't necessarily efficient. This is problematic because it means conventional finance is unsuited for selecting the correct mix of asset if you use a timing model.

The only reasonable approach in face of such uncertainty is to buy low-cost index funds and lean back.
There are so many things wrong. I really don't understand why this strategy is being taken seriously by anyone. If active managers can't beat the stock market, why do you think you can with three lines of code?

Backtests are only useful as a tool to deceive people. There is absolutely zero chance that somebody who shows you a PV timing model didn't overfit the data massively.
You and I don’t always agree on really nit picky things but boy do I agree with you on the important things like this. I also have no clue how this volatility targeting either UPRO, and then filling the rest with TMF, makes any sense 0_o

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Re: Help me figure out the Target Volatility formula

Post by kim.gold » Sat Feb 01, 2020 3:35 pm

You guys are doing great work here. Can anyone share the spreadsheet? It would save a lot of time for testers and we could get a lot of feedback.

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Re: Help me figure out the Target Volatility formula

Post by danielc » Sat Feb 01, 2020 7:49 pm

bogglizer wrote:
Sat Feb 01, 2020 3:03 am
1) The underlying distribution always has infinite variance. Variance is a flawed measure of volatility.
You know perfectly well that there are tons of distributions that don't have infinite variance. You know that what you wrote is not true.

bogglizer wrote:
Sat Feb 01, 2020 3:03 am
2) Yes, people can always go off and compute stuff. And they do. But variance goes to infinity, always, as the samples increase. So estimators of variance will do so as well.
Variance estimators do not go to infinity if the underlying distribution has finite variance. You know this. You made a ridiculous blanket statement ("If you add fat tails into the mix, then the variance is infinite") that you know perfectly well is not true.

bogglizer wrote:
Sat Feb 01, 2020 3:03 am
3) All stable distributions except the Gaussian have infinite variance. If you are using a non-stable distribution to fit a financial time series, then you have bigger problems. Of course, university professors write papers on such things, since they have to produce something.
First, you just inserted a new condition that you didn't have in your original post. You realized that you were wrong so you moved the goal post. Second, you don't get to decide that stocks have to follow a stable distribution. For example, stocks are often modeled as following a log-normal distribution. It is an empirical fact that stock market returns produce extreme events a lot more often than if they followed a Gaussian distribution. If that doesn't fit your model of how stocks should behave, the problem is with your model, not with reality.

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Re: Help me figure out the Target Volatility formula

Post by danielc » Sat Feb 01, 2020 7:56 pm

305pelusa wrote:
Sat Feb 01, 2020 10:19 am
Uncorrelated wrote:
Fri Jan 31, 2020 4:53 pm
...
There are so many things wrong. I really don't understand why this strategy is being taken seriously by anyone. If active managers can't beat the stock market, why do you think you can with three lines of code?

Backtests are only useful as a tool to deceive people. There is absolutely zero chance that somebody who shows you a PV timing model didn't overfit the data massively.
You and I don’t always agree on really nit picky things but boy do I agree with you on the important things like this. I also have no clue how this volatility targeting either UPRO, and then filling the rest with TMF, makes any sense 0_o

Question: Do you think that any kind of volatility targeting is bogus, or just the fact that people are ignoring the volatility of TMF and mis-applying academic papers that were based on T-bills?

I think that the idea of targeting volatility has an intuitive appeal in that it is kind of like targetting max draw downs. I'm not saying that there aren't flaws in the idea, but I haven't seen an alternative that obviously has fewer flaws. For example, risk parity is popular, but I think it's dumb that it ignores the correlation between assets, or the idea that UPRO is supposed to produce more returns than TMF.

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Re: Help me figure out the Target Volatility formula

Post by 305pelusa » Sat Feb 01, 2020 8:00 pm

danielc wrote:
Sat Feb 01, 2020 7:56 pm
305pelusa wrote:
Sat Feb 01, 2020 10:19 am
Uncorrelated wrote:
Fri Jan 31, 2020 4:53 pm
...
There are so many things wrong. I really don't understand why this strategy is being taken seriously by anyone. If active managers can't beat the stock market, why do you think you can with three lines of code?

Backtests are only useful as a tool to deceive people. There is absolutely zero chance that somebody who shows you a PV timing model didn't overfit the data massively.
You and I don’t always agree on really nit picky things but boy do I agree with you on the important things like this. I also have no clue how this volatility targeting either UPRO, and then filling the rest with TMF, makes any sense 0_o

Question: Do you think that any kind of volatility targeting is bogus, or just the fact that people are ignoring the volatility of TMF and mis-applying academic papers that were based on T-bills?
The latter.

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Re: Help me figure out the Target Volatility formula

Post by bogglizer » Sat Feb 01, 2020 9:25 pm

danielc wrote:
Sat Feb 01, 2020 7:49 pm
bogglizer wrote:
Sat Feb 01, 2020 3:03 am
1) The underlying distribution always has infinite variance. Variance is a flawed measure of volatility.
You know perfectly well that there are tons of distributions that don't have infinite variance. You know that what you wrote is not true.

bogglizer wrote:
Sat Feb 01, 2020 3:03 am
2) Yes, people can always go off and compute stuff. And they do. But variance goes to infinity, always, as the samples increase. So estimators of variance will do so as well.
Variance estimators do not go to infinity if the underlying distribution has finite variance. You know this. You made a ridiculous blanket statement ("If you add fat tails into the mix, then the variance is infinite") that you know perfectly well is not true.

bogglizer wrote:
Sat Feb 01, 2020 3:03 am
3) All stable distributions except the Gaussian have infinite variance. If you are using a non-stable distribution to fit a financial time series, then you have bigger problems. Of course, university professors write papers on such things, since they have to produce something.
First, you just inserted a new condition that you didn't have in your original post. You realized that you were wrong so you moved the goal post. Second, you don't get to decide that stocks have to follow a stable distribution. For example, stocks are often modeled as following a log-normal distribution. It is an empirical fact that stock market returns produce extreme events a lot more often than if they followed a Gaussian distribution. If that doesn't fit your model of how stocks should behave, the problem is with your model, not with reality.
Financial instruments like stocks are multiplicative, so are log distributed. A company going bankrupt is always a possibility, and this is a change to a value of zero, which in log-space, will always make the variance infinite. Unless your personal religious deity guarantees that loss of capital is impossible, the variance is infinite. Even a credit default swap won't save you if the insurance company goes under.

All sums of variables converge to a small subset of distributions, called attractors. These distributions are the stable distributions. If the original variables had finite variance, then the sum will converge to a Gaussian, known as the Central Limit Theorem. However, A sum of variables with infinite variance (like all non-guaranteed financial instruments) will have heavy tails, and the distribution will also have infinite variance.

People don't assume Gaussian because it is right; they assume it because it is easy to compute with. It is still wrong.

I wanted to understand this and so I went and got a masters degree in it. For fun. I have entire textbooks right now on my desk about this exact subject. I am done here.

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Re: Help me figure out the Target Volatility formula

Post by Uncorrelated » Sun Feb 02, 2020 5:08 am

danielc wrote:
Sat Feb 01, 2020 7:56 pm
I think that the idea of targeting volatility has an intuitive appeal in that it is kind of like targetting max draw downs. I'm not saying that there aren't flaws in the idea, but I haven't seen an alternative that obviously has fewer flaws. For example, risk parity is popular, but I think it's dumb that it ignores the correlation between assets, or the idea that UPRO is supposed to produce more returns than TMF.
With risk parity you're also required to make arbitrary choices of asset classes. For example, some use gold in their risk parity portfolio, and some use multiple types of bonds or equities.

Do yourself a favor and download Forecasting the Equity Risk Premium: The Role of Technical Indicators (2014 revision), go to chapter 4 "Asset Allocation Exercise" and take a good look at the first 3 equations. Then try to reverse engineer which assumptions you need to make before the equations match what target target volatility is supposed to be doing. Does it match your assumptions? Does empirical data match these assumptions? Feel free to ask if you're not getting anywhere.

The appeal of risk parity is the same as for target volatility. It sounds intuitive, but the underlying theory is bogus. For it to make any sense you need to make very specific and weird assumptions. I'm very disappointed that the target volatility literature doesn't specify these assumptions. (somewhat related: portfolio optimization humor)

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Re: Help me figure out the Target Volatility formula

Post by ocrtech » Sun Feb 02, 2020 8:43 am

danielc wrote:
Fri Jan 31, 2020 6:07 pm
ocrtech wrote:
Fri Jan 31, 2020 4:35 pm
If you set a max cap of UPRO at 60%, risk parity and target volatility (at any target) have approximately the same returns. It's only when you unbound UPRO and increase the targeted volatility past 17% that you start to see excess returns from the target volatility approach.

I originally thought about going the targeted volatility route but ultimately wasn't comfortable with having to have 100% of my dollars deployed in UPRO fairly frequently. Especially since this also increased the drawdowns by another 10 percentage points.
I'm thinking of putting a UPRO cap at 90%. I am also thinking of including TMF in the target volatility calculation. The upshot is somewhere between risk parity and the target volatility values from PV. For example, to continue using January 2019 as an example:

* PV Target Volatility = 25% (i.e. ignoring TMF) ----> UPRO = 44.3%
* Target volatility using UPRO + TMF with correlation = -0.1 ----> UPRO = 37.5%
* Risk parity using Volatility ----> UPRO = 30.7%
Take a look at incorporating Risk Budgets into Risk Parity. It utilizes covariance matrices to "influence" the percentage of each asset under Risk Parity. If you are only dealing with two assets (UPRO and TMF), the calculations are pretty straight forward. If you move towards Asset Classes from just the two assets, the calculations become more complex.

I think @Hydromod presented some results in his OP thread. I also did some initial backtesting incorporating both Risk Budgets and Asset Classes and saw slightly higher returns with slightly lower drawdowns but decided the extra time spent running the calculations every month wasn't worth my time.

Edit: This paper provides some decent reading on this approach https://papers.ssrn.com/sol3/papers.cfm ... id=2297383

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Re: Help me figure out the Target Volatility formula

Post by danielc » Sun Feb 02, 2020 10:02 am

ocrtech wrote:
Sun Feb 02, 2020 8:43 am
Take a look at incorporating Risk Budgets into Risk Parity. It utilizes covariance matrices to "influence" the percentage of each asset under Risk Parity. If you are only dealing with two assets (UPRO and TMF), the calculations are pretty straight forward. If you move towards Asset Classes from just the two assets, the calculations become more complex.

I think @Hydromod presented some results in his OP thread. I also did some initial backtesting incorporating both Risk Budgets and Asset Classes and saw slightly higher returns with slightly lower drawdowns but decided the extra time spent running the calculations every month wasn't worth my time.

Edit: This paper provides some decent reading on this approach https://papers.ssrn.com/sol3/papers.cfm ... id=2297383
Thanks!

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Re: Help me figure out the Target Volatility formula

Post by danielc » Sun Feb 02, 2020 10:12 am

bogglizer wrote:
Sat Feb 01, 2020 9:25 pm
Financial instruments like stocks are multiplicative, so are log distributed. A company going bankrupt is always a possibility, and this is a change to a value of zero, which in log-space, will always make the variance infinite. Unless your personal religious deity guarantees that loss of capital is impossible, the variance is infinite. Even a credit default swap won't save you if the insurance company goes under.
How did you go from talking about stable distributions to deities?

bogglizer wrote:
Sat Feb 01, 2020 9:25 pm
People don't assume Gaussian because it is right; they assume it because it is easy to compute with. It is still wrong.
I can't imagine what part of my posts made you think that I or anyone here thinks that Gaussians are "right". This whole discussion started because you took issue with the post where I said that stock returns are not Gaussian.

bogglizer wrote:
Sat Feb 01, 2020 9:25 pm
I wanted to understand this and so I went and got a masters degree in it. For fun. I have entire textbooks right now on my desk about this exact subject. I am done here.
A lot of people in this forum, including myself, have advanced degrees. Bragging about them does not help further the discussion. In any case, I am glad to see the end of this sub-discussion that you initiated. I don't think it was very productive. I hope you have a wonderful day.

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Re: Help me figure out the Target Volatility formula

Post by danielc » Sun Feb 02, 2020 12:29 pm

ocrtech wrote:
Sun Feb 02, 2020 8:43 am
Take a look at incorporating Risk Budgets into Risk Parity. It utilizes covariance matrices to "influence" the percentage of each asset under Risk Parity. If you are only dealing with two assets (UPRO and TMF), the calculations are pretty straight forward. If you move towards Asset Classes from just the two assets, the calculations become more complex.

I think @Hydromod presented some results in his OP thread. I also did some initial backtesting incorporating both Risk Budgets and Asset Classes and saw slightly higher returns with slightly lower drawdowns but decided the extra time spent running the calculations every month wasn't worth my time.

Edit: This paper provides some decent reading on this approach https://papers.ssrn.com/sol3/papers.cfm ... id=2297383
So I read the paper. While it's nice that it takes into account correlations, you still have to decide the fraction of the portfolio risk you want to assign to each asset. There's no guidance on how to do that. I need to think more about this.

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Re: Help me figure out the Target Volatility formula

Post by Hydromod » Sun Feb 02, 2020 1:06 pm

danielc wrote:
Sun Feb 02, 2020 12:29 pm
ocrtech wrote:
Sun Feb 02, 2020 8:43 am
Take a look at incorporating Risk Budgets into Risk Parity. It utilizes covariance matrices to "influence" the percentage of each asset under Risk Parity. If you are only dealing with two assets (UPRO and TMF), the calculations are pretty straight forward. If you move towards Asset Classes from just the two assets, the calculations become more complex.

I think @Hydromod presented some results in his OP thread. I also did some initial backtesting incorporating both Risk Budgets and Asset Classes and saw slightly higher returns with slightly lower drawdowns but decided the extra time spent running the calculations every month wasn't worth my time.

Edit: This paper provides some decent reading on this approach https://papers.ssrn.com/sol3/papers.cfm ... id=2297383
So I read the paper. While it's nice that it takes into account correlations, you still have to decide the fraction of the portfolio risk you want to assign to each asset. There's no guidance on how to do that. I need to think more about this.
One simple method would be to use the unemployment index to set risk budget for equities. Others have augmented this index with additional indicators.

I would let it ride from some maximum less than one (e.g., 0.8) to zero. One could increment the risk fraction for equities each month. I lean to decrementing by 0.25 when the index is rising and accelerating, incrementing by 0.1 to 0.25 otherwise, with a single month pause when the signal changes. The change from acceleration to deceleration seems to have timed reasonably closely to the bottom of the market. But this is only based on 1987 to present.

So most of a bull market the risk budget for equities might ride at 0.8, with it changing only in response to bears.

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Re: Help me figure out the Target Volatility formula

Post by ocrtech » Sun Feb 02, 2020 3:26 pm

danielc wrote:
Sun Feb 02, 2020 12:29 pm
ocrtech wrote:
Sun Feb 02, 2020 8:43 am
Take a look at incorporating Risk Budgets into Risk Parity. It utilizes covariance matrices to "influence" the percentage of each asset under Risk Parity. If you are only dealing with two assets (UPRO and TMF), the calculations are pretty straight forward. If you move towards Asset Classes from just the two assets, the calculations become more complex.

I think @Hydromod presented some results in his OP thread. I also did some initial backtesting incorporating both Risk Budgets and Asset Classes and saw slightly higher returns with slightly lower drawdowns but decided the extra time spent running the calculations every month wasn't worth my time.

Edit: This paper provides some decent reading on this approach https://papers.ssrn.com/sol3/papers.cfm ... id=2297383
So I read the paper. While it's nice that it takes into account correlations, you still have to decide the fraction of the portfolio risk you want to assign to each asset. There's no guidance on how to do that. I need to think more about this.
It seems like I'm not understanding what you are looking for. The approach I had researched and implemented was as follows:

1. Select the group of ETF funds that I was interested in using.
2. Run an algorithm that utilizes a covariance matrix to bin these ETFs into asset classes.
3. Within each class, apply a Risk Parity approach to determine the percentage of each ETF assigned to each class.
4. Determine the weighted variance of each class.
5. Apply Risk Parity to the group of weighted variances generated in step 4.

The end result is an allocation percentage of investment applied to each ETF. In this approach all allocations were determined by the process.

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Re: Help me figure out the Target Volatility formula

Post by Stump909 » Tue Feb 04, 2020 11:46 am

Uncorrelated wrote:
Fri Jan 31, 2020 4:53 pm

There is a backtest that goes 30 years back in another thread. The entire backtest looks like random noise except 2008. The strategy basically hinges on a single data point. Since the strategy was developed after 2008, it's not even an out-of-sample data point.
Surely there's more than one event worth reviewing


If I run a quick check of the proposed 18% TV strategy, against the SP500, there's a handful of historical moments where it appears TV did better during certain drawdown periods:

1. Black Monday (-21% vs -30%)

2. Dotcom Crash (-31% vs -45%)

3. 2008 Crash (-38% vs -51%)

4. Oct. '18 -Dec '18 (-10% v -14%)

Even though backtesting has its flaws, it's still one of the best options we have to aquire evidence. It's the reason target date funds are allocated the way they are, and the data can be extended longer than the funds have been open, by multiplying daily returns by 3 (once tracking error has been established as acceptable).

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Re: Help me figure out the Target Volatility formula

Post by Uncorrelated » Wed Feb 05, 2020 6:12 am

Stump909 wrote:
Tue Feb 04, 2020 11:46 am
Even though backtesting has its flaws, it's still one of the best options we have to aquire evidence. It's the reason target date funds are allocated the way they are, and the data can be extended longer than the funds have been open, by multiplying daily returns by 3 (once tracking error has been established as acceptable).
Glidepaths are the way they are due to Merton (1969). Vanguard does not use backtests to determine target date funds, they use a combination of mathematical modeling and monte carlo simulations (Vanguard Capital Markets Model). Conveniently, the equations from Merton are the exact equations I recommend for implementing a market timing algorithm such as TV.

If we had 100000+ monthly data points, I would agree that backtesting is a good tool. But we only have around 500 data points. To indicate what we can do with 500 data points, I think it's sufficient to conclude with 95% certainty that t-bills have lower expected returns than stocks. I don't think it's enough to conclude that stocks, long term bonds or gold have different average returns.

Rather than cherry picking data points from a carefully overfitted backtest, don't you think it would be a much better idea to figure out which assumptions are required in order to make TV work. And then test if those assumptions are true in actual data? You're in luck because I already did that: Merton's equation agrees with target volatility if predicted return follows the equation a + b * predicted_volatility, where a = 0. This is easily testable: if you look at empirical data, you will find that a != 0 and b very close to zero (academics can't even seem to agree on it's sign). You would never figure this out by running backtests.

I mentioned earlier that backtests are only useful as a tool to deceive. That wasn't a joke. It is extremely difficult to backtest correctly and even more difficult to interprent, the entire active management industry is build around misleading people with benchmarks.

Stump909
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Re: Help me figure out the Target Volatility formula

Post by Stump909 » Wed Feb 05, 2020 11:40 am

Uncorrelated wrote:
Wed Feb 05, 2020 6:12 am


I mentioned earlier that backtests are only useful as a tool to deceive. That wasn't a joke. It is extremely difficult to backtest correctly and even more difficult to interprent, the entire active management industry is build around misleading people with benchmarks.
So are simply against allocations, based on volatility signaling, or leveraged investing as a whole?

Uncorrelated
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Re: Help me figure out the Target Volatility formula

Post by Uncorrelated » Wed Feb 05, 2020 4:12 pm

Stump909 wrote:
Wed Feb 05, 2020 11:40 am
Uncorrelated wrote:
Wed Feb 05, 2020 6:12 am


I mentioned earlier that backtests are only useful as a tool to deceive. That wasn't a joke. It is extremely difficult to backtest correctly and even more difficult to interprent, the entire active management industry is build around misleading people with benchmarks.
So are simply against allocations, based on volatility signaling, or leveraged investing as a whole?
Volatility signaling, market timing and leverage are tools that can be used with great success. The problem is that the method (backtesting) is too error prone to evaluate these tools correctly, and that Target Volatility depends on strange assumptions that are provably false.

Stump909
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Re: Help me figure out the Target Volatility formula

Post by Stump909 » Wed Feb 05, 2020 5:49 pm

Uncorrelated wrote:
Wed Feb 05, 2020 4:12 pm
Stump909 wrote:
Wed Feb 05, 2020 11:40 am
Uncorrelated wrote:
Wed Feb 05, 2020 6:12 am


I mentioned earlier that backtests are only useful as a tool to deceive. That wasn't a joke. It is extremely difficult to backtest correctly and even more difficult to interprent, the entire active management industry is build around misleading people with benchmarks.
So are simply against allocations, based on volatility signaling, or leveraged investing as a whole?
Volatility signaling, market timing and leverage are tools that can be used with great success. The problem is that the method (backtesting) is too error prone to evaluate these tools correctly, and that Target Volatility depends on strange assumptions that are provably false.
So which market timing strategy are you practicing?

Uncorrelated
Posts: 374
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Re: Help me figure out the Target Volatility formula

Post by Uncorrelated » Thu Feb 06, 2020 4:45 am

Stump909 wrote:
Wed Feb 05, 2020 5:49 pm
Uncorrelated wrote:
Wed Feb 05, 2020 4:12 pm
Stump909 wrote:
Wed Feb 05, 2020 11:40 am
Uncorrelated wrote:
Wed Feb 05, 2020 6:12 am


I mentioned earlier that backtests are only useful as a tool to deceive. That wasn't a joke. It is extremely difficult to backtest correctly and even more difficult to interprent, the entire active management industry is build around misleading people with benchmarks.
So are simply against allocations, based on volatility signaling, or leveraged investing as a whole?
Volatility signaling, market timing and leverage are tools that can be used with great success. The problem is that the method (backtesting) is too error prone to evaluate these tools correctly, and that Target Volatility depends on strange assumptions that are provably false.
So which market timing strategy are you practicing?
I use a boring passive investing approach. I may have enough knowledge to design and implement a market timing approach, I simply have no need or desire to do so.

ocrtech
Posts: 47
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Re: Help me figure out the Target Volatility formula

Post by ocrtech » Thu Feb 06, 2020 9:01 am

Uncorrelated wrote:
Wed Feb 05, 2020 4:12 pm
Uncorrelated wrote:
Wed Feb 05, 2020 6:12 am


I mentioned earlier that backtests are only useful as a tool to deceive. That wasn't a joke. It is extremely difficult to backtest correctly and even more difficult to interprent, the entire active management industry is build around misleading people with benchmarks.
The problem is that the method (backtesting) is too error prone to evaluate these tools correctly, and that Target Volatility depends on strange assumptions that are provably false.
I've seen you make similar general comments on backtesting in a couple different threads. What specifically do you not like about using backtesting? This is a fundamental tool that has been and continues to be used in many different industries. Concepts and mathematical models are all well and good but without backtesting how do you validate or more importantly invalidate your concepts?

Uncorrelated
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Re: Help me figure out the Target Volatility formula

Post by Uncorrelated » Thu Feb 06, 2020 6:32 pm

ocrtech wrote:
Thu Feb 06, 2020 9:01 am
Uncorrelated wrote:
Wed Feb 05, 2020 4:12 pm
Uncorrelated wrote:
Wed Feb 05, 2020 6:12 am


I mentioned earlier that backtests are only useful as a tool to deceive. That wasn't a joke. It is extremely difficult to backtest correctly and even more difficult to interprent, the entire active management industry is build around misleading people with benchmarks.
The problem is that the method (backtesting) is too error prone to evaluate these tools correctly, and that Target Volatility depends on strange assumptions that are provably false.
I've seen you make similar general comments on backtesting in a couple different threads. What specifically do you not like about using backtesting? This is a fundamental tool that has been and continues to be used in many different industries. Concepts and mathematical models are all well and good but without backtesting how do you validate or more importantly invalidate your concepts?
I think the counter-question is more interesting: How do you plan on validating your concepts with backtesting? Here are three simple examples that appear to be "validated" by backtests:

over-weighting walmart increases returns and lowers risk
long term bonds have similar sharpe ratio as S&P500
TQQQ has higher expected returns than UPRO

The above examples are carefully chosen to be completely wrong. But there are posters that actually believe these or similar claims (often related to gold, REIT's, international investing or high-dividend stocks). The usual way backtests are used resembles a machine learning problem with more dimensions than samples, without separate train/test and validation sets. Cutting that many corners might result in acceptable results if you have 5 million data points, but with 500 data points it's a recipe for disaster. There are simply not enough data points to draw any conclusion.

Now for the eternal question: how do you validate your concepts? That is a difficult one to answer but I have seen two good approaches. The first approach is choose a simple model that depends on minimal assumptions and start calculating, lifecycle investing takes this approach with great success. Sometimes you can use this to reverse-engineer the model behind a trading strategy, I have done this with TV and it's pretty obvious that the model is garbage (If you would only use backtests, you would never notice that). The second approach is to use tools from statistics to calculate a confidence boundary for your hypothesis, this is the approach that factor investing (small/value) takes. I believe in size/value because there is a significant amount of statistical of evidence, not because of backtests.

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lock.that.stock
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Re: Help me figure out the Target Volatility formula

Post by lock.that.stock » Thu Feb 06, 2020 8:24 pm

If OP is working on putting together a target volatility calculation spreadsheet, with help from others - would you consider uploading the sheet publicly for review and for others to use?

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Re: Help me figure out the Target Volatility formula

Post by danielc » Fri Feb 07, 2020 12:51 am

lock.that.stock wrote:
Thu Feb 06, 2020 8:24 pm
If OP is working on putting together a target volatility calculation spreadsheet, with help from others - would you consider uploading the sheet publicly for review and for others to use?
Good idea! So I went ahead and made a new spreadsheet meant for public consumption:

https://docs.google.com/spreadsheets/d/ ... sp=sharing

This spreadsheet looks at 1 month of daily returns of UPRO and TMF and produces several possible asset allocations according to various criteria that I have been experimenting with:
  • Target Volatility. I actually implemented four versions of target volatility:
    • Use only the volatility of UPRO; assume that TMF has zero volatility. This is how PV works.
    • Use the volatilities of both UPRO and TMF, assuming zero correlation.
    • Use the volatilities of both, and the measured covariance between them.
    • Use the volatilities of both, and assume a permanent correlation of -0.1.
  • Inverse Volatility
  • Maximum Diversification (described on page 16 of this paper).
  • Equal Risk Contribution (described on page 19 of the same paper).
  • Maximum Sharpe Ratio
I chose those purely because they grabbed my interest and I wanted to see what they did.

EDIT: For the Maximum Sharpe Ratio I used iShares SHV as a proxy for the risk-free return rate. Something like the 3-month Treasury Bill or LIBOR would be a better choice, but I have no idea how to pull that data automatically into a Google spreadsheet.

Stump909
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Re: Help me figure out the Target Volatility formula

Post by Stump909 » Fri Feb 07, 2020 9:51 am

danielc wrote:
Fri Feb 07, 2020 12:51 am
lock.that.stock wrote:
Thu Feb 06, 2020 8:24 pm
If OP is working on putting together a target volatility calculation spreadsheet, with help from others - would you consider uploading the sheet publicly for review and for others to use?
Good idea! So I went ahead and made a new spreadsheet meant for public consumption:

https://docs.google.com/spreadsheets/d/ ... sp=sharing

This spreadsheet looks at 1 month of daily returns of UPRO and TMF and produces several possible asset allocations according to various criteria that I have been experimenting with:
  • Target Volatility. I actually implemented four versions of target volatility:
    • Use only the volatility of UPRO; assume that TMF has zero volatility. This is how PV works.
    • Use the volatilities of both UPRO and TMF, assuming zero correlation.
    • Use the volatilities of both, and the measured covariance between them.
    • Use the volatilities of both, and assume a permanent correlation of -0.1.
  • Inverse Volatility
  • Maximum Diversification (described on page 16 of this paper).
  • Equal Risk Contribution (described on page 19 of the same paper).
  • Maximum Sharpe Ratio
I chose those purely because they grabbed my interest and I wanted to see what they did.

EDIT: For the Maximum Sharpe Ratio I used iShares SHV as a proxy for the risk-free return rate. Something like the 3-month Treasury Bill or LIBOR would be a better choice, but I have no idea how to pull that data automatically into a Google spreadsheet.
Kind of you to share. :beer

Stump909
Posts: 27
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Re: Help me figure out the Target Volatility formula

Post by Stump909 » Fri Feb 07, 2020 10:01 am

ocrtech wrote:
Thu Feb 06, 2020 9:01 am
Uncorrelated wrote:
Wed Feb 05, 2020 4:12 pm
Uncorrelated wrote:
Wed Feb 05, 2020 6:12 am


I mentioned earlier that backtests are only useful as a tool to deceive. That wasn't a joke. It is extremely difficult to backtest correctly and even more difficult to interprent, the entire active management industry is build around misleading people with benchmarks.
The problem is that the method (backtesting) is too error prone to evaluate these tools correctly, and that Target Volatility depends on strange assumptions that are provably false.
I've seen you make similar general comments on backtesting in a couple different threads. What specifically do you not like about using backtesting? This is a fundamental tool that has been and continues to be used in many different industries. Concepts and mathematical models are all well and good but without backtesting how do you validate or more importantly invalidate your concepts?
I actually started looking into this, because I was curious if his argument held up to scrutiny. Apparently the entire concept of quant investing revolves around backtesting.

ocrtech
Posts: 47
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Re: Help me figure out the Target Volatility formula

Post by ocrtech » Fri Feb 07, 2020 1:17 pm

Stump909 wrote:
Fri Feb 07, 2020 10:01 am

I actually started looking into this, because I was curious if his argument held up to scrutiny. Apparently the entire concept of quant investing revolves around backtesting.
Backtesting can certainly have issues. However, these are usually a function of how it's being applied, lack of attention to the applicable biases, or insufficient data.

I personally believe that backtesting the hedgefundie approach and its several variants is appropriate. Given the amount of actual historical and simulated data available, I also believe valid conclusions can be drawn from the results.

I'm not sure where @uncorrelated came up with his estimate of 500 data points. If I recall correctly, the raw data consists of nearly 10k daily trading values. Depending upon the hedgefundie variant being tested, the volatility window being used, and the rebalance timeframe being considered, significantly more data points are utilized in the backtests.

At the end of the day, I personally would never have invested any money in the variant I selected without first fully understanding the underlying approach thesis as well as confirming its historical validity through backtesting.

Uncorrelated
Posts: 374
Joined: Sun Oct 13, 2019 3:16 pm

Re: Help me figure out the Target Volatility formula

Post by Uncorrelated » Fri Feb 07, 2020 4:05 pm

Stump909 wrote:
Fri Feb 07, 2020 10:01 am
ocrtech wrote:
Thu Feb 06, 2020 9:01 am
Uncorrelated wrote:
Wed Feb 05, 2020 4:12 pm
Uncorrelated wrote:
Wed Feb 05, 2020 6:12 am


I mentioned earlier that backtests are only useful as a tool to deceive. That wasn't a joke. It is extremely difficult to backtest correctly and even more difficult to interprent, the entire active management industry is build around misleading people with benchmarks.
The problem is that the method (backtesting) is too error prone to evaluate these tools correctly, and that Target Volatility depends on strange assumptions that are provably false.
I've seen you make similar general comments on backtesting in a couple different threads. What specifically do you not like about using backtesting? This is a fundamental tool that has been and continues to be used in many different industries. Concepts and mathematical models are all well and good but without backtesting how do you validate or more importantly invalidate your concepts?
I actually started looking into this, because I was curious if his argument held up to scrutiny. Apparently the entire concept of quant investing revolves around backtesting.
Quant investing generally revolves around shorter trade durations. One of my criticism for backtesting is that you have too little data points to work with. But if you trade on the time scale of a few seconds there is a lot more data and the probability of overfitting is a lot lower. Also, quants generally have better tools that take slippage and execution into account, which makes backtesting slightly more accurate. I also suspect that most employees at trading firms (i.e. Renaissance Technologies) spend 99% of their time on mathematical modeling and not backtesting. They mostly hire psychologists, chemists and mathematicians.

I also forgot to mention earlier that risk tolerance is an important part for an individual investor, but that is often ignored in these discussions. The paper Forecasting the Equity Risk Premium: The Role of Technical Indicators reports the certainty equivalent return for their trading strategy instead of the CAGR, that makes a lot more sense.

My criticism really isn't that backtesting is useless, just that it is extremely hard to use correctly. Go back to the fundamentals. Figure out what assumptions are used in the strategy. Determine your personal risk tolerance. Do some math. Then start backtesting. If you skip directly to backtesting, you will miss many valuable insights.
ocrtech wrote:
Fri Feb 07, 2020 1:17 pm
I'm not sure where @uncorrelated came up with his estimate of 500 data points. If I recall correctly, the raw data consists of nearly 10k daily trading values. Depending upon the hedgefundie variant being tested, the volatility window being used, and the rebalance timeframe being considered, significantly more data points are utilized in the backtests.
The target volatility backtester at portfolio visualizer changes the allocation monthly. This equates to around 500 data points for 40-50 years of data.

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