Engineering Targeted Returns and Risk

[Stefan]

1.  Risk Moderation  achievable by :
            a.    Using a risk parity allocation formula based on volatility weights
            b.    Managing the whole portfolio volatility to a target.

How does one align the whole portfolio to a target after scaling each asset's volatility to a 1% daily target?  It seems like this would be easy to do after the new weights and equity curve are generated, but not before a rebalancing.

Here is a solution to this:

Do not scale each asset to 1% volatility. Instead do this:

1. Allocate to each of the PP 3 assets (Stocks, Bonds & Gold) based on the simple formula I provided - i.e. inversely proportional to the asset volatility. Use trailing N days volatility of log of returns. Cash = 0 at this step.

2. You do not have to generate the equity curve to find the portfolio volatility after this allocation step. Instead, estimate the portfolio variance resulting from this allocation:

Variance(P) = Sum(1 <=i,j <=3, weight(i) * weight(j) * Covariance(i,j));

The estimated portfolio volatility based on the preliminary weights will be thus Sigma(P) = sqrt(Variance(P));

3. Compare Sigma(P) with your self imposed TargetSigma. If Sigma(P) > TargetSigma, reduce the portfolio by reducing all weights proportionally based on  the Target Sigma/Sigma(P) ratio. This determines the size of your cash component.

2.  Risk Containment achievable by:
          a.   Asset classes trend. Capturing most of the trend upside and avoiding most of the trend downside.
          b.   Minimizing exposure gradually in times of market turbulence.

So essentially TA is required here.  Momo is not enough.

Here is an AWP example which makes use of most of these techniques (vs PRPFX as a benchmark):
The CAGR is 11.23% and MaxDD -6%

I assume your portfolio involves more than the standard four PP assets?  The momo would do better if more assets than the standard three were included.  I suspect the same applies to yours.

The portfolio is exactly the AWP - 4 quadrants asset classes you have shown here.
Also, I wouldn't say TA, I would say TF (as in trend following).

[MachineGhost]

3. Compare Sigma(P) with your self imposed TargetSigma. If Sigma(P) > TargetSigma, reduce the portfolio by reducing all weights proportionally based on  the Target Sigma/Sigma(P) ratio. This determines the size of your cash component.

The weakness of this approach seems to be that it is highly dependent on the lookback period used for determining the volatility as well as the covariances.  I assume both should match.

It does reduce the equal weight PP down to -12.19% MaxDD and a CAR of 7.14% using a 100-bar lookback period and a 10% volatility target.  However, it doesn't work anywhere as well with other risk measurements not based on volatility.

But in the end, it seems not all that different in the resulting weights from using cash as a risk-reduction knob with the equal weight PP.

[Stefan]

3. Compare Sigma(P) with your self imposed TargetSigma. If Sigma(P) > TargetSigma, reduce the portfolio by reducing all weights proportionally based on  the Target Sigma/Sigma(P) ratio. This determines the size of your cash component.

The weakness of this approach seems to be that it is highly dependent on the lookback period used for determining the volatility as well as the covariances.  I assume both should match.

It does reduce the equal weight PP down to -12.19% MaxDD and a CAR of 7.14% using a 100-bar lookback period and a 10% volatility target.  However, it doesn't work anywhere as well with other risk measurements not based on volatility.

But in the end, it seems not all that different in the resulting weights from using cash as a risk-reduction knob with the equal weight PP.

MG: I hope you see some value in this approach and you are not extremely disappointed with these results 

To me, -12.19% MaxDD and a CAR of 7.14% look pretty good compared to the benchmark equal weight PP.  It reduced the benchmark's MaxDD by 60% while capturing 83% of the upside.

This whole investing game performance is measured by the risk adjusted return.
Hedge funds score strategies with the MAR ratio, which measures return by unit of risk. MAR = CAGR/MaxDD, taken over the whole lifetime of the strategy. Personally, this is what I use as well, and I don't care for other risk measurements.

In practice, investors start panicking over 10% MaxDD . And over 20% MaxDD they bail out.
As for CAGR, consistent 8% is about the target for institutional investors. Anything above is just a bonus.
So, a MAR = 0.8 is about where an institution will consider giving you money for your strategy.

So, for what you did here, MAR = 7.14/12.19 = 0.55 is a decent return/risk ratio.
By contrast, the return/risk for the benchmark PP is MAR = 8.6/29.84 = 0.28. This is hard to trade.

Also, based on the MAR comparison, you can say the volatility adjusted PP you just backtested is 2X better than the benchmark.

In my tests, the system is quite robust to  lookback window sizes in the 30-80 days interval. Heuristically - a smaller interval is more relevant for correlations and volatilities. But too small an interval becomes statistically irrelevant.

Now, I think that the goal should be a MAR > 1. This may be achieved if, in addition to what you did, you overlay some form of risk containment mechanism as I mentioned in my posts.

[MachineGhost]

MG: I hope you see some value in this approach and you are not extremely disappointed with these results 

I am a little because there are easier ways to get to a favorable risk-reward ratio than the kind of upkeep, math and coding all of this imposes.

To me, -12.19% MaxDD and a CAR of 7.14% look pretty good compared to the benchmark equal weight PP.  It reduced the benchmark's MaxDD by 60% while capturing 83% of the upside.

True, but it really killed the return.  We're talking about CAR so small changes make huge differences in outcomes due to the compounding.  So going from 9% to 7% is hard to swallow.

This whole investing game performance is measured by the risk adjusted return.
Hedge funds score strategies with the MAR ratio, which measures return by unit of risk. MAR = CAGR/MaxDD, taken over the whole lifetime of the strategy. Personally, this is what I use as well, and I don't care for other risk measurements.

I plan on coding that today as a portfolio target since I'm not a big believer in the volatility approach to risk measurement.  But generally, using a MaxDD just corresponds to a longer volatility lookback period as it captures a singular historical event.  Deviations from the highest point above the mean to the lowest point below the mean is essentially a peak-to-trough measurement.  One thing I've noticed about all-bars volatility or derivatives is that it tends to slowly decrease (improve) over time as the event fades further and further into the past.  MaxDD doesn't do that.  But on the other hand if you just do 1/n for MaxDD you'll won't have as good a risk-reward ratio as these other weighting schemes.

In practice, investors start panicking over 10% MaxDD . And over 20% MaxDD they bail out.  As for CAGR, consistent 8% is about the target for institutional investors. Anything above is just a bonus.

That is my gut feel too and what I believe will be the undoing of the vast majority of vanilla PPers.  I've experienced the whole gamut of losses in different vehicles all the way down to 90%.  Fun.

Now, I think that the goal should be a MAR > 1. This may be achieved if, in addition to what you did, you overlay some form of risk containment mechanism as I mentioned in my posts.

I agree.  I posted backtest results of that in some other thread but they were using more complex systems than a simple trend filter and not generated in Ami.

[MachineGhost]

Here's the results from the next batch of testing:

Code: Select all

Annual Rebalancing (6/1974+)

Equal Weight, 10% SD Target				7.45%	-20.28%
Equal Weight, 10% MaxDD Target				6.69%	-10.70%
Volatility Weight ST, 10% SD Target			7.14%	-12.19%
Volatility Weight ST, 10% MaxDD Target			6.60%	-7.75%
Volatility Weight ST, 1% Daily & 10% SD Target		7.47%	-14.71%
Volatility Weight ST, 1% Daily & 10% MaxDD Target	6.76%	-9.50%
MaxDD Weight, 10% SD Target				7.54%	-20.33%
MaxDD Weight, 10% MaxDD Target				6.68%	-10.10%
Momo, Equal Weight, 10% SD Target			6.90%	-4.11%
Momo, Equal Weight, 10% MaxDD Target			6.99%	-7.49%
Momo, Volatility Weight, 1% Daily & 10% SD Target	6.93%	-5.27%
Momo, Volatility Weight, 1% Daily & 10% MaxDD Target	7.03%	-7.49%
Dynamic							9.52%	-33.13%
Dynamic, 10% MaxDD Target				7.28%	-15.23%

[Stefan]

Here's the results from the next batch of testing:

Code: Select all

Annual Rebalancing (6/1974+)

Equal Weight, 10% SD Target				7.45%	-20.28%
Equal Weight, 10% MaxDD Target				6.69%	-10.70%
Volatility Weight ST, 10% SD Target			7.14%	-12.19%
Volatility Weight ST, 10% MaxDD Target			6.60%	-7.75%
Volatility Weight ST, 1% Daily & 10% SD Target		7.47%	-14.71%
Volatility Weight ST, 1% Daily & 10% MaxDD Target	6.76%	-9.50%
MaxDD Weight, 10% SD Target				7.54%	-20.33%
MaxDD Weight, 10% MaxDD Target				6.68%	-10.10%
Momo, Equal Weight, 10% SD Target			6.90%	-4.11%
Momo, Equal Weight, 10% MaxDD Target			6.99%	-7.49%
Momo, Volatility Weight, 1% Daily & 10% SD Target	6.93%	-5.27%
Momo, Volatility Weight, 1% Daily & 10% MaxDD Target	7.03%	-7.49%
Dynamic							9.52%	-33.13%
Dynamic, 10% MaxDD Target				7.28%	-15.23%

Momo, Equal Weight, 10% SD Target 6.90% -4.11% looks pretty good with a MAR > 1.
(You do not show a Momo, Volatility Weight 10% SD Target though...)

A hypothetical: if you were to leverage the system you show 2:1 (at Fidelity's 3.75% margin rate, available if your portfolio >$500k, let's say) you get ~10% CAGR with -8% MaxDD from this strategy, after paying margin interest. This would be a very good risk/return .

But I think you could get a CAGR > 10% without any leverage and still keep MaxDD below 7%.

Here is one more idea on how moderate risk:
The volatility weighted formula allocates weights inversely proportional with the stddev. In addition, you could also allocate weights inversely proportional with the correlation of each asset class to the rest of the portfolio.
So, if Corr(i, P) is the correlation of asset i with the other assets in the portfolio, you allocate weight(i) = k* (1- Corr(i, P))/sigma(i), where k is a normalization constant.
While I know this works well for the larger AWP portfolio, with more asset classes, I think it would also work for PP.

Now, it would be helpful to know, a bit, about how do you use momentum in your system, whatever you fell OK revealing - just to get the sense of the heuristics you use.

[frommi]

The big problem i have with the volatility approach is that it is not possible to know the volatility of the assets in the future. (even not tomorrow`s volatility)
So you can only look back. But the more you are trying to find the best portfolio for risk/reward in history, what you are really doing is curve-fitting to the past.
But the future will not repeat the same way as the last 40 or 80 years.

Thats the biggest argument i have for the pure PP, because it is a theoretical and easy to use approach which worked very well in the past.
You never know if the max-drawdown will sometimes be higher because of some black-swan event in the future, and the more you curve-fit your algorithm to the past
the more likely it is to get a higher max-drawdown in the future.

[MachineGhost]

(You do not show a Momo, Volatility Weight 10% SD Target though...)

Momo, Volatility Weight ST, 10% SD Target        6.91%    -5.18%

So, if Corr(i, P) is the correlation of asset i with the other assets in the portfolio, you allocate weight(i) = k* (1- Corr(i, P))/sigma(i), where k is a normalization constant.
While I know this works well for the larger AWP portfolio, with more asset classes, I think it would also work for PP.

I'll have to try that!

Now, it would be helpful to know, a bit, about how do you use momentum in your system, whatever you fell OK revealing - just to get the sense of the heuristics you use.

Nothing special.  Just 6-month ROC and taking the top 2 out of 3.

[MachineGhost]

Thats the biggest argument i have for the pure PP, because it is a theoretical and easy to use approach which worked very well in the past.

But the history of the PP is limited to 1987+ out of sample and it suffers from granularity problems when backtesting before that.  Annual or monthly data easily hides max drawdowns that occur intrayear or intramonth.  For instance, until I posted the stats here in the forum, no one had any idea that the PP had historically suffered at least a 25% MaxDD.  Almost all thought it was only 15% or 20% at worse (even I got confused and reverted back to 20% at one time).  Essentially, 1973-1974 was a particularly vicious bear market in stocks.  The limited history of 30-year T-Bond's unfortunately prevents backtesting the PP during that full bear market.

You never know if the max-drawdown will sometimes be higher because of some black-swan event in the future, and the more you curve-fit your algorithm to the past
the more likely it is to get a higher max-drawdown in the future.

Well, by that logic the equal weight PP will always be suboptimal because it will not be anywhere close to the 80% MaxDD in stocks that occured during the Great Depression.  We all have blinders on regarding that event and don't want to allocate with that risk in mind because it kills our greedy future expectations.  So everything is a compromise in the end.  The "surprise gap" between a larger MaxDD in the future between the equal weight PP and one based on a post-Nixon Shock MaxDD history will be far more gruesome to the equal weight.

[Stefan]

The big problem i have with the volatility approach is that it is not possible to know the volatility of the assets in the future. (even not tomorrow`s volatility)
So you can only look back. But the more you are trying to find the best portfolio for risk/reward in history, what you are really doing is curve-fitting to the past.

The usefulness of volatility measurement is a function of your timeframes in running the system. Volatility has predictability (and there is a vast literature documenting it) in the same way that price momentum predicts future returns. But price momentum is a good predictor within a timeframe. That timeframe is around 6 month. Beyond that its predictability of future returns starts fading . Over 12 month it is not predictive anymore.

Similarly, Volatility has its own momentum and timeframes which can be used to predict future volatility.

So, you should be more specific in your statement. If you mean that historical volatility over the last decade is useless, you are right. But, if you mean that last 60 days volatility will tell you nothing about tomorrow's volatility, you are wrong.

You never know if the max-drawdown will sometimes be higher because of some black-swan event in the future, and the more you curve-fit your algorithm to the past the more likely it is to get a higher max-drawdown in the future.

If you read my posts, you saw that there are methods to avoid black swans, from "depression gauges" to technical heuristics. I won't repeat those here. But, there is no risk management in PP - so you have zero control on your portfolio MaxDD during a tail event. Zero. And this is what should concern you.

Not to mention the psychological difficulty of, for, instance, taking money out of cash & putting it in assets that went down, say 15%, in the middle of a relentless bear market, only to see those assets losing another 15%. How many people will be able to withstand this when PP is their retirement portfolio, especially if they are already retired?

[Stefan]

Nothing special.  Just 6-month ROC and taking the top 2 out of 3.

So, you use relative momentum.

I think you could get better results using a simple absolute momentum rule => i.e compare N months ROC of each asset class with TBills and stay in TBills if the asset class ROC is smaller. N =6 - 12 months.
Here is an excellent paper describing this:
http://www.docyoushare.com/file/index.php?f=HbdfwVz5

Some results from the paper - performance 1974-2012 and detailed for each decade:



Parity is basically volatility weights as you use them.
These are excellent results, MAR > 1, Return ~12%, MaxDD < 10%, and should be easily reproduced, with this super simple Momo rule overlayed on the volatility weights allocation.

[MachineGhost]

What are you using to generate covariance or correlation matrxies in AmiBroker?  The RMath plugin is too outdated to work with SIT.

[Stefan]

What are you using to generate covariance or correlation matrxies in AmiBroker?  The RMath plugin is too outdated to work with SIT.

I am not using any plugin. I use the Correlation() and StDev() functions in AB.
To create a 2D matrix in AB where each element is a time series array, I use a naming convention for dynamic variables.
So, the matrix element CorrelationMatrix(i)(j)  is generated like this:

VarSet("CorrelationMatrix" + i + j, Correlation(Array_i, Array_J, Period));

[MachineGhost]

This was an off topic post, but I wanted to include it in this thread for archival purposes:

http://gyroscopicinvesting.com/forum/ht ... 322#p63322

[iwealth]

So I hate to interject a simpleton comment into this fascinating discussion, but I can't help myself. At a time where gold, stocks, and LTTs are all sitting above, at, or close to all-time highs, this risk containment element seems particularly attractive. One would think it can't go on forever and the fed actions of the past couple years seems to be a bubble-creator like those that led to the sub-prime mortgage crisis. As eager as I may be to whip out Amibroker and start coding ideas, I don't have the math or programming background. Sorry in advance ha.

Anyway, so I'm clear, one of the risk containment strategies mentioned in a paper above is to use a 12 month trailing ROC of some volatile risk parity assets and compare that to the ROC of T-bills over those same 12 months, and either enter the volatile asset if ROC > T-Bills, or exit volatile asset if ROC < T-Bills.

So if on month 1, an asset was hypothetically priced at $100, and month 13 priced at $120, that's a 20% return. If on month 2, that asset had spiked at $200, but then on month 14 priced at $150, that's actually a 25% loss, despite seeing a 25% gain from month 13 to month 14. So theoretically were you still long that asset, you'd be exiting? At least according to this strategy. Just want to make sure I'm clear how that would work.

[Stefan]

iwealth: Indeed. This is what the paper uses for absolute momentum.

If you try to visualize what you described, you get a trend down which looks like this:



And you can contain your risk by switching to TBills for this asset class until the trend reverses.

[iwealth]

Thanks Stefan. This would have had me out of GLD at $164

That said, I can see how a choppy consolidating market would cause some havoc with a momentum based strategy, but avoiding future 2008-type market collapses is appealing. Although I guess asset class and allocation %'s is just as important. Hopefully you guys will continue this conversation and share whatever details you feel comfortable with about your personal portfolio selections.

[MachineGhost]

The difference between standard deviation and drawdown minimization with MinCorr is interesting.  The optimal weights is:

Stocks 13.98%
TBonds 7.27%
Gold 2.03%
TNotes 76.72%

...for 7.20% CAR and -5.10% MaxDD.  It can be levered by 68% to reach -10% MaxDD. Without the TNotes, it is:

Stocks: 60.86%
TBonds: 33.28%
Gold: 5.86%

...for 7.86% CAR and -30.56% MaxDD.  It can be delevered by 62% to reach -10% MaxDD. 

Transaction costs and taxes are not included for the above.

The optimal standard devation MinCorr weights that I posted the other day in the other thread, can be delevered by 50% to reach -10% MaxDD.  For three assets, the weighting is essentially the same as above, except gold is a bit over 2x higher deducted from TBonds.

[Stefan]

The difference between standard deviation and drawdown minimization with MinCorr is interesting.  The optimal weights is:

Stocks 13.98%
TBonds 7.27%
Gold 2.03%
TNotes 76.72%

...for 7.20% CAR and -5.10% MaxDD.  It can be levered by 68% to reach -10% MaxDD. Without the TNotes, it is:

Stocks: 60.86%
TBonds: 33.28%
Gold: 5.86%

...for 7.86% CAR and -30.56% MaxDD.  It can be delevered by 62% to reach -10% MaxDD. 

MG: I am not sure what you mean when you show "the optimal weights"?

There are no universal optimal weights, outside the rebalancing period. So, for instance, if your rebalancing periodicity is monthly, and you rebalance for risk parity and/or minimal correlation, you can calculate each month the optimal weights => for the next month. But, these optimal weights for month T will not be valid for month T+1. Because both correlation and volatilities, which are the input parameters in the rebalancing function, will change from the previous month...So, that "optimal weights" value set was confusing for me.

Here is the StDev for S&P 500 and its correlation to TBonds. You cannot talk about an optimal weight except for a period where these numbers change is relatively small, which is rarely > 1 month.

[MachineGhost]

MG: I am not sure what you mean when you show "the optimal weights"?

It was for all of the data, i.e. back to 1973.  I'm not sure why you think short-term matters when the AWP/PP is designed around long-term economically non-correlated environments?  So it won't matter if the correlations flunctuate in the short-term.  For one thing, frequent rebalancing incurs transaction costs and taxes which pushes the net return under buy and hold at the monthly or lower granularity level.  Quarterly may just squeak by break-even.  Even annual is below TNotes's buy and hold in many (if not all) cases.

BTW, to get the vanilla PP down to -10 MaxDD requires decreasing each of the three assets to under 9% with the rest in TNotes.  It may be even worse for TBills.  The vanilla PP is very risk unbalanced.

[Stefan]

MG: I am not sure what you mean when you show "the optimal weights"?

It was for all of the data, i.e. back to 1973.  I'm not sure why you think short-term matters when the AWP/PP is designed around long-term economically non-correlated environments? 

The Risk Parity method tries to create a portfolio of equal risk contribution for each asset class.

Look at the S&P 500 volatility chart below. The risk to own stocks in the portfolio is proportional to stocks volatility. There are times when this volatility reaches 150% like during the bear market 2008, and other times when it is 20%, in a bull market.

So, a risk parity portfolio will decrease its stocks allocation gradually, as the volatility rises, towards a minimal allocation in 2008 and will increase it as volatility falls, gradually, towards a maximum  in 2009 for instance - just to use 2 recent dates.



Using an average volatility to allocate for the whole period 1973-2012 doesn't protect your portfolio and moderate your risk in any way, as it hides the immense variability of risk in each interval. The finer your interval, the better the risk based allocation.

What is the cost of doing this?
A monthly interval means 12 transactions / year / asset class. That makes 4 X 12 X $8 = $384/year in transaction costs.

What is the cost of not doing this? It is the drawdown you would have to endure in each bear market, as you have no risk management. And drawdowns have a negative effect not only on your psychology  but, as importantly, on the compounding of your portfolio equity. So, is $384 too high a price to pay to control risk?

[MachineGhost]

What is the cost of not doing this? It is the drawdown you would have to endure in each bear market, as you have no risk management. And drawdowns have a negative effect not only on your psychology  but, as importantly, on the compounding of your portfolio equity. So, is $384 too high a price to pay to control risk?

I see.  So any kind of risk parity approach is always presupposed on frequent rebalancing so the portfolio can adjust to radical changes in risk.  But in the drawdown minimization portfolio that I posted, I am using correlated singular events (the worst MaxDD) which do not change frequently.

I would disagree with $384 and you conveniently ignored taxes, including the absurd collectibles tax of 35% on gold.  Over the entire period back to 1973, commission costs were hardly that low, taxes were a heck of a lot higher, and mainstream tax-deferred accounts did not exist or had extremely low limits.  If you choose to ignore historical limitations, it is no different in concept than the survivorship bias.  No muppet could have possibly been trading as illustrated in this thread.  Even now, transaction costs are not insignificant...  .26% per trade for market impact, .51% yearly for capital gains taxes, plus brokerage commissions.

[MachineGhost]

Now that I'm moving onto the risk reduction section, does anyone have any pseudo-code or R-code or even AFL code for implementing rebalancing bands?

[Libertarian666]

What is the cost of not doing this? It is the drawdown you would have to endure in each bear market, as you have no risk management. And drawdowns have a negative effect not only on your psychology  but, as importantly, on the compounding of your portfolio equity. So, is $384 too high a price to pay to control risk?

I would disagree with $384 and you conveniently ignored taxes, including the absurd collectibles tax of 35% on gold.

I'm not familiar with that tax. As far as I'm aware, the max tax on gold held more than one year is 28%, and has been for many years.

[MachineGhost]

I'm not familiar with that tax. As far as I'm aware, the max tax on gold held more than one year is 28%, and has been for many years.

Short-term is at income tax rates, 28% is for long-term.  I was conflating the two.  But 28% doesn't come into play when rebalancing every month.  Ironically, it might be cheaper under income tax rates and I think the 28% is the maximum you would ever pay, not fixed.