Refinements to Hedgefundie's excellent approach

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Hydromod
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Re: Refinements to Hedgefundie's excellent approach

Post by Hydromod » Mon Sep 02, 2019 11:59 pm

The final plot for tonight compares three risk-parity schemes based on volatility (the standard deviation of returns), all with 63-day lookback to calculate volatility and 21-day rebalancing.

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  • SD-down calculates volatility with only downward returns (asymmetric calculation)
  • SD-tdd calculates volatility with all returns, but with upward returns set to zero (asymmetric calculation)
  • SD-sym calculates volatility with all returns (symmetric calculation)
These schemes have a similar effect, tightening up the spread in returns and systematically increasing the total return over the reference approach. The asymmetric schemes appear to be slightly better than the symmetric scheme, but all three improve the average 5-year CAGR by around 200 bps. Resetting the weights monthly would have increased average overall returns over the entire duration by almost a factor of two relative to the reference scheme.

Again, the starting day appears to have a larger influence than the scheme.

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Forester
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Re: Refinements to Hedgefundie's excellent approach

Post by Forester » Tue Sep 03, 2019 3:59 am

In Excel, would you calculate a 63-day lookback with more weight applied to recent data?

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Re: Refinements to Hedgefundie's excellent approach

Post by Hydromod » Tue Sep 03, 2019 9:19 am

Forester wrote:
Tue Sep 03, 2019 3:59 am
In Excel, would you calculate a 63-day lookback with more weight applied to recent data?
I haven't yet explicitly looked at the specific question of variable weighting of data. At this point, I personally would keep it simple to minimize the chance of coding errors. I don't think variable weighting will really move the needle much one way or the other, especially over such a long period, so it is really a question of personal taste more than anything else.

My very strong suspicion is that both the duration of lookback and the selective increase in weighting of recent data give systematic effects that are small compared to doing risk parity in the first place. Remember, volatility tends to be correlated over periods of months, and the comparison with different fixed weights shows relatively similar behavior with weights from 30/70 through 60/40.

I've done some previous tests with lookback duration and rebalancing period, and didn't notice dramatic effects from either. I will be putting up some comparisons tonight that will get at this question.

Now that my software is set up better, I intend to generate a series of similar plots to systematically illustrate the effects from a range of risk parity and risk budget schemes, then look at various methods for incorporating market and macroeconomic signals. So far I am finding that the implementation details related to rebalancing and volatility tend to be lost in the noise, and have small influence relative to decisions on how market and macroeconomic information is used to adjust the risk budget between UPRO and TMF.

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Hydromod
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Re: Refinements to Hedgefundie's excellent approach

Post by Hydromod » Tue Sep 03, 2019 10:45 pm

The following figure considers combinations of the rebalancing period (1 and 3 months) and volatility lookback period (1 and 3 months) for a standard risk parity model with symmetric volatility, compared to the reference 40/60 scheme with quarterly rebalancing.

Image

Three of the four combinations are clumped with similar expected CAGR.

The combination with 21 days volatility lookback/21 days between rebalancing outperforms the reference scheme to a lesser extent.

The two best combinations are 21/63 and 63/21 days between rebalancing/lookback days.

Based on this comparison, there isn't a compelling reason to select monthly versus quarterly rebalancing or monthly versus quarterly lookback.

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Re: Refinements to Hedgefundie's excellent approach

Post by Hydromod » Tue Sep 03, 2019 11:08 pm

It is more difficult to generate a representative band scheme comparison, because after awhile the different realizations end up in lock step.

Image

In the top left plot, the swath around the total price/nominal line equal to 1 is actually a number of realizations that are moving in lock step with the first realization. The bottom curve ends up in lock step in 2012.

A fuller example might mitigate this tendency to lock step using several approaches, such as randomly adding days to the transition signal or randomly sampling the durations for limiting bands.

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Re: Refinements to Hedgefundie's excellent approach

Post by Hydromod » Wed Sep 04, 2019 10:43 pm

The next figure shows portfolio trajectories assuming a fixed 40/60 UPRO/TMF ratio reset whenever the UPRO fraction moves outside a fixed band. The bands are 5, 10, 15, and 20 percent.

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The important takeaway from this case is that the band approach gives comparable or arguably slightly better results than the straight quarterly rebalancing for all of the bands considered.

With 5 percent bands, the average rebalancing frequency would have been approximately bimonthly, with approximately half of the rebalances performed between 10 and 25 days apart (see top plot in the right column) As the tolerance increases, the average duration between rebalances increases substantially.

Looser tolerances appear to increase CAGR slightly (third plot in the right column). I think that this is because the higher-performing asset tends to drift towards a larger allocation, increasing the returns. However, looser tolerances also tend to allow a wider spread in outcomes, both high and low.

As discussed in the previous entry, an analysis with rebalancing based on bands is arguably less representative than an analysis using periodic rebalancing because the trajectories tend to fall into lock step within a few years. So take this analysis with a grain of salt. Nevertheless, the analysis suggests strongly that using rebalance bands would be a viable approach to reducing the total number of rebalances, as has been suggested by others, with the requirement that the portfolio be monitored to some extent.

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Re: Refinements to Hedgefundie's excellent approach

Post by Hydromod » Wed Sep 04, 2019 11:07 pm

The adaptive risk parity case with rebalancing bands is generally consistent with the previous analyses. The risk parity gives a noticeable increase in expected CAGR, at least with the tighter bands, and the coarser bands allow for less frequent rebalances overall.

The benefit from adaptive risk parity disappears for the 20-percent bands, which has an average duration between rebalances of more than 400 days.

Based on this comparison, it would have been satisfactory to defer rebalancing until the UPRO fraction deviated by 10 to 15 percent, which would have been more than a year at times. The tighter bands would have reduced the spread in outcomes due to starting on different days.

Image

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Re: Refinements to Hedgefundie's excellent approach

Post by Hydromod » Wed Sep 04, 2019 11:26 pm

One more for tonight. This one considers weighting based on variance rather than volatility.

Overall the variance weighting approach tends to reduce the UPRO fraction relative to the volatility weighting, but occasionally the weighting goes up substantially. The variance approach tended to outperform in the periods with low returns. The asymmetric weighting schemes (downward returns only) overall performed better than the symmetric scheme.

The asymmetric variance schemes appear to have outperformed the volatility schemes by approximately 100 bps.

Image

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Re: Refinements to Hedgefundie's excellent approach

Post by Hydromod » Fri Sep 06, 2019 10:00 pm

In the light of Hedgefundie's switch from 40/60 to 55/45 UPRO/TMF allocation, it's of potential interest to use adaptive risk parity to meet the same overall asset weighting on average while improving the drawdown behavior.

The risk budget approach gives a simple knob to turn. I described this earlier in the thread. In the figures, the b parameter adjusts the UPRO risk budget, with b = 0.5 corresponding to no bias (straight risk parity) and b = 1 corresponding to the case with 100% UPRO.

In the figures, b = 0.7 (the aqua line) would have resulted in a time average of approximately 55/45 UPRO/TMF and has the largest return to risk ratio. The first figure uses symmetric standard deviation (volatility) and the second figure uses downward-only variance.

The figures suggest that performing adaptive risk parity without bias (b = 0.5) would have been expected to boost mean CAGR by 150 to 200 bps. Using downward-only returns to calculate variance would have boosted mean CAGR by 500 bps with b = 0.7.

Increasing b to 0.9 would not have significantly increased mean CAGR further, and would have significantly increased drawdowns.

Image

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I suggest that folks might want to consider whether adding the risk budget component to the allocation might be of use. Granted, going forward the boost from LT treasuries is expected to continually decline while interest rates drop, so the benefit may also drop over time.

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Re: Refinements to Hedgefundie's excellent approach

Post by Hydromod » Sat Sep 07, 2019 3:04 pm

Returning to the question of volatility lookback period, I offer a comparison using downward-only variance parity to determine nominal weights on a daily frequency. This is the same scheme displayed in the last figure in the previous entry. Unlike the previous example, rebalancing is performed whenever UPRO is above or below a band that is 15 percent above or below the target weights, and when 126 days have passed without rebalancing (semiannual). This allows the rebalancing frequency to arise naturally from accuracy constraints.

The nominal minimum rebalance frequency is 21 days (one month). The actual minimum rebalance frequency is allowed to vary slightly in order to maintain more independent trajectories. Remember, the initial set of trajectories tends to be "captured" into the same rebalance frequency over time. For this analysis, the nominal minimum is allowed to vary by a few days, sampled from a zero-mean Gaussian distribution with a standard deviation of 1 day.

The first figure uses a lookback duration for volatility of 21 days (monthly); the second uses 63 days (quarterly), otherwise the schemes are identical.

Image

Image

The 21-day volatility lookback results in a much choppier asset allocation sequence than the 63-day lookback. The average rebalance frequency is around 30 days with the 21-day lookback and is around quarterly with the 63-day lookback, so the 21-day lookback would generate roughly twice the number of trades. If the minimum rebalance period was 10 days instead of 21 for the 21-day lookback (not shown), the average rebalance frequency would have been about 20 days and the 21-day lookback would have generated roughly three times the number of trades than the 63-day lookback would have generated.

I expect it would likely be easier to just rebalance monthly rather than tracking violation of the bands when using 21-day lookback.

The quarterly lookback has slightly better expected CAGR and the ratio of return to risk is improved for all three risk budget parameter values.

I think that, in this case, the noise in volatility from the shorter lookback period drives a certain amount of spurious rebalancing. The longer lookback period has enough observations that the noise is smoothed out, and the performance improves.

I think that the particular parity measure of downward-only variance may be especially sensitive to noise in the daily returns. A takeaway from this example is that a short lookback period can hurt performance when the changes in volatility create large and frequent changes in the weights. This is not so much an issue when the weighting scheme is less sensitive to volatility.

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Re: Refinements to Hedgefundie's excellent approach

Post by Hydromod » Sat Sep 07, 2019 5:02 pm

Volatility limiting is another way to adaptively adjust asset weights. The idea is to define a volatility limit for the risky asset (UPRO in this case). When the lookback volatility of the risky asset is less than the limit, the portfolio is completely set to the risky asset. If the volatility is greater than the limit, the risky asset weight is set to wt = vol/vollim.

The following figure shows the returns with three annualized volatility limits, compared to the original 40/60 quarterly rebalance scheme. The expected CAGR for the scheme is larger than the original method by 200 to 300 bps, and the return/risk ratio is substantially better. Most of the out-performance comes from smaller drawdowns in 2001-2003 and 2008-2009, with slight out-performance since 2008. The volatility limit of 25% would have resulted in better CAGR and risk-adjusted returns than the other two limits. The method triggered relatively few rebalances, usually rebalancing at the default semiannual limit.

Image

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Re: Refinements to Hedgefundie's excellent approach

Post by Hydromod » Sat Sep 07, 2019 11:50 pm

All of the examples thus far have weights that are based solely on the historical volatility over various periods.

The original Hedgefundie 40/60 weights are based on backfitting over several decades, and the weights are closely tied to a risk parity balance over this period. Other examples have weights based on the volatility over the trailing one to three months. A key assumption in risk parity is that expected return is proportional to risk, with the asset allocation determined to equalize expected risks given the assets in the portfolio. To the extent that asset risks are known, then a pure risk parity is market agnostic once the assets have been selected. The argument for using trailing volatility over a relatively short period, such as a month or quarter, to determine risk is typically that the current market risks are best estimated with recent information.

Some of the examples include tilts to the risk budget for the assets, with the tilts fixed at a constant value. A tilt away from equal risk budget weights serves to emphasize some assets at the cost of others. The emphasized assets might have better growth potential or lower volatility, depending on the desires of the portfolio manager. Setting a risk budget is akin to setting an asset allocation, except with a fixed risk budget the asset allocation changes over time and with a fixed allocation the risk budget changes over time.

There are various ways that one could introduce market timing into a scheme that uses risk parity, such as swapping assets in and out of the portfolio or adaptively adjusting risk budget weights.

The approach I discussed before (using the unemployment rate index) adjusts the risk budget according to the unemployment rate. In this scheme, the risk budget allocated to UPRO is decreased each month that the unemployment rate is increasing and accelerating, and is increased each month that the unemployment rate is decreasing or the increase is decelerating. The risk budget is determined entirely by the macroeconomic indicator.

The next two figures illustrate the adaptive allocation of the risk budget for UPRO between 20 and 90 percent. The UPRO risk budget allocation drops from high to low over a minimum of four months, and increases from low to high over a minimum of 20 months.

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In these figures, the timing of the UPRO weight changes is essentially completely determined by the unemployment rate.

As before, the variance-based methods perform slightly better than the volatility-based methods in terms of expected return and risk-adjusted return. The variance methods have approximately 800 bps larger expected CAGR than the 40/60 fixed weighting scheme.

The effect of the limiting budget is shown in the next figure. Widening the UPRO limits tends to increase the overall returns, but exposes the portfolio to increased drawdown from rapid events that are not signalled.

Image

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Re: Refinements to Hedgefundie's excellent approach

Post by AlphaLess » Mon Oct 28, 2019 10:34 pm

Interesting thread, and I commend the effort.

Before trying to figure out any sort of dynamic weights (e.g., 100% TMF, or 100% UPRO, or anything in between), do please realize that you are trying to time the SP500 and the Long Bond.

Those are very hard to do. Put another way: if you find ways to time those, then you could *ALSO* apply the exact same techniques to a vanilla stock-treasury portfolio.

Of course, you can find something that 'works' over a period of 5 years, 10 years, or even 15-20 years. But likely, you are just overfitting in-sample.

With TMF+UPRO portfolio you have a bit more nuances factors than just a LongBond+SP500 portfolio (you are basically long LongBond and short the 3-month LIBOR, and you are long SP500 and short the LIBOR), but not much.

If you are going to try to market time and do other exotic stuff, you may as well consider adding Buy-Write and Put-Write concepts to your portfolio.
"A Republic, if you can keep it". Benjamin Franklin. 1787. | Party affiliation: Vanguard. Religion: low-cost investing.

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Re: Refinements to Hedgefundie's excellent approach

Post by Hydromod » Mon Oct 28, 2019 11:11 pm

AlphaLess wrote:
Mon Oct 28, 2019 10:34 pm
Interesting thread, and I commend the effort.

Before trying to figure out any sort of dynamic weights (e.g., 100% TMF, or 100% UPRO, or anything in between), do please realize that you are trying to time the SP500 and the Long Bond.

Those are very hard to do. Put another way: if you find ways to time those, then you could *ALSO* apply the exact same techniques to a vanilla stock-treasury portfolio.

Of course, you can find something that 'works' over a period of 5 years, 10 years, or even 15-20 years. But likely, you are just overfitting in-sample.

With TMF+UPRO portfolio you have a bit more nuances factors than just a LongBond+SP500 portfolio (you are basically long LongBond and short the 3-month LIBOR, and you are long SP500 and short the LIBOR), but not much.

If you are going to try to market time and do other exotic stuff, you may as well consider adding Buy-Write and Put-Write concepts to your portfolio.
I appreciate the concerns. This part of the portfolio is only 5 percent of my overall portfolio and I'm not adding to it, so I have a rooting interest but not a keep-the-house interest. The rest of the portfolio is buy and hold; even with this strategy included the overall portfolio is not particularly aggressive.

I have thought considerably about the difficulties with timing. I will not go above 70% UPRO, and I'm basically aiming for a moderate rate of change in the risk budget for UPRO (e.g., completely dropping out of the maximum UPRO position will take at least 4 months). So this is more of a sliding back and forth to tilt the odds than sudden switching based on market twitches.

The timing flag from the unemployment rate has been backtested to the 1950s, so at least that has a record of nearly 70 years.

The next year or two should be very interesting...

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Re: Refinements to Hedgefundie's excellent approach

Post by AlphaLess » Tue Oct 29, 2019 12:14 am

I would focus on a very simply approach:
- portfolio can have 3 different weight schemes, and only 3: (a) tilted towards TMF, (b) balanced, (c) tilted towards UPRO. Keep those tilts close. E.g., balanced can be 50-50, and tilted can be 30-70, or 70-30,
- have VERY simply timing rules, where timing determines which weight scheme you are in. Both SIMPLE, as well as RARE.

#2 potentially limits your in-sample overfit, and trading costs.
#1 ensures that all 3 weight schemes are NOT very far from each other, so there is a LOW penalty from being incorrect (as well as LOW reward from being correct).
"A Republic, if you can keep it". Benjamin Franklin. 1787. | Party affiliation: Vanguard. Religion: low-cost investing.

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Re: Refinements to Hedgefundie's excellent approach

Post by Hydromod » Tue Oct 29, 2019 9:10 am

AlphaLess wrote:
Tue Oct 29, 2019 12:14 am
I would focus on a very simply approach:
- portfolio can have 3 different weight schemes, and only 3: (a) tilted towards TMF, (b) balanced, (c) tilted towards UPRO. Keep those tilts close. E.g., balanced can be 50-50, and tilted can be 30-70, or 70-30,
- have VERY simply timing rules, where timing determines which weight scheme you are in. Both SIMPLE, as well as RARE.

#2 potentially limits your in-sample overfit, and trading costs.
#1 ensures that all 3 weight schemes are NOT very far from each other, so there is a LOW penalty from being incorrect (as well as LOW reward from being correct).
From my perspective, I don't think that there is a great deal of fitting going on in anything I presented. Others may consider the frequent checking for adaptive allocation to be "fitting" and "market timing", but all it is doing is explicitly maintaining the risk budget in balance (based on the recent market volatility) instead of keeping the allocation constant and allowing the risks to vary over time.

The adaptive allocation risk parity part of the scheme essentially works the way you prefer, but with finer gradations. In my favored approach, the risk budget allocation is based on the last 60 days and checked frequently, but rebalancing only occurs if the current allocation is outside bands. Straight risk parity generally can't send the portfolio to extreme allocations.

Trading costs are not much of an issue in a Roth account on M1, especially if the trading is less frequent than, say, weekly.

And since the early 1980s, constant allocation at a range from 30/70 to 60/40 would have produced almost the same CAGR (albeit with different volatility). So that's favorable with respect to low penalties as well.

I agree that changing the risk allocation to UPRO based on the unemployment rate behavior is market timing, with almost 70 years of experience, and it will send UPRO to lower and higher extremes than straight risk parity. I'm frankly more concerned about making UPRO too high than too low.

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