NAPS 2021: Optimal Portfolio vs Equal Weighting Results

14th Aug '21 by Bitbug

45 comments 1659 reads

Hi All,

I selected all stocks (except for ERGO, where there was a data problem) in Stockopedia's 2021 NAPS and using data for the previous calendar year, performed a portfolio optimization.

The resulting plot shows the position of each stock on a Returns-Volatility diagram. The red and green stars are on the efficient frontier that correspond to a maximum Sharpe Ratio and minimum volatility, respectively. The weightings corresponding to the maximum Sharpe Ratio (and thus, Optimal Portfolio) are

Purchases were made on 1 Jan 2021 according the the above allocation and the YTD results compared to purchases made with equal weighting. The results were:

2021 YTD Returns:
Optimal Allocation: 26.2%
Equal Weight Allocation: 41.6%

It comes to me as a surprise how much less the 'optimal' allocation return is compared to the equal weight allocation. Perhaps, the 2020 pandemic year data used may have thrown things out a little. Perhaps, also, the optimal portfolio approach is more suited to the allocation of more diverse assets, including bonds for example, rather than just stocks. Comments welcome!

Unlock with Email

Unlock with Google

Already have an account?
Login here

DWit199

20th Aug '21

If B=2, mean positive 20% mean negative 10% and P =.6 then f = ((2+1)*.6 -1)/2 so f=0.4 On that basis 40% of my investment capital should be allocated to this scheme. In reality roughly 20% of my equity exposure is to this system so I have intuitively allocated a half Kelly fraction!

Michael_P

Thanks, Bitbug. This has been a fascinating read. I'd be interested in having a copy of this code. What's the best way to do that? Is it on GitHub? Or on google drive, perhaps?

This is looking like a stellar year for NAPS, so perhaps it will be tough to beat this year. For example, I'm pretty sure it is beating the CyNAPS portfolio quite handily this time around after slightly losing out to it last year.

Bitbug

21st Aug '21

You can download a zipped copy of the code here. Unfortunately, I am unable to provide technical support as the code is reasonably well internally documented.

Thanks for the comment. Download details are in a previous entry. Grateful if you could provide a link to the CyNAPS portfolio. I am not familiar with it. Thanks.

Thanks. Below is a paragraph about CyNAPS that I copied from Ed's NAPS write-up in January this year. Ben and Jack from the Stockopedia team put together a portfolio that is partly based on stocks popularly selected in the Stockopedia annual competition and partly on NAPS principles.

"They have though put together a 44 page download about the new 2021 CyNAPS picks which you can find here - https://assets.stockopedia.com/topstocks2021.pdf. The CyNAPS is a NAPS-style portfolio that uses the top picks from the Stockopedia Challenge. It's a man+machine portfolio that has returned 31.5% over the last 12 months. There is a fair amount of overlap with some of the NAPS portfolio names. I do hope you enjoy the read."

Many thanks.

2nd Sep '21

Previously, I looked at optimal portfolios where shorting was not allowed. This required quadratic programming optimization because of the non-negative portfolio weight constraint. When shorting is allowed, the problem is a little simpler requiring the simultaneous solution of systems of linear equations [see Francis, J.C. & D Kim: Modern Portfolio Theory, Wiley Finance, 2013].

In the 'Optimal Portfolios for 2021 NAPS' graph below, with shorting allowed, all optimal portfolios lie on the straight line called the Capital Market Line (CML). One chooses the level of risk to take depending on market conditions/cycles. The solid straight line represents varying levels of cash reserves held in the portfolio: the risk free portfolio has 100% cash, while the tangency portfolio has all cash fully invested without borrowings. While moving along the dashed straight line from the Tangency Portfolio, the level of borrowing (i.e., leverage, or gearing) increases. I have optionally chosen an optimal portfolio with a 0.008 mean daily return which corresponds to an expected annual return of (0.008*252-1)*100 = 102%.

Below are the results for various optimal portfolios without and with shorting. When shorting is not allowed, it had been previously concluded that there was nothing to be gained by changing the weights from other than equally weighted ones. The table below demonstrates this. However, when shorting is allowed the picture appears to change dramatically yielding far greater returns (thanks to leverage!):

With shorting and leveraging allowed, the approach lends itself to the use of Contracts for Difference (CFD). Just a word of caution: the table represents just one outcome and perhaps an ensemble of results are needed to draw firm conclusions.

5th Sep '21

For your information: I performed an equal weight vs equal risk weightings analysis for the 2021 NAPS using data over the one year 2020 period (which may not be sufficiently long for your requirements) and the results do not differ significantly. Of course, one result may not be conclusive but it may be a good indication.

It appears a weighting proportional to the previous year's mean return - an equal return portfolio, rather than an equal risk one - may work better. To end of August 2021, it yields a 55% return compared to a 37% return :

The equal return portfolio has an historical standard deviation of 0.0227 which is less than the historical average standard deviation of all the stocks (0.03803) as expected and less than similar for any stock bar one (YCA.L 0.02170). The equal risk portfolio has an expected standard deviation of 0.0230, which is not too dissimilar to the equal return portfolio. With similar portfolio risks, I would be opting for the equal return portfolio over the equal risk one as the return is greater for the same risk.

I have added a lot more to the discussion since your post. Would you care to make further comment?

7th Sep '21

I thought it worthwhile to finally draw some conclusions and bring this to a close. The objective of the exercise was to investigate the merits or otherwise of alternatives to a simple equally weighted allocation as is typically done for Stockopedia’s NAPS selections.

First, I would like to thank everyone for their contributions, constructive or otherwise. There have been vague notions of using suitable ‘proxies’ in favour of returns and risks which however were short on implementation details. Then there was the assertion that I would be wasting my time pursuing this. To my mind, this was like raising a red flag to a bull!

Here are my conclusions:

There is little merit in using a risk-weighted allocation. I interpreted ‘risk-weighted’ as meaning a normalized allocation inversely proportional to a stock’s historical standard deviation of daily returns. By ‘historical’, I mean using one year of the most recent historical data. [This applies throughout this discussion.] In contrast, a ‘mean-weighted’ allocation, i.e., a normalized allocation proportional to a stock’s historical mean return appeared to show promise. Both these approaches, of course, have nothing to do with Modern Portfolio Theory (MPT). Also note that a negative historical mean return for a stock implies shorting the stock.
There was no clear advantage in using MPT optimal portfolio allocation when shorting is not allowed since a fair number of stocks ended up with a zero allocation with a consequent effect on the portfolio’s size (and hence risk) and diversity.
The MPT results when shorting is allowed showed much greater promise. The attractiveness of this approach is that an investor can choose a desired level of risk ranging from 100% encashment to being fully invested without borrowing (the Tangency Portfolio), to being highly leveraged - not for the faint-hearted!

In the above, when I say ‘showed promise’, I mean having portfolio returns significantly greater than an equally-weighted one. It should be noted that these conclusions are derived from cursory rather than exhaustive research. With this in mind, I invite anyone who has the Python programming skills (or is willing to learn) to experiment with the code. You can find the MPT Python code allowing shorting here and the Python code when shorting is not allowed here. Unfortunately, I am unable to provide any warranties or technical support.

Finally, it is worthwhile quoting part of a chapter in the book by W Kinlaw, MP Kritzman and D Turkington: A Practitioner's Guide to Asset Allocation. Wiley 2017. Its Chapter 7 is simply titled "1/N" and exposes the "Fallacy: Equally weighted portfolios are superior to optimized portfolios". Admittedly, the book is targeted at the broad range of asset classes and not just equities. Nevertheless, the quotation still rings true if one replaces 'asset class' with 'stock':

First, equal weighting is not sensitive to return and risk estimates; instead, it is entirely dependent on the choice of the asset class universe....The 1/N approach essentially transfers the risk of input estimation error to the risk of selecting the right asset classes.
Second, the 1/N heuristic offers investors only one portfolio no matter their attitude toward risk. The efficient frontier allows investors the choice of a wide variety of portfolios, each with different combinations of return and risk.
Third, equal weighting ignores the capacity of each asset class ... beyond expected return and risk, which might favour one asset class over another.

... from W Kinlaw, MP Kritzman and D Turkington, pp62-63.

9th Sep '21

There must be some of the more experienced folk amongst you who must be thinking: "When will this guy get it? MPT in its original form just simply doesn't work in this situation." Well, I have to say, thanks to a simple spreadsheet error, I have had to revise my conclusions dramatically! This error came about when I inadvertently used the difference in prices rather than the ratio of prices. The results now are even more interesting, particularly when one uses a weighting proportional to historical variances, an idea put forward by Ed Croft in a contributing post. With the 2021 NAPS stocks, the YTD results show a 25% improvement in profitability.

First, the corrected and updated spreadsheet results:

The fifth column headed VarWts are weightings proportional to the historical variance of the daily mean returns for each stock. By 'historical' I mean the 2020 calendar year. The third last column headed 'Var' shows the profit/loss on each stock and sum up to 1.51, indicating a 51% return (highlighted). This is better than the 40% based on equal weightings. Note that while the 'Optimal' returns column shows a 1.714 return, when the 42.04% borrowing is taken into account, this reduces to a 29% return.

So what does the YTD timeseries for each portfolio look like?

The Var Portfolio gives the best return, as already indicated, followed by equal weightings. The others follow, but their volatility can be punishingly high, particularly the leveraged portfolio. There is some insight to be had by looking at mean-stddev diagrams for both the analysis period (calendar year 2020) and the performance period (2021 YTD).

Every bit of information on this analysis period diagram is based on 2020 data. There are three 'optimal' portfolios (Leveraged, Tangent and Risk Free), and three 'sub-optimal' portfolios (Min Risk, Equal Wt and Var) according to Modern Portfolio Theory - six portfolios in total, each indicated by a coloured star. Although there are six portfolio weightings, there are five in practice, as the Risk Free portfolio never changes (100% cash!). When these five portfolio weightings are applied to 2021 YTD data (the performance period), we obtain the following:

Except for the six portfolio weightings, everything you see here is based on YTD 2021 data. You will notice that the Leveraged and Tangent portfolios show much weaker performances and are way off the positions they should be to be optimal, while the Equal, Min Risk and Var portfolios appear to hold their ground far better. This is reflected in the time-series performances of individual portfolios shown above. Note also that the positions of individual stocks have moved reflecting a different dataset.

So, hooray and yippee! We have an improvement on an equal weighted allocation as last! Initially I had interpreted Ed Croft's suggestion by trying a weighting inversely proportional to the risk which seemed to make better sense - i.e., by placing more capital on less risky stocks; then I resorted to direct proportionality, including combinations like mean and mean x risk. The variance direct proportionality seemed to work best. This now opens the door to a host of possibilities. Stay tuned!

11th Sep '21

You say you use QM Rank and Mkt Cap to forecast future returns. Surely a couple of the main drivers would be past returns and risks, i.e., the mean and standard deviation of historical daily returns? The end result are predicted future returns. What does the plot of actual vs predicted returns look like? In the end you still have to resort to a selection process to select which stocks to include in your portfolio.

12th Sep '21

The neural network is trained using data extracted from the Stockopedia Stockrank performance charts. Effectively, the forecast is the average annual return expected from a large number of shares with similar QM rank and market cap. To that I add the forecast dividend and then divide by the annualised volatility as reported by stocko. This "forecast Sharpe ratio" is then rescaled to give the final forecast in the range -20% - +20%.

The chart below shows the forecasts on the x-axis and the 26 week actual returns on the y axis for a snapshot of 735 companies taken in March 2021. There is a lot of randomness but overall the trend is towards better outcomes for high forecast shares.

Selection is then made in a modified NAPS style using shares with high forecast according to market sector and industry group.

14th Sep '21

I am on a steep learning curve on portfolio management at the moment. I have discovered that single-index models fare better and are more consistent than the classical Markowitz style mean-variance approach when projecting to the future. Have you considered using mean/beta ratios to rank a stock's desirability for portfolio inclusion rather than the Sharpe ratio? Stocko has betas for most stocks but I find it easier to generate them from scratch using historical data and a benchmark index such as the FTSE100.

I decided against incorporating beta into my forecaster. A high beta share must have high volatility but a low beta share could have high volatility but low correlation with the index. One of the anomalies that I am trying to exploit is low volatility. Using forecast Sharpe ratios favours low volatility shares and the correlations are taken care of NAPS style by forcing diversification across sectors and industry groups when creating the portfolio.

The chart below is generated by sorting my investment universe by ascending volatility. Average excess returns are calculated for each decile which are then combined into 9 overlapping quintiles. This shows the pattern of returns but removes some of the noise as these are relatively small samples from a snapshot in March 2021. The shape of the curve is surprisingly similar to the curve in Pam Van Vliet's low volatility book which supports the idea that high volatility is detrimental to returns.

The first chart below is my observed volatility data this year, low volatility on the left, high to the right. The second chart is taken from High Returns from Low risk by Van Vliet and De Koning.

Yes, I agree with you on the first paragraph. Regarding the Van Vliet and De Koning (VV&DK) graph: if one plots mean daily returns vs standard deviation of daily returns (i.e., 'volatility'), one generally finds that as volatility increases, the mean daily returns increases. However, instead of using mean daily returns, one uses the geometric mean of daily returns (what VV&DK call 'Compound Return' in their graph) you will obtain a graph similar to VV & DK's. I think it is that simple.

one generally finds that as volatility increases, the mean daily returns increases. However, instead of using mean daily returns, one uses the geometric mean of daily returns (what VV&DK call 'Compound Return' in their graph) you will obtain a graph similar to VV & DK's. I think it is that simple.

I'd like to understand that better. Do you have a relevant reference?

This scatter plot illustrates my point:

Just look up 'geometric mean of daily returns'. It will involve averaging log(1+r_i), for all of a stock's daily returns r_i, taking its exponent and then subtracting 1.

NAPS 2021: Optimal Portfolio vs Equal Weighting Results

Unlock the rest of this article with a 14 day trial

About the Author

Bitbug

45 comments

Unlock the metrics for all these stocks and more...

Unlock the metrics for all these stocks and more...