We all follow expert advice. Maybe you follow your own (expert) advice. Even if it is a non-discretionary quantitative algorithm created by a Ph.D. mathematician then he/she (the Ph.D) can be called an expert whose advice you are following.
There is a whole branch of mathematics that gives you expert advice on the best way to follow expert advice.
Imagine that after a bunch of backtests and research, you want to start out following the advice of 5 ‘experts’ including your dad who recommends a 60/40 allocation of stocks and bonds. Maybe you follow another ‘expert’ by buying equal amounts of his designer models made available at P123. Maybe you mix some designer models with some ETFs. Maybe you mix some of your own ports with some ETFs.
Each combination (of ports, designer models and/or ETFs) is a strategy and constitutes an ‘expert’s advice.’ And hopefully, you have designed the strategies so that it is likely that they will have acceptable risk going forward.
For simplicity lets say you start out investing equal amounts (20% of your capital) in each of the 5 strategies.
Going forward, do you “stay the course” and never alter your allocation? If you adjust the allocation of your capital based on the performance of the strategies, how do you do it?
There is a whole set of algorithms that try to give you the minimum amount of total regret when you are done. What is interesting about these algorithms is they make NO STATISTICAL ASSUMPTIONS. In essence they allow the market to do weird things–starting with not following a normal distribution (or any distribution for that matter)!!!
One of the better know algorithms for investors is the Cover Universal Portfolio. Here is a link to his original paper: Universal Portfolios
Here is the first line in the abstract:
"We exhibit an algorithm for portfolio selection that asymptotically outperforms the best stock in the universe.
Whoa!!! OMG!!! Are you serious?
So this is a serious paper and a serious claim. In real life there is the problem of trading costs and what exactly “asymptotically outperforms” means. Plus, it takes a truly great computer to run it for the 500 stocks in he SP 500 (cannot be done). There are also serious attempts to address these issues.
But the Hedge Algorithm is easier and it has some good theoretical bounds on how much you are going to regret the allocations you made on the 5 strategies example that I started with above. And it does not seem too weird intuitively.
It gradually reallocates more resources to the better performing strategies like you would want. And you can even adjust how quickly it adjusts these allocations. This is the learning rate (or η) in the Wikipedia link below.
Probably easiest to read about it here in (Wikipedia near the middle of the article): Multiplicative Weight Update Method
Skip to the formula at the end of the section on the Hedge Algorithm in Wikipedia.
The equation may look a little difficult but in reality it only has 2 main complications:
The learning rate (η). Just pick .5 and change that later if you want.
The cost of the decision (m also called regret). Each period (say a month), each of the 5 strategies will have a regret. Your best performing strategy has zero regret because you would not have regretted putting all of you money into that strategy.
If one of your strategies underperformed the best strategy by 5% that month then your regret is 0.05 for that strategy (for that month). Simple. Plug it into the formal to figure out how much weight you want to put on a strategy (maybe normalize the weights).
Is this a reasonable strategy? Better strategies? How do you readjust your asset allocation based on performance?
I can divide all of my P123 strategies and ETFs into 2 sets (2 strategies) that have good risk characteristics and call them 2 diversified portfolios, strategies or expert’s advice. And begin to adjust the weights of these 2 portfolios (strategies) according to their performance.
I can have a mathematically guaranteed limit on the total regret I will have when I retire and look back at what if have done. Cool math and kind of makes practical sense, I think. Should I use it? Uhhh…Let me think about that. And let me know what you think.