If one doesn’t want to ponder on this, one can follow the usual procedure of computing covariance matrix of the returns , leading to efficient frontier which spews out specific portfolio compositions for a level of risk….Amidst all this jargon, one quickly realizes that this is good in paper, this is good for software vendors who make plugins for mean variance optimization, good for hiding the complexities that actually take place in the real world. But if you want to really manage assets using MV approach, one needs to think about its limitations and work around it.
Molehill of garbage in and Mountain of garbage out , is the way the authors describe MV optimization. To begin with, the limitations of MV approach are the following:
- Returns are assumed gaussian
- Investor’s Utility function is not considered
- How to use it in Multi period context ?
Typically one uses the mean and variance of the historical data to construct the frontier. However the estimation errors gives rise to instability in the portfolios. Also there is an ambiguity in testing the differences between a portfolio on efficient frontier and outside of it. Authors of this book urge the investment community to follow approaches based on statistical grounding to grapple with the instability and ambiguity issues of portfolio optimization.
The books starts off with a basic introduction to classic mean variance approach which comprises applying quadratic programming technique which comprises an objective function(portfolio variance) with a set of equalities and inequalities like expected return of the portfolio, sector allocation rules, budget constraint rules, sector allocation rules , non negative weights for portfolio weights etc. The basic technique is minimization search algo that finds the best allocation across the assets in the portfolio. Concept of parametric quadratic programming is also touched upon where corner portfolios and its importance are discussed in the Markowitz framework.
Author then moves on to giving a gist of alternative approaches that have emerged :
1. Alternative measures of risk instead of second moment
2. Utility function Optimization
3. Multi period investment horizons
4. Monte carlo techniques for asset -liability flow simulations
5. Linear programming optimization
The book then explores Unbounded MV frontier and Linear Constrained MV Frontier. The first approach limitation can be seen by simply bootstrapping a sample from the historical returns and seeing that sharpe ratio of the sampled data is very different from the actual data. Thus the unbounded method clearly is of no investment value for a portfolio manager. One needs to give a great importance to estimation errors as it will turn out a different MV portfolio. Thus it looks like Unbounded MV solution, though analytically elegant is basically useless for an investment manager.
In most of the cases, the portfolio manager puts some constraints on the unbounded mv problem like sector specificity or sum of weights = 1 etc and the bounded one seems to be doing ok for a few repeated sampling. The book points out the reason being that in the unbounded case, the higher returns are usually over represented, which means the higher estimation errors are over represented and vice versa. In the bounded, the role of constraints is just to minimize that aspect. Apart from that Linear Constrained MV is also not good. Here the author explores a intensive bootstrap experiment to introduce the concept of Efficient Frontier Variance
The above 2 graphs summarize the concept of frontier variance. Efficient frontier, for the same set of data, under bootstrapping gives rise to variability in the frontier. Also efficient portfolios for a set of risk level( figure on the right) also span a specific area. If an investment manager has to use MV , then the statistical nature of MV has to be taken in to consideration. The book mentions the following approaches , each of which taken individually improve the MV portfolios, but when taken together can substantially improve the investment management procedure.
Resampled Efficient Frontier Optimization
Resampled Efficiency rebalancing, monitoring, analysis
Avoiding Optimization design errors
Resampled Efficient Frontier (REF) Optimization
“If you are 100% confident of the estimates, use Classic MV “, is what the author states!, reason being that MV is extremely sensitive to estimation errors. Book goes on to make a strong case for REF by saying that REF are average portfolios and provide safe bets on assets under the realistic case of estimation errors in the input. There is also a simulation based proof of the usefulness of REF. What is the look-back period for resampling ? This gives the flexibility which is missing in the rigid classic MV. A value based manager can take a longer history while a growth based manager might be willing to look at short term horizons. Another related concept is the Maximum return point, which is not present in classic MV approach. This Maximum return point gives higher return for a lesser sigma as compared to some assets which give lower returns with higher sigma!! Paradoxical it might seem to a person who is only thinking in terms of Classic MV !! However, considering the variability of the input estimates , this is very much a situation where there are inferior assets present in the asset universe included for optimization. A quick illustration would help one see this point.
One of the best things in this book is the clear illustration of REF Vs Classic MV approach using a story of a person trying to find his pet
Let me summarize this story , for this is a beautiful analogy to look at REF vs Classic MV approach.
The story is about Robert who has a pet hedgehog . One day he notices that it has escaped from the cage and is somewhere in the fence. The fence is sturdy and hedgehog escape proof. Fortunately, Robert had fitted a GPS device. He gets signals about the location of his pet.
Now if one were living in unbounded MV world , depending upon what the signal shows, Robert would go and search the location. For example , if A were the location displayed on GPS, he would go to A. However he knows that there is a fence that his pet cannot go out. Thus he ignores A. This is like living in a bounded MV world. If he believes that GPS is 100% accurate in its estimates , then he would go to whichever location that GPS points out with in the boundary.
However he knows that GPS has always some estimation error with it. So, he accumulates a set of points and then depending on the cluster properties , ventures out to search for his pet. This is similar to REF where you resample the input so that you are aware of estimation error and the portfolio you construct is more appropriate. So, instead of going to M which the GPS currently points out, he would probably go out and venture location R.
Portfolio Rebalancing, Analysis and Monitoring
Once a portfolio has been selected, it needs to be obviously rebalanced according to market conditions. Most of the present rules are adhoc , to say the least. Rebalancing is done quarterly, monthly,annually….for god knows why ? if asset weights exceed a specific percentage level, they are rebalanced ? why ? Most adhoc rules like technical analysis that is prevalent in todays world is completely arbitrary. There is no statistical significance attached to the decisions. This chapter makes a case for brining in statistical mindset to this process. By formulating a distance measure between the current portfolio and the previous selected portfolio, the author makes a case for portfolio rebalancing.
At this stage of the book, the book introduces meta-sampling method….I could not understand this method well enough to write about. Hope to understand this concept in the days to come.
Input Estimation and Stein Estimators
This is an important chapter of the book where there is a discussion on the sample means and the way they are used every where for multivariate case. Charles Stein(1955) was the first person to prove that sample means are not an admissable statistic for a multivariate population mean. Since then there have been a lot of work in this area, though finance community has been wary of adopting them, for no reason i guess.
The basic funda behind these estimators are that it starts off with a prior and then based on the sample means and their variances, population mean is calculated. James Stein estimator, Frost-Savarino estimator, Stein covariance estimation and Ledoit Covariance estimation are discussed in the book. Preliminary results of applying these estimators are provided, though the author admits that it is in the interest of finance community , industry to be specific, to focus / prod academic research in this area.
The last couple of chapters talk about benchmark mean variance optimization, bayes fundas and general aspects that need to be kept in mind while optimization.
My take on this book : It is a terrific book which will make you think about a lot of aspects about mean variance optimization. One of the highlights of this book is that most of the stuff is explained intuitively and through diagrams. This is a must read for anybody who is connected to asset management work !.