Solving the "Corner Solution Problem" of Portfolio Optimization

Many portfolio optimization methods (e.g., Markowitz/Modern Portfolio Theory in 1952) face the well-known predicament called the “corner portfolio problem”. When short selling is allowed, they usually give efficient allocation weighting that is highly concentrated in only a few assets in the portfolio. This means that the portfolio is not as diversified as we would like, which makes the optimized portfolio less practically useful. In [Corvalan, 2005], the author suggests to look for instead an “almost efficient” but “more diversified” portfolio within the close neighborhood of the Mean-Variance (MV) optimal solution. The paper shows that there are many eligible portfolios around the MV optimal solution on the efficient frontier. Specificially, given the MV optimal solution, those “more diversified” portfolios can be computed by relaxing the requirements for the portfolio return and risk in an additional optimization problem: where , is the Markowitz MV optimal weights, are the relaxation tolerance parameters, and is a diversification measure for the portfolio (for example, , ). In other words, the new optimization problem looks for a portfolio with the maximal diversification around the optimal solution. Corvalan’s approach can be extended to create an approximate, sufficiently optimal and well diversified portfolio from the optimal portfolio. The approximate portfolio keeps the constraints from the original optimization problem. References: SuanShu Javadoc Alejandro Corvalan (2005). Well Diversified Efficient Portfolios. Documentos de trabajo del Banco Central, no....

Trading and Investment as a Science

Here is the synopsis of my presentation at HKSFA, September 2012. The presentation can be downloaded from here. 1. Many people lose money playing the stock market. The strategies they use are nothing but superstitions. There is no scientific reason why, for example, buying on a breakout of the 250-day-moving average, would make money. Trading profits do not come from wishful thinking, ad-hoc decision, gambling, and hearsay, but diligent systematic study. • Moving average as a superstitious trading strategy. 2. Many professionals make money playing the stock market. One approach to investment decision or trading strategy is to treat it as a science. Before we make the first trade, we want to know how much money we expect to make. We want to know in what situations the strategy will make (or lose) money and how much. • Moving average as a scientific trading strategy 3. There are many mathematical tools and theories that we can use to quantify, analyse, and verify a trading strategy. We will show case some popular ones. • Markov chain (a trend-following strategy) • Cointegration (a mean-revision strategy) • Stochastic differential equations (the best trading strategy, ever!) • Extreme value theory (risk management, stop-loss) • Monte Carlo simulation (what are the success factors in a trading...

Quantitative Trading: Economist Approach vs. Mathematician Approach

Thank you Lewis for introducing me to the field of “Quantitative Equity Portfolio Management”. It opens my eyes to the other spectrum of “Quantitative Trading.” Apparently what Lewis considers quantitative trading is very different from what I consider quantitative trading. I call the former an economist approach and the latter a mathematician approach. This blog piece does a very brief comparison and points out some new research directions by taking the advantages of both. Briefly, the economist approach is a two-step approach. The first step tries to predict the exceptional excess returns alpha by examining its relationships with macroeconomic factors, such as momentum, dividends, growth, fundamentals and etc. The second step is capitals allocation. The focus in the economist approach is on identifying the “right” economic factors. The mathematics  employed is relatively simple: linear regression, (constrained) quadratic programming. The trading horizon is month-on-month, quarter-on-quarter, or even years. An example of such is factor model in QEPM. In contrast, the mathematician approach tries to predict the short-term price movement by building sophisticated mathematical models for the, e.g., price time series. The focus is on finding the right mathematics to better describe the statistical properties of price process, e.g., stochastic calculus, Markov chain. Macroeconomic and fundamental factors are not often used. The trading horizon is intra-day or seconds. An example of such is volatility arbitrage in different intraday time scales. One way to appreciate the differences is by looking at their trading horizons. When trading high frequency, the company fundamentals certainly have little relevance because, e.g., the quarterly earnings do not change second-by-second. The statistical properties of the price process dominate in these...

The Right (and Wrong) Way to Run an Algorithmic Trading Group

I would like to share with you the unique vision that Numerical Method Inc. has about running an algorithmic trading group. To get an edge over competing funds, we emphasize on 1) the research process and 2) technology rather than on hiring more intelligent people. Currently, the majority of the quant funds run like cheap arcade booths: the traders are given workstations and data. They do whatever to crank out strategies. The only contributing factor to profit is luck – luck in finding the right people and/or luck in finding the right strategies. I had a conversation with an executive from a large financial organization two years ago when they started to build an algorithmic trading group. He said, “Haksun, we need to hire some very smart people to be better than Renaissance.” I repeatedly hear something similar from various portfolio managers and hedge fund owners. Staffing “very smart” people in this ad-hoc, unmethodical, non-scientific process in search of profitable trading strategy is merely a lottery in disguise. The main reason is that there is no necessary condition between “very smart” people and “very profitable” trading strategies. If there is any relationship, it can only be a sufficient condition. It is very difficult to hire the “very smart” people because They are difficult to be identified among numerous pretentiously smart people. The competition for them is a very fierce battle. The best examples are the fight between Microsoft and Google over Dr. Lee Kai-Fu, the dispute between Renaissance Technologies and Millennium Partners over two former employees. The very best people are driven by passion rather than money, e.g., Gates,...

China ETFs, anyone?

Actually, I struggle to decide the title of this post. In fact, at one point I believe the more appropriate title will be “Will China long funds disappear like penguins?” However, I decide to send a more positive message to my readers, so hopefully this title will light up my readers’ interest in investing China equities… In case my readers are not aware of, one of the latest trend in the Hong Kong market is the rise of sector ETFs. You see, products such as China A-50 (2823 hk) and Tracker fund (2800 hk) have been in the market for quite sometime, but it’s not until now that there are more specific China ETFs available in the market. This is one of the most exciting developments in the Greater China equities market, as investors have options to maximize their return. Claymore has been the pioneer in terms of product offerings for Chinese equities market. Although Claymore cannot really break into the A-share market, it is combining the best of the Hong Kong, US and Singapore listed Chinese stocks to form specific ETFs. Performance of its ETF has been stellar. For example, in 2009 return of Claymore’s China small cap ETF (HAO US Equity) was 98% and perform better than the majority of long funds (I heard from market rumor that top 10 percentile fund performance, including long only and hedge funds, is about 90%). Probably having seen the success of HAO (In Chinese, hao means good), this year Claymore has launched 2 additional sector funds: A real estate and a technology ETF. So it’s actually pretty straight forward, like real...

Creative retailing

All right, I may bore my readers as my focus has been always on the consumer sector in China. But I do find a lot of interesting subplots going on in this sector, as I am spending my weekend shopping in Hong Kong. You see, Hong Kong actually is quite a matured city (in terms of high rental cost and as an international center), so it is not strange to see new forms of retailing evolving in such a sophisticated city, catering needs for a diversified pool of customers. Say for instance:                                                                                                       The birth of Cosway (288 HK)  in Hong Kong: I am actually quite amazed to see that direct sales is happening on street stores level now. This is a subsidiary of Berjaya Corp and so far it has already opened more than 90 little shops on the street level to recruit anyone who is interested in building a direct sales network. In fact, it’s so visible now that I can see people selling Cosway’s water filter equipments in the noisiest part of Causeway Bay. If you still believe direct sales business model is classified as “underground economy”, well I hope you will have time to visit Cosway’s stores. It’s quite small on the street, but you can find them and they work! I tell you every time I walk by a Cosway store in Hong Kong, I see customers shopping and getting new customers. So go figure. The funny shoes that are selling in department stores from Skechers (btw you know GEM, the cute little HK singer as the brand’s spokesperson?) : Okay, I know that this is...