The nature of financial world is stochastic, barely, hardly and scarcely deterministic. Phenomena are mainly subject to randomness and only statistical statements can be made about the real relationships; these statements may match the reality, or they may fail to do so. Using appropriate statistical methods with an adequate sample size, the statistical statements may be highly informative.
Aim:
Our aim is to rank the hedge fund equity portfolio managers during a period of time ( quarterly basis, for instance) using the numerical data related to their equities(stocks) in the file form 13F holdings . Their stock-picking decisions are quantified, analyzed and compared to other hedge funds in a Hedge Fund Universe (HFU).
Since the hedge fund manager's concern is long-term investing, it is the growth rate and not the rate of return that is of interest;
our second approach is based only on the logarithmic returns of the hedge fund portfolio to determine its capital accumulation process, this second approach is concise with the Campbell-Shiller framework.
A comparative capital asset pricing model, within stochastic finance theory, is
derived to note the similarity or dissimilarity between two given hedge fund
equity portfolios and their market portfolio benchmarks. The model linked the
appreciation rates of two hedge fund equity portfolios to their returns, their
volatilities and the covariance between their returns. Our analytical framework
has an explicit relation with alpha quantification.
For a given hedge fund equity portfolio, a wealth accumulation process based
on the logarithmic returns of its portfolio was formulated. The formulation
takes into consideration the presence of the empirical volatility which diminishes the
wealth accumulation that one might naively expect from a given empirical arithmetic
return. The discerned price movements in the stocks are formulated as a wealth
accumulation process that may be fostered by the hedge fund equity portfolio.
The result from the wealth accumulation analytic is much more concrete
than typical results from stochastic finance theory. The wealth accumulation
analytic used the empirical mean and variance of the actual returns, not as estimated
parameters in a stochastic process that is assumed to generate these returns.
Nevertheless, our mathematical analysis ( to be published ) showed that the first approach is
indeed a special case of the second approach when prices are following the
known stochastic process in the modern portfolio theory, for instance stochastic
portfolio framework.
In the light of facts mentioned above, the wealth accumulation analytic is also introduced to monitor the numerical results from the stochastic process modeling.
The stochastic process modeling is conceptualized under a measurable space (Ω, F), a given probability model, with a probability measure λ defined over this measurable space and under which suitable theory of stochastic calculus related to equity prices is done. Parameter estimations from historical data can be done more accurately under this actual measure λ than under a risk-neutral probability measure( this risk-neutral probability measure is mainly representing a degree of belief about forward-looking events and it may not exist for the existing financial markets as the arbitrage is a fact).
Under no arbitrage opportunities, there exists a risk-neutral probability measure equivalent to the real or actual probability measure.
If the risk-neutral probability measure allocates zero probability to an outcome (meaning that it is impossible), then the real or actual probability measure also allocates zero probability too, they agree on all impossible events and may assign different probabilities to possible events. If these two conditions are satisfied, the two probability measures are called equivalent.
Arbitrage activities in existing financial markets are The Rule, Not The Exception. Hence assuming the existence of a risk-neutral probability measure is simply a distortion, its adequate propriety to handle risky world is not an asset anymore.
As all our expectation calculations are supposed to be under the actual measure λ that takes into account the risk premium, the actual distribution of stock returns is then estimated directly from the time series of past prices. This will equip us with the necessary ingredients to investigate the relative performance of various portfolios. Note that we are mainly interested by deriving retrospectively specific insights to understand the evolution of a quantity of interest, ranking the performances of two portfolios, for instance.
Numerical results from this model uncertainty are called robust if at least they are compatible with the numerical results from a second model not supposed to follow similar framework assumptions.
An advantage of the wealth accumulation analytic modeling is that the prices are not supposed to follow any stochastic process, hence no probability space is needed to model the prices. The approach used values that are empirical quantities.
Hedge Fund Datasets:
We collected a highly representative group of hedge funds from the EDGAR
Form 13F Database to construct our Hedge Fund Universe (HFU). These
files are downloaded using Python code, only 13F-HR filings are used to ex
tract the holdings, excluding the amendments and the notice filings which do
not contain any significant information for our purposes, since they contain no
holdings. The historical stocks prices are retrieved from Yahoo! Finance . We
use daily historical data.
Implementation:
Our Mathematical formulations are implemented using C++ Language as an
analytic engine on the top of the databases EDGAR Form 13F Database and
Yahoo! Finance’s daily historical stock prices.
The C++ implementation used
Boost Asio Library to extract data from Yahoo! Finance Network. We also
used the C++ Boost libraries to handle major parts of our computational
finance formulations.
The insight analysis uses R programming language to
process the C++ results ( R is a software environment for statistical computing
and graphics supported by the R Foundation for Statistical Computing ). The interactive data visualizations in web browsers uses the NVD3 data visualization library.
Numerical Results:
The featured numerical results are obtained by processing our codes on a Laptop
with four cores: i3-core-CPU-2.10 GHz- 4GB-RAM and 64-bit-Ubuntu 14-04.
In order to verify that our analytics are useful in practice, we consider the daily
historical prices from a stock for the period from a given prior date until a
given posterior date. Often we use data from a quarter to make comparisons
and evaluate trends.