Growth Optimal Portfolio

Given a set of securities let the portfolio be {x1 to xn} for the n securities. where xi >= 0 and sum(xi) = 1. define the return of round j rj to be sum[xi * rij] .
Aim is t maximize product(pi* ln(1 ri)), where pi is the probability of scenario i.

growth optimal strategy

One of the motivations of minimizing variance is that the estimate of geometric mean as mu - sigma^2 is closer than just mu in approximating the geometric mean of return. geometric mean of the returns (in case all the wealth is bet in every round) is the growth rate. To summarize, the growth rate is g such that ln(1+g) = {Prod(i in 1:T)[1 + b Ri]} / T .
Hence approximating ln(1 + g) as g, maximizng growth rate amounts to maximizing return.

efficient market hypothesis by malkiel:

A capital market is said to be efficient if it fully and correctly reflects all relevant information in determining security prices. formally, the market is said to be efficient with respect to some information set... if security prices would be unaffected by revealing that information to all participants. Moreover, efficiency with respect to an information set implies that it is impossible to make economic profits by trading on the basis of that information.

the econometrics of financial markets

for a finance theory background of the subject:

Dynamic Asset Pricing Theory by Darell Duffe is a doctoral level book on the theory of asset pricing. In the book the asset pricing reslts are based on three increasingly restrictive assumptions, namely absense of arbitrage, single agent optimality and equillibrium. hey are unified under martingales and state price vetors.

They are applicable to term structure models, derivative pricing, etc.

Program for HMMs

The General Hidden Markov Model library (GHMM) is a freely available LGPL-ed C library implementing efficient data structures and algorithms for basic and extended HMMs.

stylized facts of daily return series

by Tobias Ryden, Timo Terasvirta and Stefan Asbrink

they considered long return series that are first differences of logarithmed price series or price indices. they established a set of temporal and distributional properties for such series, and sugested that returns are well characterized by double exponential distribution.
This paper shows that a mixture of mean zero normal variables can generate series with most of the properties Granger and Ding singled out. In that case, the temporal higher order dependence observed in return series may be described by an HMM.

statisical voodoo in technical analysis

I was earlier surprised at the disdain in academia of technical analysis. apriori, it seems like a very natural thing to do, when the phenomenon we are trying to learn, when aggregated, has no other way of aproaching it except historical data. Today i got the answer.

Technical analysis is primarily associated to "voodoo". Charters see certain patterns like cup and handle or head and shoulders and predict certain patterns int he price. Why would there be any reason for prce to behave as predicted in these patters! They do not give credibility to that question. they simply assume the existence of it.

First argument in favour of it. these patterns have something to do with investor psychoogy.

Defense: Earlier when investors acted on intuition tis was probably true. But now most investors act of econometric signals. So this psycology cannot be assumed. Remember in the first place we said tha investor psychlogy might vindicate these.

Argument against it: Simple arbitrage arguments should tell us that the number of believers in them are more than the probability of finding the. hence in aggregation they should now reverse the trend. This lends some explanation of why Andrew Lo couldnot catch the flaw in them. he tried to compare the distribution of returns after such a patters with ormal return distribution of the secuirty and found a difference in the behaviour. he did not cancel out the positive bias earlier with a negative bias later in the range of values he tested the hypotheses on.

Portfolio optmization

In portflio optimization, the relevant measures are the rewards of the strategy and the risks are the standard deviation of returns.

• Mean variance analysis was suggested by Harry Markoqitz , who won the Nobel prize in economics in 1990.
• The first method of portfolio optimization asks for a threshold rmin, and outputs a portfolio with expected return more than rmin and minimizes variance as far as possible.
• The second method computes the mean and standard deviation of each security. Note the fact that the portfolio is a convex combination of the given securities. The mean return of the porfolio is the convex combination of the means. The variance of the portfolio is a combination of elements in the variance covariance matrix of the securities. Hence computing these statistics is enough to find the variance minimizing portfolio with expected return above the threshold.
• Now we want to penalize only deviation below the mean return. Hence we define di = max(0, mean - return in this scenario) and try to minimize the sum of squares of di.
  
• Another thing to note is that when we are only concerned with downside risk we need not sqare it. So we minimize average downside risk. Note that in all of these we come with one eficient portflio. The reults are similar in most cases. For nrmal distributions they are identical.