# Portfolio Selection

When investing, putting all of your money (100% of it) into one stock means you have just one way to gain and one way to loose.   Being uncertain about the future, this is a precarious position that requires patience to ride through temporary drops in price.

An alternative is to invest in two stocks, A & B, that behave differently with different rates.  If they behave the same way, it would not make much difference in how you allocate your A:B investments as a 50:50 allocation would provide the same rate as a 20:80 allocation.  If they behave as perfect opposites, the A stock will rise while the B stock drops.  Then you can invest like a Ferris wheel with 100% in the A stock as it is going up and then selling A near the top and buying the B stock when it is near the bottom.  If this scenario exists, you could buy and sell in cycles between 0:100 and 100:0 allocations and ratchet up your total value.   Two investments define the simplest portfolio regardless of the allocation.

However, actual stocks do not smoothly increase and decrease in cycles.  Subject to multiple market forces, the price of one stock appears to have a randomness.  For a given interval of time, the rate is the change in price divided by the starting price.  The rate can be calculated by applying the natural logarithm transformation to the price and taking the difference in the log over the time interval.  This too appears to have randomness as time progresses.

For a selected window of time, a data set of rates can be collected and statistics applied to find a mean rate and standard deviation (from variance) or what is called volatility.   With two or more stocks, the variance concept is expanded to use covariance and this tells us if the stock’s rate variations are correlated (increasing or decreasing together) or anti-correlated (one increases while the other decreases).

The covariance is organized as a table (a matrix) because each investment can be paired with every  investment in the portfolio including itself.  Since most investments are not perfectly correlated, a benefit of the portfolio is diversification.  The ‘portfolio variance’, interpreted as risk, is calculated from the covariances of every investment according to its allocation and this tends to produce a smaller variance (risk) than the individual investments.  It is like comparing a square and circle of the same size, and the circle has less area.   The analogy is useful but we have to picture it in M-dimensions where M is the number of investments in the portfolio.

Modern portfolio theory indicates that the risk (indicated by the calculated portfolio variance) can be minimized by a particular allocation of each investment in the portfolio.  This usually does not yield the highest expected return for the portfolio (indicated by the returns of each investment weighed by its allocation).  Obtaining higher expected portfolio return is by accepting above-minimum risk.  Note that this is ‘expected’ because the portfolio involves today’s investment values or prices and the future return of each investment is unknown and therefore the portfolio return is unknown.  The uncertainty in the current price of each investment plays a role in decision making and future price estimation errors should be included.  In this, there is an assumed continuity in volatility, that is, we expect the past volatility to continue in the future.  Monte-Carlo simulations are often used to investigate the effects of future price variations on the portfolio’s risk and return.

The duals arithmetic offers distinct advantages over the Monte-Carlo method and this project was undertaken to demonstrate those advantages on a realistic financial application.  The project has produced a framework for duals arithmetic to handle portfolios with any number of finite investments, including short selling.  This has been tested on example portfolios of 2, 4, 6, 8 and 12 investments.  The duals arithmetic reports all components of uncertainty and this allows one to identify where information quality is lacking and where vulnerabilities or advantages may exist in each portfolio.

Another parallel effort is ‘rebalancing’ where a portfolio allocation is initialized and the value of each investment changes over time.  This effectively alters the allocation.  For example, if one investment grows while the others do not, that investment occupies more value in the portfolio.  Rebalancing is reallocating so the initial allocation is achieved again.  Two ideas to do this are, first, select investments that have no correlation to each other.  When investments are independent components of a recipe, a change-path can be charted between the current portfolio and the rebalanced portfolio.  The second idea is to select investments that are anti-correlated.  This would activate the Ferris wheel analogy but timing the changes in allocation have to be done correctly to achieve the advantage.