Investment Maths

A friend and I were arguing about whether to invest everything in 1 stock (as I tend to) or spread the investment wide. (A girl commented to me before that she thinks men are more likely to adopt the first strategy while women the second - for whatever that is worth). Anyway there is no difference looked at in the following way:

Suppose that for a sector in the markets, over some period of time, stocks experience a chance of increasing of p%.

Suppose that on average they increase by a percentage i%.

Then with a portfolio of T invested in 1 stock the return in a single period r(1) is:

(1) r(1) = T * p * i + T * (1-p) * 1 = T * (p * ( i-1) + 1)

So for n periods is:

(2) r(n) = T * (p(i-1)+1)^n

The strategy of investing in more than 1 stock basically replicates this return in each stock but the portfolio is divided between stocks. Adding the portfolio together afterwards simply reverses this. For three stocks where T is divided and invested: T1 + T2 + T3 = T

T1 * (p(i-1)+1)^n + T2 * (p(i-1)+1)^n + T3 * (p(i-1)+1)^n =

(T1+T2+T3) * (p(i-1)+1)^n =

T * (p(i-1)+1)^n

The two strategies are the same. Where they are different is with investment costs. Each investment has its own costs so the more investments then the greater the costs. But it is not linear. This affects the compound interest rate in some trading environments.

If the probability of a stock increasing reduces after a few increases (say speculating on oil exploration companies finding oil) then a move to another stock is advisable to keep the probability up. [Indeed this marginal behaviour of finding the environmental average of stock increase and then moving one's stock based on this conforms to the Optimal Foraging Strategy (follow this up and make an investment model).] In this strategy the % cost of moving the stock is subtracted from the %increase and so seriously dampens the compound returns.

The model here only works where a sector can be defined where an average probability of stock increasing can be defined. I need to look at the application of statistics to the stock market and definitely look at the hierarchical definition of markets an apply the model to that; that is within any market there will be stocks that perform in certain ways and within those subgroups there will be further sub-groups etc.