This is an overview of some of the most common weighting schemes used in asset allocation. The same concepts apply in the implementation of both statistical arbitrage and portfolio management strategies.

- Equal Weights: This is a very straightforward method. All equities (or assets in general) selected through the corresponding strategy are weighted equally. The weight for each one of the equities is equal to 1 / n where n is the number of equities. For example if the strategy involves the execution of 100 stocks then each weight is 1 / 100.
- Volatility Weights: This technique is aiming to minimize the overall risk of the portfolio by assigning larger weights on less volatile stocks and lower weights on the more volatile ones. A time series of historical daily volatility is required for each constituent whereas the covariance is being ignored. For example for n stocks with volatilities sigma(i) with i from 1 to n, the weights are calculated as following:

m = 1 / Σ(1/sigma(i))

and w(i) = m / sigma(i) where w(i) is the weight assigned to stock i (i from 1 to n)

- Value Weights: This is again a relatively straightforward method. The weights are equal to the ratio of the market cap of each constituent over the total cap of the portfolio.
- Alternative methods: We are going to discuss about more complex ways of assigning weights like using momentum scores, first principal components, etc. in a future post. All methods refer to the estimation of a score for each constituent that is then used to calculate the respective weight.

Capital restraints as well as other restraints (industry exposure, country exposure, etc.) are also taken into account in the optimizer when incorporating each one of the above mentioned methods.

Regarding the capital that is planned to be invested the following objective function captures the mechanics:

- Σ(round(w(i) * P / Ask Price(i)) * Ask Price(i) + trading fee(i)) <= P

with i from 1 to n, P the capital to be invested, Ask Price(i) the ask price for equity i as provided by the broker and trading fee(i) the trading fee for equity i.

Finally, the round function is used because the number of equities is an integer (you cannot purchase fractions of equities).

A simple example is illustrated below for a number of 10 stocks and the application of equal weighting: