How to adjust a strategy's alpha assuming a zero-value starting portfolio ($0 cash, $0 assets)?

[jabowery]

A simple paper test of a trading strategy is to assume one borrows all money to purchase assets and see if trading increases the liquidation value of the portfolio (cash + liquidation value of assets). However, in order to calculate the trading strategy's alpha, one must take the percentage change compared to the benchmark's. Since one starts with $0, this creates a divide-by-zero situation.

One can remedy this situation by assuming starting cash > $0 and use that in the denominator, but it isn't obvious what number to choose for this starting cash. To exemplify the difficulty let's say one chooses a starting cash amount that is some function of the beta -- so as to reduce the risk that the strategy will be forced to borrow money during trading, i.e. that the total portfolio value will hit $0 liquidation value. This function would also take, as input, the level of acceptable risk that the strategy will be forced borrow money during trading.

Moreover, since beta is the standard deviation, and one is attempting to avoid a $0 balance, it is inadequate input due to the symmetry of its deviation about the mean. There must be additional input to the function, e.g. mean, skewness, kurtosis, etc.

There is probably some work on this in the literature but I've been unable to find it.

 

[ph1l]

assume one borrows all money to purchase assets ... the percentage change compared to the benchmark's. Since one starts with $0, this creates a divide-by-zero situation.

One can remedy this situation by assuming starting cash > $0 and use that in the denominator, but it isn't obvious what number to choose for this starting cash.

Maybe use the cost of borrowing (interest, collateral value, fees, etc.) as the starting cash?

 

[globalarbtrader]

A simple paper test of a trading strategy is to assume one borrows all money to purchase assets and see if trading increases the liquidation value of the portfolio (cash + liquidation value of assets). However, in order to calculate the trading strategy's alpha, one must take the percentage change compared to the benchmark's. Since one starts with $0, this creates a divide-by-zero situation.

One can remedy this situation by assuming starting cash > $0 and use that in the denominator, but it isn't obvious what number to choose for this starting cash. To exemplify the difficulty let's say one chooses a starting cash amount that is some function of the beta -- so as to reduce the risk that the strategy will be forced to borrow money during trading, i.e. that the total portfolio value will hit $0 liquidation value. This function would also take, as input, the level of acceptable risk that the strategy will be forced borrow money during trading.

Moreover, since beta is the standard deviation, and one is attempting to avoid a $0 balance, it is inadequate input due to the symmetry of its deviation about the mean. There must be additional input to the function, e.g. mean, skewness, kurtosis, etc.

There is probably some work on this in the literature but I've been unable to find it.

I'd argue that this is irrelevant, since what matters is the statistical significance of the alpha (eg t-statistic), rather than the actual value. The statistical significance will be unchanged by the amount of capital chosen, whereas the value would.

In practice even a 'zero cost' strategy requires some capital; at a minimium you would need some margin, and you'll also need capital to absorb likely losses. To get a sensible idea of the annual % alpha you should come up with some figure here.

GAT