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

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.

The $0 case has not historically come up, and has not tended to be relevant to "the literature". Prime brokers and retail brokers all required at least a certain amount of cash (or other collateral) to open accounts and start trading. These days, we have some crypto trading platforms that attract clientele with a giveaway of some (small) amount of cryptotokens just for signing up, which could legitimately be considered a$0 starting point.