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.

I might suggest instead avoiding the issue by assigning a nonzero value to the 5 minutes needed to sign up as a client, by applying 1/12 times the local hourly minimum wage.

  • $\begingroup$ By "the literature" I do not necessarily mean the "quant" literature but economics literature. An example of something in the ballpark of my question: ideas.repec.org/a/eee/insuma/v85y2019icp153-172.html There is a lot of overlap. Not everyone thinks in your narrow terms. And, by the way, the recommended course of action is not even in the ballpark of answering my question otherwise the sarcasm might not be risible. $\endgroup$ Nov 13 '21 at 1:15
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    $\begingroup$ Not sure why you're so angry @JamesBowery but this is Quant.SE so we generally expect the questions and answers to be about quant. This answer seems to be very reasonable to me. $\endgroup$
    – Bob Jansen
    Dec 12 '21 at 18:45

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