Measuring Returns When Portfolio Value Starts at Zero

# Calculating returns on sequence of trades with zero starting capital


# Calculating returns on sequence of trades with zero starting capital












### Background

I am trying to calculate the returns on a sequence of trades performed by an entity, where I do not know the starting capital. Therefore I assume a starting capital of zero. From these returns I want to calculate the Sharpe ratio of a portfolio on which these trades are performed.

The initial idea of simply comparing successive portfolio values to calculate returns falls apart in this situation, for the following two numbered reasons.

A potential solution for calculating returns is to divide profits realised in a period by the gross market value of that period. The gross market value is the sum of the absolute values of all positions. I am interested in learning whether the returns calculated in this way are useful for calculating the Sharpe ratio.

### Example

Suppose the price at $t_1$ of TSLA is 1000USD. I buy 1 TSLA for 1000USD. I am now long 1 TSLA and short 1000USD. My total portfolio value remains at 0USD.

At time $t_2$, the price of TSLA is 2000USD. Therefore my portfolio value is now 2000USD minus the 1000USD that I am short, which is 1000USD total.

- Dividing the portfolio value at $t_2$ of 1000USD by the portfolio value at $t_1$ of 0USD, I realise an infinite return.

Suppose that at $t_3$ I now buy a further 999 TSLA at 2000USD. My portfolio value remains at 1000USD. At $t_4$, TSLA is now 3000USD. I am long 1000TSLA at 3000USD, so minus the 1999000USD that I am short, my portfolio value is 1001000USD.

- Dividing the portfolio value at $t_4$ by the portfolio value at $t_3$ and subtracting 1, I have realised an outsize return of 1000, specifically because I went short a large number of USD compared with the amount of USD I was short previously.

I have looked at a number of resources online and in books, and of the resources that cover short positions and leverage, they generally assume a positive starting capital. This is not useful for my situation.

### Possible Solution

One online resource I came across suggests dividing the profit in one period by what it calls the gross market value in that period. The gross market value is the sum of the absolute values of long and short positions. This takes account of the total capital at risk.

At $t_2$, our gross market value is 2000USD for the 1 TSLA, plus 1000USD for the short USD position. Therefore it is a total of 3000USD. Our profit realised between $t_1$ and $t_2$ is 1000USD, due to the 1000USD price increase. Therefore we have a return of 0.33.

The profit realised between $t_3$ and $t_4$ is 1000000USD. Dividing by the gross market value of 4999000 (value of TSLA held plus amount of USD that we are short) at $t_4$, we get a return of 0.2.

This seems like a more reasonable method of calculating returns. But questions still remain.

- Is it correct to divide the profit in the current period by the gross market value in the current period, and not for instance by the average of the gross market value in the previous period and current period? Dividing by the latest gross market value makes more sense to me, as we are comparing profit in a period to the capital at risk during that period. ChatGPT suggested taking a time-weighted average of gross market values, but that sounded too complex to me.

- Is it meaningful to calculate the Sharpe ratio from returns calculated in this way? Are there any sources in literature which recommend this method for the situation of zero starting capital?

## Answer by mark leeds (score 1)

https://quant.stackexchange.com/a/79492

Hi Jkut: First you should always specify the return in the sense of when you calculate the return, what are you calculating the return of ? In our case, we will calculate the return on the two share purchases you described.

You purchased 1 share at T1 at 1USD. During T1 to T2, the price went from 1USD to 2USD. At T2, you then purchase an additional share at 2USD.

Let us make 2 additional assumptions.

Suppose that, between T2 and T3, the price went up another 1USD. Also suppose that, at T3, you sold all of your shares at 3USD per share. So, T3 is when you completely stopped investing.

Given the information above, we can calculate

A) the return from T1 to T2 B) the return from T2 to T3. C) possibly the return from T1 to T3. ( we shall see )

A) At T1, you bought 1 share at 1USD. Beween T1 to T2, the share went up to 2USD. So, the return = (dollars made or lost over the period due to total investment at the beginning of the period )/(total investment in dollars at the beginning of the period).

So, the return from T1 to T2 due to the 1 share purchase at T1 = (2USD -1USD)/1USD = 100 percent.

B) At T2, you bought another share at 2USD. Then, from T2 to T3, the price went to 3USD.

The return from T2 to T3 due to the 1 share purchase at T2 = (3USD - 2USD)/2USD = 50 percent.

The return from T2 to T3 due to the original share ( bought at T1) still being held from T2 to T3 = (3USD - 2USD)/2USD = 50 percent.

So, each of the shares returned 50 percent from T2 to T3. Equivalently, we can also say that the return on the portfolio of 2 shares from T2 to T3 = 2USD/4USD = 50 percent. This is because you had 2 shares so the investment was worth 4USD at T2. Then, at T3, it was valued at 6USD. So, (6USD - 4USD)/4USD = 50 percent.

C) There is no conventional way that I know of to calculate the return from T1 to T3 because the investment amount is changing over the period. I would probably just calculate the returns from from (T1 to T2) and (T2 to T3).

Notice that you can calculate dollar profit from T1 to T2 as 1USD and dollar profit from T2 to T3 as 2USD. So, total dollar profit from T1 to T3 is 3USD. But it's not really possible to calculate the return from T1 to T3 because the investment amount changes. There is a way to approximate it but I'd rather not get into that unless you're interested. I don't regard it as all that useful of an approximation.

#===============================================================

You mentioned consdering the alternative senario where you sold the 1 share at T2 and ended your investing stint right there. The return on the 1 share would still be 50 percent from T1 to T2. This is because the calculation is EXACTLY THE SAME as it was when you didn't sell the share because the formula only needs ( how much did I make or lose over the period ) divided by the dollar investment. So, selling the share at T2, makes no difference in the calculation from T1 to T2. Of course, in that case, the investment is finished and nothing happens in the period from T2 to T3.

I hope this helped to clear up the return calculation. If you think of it as I defined it, then I think you can't go wrong. If anything was not clear, let me know.