Long-Short Portfolio Weights for a Self-Financing Strategy

Summary

The note explains how to define portfolio weights when a long position is fully funded by an equal-sized short position. For a $200 long in Apple and a $200 short in Google, it uses signed weights of +1 and −1, calculated relative to the portfolio’s net investment convention for a self-financing strategy. These weights sum to zero, reflecting that the long and short exposures require no net initial capital in the simplified academic setup.

The example shows how signed weights translate into portfolio return: multiply each asset’s return by its weight and add the results. A 10% rise in the long and a 5% fall in the short produce a 15% portfolio return under that calculation. The explanation is brief and assumes equal dollar positions; it does not cover leverage, margin, financing costs, or alternative definitions of gross exposure.

Key ideas

• A fully funded equal-dollar long and short can be represented with signed weights of +1 and −1.
• The weights sum to zero because the strategy is self-financing under the stated convention.
• Portfolio return is the sum of each asset return multiplied by its signed weight.
• The example omits margin, financing, and other implementation costs.

Tags

# calculating portfolio weight for long short


# calculating portfolio weight for long short












Sorry for this dumb question but what are the mathematical-finance /academic conventions for calculating portfolio weights in a long/short portfolio where the longs are fully funded by the shorts? Note that I am asking for the 'academic' conventions, I'm aware that in reality one needs to hold margin etc.

Suppose I am long USD200 of Apple, and I have shorted USD200 of Google stock. What is the portfolio weight of apple and google?

- w_apple = 200/200 = 1 and w_google = -200/200 = -1?

- w_apple = 200/(200 + abs(-200)) = 0.5 and w_google = -200/(200+abs(-200)) = -0.5?

- something else?

Thanks

## Answer by phdstudent (score 2, accepted)

https://quant.stackexchange.com/a/75219

The first one. Your net weight is zero. This is a self financing strategy. Think of it this way: if apple goes up by 10% and google goes down by 5% your return will be:

$r_p = 1 \times 10\% - 1 \times (-5\%)= 15\%$

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