I am a bit confused with the sign-related abbreviations used when we refer to long or short position on assets in a portfolio. For example denote the stock price, $S_t$ and the bond price, $B_t$ and consider the portfolio, $\Pi_t$,
According to "the abbreviations", the term $-10S_t$ means that the owner of this portfolio sells 10 stocks (similarly $+2B_t$ means "buys/owns" 2 bonds ~ lends that cash amount)
Why this abbreviations is used and not the opposite signs. E.g. since the owner sells 10 stocks he gets $+10S_t$ in cash and when he buys 2 bonds he is minus $B_t$, so that,
makes more sense to me as a value process of the portfolio. What is the mathematical/technical reason that the initial abbreviation is used to "characterise" a portfolio? Also, under the first abbreviation, what is the value of a portfolio, $\Pi_t= -10S_t$?