Long and Short Position Signs in Portfolio Value
Summary
The document asks why portfolio positions are conventionally written with positive quantities for assets held long and negative quantities for assets held short. It contrasts this with treating a short sale as positive cash proceeds and a purchase as negative cash, and asks how to interpret a portfolio consisting only of a short stock position.
The response supports the usual holdings-based convention by treating portfolio value as the marked-to-market value of the assets and liabilities still held, rather than as the cash generated when positions were opened. Its example starts with a short stock position and a bond holding whose initial values offset, then asks what happens to portfolio value if the stock price falls to zero while the bond price stays constant. This highlights that opening cash flows do not describe later portfolio value. The excerpt ends before giving the example’s conclusion, so it does not fully explain the short position’s value evolution or address financing, collateral, and transaction cash flows.
Key ideas
- Portfolio notation usually records holdings directly, with positive quantities for long positions and negative quantities for short positions.
- Portfolio value marks current positions to current asset prices rather than recording only the cash received or paid when they were opened.
- A short position contributes negative marked value, while cash proceeds from the initial sale are a separate balance-sheet component.
- The excerpt’s price-change example motivates the convention but does not include the complete answer.
Tags
Full text
# Notation of long/short positions in defining a portfolio # Notation of long/short positions in defining a portfolio 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, $\Pi_t$, $$\Pi_t= -10S_t+2B_t$$ 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, $$\Pi_t= +10S_t-2B_t$$ 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, $\Pi_t= -10S_t$? ## Answer by Bob Jansen (score 3, accepted) https://quant.stackexchange.com/a/57733 I agree with noob2 that it is a matter of convention. I do believe there is a good reason for it though and the common convention is the only right convention. First, $\Pi_t$ denotes the portfolio value at time $t$. If you bought a portfolio of just 10 stocks, surely you would expect it's value to be non-negative for every $t$, regardless of how you attained this portfolio. Second and related to the first, I believe your method doesn't work very well when prices evolve, you state: > 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, $$\Pi_t= +10S_t-2B_t$$ makes more sense to me as a value process of the portfolio. Consider the situation where $S_0 = 20$ and $B_0= 100$. Under any convention, $\Pi_0 = 0$, now an instant passes and the stock's value changes to 0 and we have $S_t = 0$ and $B_t= 100$. What should the value of the portfolio now be if you sold 10 stocks and bought 2 bonds?
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