Interpreting Long and Short Signs in Portfolio Value
Summary
The document clarifies the sign convention in a portfolio containing a riskless account and a stock. Portfolio value is defined as the number of units held in each asset multiplied by its price, then summed. A positive holding represents a long position, while a negative holding represents a short position; the signs describe holdings rather than the cash received or spent when the original trades were opened.
For the example holding one unit of the money market account and short three shares, the portfolio value is the bond value minus three times the stock price. The response explains liquidation from the portfolio’s perspective: selling the bond yields cash, while closing the short requires buying the shares. This is a sign-convention explanation, not a discussion of how to construct a replicating strategy or account for financing, transaction costs, or margin.
Key ideas
- Portfolio value is the sum of each asset’s price multiplied by its signed holding.
- A positive holding denotes a long position, and a negative holding denotes a short position.
- Closing a short position requires buying the borrowed asset back.
- The portfolio value reflects the marked value of holdings, not the initial cash-flow sign of each trade.
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# Signs for the assets in a portfolio and definition of portfolio value
# Signs for the assets in a portfolio and definition of portfolio value
Suppose that we have a market with a stock, modelled by $\{S_t\}_{t>0}$ and a riskless money market account $\{B_t\}_{t>0}$.
Consider a strategy $\{ H_t^B,H_t^S\}_{t>0}$ be a portfolio over time. (It matches the option price at maturity $T$).
We say that the value of the portfolio at time $t >0$ is :
$\Pi_t = H_t^B B_t + H_t^S S_t$.
Take the scenario of $H_t^B = 1, H_t^S = -3$. From what I've seen in the bibliography we make the abbreviation that $-$ corresponds to a short(sell) position and $+$ to a long(buy) position.
In other words for this scenario the value of the replicating portfolio would be :
$\Pi_t = +1 B_t - 3 S_t$.
Under this abbreviation, I get confused with the definition of the portfolio value. I think in terms of "when i buy, I spend money, so I assign $-$ to the total value, when I sell, I get money, so that my value is positive $+$.
What is the meaning of using $-$ for selling the asset. What is the meaning of the portfolio value defined under the first abbreviation.
The only explanation I have for this abbreviation comes from the convention that we see asset prices with positive numbers. Eg when we say a price of a stock is $(+)5$ then it means that to "buy" the asset we need $+5$.
Anyone could help? Thank you.
## Answer by Mats Lind (score 2)
https://quant.stackexchange.com/a/29650
To sell the entire portfolio you need to sell one bond and buy three stocks. You have to buy the three stocks to get out of your short position held in the portfolio. The price of the portfolio is the cash you receive when you sell it: +1B - 3SShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.