# 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.

• In financial jargon we have a long position in the bond and a short position in the stock. It is an important assumption of the model that short positions are allowed. Indeed Short positions, as well as fractional holdings, are allowed. In mathematical terms this means that every $H_t^S\in \mathbb{R}$ is an allowed portfolio.
– user16651
Aug 15 '16 at 16:58
• Short selling can be confusing and takes some getting used to. The convention is to treat the short position as a negative value. Aug 15 '16 at 16:59
• For more details check it: investopedia.com/terms/s/short.asp
– user16651
Aug 15 '16 at 17:04

## 1 Answer

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 - 3S

• "The price of the portfolio is the cash you receive when you sell it". Exactly. As they used to say: a thing is worth what someone else will pay you for it. toquotelatin.wordpress.com/2011/04/06/… Aug 15 '16 at 17:55
• I've got a true story on that: A quant head invented and marketed a new product, actually made market in it. Old fox client asked curiously about it and bought 100. Called in the next day heard the price was up, bought another 100. Third day, up again and 100. Now the risk manager entered asking the quant of his portfolio: -300. And the price development? Up a lot during only three days! So the P/L so far pretty bad and the portfolio big and illiquid. Risk manager ordered the quant to close his position. Poor quant could only call the client and buy back at the clients term with a huge loss. Aug 15 '16 at 18:18
• @MatsLind I may be too stupid to understand what you mean but since the quant sold 300, his position should be -300, right? I don't see where the problem is.... Aug 27 '20 at 18:08