# Notation of long/short positions in defining a portfolio [closed]

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, $$\Pi_t= -10S_t$$?

• It is a matter of convention (and you could do the opposite convention if you wish, as long as you are consistent in your signs). It may be easier to just accept the author's choice, at least for the time being (until you write your own textbook). Sep 2 '20 at 0:57

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?

• I get your point but I see prons and cons in each abbreviation. For instance, as you say "surely, we expect the portfolio value to be non negative". If you use the common abbreviation (minus for sell) then consider the case at $t=0$ that $S_0=100$ and $B_0=100$. In that case by selling the stock I make 1000 and by buying the 2 bonds I spend 200. Hence the value of my portfolio should be 800 and not -800 (which is what you get under the common abbreviation). The first abbreviation implies that the value of the portfolio is negative. Any ideas? Sep 2 '20 at 11:17
• I think you're confusing cash in hand with portfolio value. You seem to be claiming you have a portfolio worth 800 and cash in hand 800 while you started with nothing! Sep 2 '20 at 11:42
• Thank you for your reply. Indeed I confuse portfolio value with cash at hand - need some help here and please elaborate if you know how. My view is that portfolio value is the money one needs to spend to create it or how much money one gets if he sells it. Is it valid to say that the "cost" of short selling 10 stocks now is -800? In other words, we don't spend money to short sell now - on the contrary we have gained cash => { cost of -800 = the +800 cash gained}. In fact, not only it doesn't cost to short sell but the negative cost reflects the cash gained? Is that right? Sep 2 '20 at 13:58
• I see that on none of your accounts you have accepted any answers. You can accept answers by clicking the green check mark. Sep 2 '20 at 14:19
• Thanks but there are definitely more deserving answers. Sep 2 '20 at 15:10