Reconciling Holding-Period PnL with Summed Interval PnL

# Which PnL is correct?


# Which PnL is correct?












Suppose that

- There is a risky asset whose value at time $t$ is $S_t$.

- The interest rate is constant equal to $r$.

- There are three timestamps $t_0=0$, $t_1=\delta t$ and $t_2=2\delta t$.

I am interested in knowing the PnL between $t_0$ and $t_2$ of being long one unit of risky asset. However I have two contradictory reasonings:

I_ What I was told at school

At time $t$, your portfolio has value $V_t$. Then your PnL between $t_0$ and $t_2$ is $V_{t_2}-V_0$. Here:

- At time $t_0$ I borrow $S_0$ dollars to buy one unit of risky asset, hence $V_0=S_0-S_0=0$.

- At time $t_2$, my risky asset is worth $S_{t_2}$, but I paid interest $S_0(1+r\delta t)^2$ on the money I borrowed, hence $V_{t_2}=S_{t_2}-S_0(1+r\delta t)^2$.

The PnL is therefore $S_{t_2}-S_0(1+r\delta t)^2$.

II_ What I am told at work

The PnL between $t$ and $T$ is the sum of all incrementals PnLs. That is if we denote by $PnL_{u\to v}$ the PnL between times $u$ and $v$, then $$ PnL_{t\to T}=\sum_{s=t}^{T-\delta t}PnL_{s\to s+\delta t}. $$

And the incremental PnL of a long strategy between $t$ and $t+\delta t$ is calculated as the profit made by borrowing the money to buy the risky assets at $t$, then selling out your position at $t+\delta t$. So in my example:

- PnL between $0$ and $\delta t$: at 0 you borrow $S_0$ dollars to buy one unit of risky asset. At $\delta t$ you sell out. The unit of risky asset is worth $S_{t_1}$, and the interest you paid is $S_0(1+r\delta t)$. Hence $PnL_{0,\delta t}=S_{t_1}-S_0-S_0r\delta t$.

- PnL between $\delta t$ and $2\delta t$ : at $\delta t$ you borrow $S_{t_1}$ dollars to buy one unit of risky asset. At $t_2$ you sell out. The unit of risky asset is worth $S_{t_2}$, and the interest you paid is $S_{t_1}(1+r\delta t)$. Hence $PnL_{t_1\to t_2}=S_{t_2}-S_{t_1}-S_{t_1}r\delta t$.

We deduce the global PnL: $$ PnL_{0\to t_2}=S_{t_2}-S_0r\delta t-S_{t_1}r\delta t. $$

Question

Those two PnLs do not coincide. Which one do you believe makes more sense? And is there a way to connect the two? Any insight and/or reference on PnL computation would be much appreciated.

## Answer by Kurt G. (score 1, accepted)

https://quant.stackexchange.com/a/81238

Quite naturally the two PnLs do not necessarily coincide. In the "school case" you don't touch the portfolio at $t_1=t+\delta t$ and liquidate it only at $t_2=t+2\delta t\,.$ In the "work case" you liquidate the portfolio at $t_1$ realising its PnL (let me simplify the notation a bit) $$ PnL_1=S_{t_1}-S_{t_0}-S_{t_0}\,r\,\delta t\,. $$ When you then set up the portfolio again by borrowing $S_{t_1}$ at rate $r$ you can realise a PnL at $t_2$ of $$ PnL_2=S_{t_2}-S_{t_1}-S_{t_1}\,r\,\delta t\,. $$ The question is what you did with the money $PnL_1\,.$ Did you spend it when it was positive? I do not recommend that :).

To make the two methods comparable you should think of investing/borrowing $PnL_1$ at rate $r$ so that it stays in the system until $t_2\,.$ At that time your net PnL is not $PnL_1+PnL_2$ but rather $$\require{cancel} PnL_1(1+r\delta t)+PnL_2=S_{t_2}-\cancel{S_{t_1}(1+r\delta t)}+\cancel{S_{t_1}(1+r\delta t)} -S_{t_0}(1+r\delta t)^2\,$$ as expected from the "school case."

- I should probably mention that I did not say which method is correct. Just wanted to give the reason why they are different.