I have a portfolio of three assets A, B and C, each with a beginning value of 100. I trade the portfolio over two days. On day 1, asset A returns 50, B returns 20, and C returns -10. On day two, A returns 0, B returns 0, and C returns 30. These are dollar returns.
When I calculate the returns of each stock over each day, the contribution to return as a percentage of the portfolio's beginning value is right. So, on day 1, A returns 50% (50/300), B returns 20% and C returns -3.3%. And on day two, A returns 0%, B gives 0% and C gives -8.3% (-30/(150+120+90)).
The return of the portfolio for two days is 10%. The profit in dollar terms is 30, and the daily return for the portfolio is 20% on day 1 and -8.3% on day two, which compounds to 10% using the time weighted rate of return. But, if I calculate the returns for each stock and then sum those returns (as a percentage), it does not add up. So, for stock A the return is 16.7%, B gives 6.7%, and C gives -11.4%, and these sum to 11.9%. Why does 11.9% differ from 10%, and what is the right way to adjust my stock level returns to match the 10% of the portfolio? Please see below the screenshot of the daily calculation. Essentially, the cells highlighted in green should sum up to the blue cell. In the picture below, Opening NAV = value of each stock at the beginning of day, Weight = weight of each stock in the portfolio at beginning of day, PnL = profit of each stock for the day, %ctr 1 = the stock's contribution to return of the portfolio (PnL divided by the value of the portfolio at the beginning of day). Portfolio returns over two days
Edit: Please see a clarification of the problem below.
So at the end of day 1 the market value of A is 150, B is 120, and C is 90. Then on day two A has a profit of zero (the stock price doesn't move), B has a profit of zero, and C has a profit of -30. So the market values of A,B, and C at the end of day two are 150, 120, and 60 respectively. The returns of A,B and C for day two are 0% (0/360), 0% (0/120), and -8.3% (-30/360). So, over two days, what is the return of the portfolio? The return of the portfolio on day 1 is 60/300 (20%), and on day two it is -30/360 (-8.3%). So the return for the portfolio for the entire period is (1+20%)(1-8.3%)-1 = 10%. This is the time weighted rate of return formula and makes sense because your closing NAV is 330 and you began with 300 so you made 10%. BUT, AT A STOCK LEVEL, WHAT WAS YOUR RETURN OVER THE TWO DAY PERIOD? For stock A, I calculate (1+16.7%)(1+0%)-1 = 16.7%, For B, I calculate (1+6.7%)(1+0%)-1 = 6.7%, and then for C I calculate (1-3.3%)(1-8.3%)-1 = -11.4%. If I sum 16.7%+6.7%-11.4% = 12%. This differs from my portfolio return of 10%!
Edit: The key part of the question is to link the stock level returns to the portfolio return over multiple periods. So, if I have a portfolio total return of 10% over multiple periods, how do the stocks contribute to that return ie stock A contributed 9%, B contributed 5% and c contributed -4% ie 9% + 5% - 4% = 10%.
Here you have a portfolio of 3 assets ($A,B,C$) and $2$ days. Firstly you calculate the $\textrm{growth factor}$ of the first day of the portfolio, from time $0$ to time $1$:
$$1+r_{p,01}=\frac{X_1^A+X_1^B+X_1^C}{X_0^A+X_0^B+X_0^C}=\frac{150+120+90}{100+100+100}=\frac{360}{300}=\frac{12}{10}=1.2,$$
where $X_t^A,X_t^B,X_t^C$ are the net asset values at time $t$. Next you calculate the growth factor of the second day of the portfolio, from time $1$ to time $2$:
$$1+r_{p,12}=\frac{X_2^A+X_2^B+X_2^C}{X_1^A+X_1^B+X_1^C}=\frac{150+120+60}{360}=\frac{330}{360}=\frac{11}{12}\approx 0.9167$$
Then the growth factor for the portfolio in two days is $1+r_{p,02}=(1+r_{p,01})\cdot (1+r_{p,12})$ $=\frac{12}{10}\cdot \frac{11}{12}=1.1$. Thus the growth rate of the portfolio in two days is $r_{p,02}=1.1-1=10\%$
And the average daily growth rate of the portfolio is $\sqrt{1+r_{p,02}}-1=\sqrt{1.1}-1\approx 4.88\%$
Update
The contribution (in percent) of asset A at day 1 to the return of the portfolio is $r_{A1}^c=\frac{150-100}{300}=\frac16=16.\overline 6\%\approx 16.7\%$
The contribution (in percent) of asset C at day 1 to the return of the portfolio is $r_{C1}^c=\frac{90-100}{300}=\frac{-1}{30}=-3.\overline 3\%\approx -3.3\%$
The contribution (in percent) of asset C at day 2 to the return of the portfolio is $r_{C2}^c=\frac{60-90}{360}=\frac{-1}{12}=-8.\overline 3\%\approx -8.3\%$
Finally the ontribution (in percent) of asset C from time $t=0$ to time $t=2$ to the return of the portfolio in that period is
$$(1+r_{C1}^c)\cdot (1+ r_{C2}^c)-1=\left(1-\frac{1}{30} \right)\cdot \left(1-\frac{1}{12}\right)-1$$ $$\approx 0.9666666\cdot 0.9166666-1=-0.11388901444444\approx -11.4\%$$
$\underline{\textrm{Important to mention}}$: Consecutive growth rates $r_i$ of $n$ periods are never summed up. You multiply the the growth factors $(q_i=1+r_i)$ and subtract 1 to obtain the growth rate in $n$ periods:
$$r_{0,n}=\prod\limits_{i=1}^n (1+r_i)-1=\prod\limits_{i=1}^n q_i-1$$