3

Let's consider two stocks with the following prices for a three days period:

Stock 1:
 day 1: 100
 day 2: 105
 day 3: 110

Stock 2:
 day 1:100
 day 2:100
 day 3:110

The increase percentage each time for stocks 1 and 2 are respectively:

Stock 1: [5% , 4,761904761904761904761904762%]
Stock 2: [0% , 10%]

When I average the two stocks increase percentage, I get:

  Avg Stock growth: [2,5% , 7,38095238095238095238095238%]

The problem is when I try to compute back the price of both stock combined together, with an initial price of 100, to reflect the scenario of investing equally in both stocks:

Day 1: 100
Day 2: 102.5 (100 + 100*2.5%)
Day 3: 110.0654761904761904761904762 (102.5 + 102.5*7,38095238095238095238095238%)
The end result does not add up to 110 but to 110.06 (...) which is not something I expected.

I'm not sure what I'm doing wrong here. Note that I'm using python to compute results

JMP
  • 21,771
Amine Hajyoussef
  • 133
  • To see why this kind of naïve averaging of each day's interest rates in isolation cannot possibly work, consider the following two scenarios: in scenario A, stock 1 goes up by +900% each day while stock 2 goes down by -90% each day; in scenario B, the same thing happens on the first day but then the rates reverse, so that stock 1 now goes down by -90% while stock 2 goes up by +900%. In scenario A, an initial investment of $100 in each stock on day 1 will have grown to $10000 + $1 = $10001 by day 3; in scenario B, you'll just get your original $200 back. – Ilmari Karonen Feb 05 '23 at 17:00
  • Averaging percentages when the base of each percentage is different is meaningless and can't give the correct result. – matt_black Feb 05 '23 at 18:52

2 Answers2

4

When taking average of the two stocks increases, you should also consider the investment ratio by value between the two stocks.

After day 2, the investment ratio is no longer $1:1$ as in day 1, but becomes $105:100 = 21:20$. Then when taking weighted average of the stock increases from day 2 to day 3,

$$\begin{align*} \text{Weighted avg stock growth} &= \frac{21\cdot \frac{110-105}{105} + 20\cdot \frac{110-100}{100}}{21+20}\\ &= \frac{(110-105)+(110-100)}{105+100}\\ &\approx 7.317\% \end{align*}$$

This matches the combined portfolio growth from day 2 to day 3:

$$\begin{align*} \text{Portfolio growth} &= \frac{\frac12 (110+110)-\frac12 (105+100)}{\frac12 (105+100)}\\ &= \frac{110-102.5}{102.5}\\ &\approx 7.317\% \end{align*}$$

peterwhy
  • 22,256
3

With combined stocks, the two stocks become as one, so the data becomes

200
205
220

Your required percentage is now $\frac{220-205}{205}=7.317073170\dots\%$.

JMP
  • 21,771