Equally weighted index Calculation

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allalongthewatc

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Hello All,
I have a question about this one:
Let’s say initial stock price of A = $15, B = $40, and C = $55. Initial value of equally weighted index = 136.
Now, if the current stock price is A = $15, B = $48, and C = $45
then, how can I calculate the new Equally Weighted Index ? Can someone please help me?
Thanks in advance.
 
A’s return is ($15 – $15) / $15 = 0%, B’s return is ($48 – $40) / $40 = 20%, and C’s return is ($45 – $55) / $55 = -18.18%. The average return, therefore, is (0% + 20% – 18.18%)/3 = 0.61%.
Thus, the new value for the index is the old value increased by 0.61%: 136 × 1.0061 = 136.82.
 
S2000magician,
Thank you for your response. Can you please explain why you took the average of the growth rates and why 136 was multiplied by 1.0061?
If the weights of individual stocks A, B and C are say x, y and z, then I believe that 15x + 40y + 55z = 136. Can you please correct my understanding? I really appreciate your response and your help.
Thank you
 
For an equal-weighted (also known as an unweighted) index, it’s as if you’re buying $1 of each security: they all have equal weight. Thus, the return on the portfolio is the equally-weighted (or unweighted) average of the returns on the constituent securities. If the portfolio return is r%, then the new value is the old value times (1 + r%).
 
Thank you S2000magician, for your response. I have a follow-up question, if you don’t mind.
If all have equal weights, therefore,
(A + B + C)*weight /3 = 136 Here, A = $15, B = $40, and C = $55 => A+B+C = 110
Therefore, 110*weight / 3 = 136 => weight = 3.70909
Now, for the current price, (15+48+45)*weight/3 = 133.52 – this is not equal to what you have calculated. Can you please explain what I am doing wrong. I am not able to understand this. I am a bit frustrated.

I would really appreciate your help.
 
You’re not doing anything wrong (well, not much); you’ve stumbled onto the big problem with equal-weighted indices: every time there is a change in the price of a security, the weights change. Therefore, the index is, essentially, rebalanced daily: selling some of each security whose price has increased, and buying some of each security whose price has decreased.
Of course, if you were trying to run a fund that tracks the performance of an equal-weighted index, there’s no way that you would rebalance it daily: the transaction fees would be enormous. In practice, these funds usually rebalance quarterly, which means that they will have some (perhaps considerable) tracking error between rebalancing dates.
The one thing you’re doing wrong is trying to set weights on each security so that the weighted-average of the prices equals the index value. Those weights will change daily, and the calculation of the index value never explicitly uses those weights. The calculation is done exactly as I did above: compute the return of each security, average those returns, then increase the index value by the average return value.
 
Thanks S2000magician for your response. I have another follow-up question on the last part. I am not sure why you are saying that my calculation of equal weights is wrong. Let’s take an example: (I believe that when we say “equal” weighted average, it means arithmetic average)
Three numbers A = 10; B = 15; C = 20 => Equally weighted average = (A + B + C )/3 = 45/3 = 15
Now, if A increases to 15, B increases to 25 and C increases to 20 then the new equally weighted average = arithmetic average = (15+25+20)/3 = 20
However, using your method, the answer will be different because you are averaging the returns of each security. I could be wrong, but this is not how weighted average with equal weights is computed in mathematics. (I could be wrong, considering that I am getting old, and I have been out of school for too many years.) Please help me and correct my understanding.
I am a little bit confused and also a bit frustrated with this “equally likely” thing. It will really help me if you could share any insight on this. I also went to the library (I can’t afford any book because I am not working at the moment) to see whether I can find any justification. However, I couldn’t find any book at all.
Thanks in advance.
 
The idea of an equal-weighted index is that you buy the same dollar amount of each security. You’re trying to equate the weights to the prices (as if you bought the same number of shares of each security). Let’s work with your example; the starting share prices for A, B, and C are:
  • A, $10
  • B, $15
  • C, $20
We buy 30 shares of A, 20 shares of B, and 15 shares of C: $300 worth of each. (That’s the equal-weighting part: the dollar amount held in each security is the same.) Our portfolio is now worth $900. Tomorrow, the share prices are:
  • A, $15
  • B, $25
  • C, $20
Our investment in A is now worth 30 × $15 = $450, our investment in B is now worth 20 × $25 = $500, our investment in C is now worth 15 × $20 = $300, and our portfolio is now worth $450 + $500 + $300 = $1,250. Our return on A was 50% (= ($15 – $10) / $10), our return on B was 66.67% (= ($25 – $15) / $15), and our return on C was 0% (= ($20 – $20) / $20). The average return was 38.89% (= (50% + 66.67% + 0%) / 3. And the return on our portfolio was 38.89% (= ($1,250 – $900) / $900), as it should be.
Unfortunately, our portfolio is no longer equal-weighted: we have too much A and B. So we sell 10 shares of A, leaving us with 20 shares at $15, or $300 total, and we sell 8 shares of B, leaving us with 12 shares at $25, or $300 total. We already have $300 total in C. Now we move on to the next day.
 
One way to think of it is that the index value changes by an equally weighted return. So, you’re calculating returns and then equally weighting them to move from one index value to another.
The simple reason that you can’t get the same answer by calculating the weighted average prices and then calculating returns on these prices is that “it just doesn’t work that way” in the same way that A x (B+C) usually doesn’t give you the same answer as (A x B) + C.
Mathematically, when you’re working with combinations of ratios (which is what you have here), the order of operation almost always matters.
 
S2000magician wrote:
The idea of an equal-weighted index is that you buy the same dollar amount of each security. You’re trying to equate the weights to the prices (as if you bought the same number of shares of each security). Let’s work with your example; the starting share prices for A, B, and C are:
  • A, $10
  • B, $15
  • C, $20
We buy 30 shares of A, 20 shares of B, and 15 shares of C: $300 worth of each. (That’s the equal-weighting part: the dollar amount held in each security is the same.) Our portfolio is now worth $900. Tomorrow, the share prices are:
  • A, $15
  • B, $25
  • C, $20
Our investment in A is now worth 30 × $15 = $450, our investment in B is now worth 20 × $25 = $500, our investment in C is now worth 15 × $20 = $300, and our portfolio is now worth $450 + $500 + $300 = $1,250. Our return on A was 50% (= ($15 – $10) / $10), our return on B was 66.67% (= ($25 – $15) / $15), and our return on C was 0% (= ($20 – $20) / $20). The average return was 38.89% (= (50% + 66.67% + 0%) / 3. And the return on our portfolio was 38.89% (= ($1,250 – $900) / $900), as it should be.
Unfortunately, our portfolio is no longer equal-weighted: we have too much A and B. So we sell 10 shares of A, leaving us with 20 shares at $15, or $300 total, and we sell 8 shares of B, leaving us with 12 shares at $25, or $300 total. We already have $300 total in C. Now we move on to the next day.
Click to expand...
Thank you S2000magician. I see your point. I now understand what they mean by equally weighted. It’s not the prices that are equally weighted, but (price*shares). I worked out the algebra, and it makes perfect sense now.
For others, if you are interested, please let me know. I can post the algebra.
Thank you so much.
 
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