Market Cap Weighted - Mock Exam Burn

CFA Level III
#1

In the mock for this year lvl 3, 5 securities are given and we must select the weighting that gives the highest return. Here is my understanding of how to do the calculations but apparently I am wrong.

NOTE: the question specified "assuming no stock splits or stock dividends for the stock components and no rebalancing"

Equal-weighted -> Easy, just average the returns. [It is, infact, required to calculate the returns for each security in the index individually for this method.]

Price-weighted -> Either A) calculate the returns for each security individually (in this case, returns were given) and then {sum ( product (price_start, return) ) } / sum(price_start) OR B) rather than using the individual returns, sum(price_end)/sum(price_start) -1

Market-cap weighted -> Either A) calculate the returns for each security individually and then {sum(product(marketCap_start, return))} / sum(marketCap_start) OR B) sum(marketCap_end)/sum(marketCap_start)

Now, in trying to find my error, I used both methods of calculation for price-weighted and market-cap weighted. Price-weighted gets the same answer with both approaches, as expected. However, given the numbers in the problem, the market cap weighted approachs give VERY DIFFERENT answers.

So, I dug into the problem a little further. The first security provides the following:

starting price = $2.35

ending price = $2.53

market cap start = $48.5

market cap end = $52.5

Assuming no stock splits or stock dividends for the stock components and no rebalancing

market cap = nshares x price -------> nshares = market cap / price

nshares (starting) = 48.5 / 2.35 = 20.64

nshares (ending) = 52.5 / 2.53 = 20.75

So … I feel this questions is a huge error. The ratio of market cap to share price _ CAN NOT CHANGE _!!! Isn’t this why a market-cap weighted index is self rebalancing?!?

In practice, if one was weighting the individual securities at time 0, the ending market cap would not be known. Therefore, the weights on the individual securities would be based solely on (market cap starting for company 1 / sum of all market caps).

#2

Price weighted is the same as buying 1 of each security.

Equal weight is the same as buying £10 of each security.

Market cap weighted is the same as buying £10 of the £10bn security, £5 of the £5bn security, £1 of the £1bn security etc.

Price weighted = you will need to calculate the weight of each security compared the total price and multiply that by the % change in price of said security.

Equal weighted = mean average of the increases in price.

Market Cap weighted = you will need to calculate the weight of each security in terms of market cap over sum of all market caps and the multiply this by the change in price of said security.

The answer is correct.

#3

Here is the information given:

Starting Market Caps (Given)

Security 1: 48.5

Security 2: 32.7

Security 3: 26.6

Security 4: 25.3

Security 5: 23.9

Total Starting Market Cap: 157

/

Price Appreciation (Given)

Security 1: 7.70%

Security 2: 17.90%

Security 3: 14.30%

Security 4: 11.60%

Security 5: 10.50%

/

Ending Market Caps (Given)

Security 1: 52.5

Security 2: 41.2

Security 3: 30.1

Security 4: 27.6

Security 5: 26.8

Total Ending Market Cap: 178.2

/

HERE IS MY APPROACH:

Calculated Weightings

Security 1: 48.5 / 157 = 30.892%

Security 2: 32.7 /157 = 20.828%

Security 3: 26.6 / 157 = 16.943%

Security 4: 25.3 / 157 = 16.115%

Security 5: 23.9 / 157 = 15.223%

Total appreciation = (30.892% * 7.70%) + (20.828% * 17.90%) + (16.943% * 14.30%) + (16.115% * 11.60%) + (15.223% * 10.50%) = 11.997%

/

HERE IS THE ANSWER GIVEN

Total appreciation = (Ending total market cap / Starting total market cap) -1 = (178.2/157)-1 = 13.503%

/

The two approaches should agree but they do not. Hence, there is an error in the problem, unless I am misunderstanding something.

#4

I think you guys are overthinking it a bit. The CFAI doesn’t want you to spend 10 minutes on this 3 minute question. Yes, you can go about it the long way and weight everything like you mentioned or you can do it the easy way and use the summary values they provided. Equal weighted should be the only one you have to do a little more math on.

In the real world the price and market/value weighted returns should and will be different. For a price weighted index you have to deal with stock splits among other things.

For a market cap index, share repurchases, theoretically the price wouldn’t change but this doesn’t happen in the real world, and free float adjustments will alter the index value of the market cap during the course of the measurement period.

#5

Apologies, you are doing the change in price. But you need to multiply by the change in market cap.

#6

The point is, Chuck, that both methods of calculation are correct and one of them leads the candidate to the wrong answer.

The reason I did it “the long way” is because at time 0, when the market index is constructed, the ending market cap is not known and therefore this is the only way to get the right answer if there have been secondary offerings and/or float adjustments, etc.

I agree 100% that alot of the material is purely academic in nature and can’t be directly applied to the real world. The problem is, the next time I run into this problem, it could very well be the case that the ‘right’ answer is using the other method than the one used in this question.

#7

No, sir - you are absolutely wrong about that.

Are the weights of the individual positions determined correctly? Once the weights have been properly determined, the portfolio return is the sum of the position weights times the position returns. There has been no rebalancing so what would the change in market cap have to do with anything?

Edit: Just want to apologize if I’m coming off a bit short. I’m frustrated with this question. If I’m missing something, I’d very much appreciate an explanation but I honestly feel I understand all the salient aspects of the calculation and this question is just flat wrong. If you think the question is correct, please help me understand how.

#8

Because it’s a value weighted index. Do the math and I’m sure you will come up with right answer…

#9

Are you just messing with me now? You’re messing with me right? Either that or you simply don’t understand the mechanics of the question that I’ve described above.

#10

I see what you’re after now. Yeah, based on all else being equal the % change in price for the stock should be equal to the % change in mkt cap. The share price change multipled by the beginning market caps should be the same as the ending market caps relative to the starting market caps.

#11

Probably an error in the question. If that would happen in the exam, they’d definitely accept your method. Either method should be fine to use…

#12

Thank you both! I thought I was starting to lose my mind. Hopefully they recognize that a different answer would be correct on the M/C if the other method was used.

It is worth noting that a company that has issued, for example, 1000 shares of stock at $10 would have a market cap of 1000 * $10 = $10,000 . If that company then wanted to raise equity capital by issuing new shares, assuming no dividends, etc. - they could issue 300 new shares at $10, adding $3000 to the market-cap of the firm but by increasing the number of shares from 1000 to 1300, rather than by having a price appreciation in the stock. Note further that in this case, the market-cap weighted index is NOT self-rebalancing NOR does the second method provide an accurate measure of return!

For these reasons, I chose to use the more accurate method of calculatin individual security weights.

#13

im not messing, u pleb! The question is what it is. The price change is differnt from the market cap change…

#14

Right. But it shouldn’t be. That’s my point.

#15

They should have given us three different sets of stocks each with a different index methodology. This is why the CFAI never publishes the multiple choice exams.

#16

I emailed CFAI about this. It is errata and should be published as such.

#17

**UPDATE**

So, I emailed CFAI to let them know about the error and they said,

"The question says to assume there are no stock splits or stock dividends, but it does not assume away cash dividends. Your method of calculation (summing the products of the starting market cap weights and the price returns) assumes no cash dividends which is clearly not the case because the market cap returns are different than the price returns.

The sum of the products of the initial market cap weights and the change in market cap (the market cap return) is 13.5%, the same as the ending market cap divided by the beginning market cap minus 1."

/

To which I replied:

"Thank you for the quick response but there remains a problem. As you aptly pointed out, the question does not ‘rule out’ the possibility of a cash dividend. However, the impact of a cash dividend is that wealth is transferred from the company to the shareholder and, consequently, the market cap of the company is reduced by the cash that is distributed. {Example: If a company with 1,000,000 shares outstanding and a share price of $10 distributes a $1 cash dividend, all else equal, the new share price is $9 and the market cap of the company goes from $10M -> $9M}

In short, price appreciation EQUALS market cap appreciation, in the case of a cash dividend and all else equal.

The only way that the price appreciation could be less than the change in market cap is if 1) there is a stock dividend (we are told to assume this has not happened) or 2) the company raises equity capital via a secondary offering. In the case of the latter, the performance of an index that is constructed using the initial market cap weights and is not rebalanced (again, question said to assume no rebalancing) will be equal to the initial weights times the price appreciation.

On a related note, if a cash dividend has been distributed from the individual securities but that information has not been explicitly provided in exhibit 1, then there is no way to calculate the price-weighted or equal-weighted return either." / / What am I missing? First of all, when the question said to assume that there was “no stock dividend”, I was assuming no dividends whatsoever. If they allow for a cash dividend, but that dividend distribution information is not provided, how can one calculate the total return of the index???

#18

Just let it go

#19

Never.

#20

ok then do it the long way, and spend maybe 25 mins on a 3 min problem.

let it go, do all your research and earn a PHD in this problem 5 days later…