Probability: percentages of losers and winners do not match

[Oleg]

Hi colleagues,
Please help to solve some sort of mistery in Reading 8, p. 387.
The curriculum describes a research when investment funds’ performance was compared for 2 consecutive years, and each of these 2 years half of them were named winners, and half - losers. The case states that 65% of Year 1 winners became winners in Y2, and the rest (35%) were losers in Y2. But the proportions for losers are not the same: 15% for losers in Y1 who became winners in Y2, and the rest (85%) for L1L2. How can this be possible? If the funds and their quantity are the same in Y1 and Y2, shouldn’t W1L2 “vacate the place” for the same number of L1W2?

 

[studyguy18]

I only have the electronic version of the CFAI text so there are no page numbers. Would you be able to point out which question or BB example in particular you are looking at? Just a little hard to fully understand what you mean without being able to read the question.

 

[Oleg]

studyguy18 wrote:
I only have the electronic version of the CFAI text so there are no page numbers. Would you be able to point out which question or BB example in particular you are looking at? Just a little hard to fully understand what you mean without being able to read the question.

Example 2, the first review of Vidal-Garcia research.

 

[cpk123]

There is a mis-statement here

Quote:and each of these 2 years half of them were named winners, and half - losers

What the question stem states is

Quote:The top 50 percent of funds by benchmark-adjusted return for a given year were labeled winners; the bottom 50 percent were labeled losers

and what they have provided is % of winners in year 1 who also continued to be winners in Year 2. etc. etc.
So a Winner in Year 1 - could continue to be a winner or become a Loser in Year 2. That is the only 2 possible paths for a Winner in year 1. And that is borne out by 65.5% (W1->W2) + 34.5% (W1->L2) = 100% (1).
Similarly 15.5% (L1->W2) + 84.5% (L1->L2) = 100% = 1.
And these are CONDITIONAL Probabilities….

 

[Oleg]

cpk123 wrote:
There is a mis-statement here

Quote:and each of these 2 years half of them were named winners, and half - losers

What the question stem states is

Quote:The top 50 percent of funds by benchmark-adjusted return for a given year were labeled winners; the bottom 50 percent were labeled losers

Why misstatement? The case states: “For each style of fund in each country, funds were classified as winners or losers for each of two consecutive years.”

 

[Oleg]

cpk123 wrote:
and what they have provided is % of winners in year 1 who also continued to be winners in Year 2. etc. etc.
So a Winner in Year 1 - could continue to be a winner or become a Loser in Year 2. That is the only 2 possible paths for a Winner in year 1. And that is borne out by 65.5% (W1->W2) + 34.5% (W1->L2) = 100% (1).
Similarly 15.5% (L1->W2) + 84.5% (L1->L2) = 100% = 1.
And these are CONDITIONAL Probabilities….

Let’s say we have 100 funds, 50 of them win in Y1, then 65% of those 50 win in Y2 (i.e. about 33 funds). Question: how may funds that did not win in Y1 won in Y2? Answer is 17, i think - and this represents about 35%. This contradicts to what we see in the case. Maybe the case description contains any other hint, but I don’t see it.
Generally, if:
a) the number of funds is the same in Y1 and Y2,
b) the funds are the same in Y1 and Y2, and
c) winners and losers are divided in 50/50 proportion each year,
then:
W1L2 = L1W2. Is this statement correct?

 

[cpk123]

The mistake you are again making is twisting the original statement.
Someone else - possibly S2000 could explain better. But in my mind
Top 50% of benchmark adjusted returns IS DEFINITELY NOT EQUAL TO what you are saying viz. winners and losers are divided in 50/50 proportion each year,

 

[Oleg]

Searching for explanations, I found the original paper by J. Vidal-Garcia, which was a basis for the curriculum case:
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1997474
The sample included terminated funds - the funds which disappeared from the market, apparently due to low performance. In the case of French funds discussed in the curriculum, about 22% of funds were terminated during the sample period.
So, for each 2-year comparison period, once some of the”losing” funds leave the scene after Y1, the remaining quantity realigns so that at the start of the 2-nd year Y1 winners have a larger share in total quantity than Y1 losers. And that explains what’s going on pretty well. Small example:
1. We have 100 funds at the start of Y1.
2. At the end of Y1 we divide them 50/50 by performance criteria.
3. Of the bottom 50%, 10 funds leave the market.
4. We start Y2 with 90 funds.
5. At the end of Y2 we divide them 45/45 by performance criteria.
6. Let’s assume 30 Y1 winners won in Y2. This gives W1W2 = 30/50 = 60%.
7. Then, W1L2 = 20/50 = 40%.
8. Then, L1W2 = (45-30)/(50-10) = 15/40 = 37.5%, and L1L2 = (45-20)/40 = 62.5%.
I think, it is perfectly clear now. The only thing I still don’t understand is why the proportions in the case are so different, they must indicate a very high share of terminated funds, much more than the stated 22%.