ROI Accuracy Calculation (long)

[Paul2432]

[ QUOTE ]

Can you please start a separate post on this and show the math? Thank you.

Irieguy

[/ QUOTE ]


The purpose of this calculation is to find the confidence interval around your ROI. The place to start is to calculate the standard deviation. This is the one part of the calculation that requires making an assumption, but it does not effect the result that much.

I assumed a typical winning playing will have a 30% ROI and results distribution like this:

1st 15%
2nd 15%
3rd 10%
4th-10th 60%

Assuming 10% fee, this player will have an ROI of:

ROI = 0.15 (3.9) + 0.15 (1.9) + 0.1 (0.9) - 0.6 (1.1) = 0.3

The standard deviation is then:

Square root ( 0.15 (3.9-0.3)^2 + 0.15 (1.9-0.3)^2 + 0.1 (0.9-0.3)^2 + 0.6 * (-1.1-0.3)^2)

Solving:

SD = 1.88

I have experimented with different results distributions and with anything reasonable the SD is around 1.7 to 2 (assuming each finishing position is equally likely gives an SD of 1.67). For the rest of the calculation, I'll use SD=1.9.

Now that we know the SD we can determine the standard error (SE). This is very easy. It is simply:

SE = SD / sqrt(N)

N = number of trials.

Finally, the confidence interval is found as follows:

Confidence interval = ROI +/- z (SE)

Z is found using standard statistics tables. A ~95% confidence interval corresponds to a z-value of 2.

Thus, for example after 400 tournaments the 95% confidence interval is:

+/- 2 * 1.9 / sqrt(400) = +/- 0.19

Please let me know if you see any errors.

Here is a good web-site that covers a lot of these calculations in detail.

http://davidmlane.com/hyperstat/index.html

Paul

[Irieguy]

Sorry if i'm missing something, but in your first example you listed the formula as 3.8/sqrt(N).

In this example you say the SDs are always around 1.6-2.0.

Is the formula supposed to be SD/sqrt(N)?

Irieguy

[Bigwig]

I'm certainly not a mathematician, but this seems like the it produces much too high a variance. In the other thread you used 3.8. Which means after 2000 tourneys your ROI is +/- 8.5%. That seems to be an enormous range to me for that size a sample. Could you explain why I'm wrong?

[Irieguy]

Never mind, I see. Missed the Z.

[Irieguy]

Paul,

I hope this is wrong:

For 95% confidence interval, solving for N, if I want to know how many SNGs I need to play to know my ROI +/- 1% (my SD is 1.7)

sqrt(N)= 2 * 1.7/0.01

=115,600!

I'd have to play 115,600 SNGs to know with 95% certainty that my ROI is +/- 1% of what I think it is. That could take months! /images/graemlins/wink.gif

Irieguy

[stupidsucker]

I knew it was high, but I didnt think it was that high.


I dont know if it makes any difference, but the structure of 15/15/10 % is not even close to how it should map out for a winning sng player.

Should be more firsts then seconds, and more thirds then seconds.

[Paul2432]

Thanks for the comments everyone. I was fairly surprised at the calculations myself, and was somewhat hoping I made a mistake somewhere.

Stupidsucker, you asked about the effect of more heavily weighting first place finishes. In general, if ROI is constant, more first place finishes will mean higher SD.

I am going to run a simulation to test the results. I'll report back with the findings.

Paul

[stupidsucker]

Questions about the formulas

[ QUOTE ]
ROI = 0.15 (3.9) + 0.15 (1.9) + 0.1 (0.9) - 0.6 (1.1) = 0.3


[/ QUOTE ]

Isnt it Netprofit/totalbuyin=roi an easier equation for roi?

I dont understand where the 3.9,1.9,.9 and 1.1 come from for this equation.

[ QUOTE ]
Square root ( 0.15 (3.9-0.3)^2 + 0.15 (1.9-0.3)^2 + 0.1 (0.9-0.3)^2 + 0.6 * (-1.1-0.3)^2)


[/ QUOTE ]

What does the ^ mean?


sorry to ask stupid questions, but I am curious, so I can figure out mine acording to my own stats.

[Bigwig]

[ QUOTE ]
What does the ^ mean?

[/ QUOTE ]

To the power of.

[eastbay]

Sounds about right, actually. 95% confidence on +/-1% is a pipe dream. 75% conf. on +/-2% is asking too much.

However, all of this analysis is only strictly valid for normally distributed variables. That assumption is not entirely correct here.

eastbay

[AleoMagus]

Yeah.

My spreadsheet does a similar calculation on the Conf. sheet, but I figure a confidence to +/- X$/hr. This is easy enough to convert to a %ROI figure I guess. You can input whatever confidence values you want and it will calculate automatically.

To anyone having a hard time with these calculations, I just recommend downloading the 2+2SNG spreadsheet.

Also, I recently made a post HERE (http://forumserver.twoplustwo.com/showthreaded.php?Cat=&Board=singletable&Number=124 1103&Forum=,All_Forums,&Words=&Searchpage=0&Limit= 25&Main=1241103&Search=true&where=&Name=5278&dater ange=&newerval=&newertype=&olderval=&oldertype=&bo dyprev=#Post1241103) that is at least the start of a combinatoric approach to these calculations that does not rely on a normal distribution assumptions. Actually the link in that post should also have my confidence calculator if you do not know what I am talking about with the Spreadsheet comments.

Regards
Brad S

[Paul2432]

[ QUOTE ]
Questions about the formulas

[ QUOTE ]
ROI = 0.15 (3.9) + 0.15 (1.9) + 0.1 (0.9) - 0.6 (1.1) = 0.3


[/ QUOTE ]

Isnt it Netprofit/totalbuyin=roi an easier equation for roi?

[/ QUOTE ]

I think this is a matter of preference. I used ROI = netprofit/buy-in where buy-in does not include the fee.

[ QUOTE ]
I dont understand where the 3.9,1.9,.9 and 1.1 come from for this equation.

[/ QUOTE ]

Assuming a buy-in of 1 with a 10% fee, 3.9, 1.9, 0.9 and -1.1 are the net profit or loss with a 1st, 2nd, 3rd, or 4th-10th place finish respectively. Because I used a buy-in of 1, these values are also the ROI for each tournament.

Paul

[stupidsucker]

[ QUOTE ]
I think this is a matter of preference. I used ROI = netprofit/buy-in where buy-in does not include the fee.


[/ QUOTE ]
[ QUOTE ]
ROI = 0.15 (3.9) + 0.15 (1.9) + 0.1 (0.9) - 0.6 (1.1) = 0.3


[/ QUOTE ]

I think I'll stick to mine A/B=C. It just seems easier.

[Paul2432]

I ran some simulations using microsoft excel that essentially confirmed my calculations.

The simulation shows that if 100,000 players each play 400 tournaments with an expected ROI of 0.3 about 4.6% of them will experience ROI < 0.111 or ROI > 0.48.

The code is below if anyone is interested.

Paul

Option Explicit

Sub roitest()

Dim i, j, count As Integer


Dim first, second, third, test, roi, tot As Single

first = 0.85
second = 0.7
third = 0.6
count = 0

For i = 1 To 100000
tot = 0
For j = 1 To 400
test = Rnd()
If test > first Then
tot = tot + 3.9
ElseIf test > second Then
tot = tot + 1.9
ElseIf test > third Then
tot = tot + 0.9
Else
tot = tot - 1.1
End If

Next

roi = tot / 400
If roi < 0.111851 Or roi > 0.488149 Then count = count + 1

Next


MsgBox (count)


End Sub

[stupidsucker]

This only goes to prove that short runs mean absolutly NOTHING. Sure helps the tilt factor along.

[eastbay]

[ QUOTE ]
[ QUOTE ]
Questions about the formulas

[ QUOTE ]
ROI = 0.15 (3.9) + 0.15 (1.9) + 0.1 (0.9) - 0.6 (1.1) = 0.3


[/ QUOTE ]

Isnt it Netprofit/totalbuyin=roi an easier equation for roi?

[/ QUOTE ]

I think this is a matter of preference. I used ROI = netprofit/buy-in where buy-in does not include the fee.


[/ QUOTE ]

FWIW, at least on these forums that is not the usual way to calculate ROI. It strikes me as a little dishonest. You're paying that 10% vig to get your earn. Whether it's a "fee" or not is completely irrelevant.

eastbay

[stupidsucker]

[ QUOTE ]
FWIW, at least on these forums that is not the usual way to calculate ROI. It strikes me as a little dishonest. You're paying that 10% vig to get your earn. Whether it's a "fee" or not is completely irrelevant.



[/ QUOTE ]

Actually his way = the exact same roi as my way. Its just an elaborate way of doing so. Is it needed that way because of the rest of the formula?

.5(prize award)-.1(buy in)-.01(fee)=3.9(1st)
.3(prize award)-.1(buy in)-.01(fee)=1.9(2nd)
.2 -.1-.01 =0.9(3rd)

Then plug these numbers in with the rest for the roi equation.

[stupidsucker]

ok I think I am getting it now.

You need to figuer the roi in pieces like that because you need to know the exact roi for each seperate outcome. Then by using the frequency of each outcome you can figure out your SD.

[ QUOTE ]
Square root ( 0.15 (3.9-0.3)^2 + 0.15 (1.9-0.3)^2 + 0.1 (0.9-0.3)^2 + 0.6 * (-1.1-0.3)^2)


[/ QUOTE ]

ok now what is the -0.3 for?


Sorry to ask so many questions.