Commissions are breaking the bank (it's not the agreed 20%; what went wrong?)

[Andy Galloway]

I have developed a spreadsheet to calculate how much commission I should pay my broker based on how much money he has made for me. Although the formulae are simple, the results are unexpected and I thought I would get some feedback on this conundrum before I commit to it.

His commission is 20% of whatever he makes for me per month.

If he loses me money, that loss is carried forward until such time as his earned commissions wipe of the debt and his "account" is back in the black.

All simple so far. So let's look at a possible scenario:
Month 1 my account is at $1,000 and he increases my account to $2,000. I pay the broker his commission of $200 (20% of the $1,000)
Month 2 my account is now at $1,800. If the broker now reduces my account back to $1,000, he creates commission loss of $160 (20% of the -$800)
Month 3 my account now increases to $2,000. The commission earned is $200 but last month's loss must be made up, so the broker gets the remainder of the $200 - $160 - $40.

What this means is that the broker has earned me £1,000 and I have paid him $240 to do so. This is not the 20% commission that we had agreed. What went wrong?

 

[stapel]

I have developed a spreadsheet to calculate how much commission I should pay my broker based on how much money he has made for me. Although the formulae are simple, the results are unexpected and I thought I would get some feedback on this conundrum before I commit to it.

His commission is 20% of whatever he makes for me per month.

If he loses me money, that loss is carried forward until such time as his earned commissions wipe of the debt and his "account" is back in the black.

All simple so far. So let's look at a possible scenario:
Month 1 my account is at $1,000 and he increases my account to $2,000. I pay the broker his commission of $200 (20% of the $1,000)
Month 2 my account is now at $1,800. If the broker now reduces my account back to $1,000, he creates commission loss of $160 (20% of the -$800)

I don't understand what you mean by "the broker now reduces your account back to $1,000"...? Also, when you say "my account is now at $1,800", does this mean that his commission is taken directly from your account?

Month 3 my account now increases to $2,000.

Does your account increase to $2,000 from the $1,800 or from the $1,000?

The commission earned is $200 but last month's loss must be made up, so the broker gets the remainder of the $200 - $160 - $40.

From the commission value you give, would it be correct to assume that you are counting the increase from $1,000, rather than the ending amount of $1,800? If so, why did the broker "reduce" your account to this value? On what grounds did he withdraw $800 in Month 2, in addition to the $200 that he'd taken out in Month 1?

What this means is that the broker has earned me £1,000 and I have paid him $240 to do so. This is not the 20% commission that we had agreed. What went wrong?

Should the "pound" symbol be a "dollar" symbol?

Thank you!

 

[JeffM]

I have developed a spreadsheet to calculate how much commission I should pay my broker based on how much money he has made for me. Although the formulae are simple, the results are unexpected and I thought I would get some feedback on this conundrum before I commit to it.

His commission is 20% of whatever he makes for me per month.

If he loses me money, that loss is carried forward until such time as his earned commissions wipe of the debt and his "account" is back in the black.

All simple so far. So let's look at a possible scenario:
Month 1 my account is at $1,000 and he increases my account to $2,000. I pay the broker his commission of $200 (20% of the $1,000)
Month 2 my account is now at $1,800. If the broker now reduces my account back to $1,000, he creates commission loss of $160 (20% of the -$800)
Month 3 my account now increases to $2,000. The commission earned is $200 but last month's loss must be made up, so the broker gets the remainder of the $200 - $160 - $40.

What this means is that the broker has earned me £1,000 and I have paid him $240 to do so. This is not the 20% commission that we had agreed. What went wrong?

What has gone wrong is that your agreement focuses on only a single month. In month one, we hypothesize that he increases your investment from 1000 to 2000.Thus he is owed 200 for that month, leaving you with 1800 at risk. In month two, we hypothesize that he loses 800, thus he is "in the hole" by 160, but the agreement does not require him to reimburse you for what you have already paid. Over the two months combined, you have no gain at all, but you have paid the broker 200 for a net of nothing. Now we hypothesize that in the third month, he runs the fund up to 1000 again so you owe him another 200 less the 160 for a net of 40, leaving you at the end with 1960, a net gain of 960 for which you have paid 240, or 25%. When you pay real money for each month of gains and the broker is not out any real money for each month of losses, you are going to get hosed.

Consider this scenario. Suppose the broker makes you 1000 every odd numbered month, loses 800 in the second month, and loses 1000 in all the other months. The broker will make 200 in the first month and 40 in the third month and nothing thereafter for a total of 240. But the total gain before fees was only 200. So over the course of the year, the broker will make 240 while you lose 40.

May I please be your broker?

 

[Dr.Peterson]

His commission is 20% of whatever he makes for me per month.

If he loses me money, that loss is carried forward until such time as his earned commissions wipe of the debt and his "account" is back in the black.

All simple so far. So let's look at a possible scenario:
Month 1 my account is at $1,000 and he increases my account to $2,000. I pay the broker his commission of $200 (20% of the $1,000)
Month 2 my account is now at $1,800. If the broker now reduces my account back to $1,000, he creates commission loss of $160 (20% of the -$800)
Month 3 my account now increases to $2,000. The commission earned is $200 but last month's loss must be made up, so the broker gets the remainder of the $200 - $160 - $40.

As I understand it, you have something like this (copied from my spreadsheet):

week	start	end	increase	commission	paid	unpaid	net	net increase
1	1000	2000	1000	200	200	0	1800	800
2	1800	1000	-800	-160	0	-160	1000	-800
3	1000	2000	1000	200	40	0	1960	960
total			1200	240	240			960

The negative commission is not paid immediately; if it were, you would have had $160 more in your account at the start of the third month, which would change the calculation of the increase and therefore of the subsequent commission! His total commission is 20% of the total amount of increase, which is $1200 by this calculation because it includes the first $200 commission that he got back for you. Now, the actual net increase is in the last column; after the last month's commission is paid, you really have only $1960. As a result, the total commission turns out to be 25% of your net increase; you may be familiar with the fact that a 20% commission is 25% of the amount remaining after deducting that commission.

Now, if negative commissions were paid immediately (that is, he paid his own money into your account), it would look like this:

week	start	end	increase	commission			net	net increase
1	1000	2000	1000	200			1800	800
2	1800	1000	-800	-160			1160	-640
3	1160	2000	840	168			1832	672
total			1040	208				832

You have a smaller total increase (the $1000 actual increase, plus the $40 net commission from the first two months), and he got 20% of that. In terms of net increase, the commission is again 25% of your $832 net gain. (Now, this is really a different scenario, because I am assuming the same value each month even though you started the third month with more money, so the actual increase is smaller.

What this means is that the broker has earned me $1,000 and I have paid him $240 to do so. This is not the 20% commission that we had agreed. What went wrong?

As I've shown, your actual gain in your scenario, excluding commissions, is $1200; he got 20% of that. Your error is in sometimes taking the commission into account, and sometimes not. If you calculate based only on your net gain, he gets 25%, which is a well-known phenomenon.

There is also the issue that by deferring hix repaying a negative commission, he is inflating your increase, which may not be what you intended.

 

[JeffM]

In light of Dr. Peterson's post, I'd like to clarify my previous post.

Mathematically, he is absolutely correct that if the broker is getting paid 20% of the overall gain, the broker will receive 25% of your net gain if there is a gain during the course of the contract.

If the sum of the gross monthly gains and losses is x > 0, you will have paid the broker 0.2x and remained with 0.8x. What percentage of your net receipts will you have paid the broker?

\(\displaystyle \dfrac{0.2x}{0.8x} = \dfrac{0.2}{0.8} = 0.25\%.\)

The percentage calculated against net gains will always exceed the percentage calculated against gross gains.

My post did not contradict that point though it may have seemed to do so. My post was fundamentally about a completely different point. Making the contract revolve around actual payments when there are monthly gains but notional payments when there are monthly losses may entail that you pay a fee when there are no gains cumulatively. (For mathematicians, this is a variant on the St. Petersburg Paradox.)

Suppose you invest 4000 initially, there is a gain of 1000 in the first month from which the broker takes 20%, or 200, and a loss of 4800 in the next month. Future gains will be zero because there is nothing left to invest, but you will have paid 200 to lose 4000. This too is a mathematical fact.

Moreover, this mathematical fact creates very perverse incentives for the broker. Suppose he has two customers with identical amounts to invest and identical contracts. He buys stock X for A and sells an equal amount short for B. If the price of X goes up, he gets a fee from A while paying nothing to B. If the price of X goes down, he pays nothing to A while getting a fee from B. No matter what happens (except no price change at all), the broker gets paid although his customers overall lose.

 

[Dr.Peterson]

My post did not contradict that point though it may have seemed to do so. My post was fundamentally about a completely different point. Making the contract revolve around actual payments when there are monthly gains but notional payments when there are monthly losses may entail that you pay a fee when there are no gains cumulatively. (For mathematicians, this is a variant on the St. Petersburg Paradox.)

That's right. I was actually working on mine while you did yours, and I was focusing solely on checking my interpretation of the situation, without looking deeper into the business aspects (which is not my specialty) or further implications. I was glad to see you had supplemented it.