In a proposed investment, the amount of profit X is uncertain, but a
probabilistic estimate gives a mean of $3 million and a standard
deviation of $25 million. The investors owe the source of capital
(e.g., a bank) a fee of $200,000 plus 10% of the profits. What is the
mean and standard deviation of the amount of money the investors
actually retain?
Would like to see not only solution, but HOW to compute it.

---

Request for Question Clarification by davidmaymudes-ga on 18 Jan 2003 00:07 PST

The answer depends on whether the bank also gets 10% of the losses, or
if the 10% only comes into play if there are profits.

If the bank is responsible for 10% of the losses as well, then the
problem is quite easy; if, as is likely in the real world, they expect
to only share in the profits, then the calculation is more difficult,
and in fact I think would have to be approximated by numerical
integration.

---

Clarification of Question by thecuriousone-ga on 18 Jan 2003 02:25 PST

Thank you for your question. I don't know which of the 2 assumptions
is valid. Therefore, could you show how to solve the "easy" assumption
and could you describe a methodology (if not a formula) of how to
solve the more likely "real world" assumption?

OK, if the bank is sharing in losses:

the shape of the probability distribution isn't really changed, just
reduced by 10% for the bank's share, and slid left by $200,000 for the
fee.

so the mean after the bank's cut is $3,000,000 - $300,000 - $200,000 =
$2.5 million.

the standard deviation is reduced by 10%, and not affected by the fee,
so it's $22.5 million.

(to see why this is, consider the part of the probability distribution
thats, say, >1 standard deviation on the high side.  before the bank's
cut, in that case, the profit would be $3 + $25 = $28 million.  After
the cut and fee, it would be $28 million - 200K - $2800K = $25
million, or $22.5 million above the mean.)

for the case where the bank is only sharing in profits, but not
losses, the only way I can think to solve the problem is to
approximate it numerically.  I would chop the probability distribution
for the original profit up into pieces so that instead of having a
continuous probability distribution, you would have, say, 100 discrete
profit scenarios, each equally likely and chosen so that they have the
correct mean and standard deviation.  Then, you would adjust them for
the bank's profit and fee, and compute the standard deviation of the
resulting numbers.

A few things are easy to say: the calculated mean would be somewhat
lower than the answer of $2.5 million, since the losses would go to
you and not to the bank, and, similarly, the standard deviation would
be somewhat higher than the previously calculated $22.5 million, again
because of the additional possible losses.

I hope this makes sense, please ask for a clarification if not.

--David

Thank you for a clear and thorough explanation.

I think the standard deviation that davidmaymudes-ga is not 100%
correct. It should be 20250000.