how can you calculate the sweet spot for a risk / reward ratio in a partnership

Question

Let's take 2 parties, A and B.

A has a tool that can raise the value of an investment by a known quantity (Q) with an estimated risk factor (R). A failure would incur a loss of (L).

B has money (M) to put in the tool.

let's plug some numbers to illustrate:

• R = 0.8 (80% chance of success)
• Q = 0.2 (20% profit if successful)
• L = 0.1 (10% loss if failure)
• M = $100 invested by B

The following scenarios are possible:

• The process is successful, M becomes M * (1+Q) = $120
• The process is a failure, M becomes M * (1-L) = $90

A is not taking any risks, but is providing an essential part of the process. A cannot lose money. B can lose the money put in the process

B can earn $20, or loose $10. If B earns money, he has to share with A.

How one could express the ratio at which the money is split using R, Q and L?

Answer 1

There's not much to do here:

Think in terms of ten separate plays of $100 each and the gain is (8 * $20) while the loss is (2 * $10) such that the average profit is ($140 / 10) and as 14% .

The percentage that Person A takes is just competitive marketing. Hedge funds take 20% of profit plus 2% of each year's beginning balance. Asset managers take 1% of each year's beginning balance and 0% of profit.

If Person A could guarantee a 7% profit then Person A could possibly earn 7%. But that's an insurance company endeavor.

Answer 2

This can be split into two issues: how much is this worth to B, and of that value that A is supplying to B, how much can A claim? The second question is a matter of negotiation, and is going to depend on issues such as supply and demand, how much confidence people have that those probabilities are correct, and other factors such as possibly their relationship.

As for how much value A is supplying to B, that is difficult to quantify, since you need some risk discounting factor. One method of modeling risk is to assume that the more money a person has, the less each additional dollar adds value. This then justifies using some diminishing returns function to measure the value of a person's net worth. What function to use is somewhat arbitrary, but the log function is often used. Using the log function, we would have that this investment results in some value X, and the log of X is equal to the weighted logs of the possibilities:

log(X) = 0.8(log(120))+0.2(log(90))

This gives us that X = 113.29. So this calculation suggests that B is getting $13.29 of value from this arrangement. However, it's not quite as simple as saying "Okay, then A can't charge more than $13.29"; now that we've introduced log, the math isn't linear any more. A could actually charge $17 and B would still be getting value from it.

A further complication is that B's investment of $100 probably isn't their entire net worth. If they have $1000 total, then rather than log(X) = 0.8(log(120))+0.2(log(90)), you'd have log(X) = 0.8(log(1020))+0.2(log(990)), which increases X significantly. But they're probably putting that other money in further investments, so to calculate their total risk, you'd have to look at what those investments are and how much correlation they have with this one.

You'd also have to look at further factors such as the time value of money.