More than \$40.

\$10 for \$50 with a 50/50 chance.

Technically if I have the box and I am offering you the chance to make this investment, then it depends on your risk appetite. Without me, you have no chance to make money apart from your \$10 in cash. With me, I could offer you \$x of the \$100 in which case you expected return is \$(x/2).

If you were risk-neutral, I could offer you \$20.01 of the \$100 as you'd then have an expected return above the initial investment. To a very risk averse investor, you may have to offer all the way up to \$79.99. It also depends on your own risk aversion.

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This^

Bingo. If we assume a somewhat efficient market for investment into this box, it really wouldn't matter if there's \$100 or \$100,000 inside. If you control the box and get to choose your partner, the return you have to offer would be bid down to the lowest acceptable return for an investor willing to lose his principle 50% of the time.

Sophisticated investors would at least require an EV greater than their principle - how much more would depend on risk appetite, opportunity cost and how often they could duplicate this investment. But casinos offer this type of "investment" with EVs lower than the invested amount all the time, and they rarely struggle to find investors.

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HighlyClevered:

But casinos offer this type of "investment" with EVs lower than the invested amount all the time, and they rarely struggle to find investors.

Thank you, I'm glad someone understands all the investing I do while in Vegas. My wife keeps calling it "gambling".

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Mathematically, wouldn't an offer of \$10.01 of the \$100 be the minimum the other investor would accept for an expected return above their initial investment, which was \$10, or is there something I'm not understanding?

Sure, a 100% chance a \$10.01 would suffice. But remember, in this investment there is a 50% chance of winning 0.

So the 50% chance of winning \$10.01 would only mean an expected return of \$5.005, below the \$10 initial investment.

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Let's look at it completely from "Other Investor" perspective.

I invest \$10 so my expected value would need to be \$10 to be indifferent, and marginally greater than \$10 to be willing to invest.

The expected value is .5 x 100 + 0 x 100 = 50 for the box.

I would require getting PROMISED \$20 of the \$100 to be indifferent because then my EV would be .5 x 20 + .5 x 0 = 10. and require marginally over \$20 or as mentioned above, \$20.01 to be willing to invest.

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are there real life investments / situations where people / groups provide 50% of the capital and expect less than 50% of the proceeds where all else is being equal?

its not like this is an investment in a property where the other party provides additional value of e.g. managing it subsequently.

id want 50% - a reasonable "fee" for bringing the deal to me.

but logically /mathematically i understand the marginally above 20.01 depending on risk appetite to be willing to "invest".

Yes obviously agreed in the real world.

It would matter on a lot of things, the split of the initial capital, the "fee" for bringing in the deal, perhaps man with the box has incurred expenses getting the deal / spent time bringing it to life, lots of things matter. (Also have to consider information parity, do both parties know how much who is investing or are you just being approached with a man with an investment opportunity and potential return, in which case all that would matter is IRR).

But purely from an Expected Value perspective the above math would apply.

I would make my investor post a minimum of \$110 so we open the box 5 times (10% equity placement Fee) so he really invested \$100. the expected value is \$250 but could be \$0 3.1% of the time and -\$10 around 6.3% of the time. If we beat those 2 then I would give him all of the invested capital pari passu alongside myself (90/10 split) on all money until we have both of our capital returned and then split the profits 50/50. Then for box 6 I would open it with my placement fee and initial equity for myself as some gravy on my profit.

How can it be \$0.00 3.1% of the time?

There's a 50% of getting \$0 each time you open it. If you open 5 boxes, the odds of getting \$0 every time are 3.125% (0.5^5).

The expected value of the box is \$50 but there is a sunk cost of -\$20 regardless of the outcome of opening the box. So you really have a 50% change of winning \$80 and a 50% chance of losing -\$20, taking the expected value to \$30. Given that we are splitting the cost of the box equally, I would want at least an equal share of the expected value, or \$15.

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Anecdotally that's fine but I don't think you can look at it that way and still be statistically sound.

The \$20 isn't really a sunk cost, its a cost of entering the scenario in which the EV is 50. You can't apply that \$20 as a sunk cost and then also reduce the outcome. You're basically double counting the implied \$20 fee and getting to \$15. The impact of double counting the \$20 would be:

20 x 1/2 (probability of loss) x 1/2 (split of EV due to duality of the situation). Thus you're impact is 20 x 1/4 or \$5. Adding this back in correctively would bring you to 15+5 or \$20, as stated in above answers.

Secyh62:

The expected value of the box is \$50 but there is a sunk cost of -\$20 regardless of the outcome of opening the box. So you really have a 50% change of winning \$80 and a 50% chance of losing -\$20, taking the expected value to \$30. Given that we are splitting the cost of the box equally, I would want at least an equal share of the expected value, or \$15.

This is my thought process. Except, I counted just your own \$10 as your own personal risk. So, you would want 2x or \$20 in addition to your capital.

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\$25