The Investor’s Gift Exchange Paradox

One fine day, an angel descends from heaven and picks you to offer a gift.

She displays two envelopes in front of her and tells you that the two envelopes both contain money.  However, she cannot tell you how much money is contained within the two sealed envelopes; all she can say is that one envelope contains 10,000 times the amount of money the other envelope contains.  To put that in another way, one envelope contains 1/10,000 of what the other envelope contains.  Which envelope you pick depends entirely upon your luck.

So you close your eyes and pick one of them.  You discover a $100 bill inside the envelope.  Now, the angel makes another offer to you.  “Would you like to exchange your envelope with the $100 bill for the other envelope which I’ve previously offered you but you’ve not chosen?”

Now, some of you might get incredulous.  You might ask, “Wouldn’t the two envelopes be the same?  If I had chosen the other envelope, wouldn’t I now be offered the current envelope?”

However, the answer is not that simple.  First of all, I need you to think like a truly rational, risk-taking investor.  Take a moment to ponder upon the question before you look at the answer.

The answer is that you should exchange the $100 bill for the other envelope.

You must remember that one envelope contains 10,000 times the amount of money that the other envelope contains.  So, the other envelope will contain either a meager 1 cent ($100/10000) or a whopping $1 million ($100X10000).  As an enterprising investor, surely you would risk a 50% chance of losing only nearly all of $100 or a 50% chance of winning the million-dollar jackpot.  In mathematical terms, the expected outcome of the exchange, E(x), is ($1,000,000+$0.01)/2 = $500,000.005.  This is a tremendously good offer by the benevolent angel.

Now, the question – do you see the contradictory paradox in the above passage?

3 thoughts on “The Investor’s Gift Exchange Paradox”

  1. On the one hand, the pure common sense of picking up one of two envelopes and immediately exchanging it for the other envelope would tell you that you would not stand to gain from the exchange.

    However, the maths I illustrate above points to a profit.

    Won’t the 2 views be contradictory?

    1. Haha, yes, I see it now. The plight of the industry is that average returns are taken as fact without more than a cursory glance. I’ve seen NPV calculations which incorporate real options whose values were higher than the project in question. Really mind boggling.

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