A bank account is guaranteed to earn a fixed rate of 6% on a $10,000 deposit over the next year.?

A speculative investment offers the possibility of 15% earnings on the $10,000 if the investment succeeds and a loss of 5% of the $10,000 if the investment fails. Using what you know about probability and expectation, find the necessary probility of success for the speculative investment to be the better choice.

The value of $10,000 at 6% is $600.

That determines the expected value of the speculative investement must be greater than $600 for it to be the better choice.

(10,000 * 0.15) (x) + (10,000 * -0.05) (1-x) = 600
1500x + (-500+500x) = 600
2000x = 1100
x = 1100/2000 = 0.55 or 55%

So if you knew the probability of the 15% earnings outcome occurs 55% of the time, then it would be equivalent to the bank account with the 6% interest.

However, it would be prudent to add a margin of safety to the probability. Something like 25%, so if the positive outcome of 15% return occured 55% * 1.25 = 68.75% it would be a good decision.

You may want to read up on: expected value and modern portfolio theory.

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