A trading firm has the utility function $U(w) = \sqrt{w}$ where $w$ is the wealth of the firm, in dollars. Currently, the firm is worth $100,000,000, so their happiness is $U(100,000,000) = \sqrt{100,000,000} = 10,000,$ and they always want to maximize their expected happiness.

They are offered a risky bet which will succeed with probability $p,$ doubling the wealth of the firm. However, if it fails, the firm will go bankrupt.

Find the smallest probability under which they would take this bet, and then round that probability to the nearest 10%.

**Hint:** Find their expected utility based on the possibilities after this bet, and determine the conditions under which it exceeds their current utility of $10,000.$

As the last few questions have illustrated, there is an aspect of *diminishing returns* for wealth; e.g., the happiness gained from each additional $1 tends to decrease as a person becomes wealthier.

Similarly, there is also an effect of **time** on the value of money. The last two questions begin to explore this effect.

The previous question illustrated the idea that “money now is worth more than money later”. A fundamental reason behind this is that money can be invested - in financial assets or elsewhere - so that it grows over time. This is why lenders are paid **interest**.

If money is lent with an annual interest rate of 1% compounded continuously, about how long would it take for the money to double?