# Value and Risk

Most people think of value as being measured in terms of money (e.g., dollars). In other words, people and investors make decisions to maximize the expected value of their money. While this is generally true, it is potentially misleading because it does not account for risk.

# Value and Risk

In a bet, a fair coin is flipped. If it is heads, the player doubles their life savings and gets an additional $1. If the coin is tails, they lose all of their assets (their entire life savings, home, etc.). Would the average adult human take this bet? ## Value and Risk ### Intro to Quant Finance # Value and Risk The last question illustrated that “value” is often more complicated than an expected value calculation. Which of the following curves is a good depiction of an average individual’s “happiness” as a function of their wealth? ## Value and Risk ### Intro to Quant Finance # Value and Risk A trading firm has the utility function $$U(w) = \sqrt{w}$$ where $$w$$ is the the wealth of the firm, in dollars. Currently, the firm is worth$100,000,000, so their happiness is 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. What is the approximate value (rounded down to the nearest percent) of the smallest $$p$$ for which they would take this bet?

# Value and Risk

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?

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