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Math for Quantitative Finance

# 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.

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? 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? 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?

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. Under typical circumstances, what is the most that an individual would pay right now to receive$1,000 one year from now?

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

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