Math for Quantitative Finance
# Expected Value

A utility function is a way of measuring one’s “happiness” as a function of wealth, often denoted as \(U(w).\) Which of the following functions is the most reasonable approximation for a human's utility function in terms of their total wealth in dollars, \(w?\)

Hint: Do millionaires typically save up their loose change?

Tony has the utility function \(U(w)=\sqrt{w}\) and an initial wealth of $100. A coin will be flipped a single time; if it is heads, he will get $21, and if it is tails, he will lose $20. What is the expected value of Tony’s utility after playing this game?

Bonus: What does this calculation say about whether or not he would want to play this game?

The **expected utility hypothesis** states that an individual (or firm) will try to maximize the expected value of their utility, rather than simply the expected value of their wealth.

The previous problem illustrated how a specific utility function could lead someone to want to pass up a bet which has positive expected value. "Charging for risk," i.e. ensuring that the bet has enough positive expected value to compensate for the risk of loss, is an important part of most individuals' (and firms') utility functions. Many individuals (or firms) will not take bets that have the potential to bankrupt them no matter the return. While others are only likely to do so for potential return that is very large, \(1000\) times their current net worth for instance.

The next question shows how things can go awry with a risk-neutral utility function.

A trader is presented with the following game as a one-time offer:

*A fair coin will be flipped until a heads appears, and you will be paid \( \, $2^n\) if there are \(n\) flips.*

If her manager tells her to optimize the utility function \(U(w)=w\) with an initial wealth of $100, what is the most she would pay to play this game?

Assume, unrealistically, that there is no counterparty risk; i.e., the person offering this game can actually pay out any sum of money.

Once again, a trader is presented with the following game as a one-time offer:

*A fair coin will be flipped until a heads appears, and you will be paid \( \, $2^n\) if there are \(n\) flips.*

This time, her manager tells her to optimize the utility function \(U(w)=\log_e(w)\) with an initial wealth of $100. What is the most she would pay to play this game, rounded to the nearest dollar?

You can make use of this approximate graph of \(\displaystyle f(x) = \sum_{n=1}^{\infty} \frac{1}{2^n} \cdot \log_e\left(x+2^n\right)\) below.

The green line is the approximation of \(f(x)\). The pink line is the value \(4.61\) which is \(\log_e(100) \approx 4.61.\) That is, her utility with her existing \(\$100\) is approximately \(4.61\). To play this game, the loss in utility from her payment \(c\) must be offset by the gain in utility from her expected payout.Note that \(\log_e(\cdot)\) is the natural logarithm.

Our trader still has $100 and is optimizing the utility function \(U(w)=\log(w),\) and she is offered two favorable bets:

**Offer 1:** She will either lose $10 or get $20 with 1:1 odds.

**Offer 2:** She will either lose $X or get $(5X) with 1:1 odds.

What is the smallest integer dollar amount $X, greater than $10, where she would prefer Offer 1 over Offer 2, despite Offer 2 giving her higher expected wealth?

Utility functions inherently define a measure of “risk tolerance” for an individual or firm. For example, in the previous question, \(U(w)=\log(w)\) played a key role in determining when the trader would prefer the lower risk bet over the higher risk bet.

Even utility functions which behave somewhat similarly, such as \(U(w)=\log(w)\) and \(U(w)=\sqrt{w}, \) indicate different levels of risk tolerance. The next question directly compares the risk tolerance of these two utility functions.

A very friendly casino manager is running a promotion: for a payment of $1, he will flip a fair coin, and if it is heads you get your $1 back plus a bonus of $X.

Alice has utility function \(U_A(w) =\log(w)\) and Bob has utility function \(U_B(w)=\sqrt{w}.\) If they each have wealth $100, who is more likely to play the game (i.e., who will be willing to play for a smaller value of X)?

Stella has a car worth $22,500 and additional assets worth $40,000. Her utility function is \(U(w)=\sqrt{w}.\) She believes there is a 10% chance of being in an accident which destroys her car, but an insurer will offer her a policy which gives her the full value of her car ($22,500) if that happens.

What is the most that Stella would be willing to pay (as a one-time payment) for this insurance?

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