## What is the standard deviation?

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1. Download the historical daily prices data for the S&P 500 index (∧GSPC) from Yahoo Finance (https://finance.yahoo.com), and calculate the daily returns from 1928 to 2019.

(a) What is the mean?

(b) What is the standard deviation?

(c) If we assume daily returns are normal distributed, what is the probability of a market crash of Oct 19, 1987?

2. You have 1 million dollars of wealth. You build \$800, 000 house at the beginning of the year on the east coast of Florida. During the year there is a 2% chance that the house will be destroyed by a hurricane-related flood. However, you are able to purchase flood insurance for a premium of \$2 per \$100 dollars of insured house value for the year. In the event that your home is destroyed by a flood, the insurance company makes a payment equal to the insured house value. You are a risk averse agent with utility of end-of-year wealth given by U(W ) = −W−1. Assume that the interest rate for the year is 0% and that the value of the house, if unflooded, remains unchanged at the end of the year.

(a) Let x be the amount of insurance that you purchase. In terms of x, what is wealth at the end of the period if the house is not destroyed by the flood?

(b) Let x be the amount of insurance that you purchase. In terms of x, what is wealth at the end of the period if the house is destroyed by the flood?

(c) What is expected utility in terms of the quantities given in parts (a) and (b)?

(d) How much insurance should you purchase? You wish to maximize expected utility.

(e) What percentage of the house value did you choose to insure?

(f) Given the relationship between the premium and the insured value, is the ex- pected profit for the insurance company positive? Is the insurance contract actuarially fair?

(g) Would the insurance company be more profitable if the premium was \$3 per \$100 of insured house value? Why or why not?

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