Suppose an investor with $2 in capital has a logarithmic utility of wealth function: u(w) = In (w). The investor has the opportunity to buy a gamble similar to that described in the St. Petersburg paradox. The gamble is the same as that described in the St Paradox except that the maximum number of coin tosses is set to 10. That is, the gamble ends and the payoff is $0 if a tails lands on the 10th coin toss. Moreover, let us define i as the coin toss indicator. If heads lands on the ith coin toss, the pay-off is $2' and the gamble ends. If a tails lands on coin tosses one through to nine, the coin is tossed again. (a) What is the investor's initial utility of wealth? Report your response to four decimal places. (b) How much would the investor be willing to pay for the gamble? Report your response to two decimal places. Show all working. You may use MS Excel. You may copy and paste a table from MS Excel as part of your working. (c) If the investor's initial capital was $50 rather than $2, would the investor be willing to pay a higher price or lower price for the gamble? How does your answer relate to the investor's level of risk aversion? You may use MS Excel. You may copy and paste a table from MS Excel as part of your working. (d) Explain the St Petersburg Paradox and how the Paradox can be resolved with the consideration of utility functions such as: u(w) = ln (w). Suppose an investor with $2 in capital has a logarithmic utility of wealth function: u(w) = In (w). The investor has the opportunity to buy a gamble similar to that described in the St. Petersburg paradox. The gamble is the same as that described in the St Paradox except that the maximum number of coin tosses is set to 10. That is, the gamble ends and the payoff is $0 if a tails lands on the 10th coin toss. Moreover, let us define i as the coin toss indicator. If heads lands on the ith coin toss, the pay-off is $2' and the gamble ends. If a tails lands on coin tosses one through to nine, the coin is tossed again. (a) What is the investor's initial utility of wealth? Report your response to four decimal places. (b) How much would the investor be willing to pay for the gamble? Report your response to two decimal places. Show all working. You may use MS Excel. You may copy and paste a table from MS Excel as part of your working. (c) If the investor's initial capital was $50 rather than $2, would the investor be willing to pay a higher price or lower price for the gamble? How does your answer relate to the investor's level of risk aversion? You may use MS Excel. You may copy and paste a table from MS Excel as part of your working. (d) Explain the St Petersburg Paradox and how the Paradox can be resolved with the consideration of utility functions such as: u(w) = ln (w)
