12. Consider an increasingly popular deposit contract with payoffs linked to the performance on the S&P 500 Index on the U.S. stock market. For every dollar invested in the contract, the rate of return in one year is equal to 60% of the realized rate of return of the S&P 500 Index during this year if this rate of return is positive; otherwise, you get your money back. In essence, you are protected from the downside risk of the S&P 500 Index, while you are still able to participate in the upside potential of the stock market. The one year riskless interest rate is 10%. For simplicity, assume that the stocks in the index do not pay dividends. (a) Draw a graph for the payoff one year from now for a one dollar investment in the contract with the horizontal axis being the re- alized rate of return on the S&P 500 Index. Also write down the payoff symbolically. (b) Show that the payoff one year from now for a one dollar investment in this contract is the payoff to a portfolio of a default-free bond and a European call option on the S&P 500 Index. (c) Suppose that the rates of return on the S&P 500 Index can take two possible values one year from now, 20% and -20% with prob- abilities 60% and 40%, respectively. Do you make money or lose money investing in this contract? If so, how much? 12. Consider an increasingly popular deposit contract with payoffs linked to the performance on the S&P 500 Index on the U.S. stock market. For every dollar invested in the contract, the rate of return in one year is equal to 60% of the realized rate of return of the S&P 500 Index during this year if this rate of return is positive; otherwise, you get your money back. In essence, you are protected from the downside risk of the S&P 500 Index, while you are still able to participate in the upside potential of the stock market. The one year riskless interest rate is 10%. For simplicity, assume that the stocks in the index do not pay dividends. (a) Draw a graph for the payoff one year from now for a one dollar investment in the contract with the horizontal axis being the re- alized rate of return on the S&P 500 Index. Also write down the payoff symbolically. (b) Show that the payoff one year from now for a one dollar investment in this contract is the payoff to a portfolio of a default-free bond and a European call option on the S&P 500 Index. (c) Suppose that the rates of return on the S&P 500 Index can take two possible values one year from now, 20% and -20% with prob- abilities 60% and 40%, respectively. Do you make money or lose money investing in this contract? If so, how much