Problem 1. In North America, it is common to issue investment products with return guarantees. For instance, a product may offer the returns of the S&P 500 index except the investor is guaranteed a 0% return over the investment horizon and the maximum total return the investor will be credited over the investment horizon is 30%. You are the pricing actuary for a large insurance company. Your company has decided to sell an investment product for which the investments will be credited a return equal to that on an index that will experience either a 20% gain or a 10% loss at the end of one period (i.e. the single-period binomial model). The continuously compounded interest-rate (force of interest) for the period is 0.08. The customer is guaranteed a 5% return and you are to set the maximum total return the customer will receive over the period (denoted a) so that your insurance company will break even on the product. Determine a.