1. (30 Pts) Frontier Airlines signed a contract with BP to hedged the cost of jet fuel by purchasing options that allowed the airline to purchase fuel at a fixed price for 2 years. The savings in fuel costs were 100,000 in quarter 1, increasing by 1% per quarter through the 2-year option period. In order to keep this contract active Frontier Airlines should pay 30,000 per month. What is the present worth of the contract at an interest rate of 16% persemiannual compounded quarterly?

To find the present worth of the contract, we need to calculate the present value of the savings in fuel costs and the present value of the monthly payments that Frontier Airlines needs to make to keep the contract active.<br /><br />1. Present value of the savings in fuel costs:<br />The savings in fuel costs are increasing by 1% per quarter, so we can represent them as a geometric series. The first term of the series is $100,000, and the common ratio is 1.01 (since the savings increase by 1% each quarter).<br /><br />The present value of the geometric series can be calculated using the formula:<br /><br />PV = a * (1 - r) / (1 - r)<br /><br />where PV is the present value, a is the first term, and r is the common ratio.<br /><br />PV = $100,000 * (1 - 1.01) / (1 - 1.01) = $100,000<br /><br />2. Present value of the monthly payments:<br />Frontier Airlines needs to pay $30,000 per month to keep the contract active. We need to find the present value of these payments, which can be represented as an annuity.<br /><br />The present value of an annuity can be calculated using the formula:<br /><br />PV = P * (1 - (1 + r)^(-n)) / r<br /><br />where PV is the present value, P is the payment amount, r is the interest rate per period, and n is the number of periods.<br /><br />In this case, the interest rate is 16% per semiannual, compounded quarterly. So, the interest rate per period is 16% / 2 = 8% per quarter.<br /><br />PV = $30,000 * (1 - (1 + 0.08)^(-24)) / 0.08 = $571,429.79<br /><br />Now, we can calculate the present worth of the contract by adding the present value of the savings in fuel costs and the present value of the monthly payments:<br /><br />Present worth of the contract = PV of savings in fuel costs + PV of monthly payments<br />Present worth of the contract = $100,000 + $571,429.79 = $671,429.79<br /><br />Therefore, the present worth of the contract at an interest rate of 16% per semiannual compounded quarterly is $671,429.79.