Imagine getting a bill for $$125$ billion that you did not know you owed! The case involves a real$-$world situation. That happened to the residents, called $$burghers$,$ of Ticino, Switzerland. The New York Supreme Court in Brooklyn, New York, ordered Ticino to pay a group of American investors. The investors had sued in the Brooklyn court over a loss they claimed concerning the failure $27$ years earlier of InterChange Bank, a tiny bank in Ticino. The burghers had known about the suit but thought the matter was trivial and were naturally stunned by the bill. Their lead lawyer quipped that $$if the judgment were upheld by the higher courts, all of Ticino$$s citizens would have to spend the rest of their lives flipping real burgers $($the kind you eat$)$ at McDonald$$s and Burger King to pay off the debt.$$

The root of Ticino$$s problem was a deposit made $28$ years ago, one year before the bank failed. The estate of one Sterling Granville Higgins deposited $$600$ million of options on Venezuelan oil and mineral deposits in the InterChange Bank. The bank agreed to pay an interest rate of $1\%$ per week. $($No wonder the bank failed the next year!$)$ The Brooklyn court ruled that Ticino had to pay $1\%$ interest per week compounded weekly for the seven years between the date of deposit and the date Ticino had the bank liquidated and interest at the rate of $8\mathrm{.}54\%$ APY for the remaining $21$ years since the bank$$s liquidation.

Case Questions:

$1.$ The $$125$ billion reported in the press was rounded, but the original amount was precisely $$600$ million. Assuming the time periods are exactly seven years at a periodic interest rate of $1\%$ per week and exactly $21$ years at $8\mathrm{.}54\%$ APY, how much was the bill to the nearest dollar?

Correct Answer: About $$125\mathrm{.}46$ billion, how do you get this answer?

Hint: You have an initial amount today that is first compounded at a weekly rate to find the amount in seven years, then the amount in seven years is compounded at an annual rate for another $21$ years. Problem B$6$ in the end$-$of$-$chapter problems shows you how to find the future value of a lump sum with an interest rate other than annual. For the first seven year calculation, you can either use the weekly rate and enter the number of periods in weeks to finds the future value, or alternatively, convert the periodic rate to an APY, and use the APY for the seven years.