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利息理论（Interest Theory）是精算师们的必修课之一，而年金（annuity）作为利息理论中的主要研究对象，掌握程度好坏直接关系到金融数学（Financial Mathematics）这一门考试的通过与否。本期题目的主题就是年金，题目不算特别难，但需要大家细心和耐心才能得出最后的正确答案。
1、1000 is deposited into Fund X, which earns an annual effective rate of 6%. At the end of each year, the interest earned plus an additional 100 is withdrawn from the fund. At the end of the tenth year, the fund is depleted. The annual withdrawals of interest and principal are deposited into Fund Y, which earns an annual effective rate of 9%.
Determine the accumulated value of Fund Y at the end of year 10.
2、Victor invests 300 into a bank account at the beginning of each year for 20 years. The account pays out interest at the end of every year at annual effective interest rate of i%. The interest is reinvested at an annual effective rate of (i/2)%. The yield rate on the entire investment over the 20 year period is 8% annual effective.
Determine i.
3、Sally lends 10,000 to Tim. Tim agrees to pay back the loan over 5 years with monthly payments payable at the end of each month. Sally can reinvest the monthly payments from Tim in a savings account paying interest at 6%, compounded monthly. The yield rate earned on Sally's investment over the five-year period turned out to be 7.45% compounded semiannually.
What nominal rate of interest, compounded monthly, did Sally charge Tim on the loan?
4、Bill purchases an annuity at a price of 10,000. The annuity makes payments of 500 at the beginning of every 6 months for 20 years. The payments are reinvested in a fund which earns interest at an annual effective rate i. Interest payments are received every 6 months and reinvested at a nominal rate of 6% convertible semiannually. Bill realizes an overall effective annual yield of 7% on his original investment over the 20-year period.
Determine i.
5、A small business takes out a loan of 12,000 at a nominal rate of 12%, compounded quarterly, to help finance its start-up costs. Payments of 750 are made at the end of every 6 months for as long as is necessary to pay back the loan. Three months before the 9th payments is due, the company refinances the loan at a nominal rate of 9%, compounded monthly. Under the refinanced loan, payments of R are to be made monthly, with the first monthly payment to be made at the same time that the 9th payment under the old loan was to be made. A total of 30 monthly payments will completely pay off the loan.
Determine R.
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