1. Consider the following retirement plan. An initial payment of $50, with subsequent payments growing geometrically at a rate of 1.2%, are made for a total of 12 years. The first payment is made 7 years from today. The annual effective rate of interest is 6.3% throughout. How much must be deposited today to achieve this?

2. A perpetuity-immediate pays $100 per year. Immediately after the fifth payment, the perpetuity is exchanged for a 25-year annuity-immediate that will pay X at the end of the first year. Each subsequent annual payment will be 8% greater than the preceding payment. Immediately after the 10th payment of the 25-year annuity, the annuity will be exchanged for a perpetuity-immediate paying Y per year. The annual effective rate of interest is 8% throughout. Calculate Y.

3. Annual end-of-year deposits are made to a fund paying an annual effective rate of interest of 6%. The first deposit is $1,000 an then they go up by 3% annually. What is the accumulated value of the account immediately after the 30th deposit.

4. Consider an annuity-immediate with monthly payments for twenty years. The payments are level in the course of each year, then increase by 2% for the next year. Find the present value of this annuity if the initial payment is $1,000 and the annual effective interest rate is i = 0.04.