1. An account pays an annual rate of 8% percent compounded monthly. What lump sum must you deposit into the account now so that in 10 years you can begin to withdraw $4000 each month for the next 20 years, drawing down the account to zero?
2. Suppose you take out a 20-year, $300,000 mortgage at 7% and decide after 15 years to pay off the mortgage. How much will you have to pay?
3. You can retire a loan either by paying off the entire amount $8000 now, or by paying $6000 now, $2000 at the end of 5 years, and an additional $2000 at the end of 10 years. Find a cutoff value r0 such that if the nominal rate r is < r0, then you should pay off the entire loan now, but if r > r0, then it is preferable to wait. Assume that interest is compounded continuously.