HW3 Problems, due Wednesday 3/8/06
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1. Find the present value at interest rate i in terms 
of actuarial annuity-due or annuity-immediate notations 
of a payment stream of  $100 at the end of 1 years, 
4 years, 7 years, ..., with the last payment at the 
31th year. Your answer may also include a discounting 
factor v^t , but should not contain any summations or 
changed interest rates.

2. (a) Suppose that you borrow $20,000 for 10 years at 
5% APR, to be repaid in level payments every 3 months 
(4 evenly spaced payments yearly), with the first payment 
to begin exactly 3 months after the loan is taken out.
Find the amount of each payment.
   (b) Same question if the payments are to be made every 
three months beginning exactly 2 years from the date of 
the loan, with last payment to be made at the 10-year mark.
   (c) Same question as (b) if the 2 payments at exactly
5 and 6 years from the beginning of the loan are to be 
omitted. (So you are being asked for the level payment P
such that the payments actually made have present value at 
time 0 equal to the initial cash value of the loan.)

3. Suppose that a homeowner borrows $200,000 from a bank 
at 6% APR, to be repaid in 15 years with monthly payments, 
where the first payment is to be made 1 month after taking 
out the loan
(a). Find the amount of the level payment and the amount 
of interest paid during the second year.
(b). Find the balance owed on the loan after exactly 7 years.
(c). Suppose that the terms of the loan could have been 
altered at time 0 so that the amount borrowed is changed 
to $205,000, in return for lowering the interest rate from 
6% to 5.25%. Would that have been advantageous to the homeowner ?

4. (a) Find the constant C and the force of mortality 
  associated with the lifetime density
  f(t) = C * (t-20) * exp(-.05*(t-20))      for 20 < t 
  (with f(t) = 0 for t < 20).
   (b) What is the expected lifetime for an individual 
  governed by this lifetime density who is known to be 
  alive at T=40 ?

5. Do problems # 2 and 3 at the end of Chapter 2. 
   (Each counts as one full problem.)