Homework 1, Due 2/8/06:
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IN THIS AND ALL FUTURE HOMEWORKS, MAKE SURE THE NUMERICAL
ANSWERS YOU GIVE ARE ACCURATE TO AT LEAST 3 SIGNIFICANT 
FIGURES OR 1-PENNY ACCURACY, WHICHEVER IS FINER.

1. Assume that a population's survival density function is 
given by f(x) = C*exp(-.02 *x)  for  0 <= x < 90, and by 0 
for x > 90, where  C is a constant.
  (a) Find  C .
  (b) Find the probability that a member of the population 
dies between his 40th and 50th birthdays (ie between exact 
ages 40 and 50).
  (c) Find the expected (exact) age at death in this population.

EXTRA CREDIT (d)* Suppose that only the number of the last 
birthday is recorded on each population member's birth 
certificate. What is the expected value of this number for 
a newborn member of this population ?

2. In a certain population, you are given the following facts.
(i) The probability of surviving from 20 to 40 is 0.8 .
(ii) The probability of surviving from 40 to 60 is 0.7 .
(iii) The probability of surviving from 20 to 30 is 0.95 .

(a) Find the probability that a life aged 30 in this population
survives to age 60.
(b) Suppose that two members of this population, aged 20, have 
lifetimes that can be assumed independent. What is the 
probability that at least one survives to age 60 ? What is the
probability both that one dies before 40 and the other 
between 30 and 60 ?


3. Use the illustrative life table (Table 1.1) to calculate (or 
more precisely, to estimate) the probabilities of the following 
event which refer to exact ages at death, for individuals on 
their birthdays:
(a) that a life aged 23 will live at least 27 more years;
(b) that a life aged 25 will die between exact ages 46 and 52;
(c) that a life aged 30 will either die before 40 or live 
past 70.


   Use a calculator or computing platform to answer the 
following problems numerically.

4. (a) $200 deposited 15 years ago has grown at a constant 
interest rate, compounded every 6 months, to $350. What 
were the nominal and effective annualized interest rates ?
   (b) At what annual effective interest rate does your 
money exactly double in 7 years ? 
   (c) At what annual effective interest rate does your 
money exactly triple in 10 years ? 


5. Suppose you sell for $5,000 the right to receive for 
10 years the amount of $1,000 per year payable quarterly, 
with first payment to be in 3 months. What effective rate 
of interest (assumed constant over the next 10 years) would 
make this a fair sale price ?
   

6. (a) How much (a constant amount $x) should be deposited 
in a bank account at the beginning of each year for 5 years 
to amount to $12,000 at the end of the 5th year, at the 
constant annual interest rate of 5% ?
   (b) Suppose the amount you were to deposit was twice as 
large (2x) in years 3, 4, and 5 as in the 1st and 2nd year.
In this case, what is the amount x if the accumulated 
value of the account at 5% is to be 12,000 at the end of 
the 5th year ?