Homework 2, Due 2/20/06:
=======================

YOU MAY TRY AGAIN ON PROBLEMS 4--6 IN LIGHT OF HINTS IN
CLASS  2/20: ANY ADDITIONAL PROBLEMS MUST BE HANBDED IN 
WEDNESDAY 2/22 IN CLASS. 


1. Suppose that you have found an investment which will
earn 5% nominal quarterly-compounded interest for 4 years, 
and two-periods-per-year nominal discounted interest rate 
3% for the next 5 years. What is the effective rate 
(= internal rate of return) of this investment over its 
9-year life ?

2. Suppose that the investment in Problem 1 was completely 
tax-free, and you want to compare it to a taxable bond 
investment which earns 6% effective yield each year but 
on which you must pay 25% tax on the interest you earned
each year (with tax paid in a lump sum at the end of the 
year). What is the internal rate of return of this 
second investment, taking taxes into account ?

NOTE: I DID NOT EXPLICITLY DEFINE IN CLASS THE DIFFERENCE 
BETWEEN A BANK ACCOUNT AND A BOND INVESTMENT: IN A BANK 
ACCOUNT, THE INTERESTYOU EARN STAYS IN THE ACCOUNT AND 
LATER EARNS MORE INTEREST ON ITSELF; WHILE A BOND PAYS 
OUT INTEREST IN CASH DURING THE YEAR AND THAT INTEReST 
IS NOT REINVESTED. In Problem 2, though, we do not say 
what happens to the bond interest payments, so it makes 
most sense to treat all money as being reinvested as 
though the investment were a bank account.


3. For the following small cohort life table with 10 age 
categories, find the probabilities for all values of [T], 
both unconditionally and for lives aged 3, and find the 
expectation of both $1000*(10-[T])  and  $10,000*(1.04)^{-[T]-1}.
These are the expectations  respectively of the cash payout 
on a decreasing-value insurance policy and of the present 
value at time 0 under 4% interest of the cash payout of an 
insurance which pays at the end of the year of death.
    
           x        l_x         d_x
	 ---------------------------
           0       1000         150
           1        850          50
           2        800          40
           3        760          60
           4        700          80
	   5	    620		100
	   6	    520		120
	   7	    400		150
	   8	    250		175
	   9	     75		 75
	  10	      0		  0


4. Suppose that a population of people 
aged 40 and older has survival function
S(t) = P(T >= t) 

     =   1  if  0<t<40

     =  (t-39)^(-3)   if  40 < t <= 90

     =   0  if  90 < t

(a) Find the probability that a 50-year old
in this population dies between 60 and 80.

(b) Find the expected age at death  T  for a 
60-year-old in this population.

(c) Find the expected WHOLE-number age at death [T]
for a 60-year-old in this population.


5. Assume that the instantaneous-compounding 
time-varying interest rate  r(t)  will fluctuate
between 4% and 6% over the next 10 years according 
to the schedule: 

r(t)  =  (5 - (0.2)*t)*.01      for 0 <= t <= 5

      =  (2 + (0.4)*t)*.01      for 5 < t <= 10

(a) Find the internal rate of return of $10000 
deposited in a bank account earning interest at the 
specified instantaneous rate over the next 10 years.

(b) If you deposit $2000 at time 0, $2000 at time 3, 
and $2000 at time 6 into the same bank account earning 
interest at the indicated time-varying rate, then what 
will the accumulated bank balance be at time 10 ? 
(All times are measured in years.) 

(c) Suppose that in the setting of part (a), the bank 
will collect a fee of $400 from your account at the end 
of exactly 5 years. In this case, what is the internal 
rate of return earned over the full 10years by your 
bank deposit ?


6. In continuous time, payment is made into a bank 
account at a time-varying rate per unit time (with t 
measured in years) given by 

D(t) = 3000     for    k   <= t <  k+0.5

     = 2000     for  k+0.5 <  t <= k+1

for all integers k=0,1,...,9. (That is, the payment stream 
goes on continuously for 10 years, and the payment rate 
is 50% higher for the first half of each year than for the 
second half.)

(a) Find the accumulated balance at the end of the 10 years
if interest accrues at the constant effective annual rate 
of  8% .

(b) Find the accumulated balance at the end of the 10 years
if interest accrues at the time-varying instantaneous 
interest rate (= "force of interest") r(t) =  (6 + t/2)%. 

(c) Suppose that $500 is withdrawn from the account at 
exactly the midpoint of each year, but that continuous-time 
deposits at rate D(t) are made as above. Find the 
accumulated balance in the account at the end of the 10 years 
under each of the interest rate assumptions of (a) and (b).