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					Annuities: Math 360 Chapter 2




        Lecture 2

          Annuities
               Introduction
• An annuity is a series of periodic
  payments
• For us, the payments are contingent only
  on the passage of time, not on certain
  events (i.e. all annuities are annuity certain)
• We will need a simple, but very useful,
  result from algebra:
                Geometric Series
1  x  x  .... x  X
           2       n


and      so
                           
(1  x) X  1  x  1  x  x  .... x     2              n
                                                                
                               n
                                     
 1  x  x  .... x  x 1  x  x  .... x
                2                                       2               n
                                                                            
 1  x  x    2
                     .... x   n
                                      x  x   2
                                                      .... x  x  n   n 1
                                                                                
         n 1
 1 x
                                      n 1
                 1 x
 X  k  0 x 
                n      k

                  1 x
               Example 2.1
• The federal gov’t sends Smith a family
  allowance payment of 30 every month for
  Smith’s child. Smith deposits the payments in a
  bank account on the last day of each month. The
  account earns interest at the annual rate of 9%
  compounded monthly and payable on the last
  day of each month, on the minimum monthly
  balance. If the first payment is deposited on May
  31, 1998, what is the account balance on
  December 31, 2009, including the payment just
  made?
                       Ex 2.1 Solution
140 total       deposits
first deposit  1
imonthly  i    .09     .0075
        12   12
value at time of            last   deposit  30(1.0075)139
TotalValue  30  30(1.0075)  30(1.0075)  ... 30(1.0075)
                                           2                 139


     1  1.0075140 
     1  1.0075   7385.91
 30               
                   
     Level Payment Annuities
• Number of payments in series of
  payments is called the term of the annuity
• Time between the successive payments is
  called payment period, or frequency
• A series of payments whose value is found
  at the time of the final payment is known
  as an accumulated annuity immediate
                  More Notation

                        1  (1  i )
                                  n
                                       (1  i )  1
                                             n
sn|i  k 0 (1  i ) 
           n 1       k
                                     
                        1  (1  i )         i
or
(1  i )  1  i  sn|i
       n
             Example 2.2
• What level amount must be deposited on
  May 1 and Nov 1 each year from 1998 to
  2005, inclusive, to accumulate to 7000 on
  November 1, 2005 if the nominal annual
  rate of interest compounded semi-annually
  is 9% ?
                    Ex 2.2 Solution
16 total deposits
i   ( 2)
            0.09
X  level amount                deposited / year
EoV :
      
X (1.045)  (1.045)  ..  (1.045)  1 
                15         14
                                              
X  s16|0.045  22.719337 X  7000
 X  308.11
                Example 2.3
• Suppose that in Example 2.1, Smith’s child is
  born in April 1998 and the first payment is
  received in May (and deposited at the end of
  May.) The payments continue and the deposits
  are made at the end of each month until (and
  including the month of) the child’s 16th birthday.
  The payments stop after the 16th birthday, but
  the balance continues to accumulate with
  interest until the end of the month of the child’s
  21st birthday. What is the balance in the account
  at that time?
           Ex 2.3 solution



X  (1.0075)  30  s192|.0075  12792.31
             60
               Some Arithmetic

Value@ time _ n  growth                from n  n  k
                      (1  i ) n  1
 sn|i  (1  i )                    1  i 
                    k                          k

                             i
  (1  i ) n  k  (1  i ) k (1  i ) n  k  1 (1  i ) k  1
                                               
                i                        i             i
 sn  k |i  sk |i
or
sn  k |i  sn|i  sk |i (1  i ) n
             Example 2.4
• Suppose that in Example 2.1, the nominal
  annual interest rate earned on the account
  changes to 7.5 %, still compounded
  monthly, as of January 1 2004. What is
  the accumulated value of the account on
  December 31 2009?
             Ex 2.4 Solution

             
Value  30  s68|.0075  (1.00625)  s72|.00625
                                  72
                                                  
               Example 2.5
• Suppose 10 monthly payments of 50 each are
  followed by 14 monthly payments of 75 each. If
  interest is at an effective monthly rate of 1%,
  what is the accumulated value of the series at
  the time of the final payment?
• Answer:

 Value  50s24|.01  25s14|.01  1722.36
      Present Value of an Annuity
              Immediate
• Make a lump sum payment X now to receive
  periodic payments C , starting one period from
  today. If the market bears a constant interest of i,
  then Present Value of this Annuity Immediate is
  calculated as
      C         C              C
X                   ...
    1  i (1  i ) 2
                            (1  i ) n
                                                         1  n
       n
                       
 C k 1 k  C   1    ..   n 1       C  
                                                         1 
   1  n
C         Can|i
      i
    Loan Repayment – Ex 2.7
• Brown has bought a new car and requires a loan
  of 12000 to pay for it. The car dealer offers
  Brown two alternatives on the loan:
• A.) Monthly payments for 3 years, starting one
  month after purchase, with an annual interest
  rate of 12% compounded monthly, or
• B.) Monthly payments for 4 years, also starting
  one month after purchase, with annual interest
  rate 15% compounded monthly.
• Find Brown’s monthly payment and the total
  amount paid over the course of the repayment
  period under each of the two options
         Ex 2.17 Solution

a.)12000  P  a36|.01  P  398.57
            1             1

b.)12000  P2  a48|.0125  P2  333.97
TotalValue( A)  36  P  14348.52
                       1

TotalValue( B)  48  P2  16030.56
Present Value of an Annuity Some
  Time Before Payments Begin
• Example 2.8:
• Suppose that in Example 2.7 Brown can repay
  the loan, still with 36 payments under option (a)
  or 48 payments under option (b), with the first
  payment made 9 months after the car is
  purchased in either case. Assuming interest
  accrues from the time of the car purchase, find
  the payments required under options (a) and (b).
• This is known as a deferred annuity.
                  Ex 2.8 Solution

         1
          '
                 9
12000  P  1.01  1.01  10             44
                                ..  1.01       P 
                                                   1
                                                    '
                                                          1
                                                           8
                                                                a36|.01
so
       12000
P 
  '
        8
                    431.60
    1.01  a36|.01
 1


        12000
P 
  '
          8
                        368.86
    1.0125  a48|.0125
 2
Duality of Total Value an Present
       Value for Annuities


  sn|i  1  i  an|i
                    n



  an|i   sn|i
              n
                Perpetuities
• What happens as the term approaches infinity?

                               1 n
                                      1
     a|i    lim an|i  lim        
               n        n     i   i
                 Ex. 2.10
• A perpetuity immediate pays X per year.
  Brian receives the first n payments,
  Colleen receives the next n payments, and
  Jeff receives the remaining payments.
  Brian’s share of the present value of the
  original perpetuity is 40%, and Jeff’s share
  is K. Calculate K.
               Ex. 2.10 Solution
                              1  n 
PV ( Brian)  X  an|i  X  
                              i       0.4 X  a|i
                                     
       X
 0.4   1  n  0.4
       i
                                           X           X
PV (Colleen)    X  an|i  0.4  0.6   0.24 
                  n

                                           i           i
X
    PV  PV ( Brian)  PV ( Brian)  PV ( Jeff )
 i
                                         X          X
 K  PV ( Jeff )  (1  0.4  0.24)  0.36
                                         i           i
             Annuity Immediate
                                      n 1     (1  i )  1
                                                      n
S _  1  (1  i )  ...  (1  i )          
  n|i                                                i
                        1 v           n
a _  v  v  ...  v 
             2           n
 n|i                      i
S _  Payment Accumulation at Final Time
 n|i

a _  Present Value one period before
 n|i

        first payment
                                    Annuity Due


S n|i  (1  i )  ... (1  i ) n  (1  i ) S n|i
     _                                           _



           value 1 period after final payment

a n|i  (1  i )a _  Present Value at time of first payment
     _

                        n|i

0          1        2         ...      n -1           n
                                                           
                                                            
                _                                         _        _
a_         a n|i                                      S n|i   S n|i
    n|i
Level Payment Annuities-Some
       Generalizations
• What happens when compounding interest
  period and the annuity payment period do
  not coincide ?
                  Ex 2.11
• Smith deposits quarterly of 1000 each
  over 16 years. Find balance after last
  payment.
  a) Interest quoted at 9% nominal annual
  rate compounded monthly
  b) Interest quoted at effective annual rate 10%
  Compounded interest paid for fraction of years.
               Ex 2.11 Soln
64 total deposit. Suppose effective quarter-
  rate is j.                  (1  j )  1
                                       64
                   _
value = 1000 S 64| j =1000           j
a)               0.09 3
    1  j  (1      )  (1  0.0075)     3

                  12
    so j  0.02266917.
b)
 1  j  (1  0.01)1/ 4 so j  0.02411369 .

				
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