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1 Deferred Annuities Certain
2 General terminology A deferred annuity is an annuity whose first payment takes place at some predetermined time k + 1 k n a... the present value of a basic deferred annuity-immediate with term equal to n and the deferral period k; it can be readily expressed as k na = v k a n = a k+n a k It makes sense to ask for the value of a deferred annuity at any time before the beginning of payments and also after the term of the annuity is completed; here we mean more than one period before and more than one period after since these two cases are easily reduced to annuities immediate and annuities due It will be clear what we mean after some examples...
3 General terminology A deferred annuity is an annuity whose first payment takes place at some predetermined time k + 1 k n a... the present value of a basic deferred annuity-immediate with term equal to n and the deferral period k; it can be readily expressed as k na = v k a n = a k+n a k It makes sense to ask for the value of a deferred annuity at any time before the beginning of payments and also after the term of the annuity is completed; here we mean more than one period before and more than one period after since these two cases are easily reduced to annuities immediate and annuities due It will be clear what we mean after some examples...
4 General terminology A deferred annuity is an annuity whose first payment takes place at some predetermined time k + 1 k n a... the present value of a basic deferred annuity-immediate with term equal to n and the deferral period k; it can be readily expressed as k na = v k a n = a k+n a k It makes sense to ask for the value of a deferred annuity at any time before the beginning of payments and also after the term of the annuity is completed; here we mean more than one period before and more than one period after since these two cases are easily reduced to annuities immediate and annuities due It will be clear what we mean after some examples...
5 General terminology A deferred annuity is an annuity whose first payment takes place at some predetermined time k + 1 k n a... the present value of a basic deferred annuity-immediate with term equal to n and the deferral period k; it can be readily expressed as k na = v k a n = a k+n a k It makes sense to ask for the value of a deferred annuity at any time before the beginning of payments and also after the term of the annuity is completed; here we mean more than one period before and more than one period after since these two cases are easily reduced to annuities immediate and annuities due It will be clear what we mean after some examples...
6 An Example: Accumulated value after the last payment date On January 1 st, 2009, you open an investment account. If an annuity such that twelve annual payments equal to $2, 000 are made starting December 31 st, 2009 is going to be credited to the account, find the account balance on December 31 st Assume that i = 0.05.
7 An Example: Accumulated value after the last payment date (cont d) The payments are level, so let us start by considering a basic deferred annuity-immediate. The accumulated value of the 12 year long annuity-immediate at the time of the last payment, i.e., on December 31 st, 2020, is s During the following four years this value will grow to ( ) 4 s Finally, recall that each level payment equals $2,000. So, the accumulated value we seek is 2000 ( ) 4 s = 2000 ( ) 4 ( ) = 38, Assignment: For a similar story, see Example 3.5.2
8 An Example: Accumulated value after the last payment date (cont d) The payments are level, so let us start by considering a basic deferred annuity-immediate. The accumulated value of the 12 year long annuity-immediate at the time of the last payment, i.e., on December 31 st, 2020, is s During the following four years this value will grow to ( ) 4 s Finally, recall that each level payment equals $2,000. So, the accumulated value we seek is 2000 ( ) 4 s = 2000 ( ) 4 ( ) = 38, Assignment: For a similar story, see Example 3.5.2
9 An Example: Accumulated value after the last payment date (cont d) The payments are level, so let us start by considering a basic deferred annuity-immediate. The accumulated value of the 12 year long annuity-immediate at the time of the last payment, i.e., on December 31 st, 2020, is s During the following four years this value will grow to ( ) 4 s Finally, recall that each level payment equals $2,000. So, the accumulated value we seek is 2000 ( ) 4 s = 2000 ( ) 4 ( ) = 38, Assignment: For a similar story, see Example 3.5.2
10 An Example: Accumulated value after the last payment date (cont d) The payments are level, so let us start by considering a basic deferred annuity-immediate. The accumulated value of the 12 year long annuity-immediate at the time of the last payment, i.e., on December 31 st, 2020, is s During the following four years this value will grow to ( ) 4 s Finally, recall that each level payment equals $2,000. So, the accumulated value we seek is 2000 ( ) 4 s = 2000 ( ) 4 ( ) = 38, Assignment: For a similar story, see Example 3.5.2
11 An Example: Accumulated value after the last payment date (cont d) The payments are level, so let us start by considering a basic deferred annuity-immediate. The accumulated value of the 12 year long annuity-immediate at the time of the last payment, i.e., on December 31 st, 2020, is s During the following four years this value will grow to ( ) 4 s Finally, recall that each level payment equals $2,000. So, the accumulated value we seek is 2000 ( ) 4 s = 2000 ( ) 4 ( ) = 38, Assignment: For a similar story, see Example 3.5.2
12 An Example: Present value of a deferred annuity - The value before the term of the annuity Today is January 1 st, An annuity-immediate pays $1,000 at the end of every quarter. The first payment is scheduled for March 31 st, 2011 and the last payment for December 31 st, Assume that the rate of interest is equal to i (4) = Find the present value of the annuity.
13 An Example: Present value of a deferred annuity - The value before the term of the annuity Today is January 1 st, An annuity-immediate pays $1,000 at the end of every quarter. The first payment is scheduled for March 31 st, 2011 and the last payment for December 31 st, Assume that the rate of interest is equal to i (4) = Find the present value of the annuity.
14 An Example: Present value of a deferred annuity - The value before the term of the annuity (cont d) It is more convenient to be thinking in terms of quarter-years. The interest rate per quarter is j = i (4) /4 = The value on January 1 st, 2011 of a basic annuity-immediate corresponding to the one in the example is a = 1 v 24 = i So, the present value of a basic annuity-immediate is ( ) 4 1 a = Finally, the present value of our level annuity-immediate is ( ) a = 17, 473.5
15 An Example: Present value of a deferred annuity - The value before the term of the annuity (cont d) It is more convenient to be thinking in terms of quarter-years. The interest rate per quarter is j = i (4) /4 = The value on January 1 st, 2011 of a basic annuity-immediate corresponding to the one in the example is a = 1 v 24 = i So, the present value of a basic annuity-immediate is ( ) 4 1 a = Finally, the present value of our level annuity-immediate is ( ) a = 17, 473.5
16 An Example: Present value of a deferred annuity - The value before the term of the annuity (cont d) It is more convenient to be thinking in terms of quarter-years. The interest rate per quarter is j = i (4) /4 = The value on January 1 st, 2011 of a basic annuity-immediate corresponding to the one in the example is a = 1 v 24 = i So, the present value of a basic annuity-immediate is ( ) 4 1 a = Finally, the present value of our level annuity-immediate is ( ) a = 17, 473.5
17 An Example: Present value of a deferred annuity - The value before the term of the annuity (cont d) It is more convenient to be thinking in terms of quarter-years. The interest rate per quarter is j = i (4) /4 = The value on January 1 st, 2011 of a basic annuity-immediate corresponding to the one in the example is a = 1 v 24 = i So, the present value of a basic annuity-immediate is ( ) 4 1 a = Finally, the present value of our level annuity-immediate is ( ) a = 17, 473.5
18 Assignment Examples 3.5.3,4 Problems 3.5.1,2
19 Assignment Examples 3.5.3,4 Problems 3.5.1,2
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