Deferred Annuities Certain, Study notes of Financial Accounting

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Deferred Annuities Certain

General terminology

• (^) A deferred annuity is an annuity whose first payment takes place at some predetermined time k + 1
• (^) k|na... 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 =^ ak+n −^ ak

• (^) 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...

General terminology

• (^) A deferred annuity is an annuity whose first payment takes place at some predetermined time k + 1
• (^) k|na... 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 =^ ak+n −^ ak

• (^) 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...

General terminology

• (^) A deferred annuity is an annuity whose first payment takes place at some predetermined time k + 1
• (^) k|na... 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 =^ ak+n −^ ak

• (^) 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...

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 31st^ , 2020 , is

s12 0. 05

During the following four years this value will grow to

(1 + 0.05)^4 · s12 0. 05

Finally, recall that each level payment equals $2,000. So, the accumulated value we seek is

2000 · (1 + 0.05)^4 · s12 0. 05 = 2000 · (1 + 0.05)^4 ·

(1 + 0.05)^12 − 1

• (^) Assignment: For a similar story, see Example 3.5.

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 31st^ , 2020 , is

s12 0. 05

During the following four years this value will grow to

(1 + 0.05)^4 · s12 0. 05

Finally, recall that each level payment equals $2,000. So, the accumulated value we seek is

2000 · (1 + 0.05)^4 · s12 0. 05 = 2000 · (1 + 0.05)^4 ·

(1 + 0.05)^12 − 1

• (^) Assignment: For a similar story, see Example 3.5.

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 31st^ , 2020 , is

s12 0. 05

During the following four years this value will grow to

(1 + 0.05)^4 · s12 0. 05

Finally, recall that each level payment equals $2,000. So, the accumulated value we seek is

2000 · (1 + 0.05)^4 · s12 0. 05 = 2000 · (1 + 0.05)^4 ·

(1 + 0.05)^12 − 1

• (^) Assignment: For a similar story, see Example 3.5.

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 31st^ , 2020 , is

s12 0. 05

During the following four years this value will grow to

(1 + 0.05)^4 · s12 0. 05

Finally, recall that each level payment equals $2,000. So, the accumulated value we seek is

2000 · (1 + 0.05)^4 · s12 0. 05 = 2000 · (1 + 0.05)^4 ·

(1 + 0.05)^12 − 1

• (^) Assignment: For a similar story, see Example 3.5.

An Example:

Present value of a deferred annuity -

The value before the term of the annuity

• Today is January 1st^ , 2010. 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 31st^ , 2016. Assume that the rate of interest is equal to i(4)^ = 0.08. Find the present value of the annuity.

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 = 0. 02. The value on January 1 st^ , 2011 of a basic annuity-immediate corresponding to the one in the example is

a24 0. 02 = 1 − v 24 i

So, the present value of a basic annuity-immediate is ( 1

1. 02

a24 0. 02 = 17. 4735

Finally, the present value of our level annuity-immediate is

1000 ·

· a24 0. 02 = 17, 473. 5

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 = 0. 02. The value on January 1 st^ , 2011 of a basic annuity-immediate corresponding to the one in the example is

a24 0. 02 = 1 − v 24 i

So, the present value of a basic annuity-immediate is ( 1

1. 02

a24 0. 02 = 17. 4735

Finally, the present value of our level annuity-immediate is

1000 ·

· a24 0. 02 = 17, 473. 5

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 = 0. 02. The value on January 1 st^ , 2011 of a basic annuity-immediate corresponding to the one in the example is

a24 0. 02 = 1 − v 24 i

So, the present value of a basic annuity-immediate is ( 1

1. 02

a24 0. 02 = 17. 4735

Finally, the present value of our level annuity-immediate is

1000 ·

· a24 0. 02 = 17, 473. 5

Assignment

• Examples 3.5.3,
• Problems 3.5.1,