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Section 3.4. Evaluating annuities thus far has always been done at the beginning of the term or at the end of the term. We shall now consider evaluating the (1) (2) (3). present value of an annuity more than one period before the first payment date,.

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Present value of an annuity more than one period before the first payment date

Section 3.4

Evaluating annuities thus far has always been done at the beginning of the term or at the end of the term. We shall now consider evaluating the

(1)

(2)

(3)

present value of an annuity more than one period before the first payment date,

accumulated value of an annuity more than one period after the last payment date,

current value of an annuity between the first and last payment dates.


Present value of an annuity more than one period before the first payment date

Payments

1

1

1

1

Periods

0

1

2

n– 1

n

m periods below 0

The present value of an n-period annuity m + 1 periods before the first payment date (called a deferred annuity) is the present value at time 0 discounted for m time periods, that is,

1 – vn

vm —— =

i

vm– vm+n

———— =

i

1 – vm+n– (1 – vm)

——————— =

i

vm =

a –

n|

a ––––

m+n|

a –

m|

Exactly 3 years from now is the first of four $200 yearly payments for an annuity, with an effective 8% rate of interest. Find the present value of the annuity.

200 =

200(4.6229 – 1.7833) =

$567.92

a ––

6 | 0.08

a ––

2 | 0.08


Present value of an annuity more than one period before the first payment date

Payments

1

1

1

1

Periods

0

1

2

n– 1

n

m periods above n

The accumulated value of an n-period annuity m periods after the last payment date is the accumulated value at time n accumulated for m time periods, that is,

(1 + i)n– 1

————–(1 + i)m =

i

(1 + i)m =

s –

n|

(1 + i)m+n– (1 + i)m

———————– =

i

(1 + i)m+n– 1 – [(1 + i)m–1]

——————————— =

i

s ––––

m+n|

s –

m|

For four years, an annuity pays $200 at the end of each year with an effective 8% rate of interest. Find the accumulated value of the annuity 3 years after the last payment.

200 =

200(8.9228 – 3.2464) =

$1135.28

s ––

7 | 0.08

s ––

3 | 0.08


Present value of an annuity more than one period before the first payment date

Payments

1

1

1

1

1

Periods

0

1

2

n– 1

n

time for mth payment

The current value of an n-period annuity immediately upon the mth payment date is the present value at time 0 accumulated for m time periods which is equal to the accumulated value at time n discounted for n–m time periods, that is,

(These equations are easy to derive.)

(1 + i)m=

vn–m =

+

a –

n|

s –

n|

s –

m|

a ––––

n–m |

For four years, an annuity pays $200 at the end of each half-year with an 8% rate of interest convertible semiannually. Find the current value of the annuity immediately upon the 5th payment (i.e., middle of year 3).

200 =

+

200(5.4163 + 2.7751) =

$1638.28

s ––

5 | 0.04

a ––

3 | 0.04


Present value of an annuity more than one period before the first payment date

For four years, an annuity pays $200 at the end of each half-year with an 8% rate of interest convertible semiannually. Find the current value of the annuity three months after the 5th payment (i.e., nine months into year 3).

The current value of the annuity immediately upon the 5th payment (i.e., six months into year 3) is

200 =

+

200(5.4163 + 2.7751) =

$1638.28

s ––

5 | 0.04

a ––

3 | 0.04

The current value of the annuity three months after the 5th payment (i.e., nine months into year 3) is

$1638.28(1 + 0.04)1/2 =

$1670.72