Chapter 10 Section 2

Annuities

Definitions

• Annuity – A sequence of equal ‘payments’ made at regular intervals of time.

• Rent – The amount of each equal ‘payment’ made at each compounding period.

– Note that Rent can be deposits or withdrawals

Diagram for an Increasing Annuity

…

B1 B2 B3 B4B0

• B = Balance • R = Rent• R is the regular deposits at the end of each compound period. • Each tick mark represents a compound period.• Notice that the initial value is 0. This is important!

Balances:

Interest:

Depositsor

Withdraws

i ·B0 i ·B1 i ·B2 i ·B3

0 R R R R

Diagram for a Decreasing Annuity

…

B1 B2 B3 Bn-1B0

• B = Balance • R = Rent (which is the regular withdrawals at the end of each compound period. • Each tick mark represents a compound period.)• It is assumed that when the very last withdraw is made (at compound period n), the balance in the account is 0, unless stated otherwise

Balances:

Interest:

Depositsor

Withdraws

i·B0 i·B1 i·B2 i·Bn-2

P – R – R – R – R

…

…

…

Bn

i·Bn-1

– R

Increasing vs. Decreasing Annuities

• Increasing annuities:

– Start an account.

– At the end of each compounding period, you deposit the rent into the account.

• Decreasing annuities (Lotto Scenario)

– Start off with money in the account.

– At the end of each compounding period, you withdraw the rent from the account (and in most cases, until you run out of money).

Notes

• For increasing annuities, we treat the initial deposit/balance to be $0 (unless stated otherwise).

• For decreasing annuities, it is assumed that when the last withdrawal is made, then there is no more money in the account (unless stated otherwise).

New Balance for an Increasing Annuity

At any time:

Bnew = Bprevious + i·Bprevious + R

which simplifies to

Bnew = ( 1 + i )Bprevious + R (Note that this is in the form

of a difference equation)

Note that i·Bprevious represents the interest earned at the end of the compound period.

New Balance for a Decreasing Annuity

At any time:

Bnew = Bprevious + i·Bprevious – R

which simplifies to

Bnew = ( 1 + i )Bprevious – R (Note that this is in the form

of a difference equation)

Note that i·Bprevious represents the interest earned at the end of the compound period.

TVM Solver

• Recall :

PMT = Payment per compounding period

so

PMT = R (i.e. Rent)

• Remember : When using the TVM Solver;

– When you deposit rent into the account, you have a NEGATIVE cash flow. (Increasing Annuities).

– When you withdraw rent from the account, you have a POSITIVE cash flow. (Decreasing Annuities).

Exercise 3 (page 487)

• Given:– 6% interest– Interest compounded quarterly – Increasing annuity– For 5 years– Rent = $1,000

• Calculate the Future Value

Exercise 3 Formula Solution (slide 1)

·Ri

(1 + i )n – 1 F =

• i = r/m = 0.06/4 = 0.015 • n = (5)(4) = 20• R =1000• So

The formula:

·10000.015

(1 + 0.015 )20 – 1 F =

Exercise 3 Formula Solution (slide 2)

·10000.015

0.3468550065F =

(23.1236671) ·1000F =

F = 23123.6671

The account will have a balance of $23,123.67 at the end of the 5 years.

Exercise 3 TVM Solver Solution

• TVM Solver:

N = (5)(4) = 20

I% = 6

PV = 0

PMT = –1000

FV = 23123.6671

P/Y = C/Y = 4

The rent is $23,123.67 per quarter-year

Exercise 5 (page 487)

• Given:– 8% interest– Interest compounded quarterly – Decreasing annuity– For 7 years– $100,000

• Calculate the Rent

Exercise 5 Formula Solution (slide 1)

·R1 – (1 + i )– n

P =

• i = r/m = 0.08/4 = 0.02 • n = (7)(4) = 28• P =100000• So

The formula:

·R1 – (1 + 0.02 )–28

100000 =

i

0.02

Exercise 5 Formula Solution (slide 2)

·R0.02

0.4256254471100000 =

R = 4698.967164

The rent will be $4,698.97.

100000 = 21.28127236 ·R

Exercise 5 TVM Solver Solution

• TVM Solver:

N = (7)(4) = 28

I% = 8

PV = – 100000

PMT = 4698.97

FV = 0

P/Y = C/Y = 4

The rent is $4,698.97 per quarter-year