1. Chapter 10 Section 2 Annuities

2. 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

3. Diagram for an Increasing Annuity … B1B1 B2B2 B3B3 B4B4 B0B0 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: Deposits or Withdraws i ·B 0 i ·B 1 i ·B 2 i ·B 3 0RRRR

4. Diagram for a Decreasing Annuity … B1B1 B2B2 B3B3 B n-1 B0B0 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: Deposits or Withdraws i·B 0 i·B 1 i·B 2 i·B n-2 P – R … … … BnBn i·B n-1 – R

5. 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).

6. 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).

7. New Balance for an Increasing Annuity At any time: B new = B previous + i·B previous + R which simplifies to B new = ( 1 + i )B previous + R (Note that this is in the form of a difference equation) Note that i·B previous represents the interest earned at the end of the compound period.

8. New Balance for a Decreasing Annuity At any time: B new = B previous + i·B previous – R which simplifies to B new = ( 1 + i )B previous – R (Note that this is in the form of a difference equation) Note that i·B previous represents the interest earned at the end of the compound period.

9. 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).

10. Exercise 3 (page 487) Given: –6% interest –Interest compounded quarterly –Increasing annuity –For 5 years –Rent = $1,000 Calculate the Future Value

11. Exercise 3 Formula Solution (slide 1) ·R·R i (1 + i ) n – 1 F = i = r/m = 0.06/4 = 0.015 n = (5)(4) = 20 R =1000 So The formula: ·1000 0.015 (1 + 0.015 ) 20 – 1 F =

12. Exercise 3 Formula Solution (slide 2) ·1000 0.015 0.3468550065 F = (23.1236671) ·1000F = F = 23123.6671 The account will have a balance of $23,123.67 at the end of the 5 years.

13. 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

14. Exercise 5 (page 487) Given: –8% interest –Interest compounded quarterly –Decreasing annuity –For 7 years –$100,000 Calculate the Rent

15. Exercise 5 Formula Solution (slide 1) ·R·R 1 – (1 + i ) – n P = i = r/m = 0.08/4 = 0.02 n = (7)(4) = 28 P =100000 So The formula: ·R·R 1 – (1 + 0.02 ) –28 100000 = i 0.02

16. Exercise 5 Formula Solution (slide 2) ·R·R 0.02 0.4256254471 100000 = R = 4698.967164 The rent will be $4,698.97. 100000 = 21.28127236 ·R

17. 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