1. Chapter 3 ANNUITIES
Basic Results
Perpetuities
Unknown Time and Unknown Rate of Interest
Continuous Annuities
Varying Annuities

2. 3.2 Basic Results
Definition: Annuity is a series of payments made at regular intervals
Practical applications: loans, mortgages, periodic investments
Basic model: consider an annuity under which payments of 1 are made at the end of each period for n equal periods

3. Series of n payments of 1 unit
Formulas
Series of n payments of 1 unit
present value of annuity
accumulated value of annuity
an|
sn| 1 1 1 1
….. n 1 2 3
The present (accumulated) value of the series of payments is equal to the sum of present (accumulated) values of each payment

4. Examples (p. 46)
(Loan) John borrows 1500 and wishes to pay it back with equal annual payments at the end of each of the next ten years. If i = 17% determine the size of annual payment
(Mortgage) Jacinta takes out 50,000 mortgage. If the mortgage rate is 13% convertible semiannually, what should her monthly payment be to pay off the mortgage in 20 years?
(Investments) Eileen deposits 2000 in a bank account every year for 11 years. If i = 6 % how much has she accumulated at the time of the last deposit?

5. One more example… (p. 47)
Elroy takes out a loan of 5000 to buy a car. No payments are due for the first 8 months, but beginning with the end of 9th month, he must make 60 equal monthly payments. If i = 18%, find:
the amount of each payment
the amount of each payment if there is no payment-free period

6. Annuity-immediate and Annuity-due
Annuity-immediate (payments made at the end)
sn| 1 1 1 1
….. n 1 2 3
Annuity-due (payments made at the beginning)
än|
.. sn| 1 1 1 1
….. n
n+1 1 2 3

7. Example (p. 50)
Henry takes out a loan of 1000 and repays it with 10 equal yearly payments, the first one due at the time of the loan. Find the amount of each payment if i = 16%

8. 3.3 Perpetuities
Definition: Perpetuity is an annuity whose payments continue forever
Practical applications: perpetual bonds
Basic model: consider perpetuity under which payments of 1 are made forever

9. Formulas
present value of perpetuity
a∞| 1 1 1
….. 1 2 3

10. 3.4 Unknown Time and Unknown Rate of Interest
Examples – Unknown Time
3.4 Unknown Time and Unknown Rate of Interest
A fund of 5000 is used to award scholarships of amount 500, one per year, at the end of each year for as long as possible. If i=9% find the number of scholarships which can be awarded, and the amount left in the fund one year after the last scholarship has been awarded
A trust fund is to be built by means of deposits of amount 5000 at the end of each year, with a terminal deposit, as small as possible, at the end of the final year. The purpose of this fund is to establish monthly payments of amount 300 into perpetuity, the first payment coming one month after the final deposit. If the rate of interest is 12% per year convertible quarterly, find the number of deposits required and the size of the final deposit

11. Example – Unknown Rate of Interest
At what effective yearly rate of interest is the present value of 300 paid at the end of every month, for the next 5 years, equal to 15,000?
1st method: linear interpolation
2nd method: successive approximations

12. Series of n∙m payments of 1/m
3.5 Continuous Annuities
Let effective period be 1/m part of the year and i(m)/m
be the effective rate of interest: (1+ i(m)/m)m = 1 + i
Series of n∙m payments of 1/m
1/m
1/m
1/m
1/m
…..
1/m
2/m
3/m
n = nm(1/m)
Formula:

13. Let m approach infinity…
Present value of continuous annuity
Accumulated value of continuous annuity
Formulas:

14. 3.6 Varying Annuities Arithmetic annuities
increasing
decreasing
Arithmetic increasing perpetuities

15. Arithmetic annuity
Definition: Arithmetic annuity is a series of payments made at regular intervals such that
the first payment equals P
payments increase by Q every year
Thus payments form arithmetic progression P, P+Q, P+2Q, …, P+(n-1)Q

16. Series of n payments k-th payment = P+ (k-1)Q
Formulas
Series of n payments k-th payment = P+ (k-1)Q
k=1,2,…, n
accumulated value S
present value A P
P+Q
P+2Q
P+(n-1)Q
….. n 1 2 3

17. 1) Increasing annuity with P = 1, Q = 1
Series of n payments k-th payment = k
k=1,2,…, n
accumulated value (Is)n|
present value
(Ia)n| 1 2 3 n
Two Special Cases
….. n 1 2 3
We have:
P = 1
Q = 1

18. 2) Decreasing annuity with P = n, Q = -1
Series of n payments k-th payment = n – k + 1
k=1,2,…, n
accumulated value (Ds)n|
present value
(Da)n| n
n-1
n-2 1
….. n 1 2 3
We have:
P = n
Q = -1

19. Increasing perpetuity
k-th payment = k continues forever
present value
(Ia)∞| 1 2 3
….. 1 2 3

20. Examples (p )
Find the value, one year before the first payment, of a series of payments 200, 500, 800,… if i = 8% and the payments continue for 19 years
Find the present value of an increasing perpetuity which pays 1 at the end of the 4th year, 2 at the end of the 8th year, 3 at the end of the 12th year, and so on, if i = 0.06
Find the value one year before the first payment of an annuity where payments start at 1, increase by annual amounts of 1 to a payment of n, and then decrease by annual amounts of 1 to a final payment of 1
Show both algebraically and verbally that (Da)n| = (n+1)an| - (Ia)n|

21. More examples: Geometric annuities
(Geometric annuity) An annuity provides for 15 annual payments. The first is 200, each subsequent is 5% less than the one preceding it. Find the accumulated value of annuity at the time of the final payment if i = 0.9
(Inflation and real rate of interest) In settlement of a lawsuit, the provincial court ordered Frank to make 8 annual payments to Fred. The firs payment of 10,000 is made immediately, and future payments are to increase according to an assumed rate of inflation of 0.04 per year. Find the present value of these payments assuming i = .07