This document discusses annuities and formulas for calculating periodic payments, present value, and accumulated value for different types of annuities. It provides examples of using the annuity formulas to solve problems related to loans, investments, and determining interest rates. Key formulas presented include calculating the periodic payment as the principal divided by the annuity factor, and calculating present value or accumulated value as the periodic payment multiplied by the annuity factor. Interpolation is also described as a method to find unknown interest rates or time periods in annuity calculations.

1. •Finding The Periodic Payment
Interpolation to Find Unknown Rate or Time
Annuities Due

2. Finding The
Periodic Payment
• Formula:
A = R a n i
S = R s n i – I
R = A
a n i
R = S
s n i
 WHERE:
 R = the periodic payment of
the annuity
 N = number of payment of
the annuity or length of term
expressed in interest period
 i = interest rate per
conversion period

3. Sample Problem
 Mr. Santos borrowed 30,000 and
promised to cancel his debt 6 years by
paying equal sums at the end of each
month with interest at 5 % compounded
monthly . How much is Mr. Santos
monthly payment?

4. GIVEN
R = ?
A = 30,000
t = 6 years compounded monthly
n = 72
r = 5%
i = 5 %
12

5. SOLUTION
R = A
a n i
R = 30,000
a 72 5/12%
R = 30,000
62.0928
R = 483.15 ( the monthly payment )

6. Exercise Problem
 If money is worth 7% compounded
semiannually , how much shall be
invested at the end of every6 months to
accumulate a fund of 120,00 pesos at the
end of the years? ( the full amount is
given)

7. GIVEN
 R = the unknown semiannual deposit
 S = 120,000
 r = 7%
 i = 3.5%
 t = 10 years
 n = 20

8. SOLUTION
R = S
s n i
R = 120,000
S 20 3.5%
R = 120,000
28.2797
R = 4243.33

9. INTERPOLATION TO FIND
UNKNOWN RATE OR TIME
 Interpolation = is finding the value
of the conversion period i and n to get
the exact rate and time respectively.
= is also necessary when the nearest
value of i and n is desired.

10. SAMPLE PROBLEM
• What nominal rate compounded
semiannually is 8,000 the present value of
400 annuity payable semiannually for 15
years?

11. GIVEN
 i = the unknown rate per interest period
 R = 400
 r = ?
 n = 30 ( 15 years semiannually )

12. SOLUTION
 Substitute in the formula : R = A Substitute in the formula : R = A
a n i
 400 = 8,000 = 800 = 20
a n i 400

13. SOLUTION
 Referring to table V . Under column
n=30 , look for a value of n which is nearly
less, and little bit more than 20 .
TABLE V n=30
2-3/4 % 20.249
i 20.000
3% 19.600

14. SOLUTION
 20.249 – 20.000 = 0.249
 20.249 – 19.600 = 0.649
 Applying the principle of Ratio and
Proportions we obtain:
d = 0.249
¼% 0.649
d = ¼ x (0.249) d= 0.096
4(0.649)

15. SOLUTION
 From Table V we have :
 i = 2 ¾ % + d which is ; i = 2.75 + 0.096
 i – 2.846%
The nominal rate semiannually is 2 (i) ;
substitute the value if i ; 2(2.846) = 5.69
Therefore: the Nominal rate is = 5.69%

16. ANNUITIES DUE
 Annuity Due = is an annuity whose first payment
occurs immediately on a day to be called present.
 Term of an Annuity = is also defined a the time
from the beginning of the first payment interval to the
ends of the last one.
 First Payment = is made at the beginning of the term
and ends one payment interval after the last payment.
 Present Value of an Annuity due = is the sum
of the present values of the payments.
 Amount of an Annuity due = is the sum of the
accumulated values of the payments at the end of the
term.

17. SAMPLE PROBLEM
 Find the present value and the amount of
an annuity due paying 2,000 pesos
semiannually for a term of 9 ½ years if
money is worth 6%.

18. GIVEN
• R = 2000 every 6 months
• n = 19 ( for 9 ½ years)
• i = 3% from 6% / 2 semiannually

19. SOLUTION
• 1. Since the end of an interval is the
beginning of the next interval, the
annuity due will be 2,000 cash plus an
ordinary annuity of 2,000 payable at the
end of each 6 months for 9 ½ years .
Point 0 indicates the present and the
block dots indicates payment dates.

20. SOLUTION
• 2. The present value A is found by adding
the first payment at point 0 to the present
value of the last 18 payments which form
an ordinary annuity whose term at point 0.
Hence:
• A = Down payment + 2,000 a n i
• A = 2,000 + 2,000 ( a 18 3% )

21. SOLUTION
• 3. Refer to Table V , under column n=18
and i = 3% the value is 13.753513
• A= 2,000 + 2,000 ( 13.753513 )
• A = 2,000 + 27.507
• A = 29,507

22. SOLUTION
• S = ( Accumulated 20 payments ) –
(Fictitious Payment )
• S = R s n i – ( one fictitious payment)
• S = 2,000 ( s 20 3 %) – 2,000
• 4. Refer to Table IV . Under column n= 20
and i = 3%, the value is 26.870374 .
Substitute:
• S = 2,000 ( 26.870374 ) – 2000 =
51,740.75

23. SUMMARY OF ANNUITIES
DUE FORMULA
 a n i = used to represent the Present
Value
 s n i = used to represent the Amount of
the Annuity Due
 A = R + R a n – 1 I
 S = R s n + 1 i - R

24. SAMPLE ROBLEM
 To discharge a debt amounting to 80,000
pesos , Mr. D. Cruz agreed to make equal
monthly deposit at the beginning of each
month for 5 years . How much should he
deposit monthly if money is worth 4%
compounded monthly?

25. GIVEN
 R = the periodic deposit
 A = 80,000
 n = 60 ( monthly for 5 years )
 i = 1/3 ( from 4% compounded monthly)

26. SOLUTION
 A = R = R a n-1 1/3 %
 80,000 = R + R a 60-1 1/3%
 Table V , under column n = 59 and I = 1/3% the
entry is : 53.480065
 SUBSTITUTE:
 80,000 = R + R ( 53.480065)
 80,000 = R ( 1 + 53.480 )
 R = 80,000 = 1.468.43 (PERIODIC PAYMENT)
 54.480