•Finding The Periodic PaymentInterpolation to Find Unknown Rate or TimeAnnuities Due

Finding The Periodic Payment

• Formula:A = R a n iS = R s n i – IR = A a n i

R = Ss 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

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?

GIVEN

R = ?A = 30,000t = 6 years compounded monthlyn = 72r = 5%i = 5 % 12

SOLUTIONR = A

a n i

R = 30,000a 72 5/12%

R = 30,000

62.0928

R = 483.15 ( the monthly payment )

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)

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

SOLUTIONR = S

s n i

R = 120,000S 20 3.5%

R = 120,000 28.2797

R = 4243.33

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.

SAMPLE PROBLEM

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

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

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

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

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)

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) = 5.69Therefore: the Nominal rate is = 5.69%

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.

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%.

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

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.

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% )

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

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

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

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?

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

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