Business And Financial Mathematics-annuity

  • June 2020
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Annuity Sub Topics I Definition of annuity II Annuity Formula III Application of annuity IV. Problems

I

Definition of annuity An Annuity is a sequence of payments made at equal intervals of time and usually equal in amount.

II Formula Annuity Formula Amount of Annuity n

Sn =A

(1 + i ) – 1 i

Where: Sn = Amount of annuity A = periodic payment (usally every month) i = interest rate (usually per month) n = number of periods The formula above can be written as follows: Sn = A Sni Where: n (1 + I ) – 1 Sni = i The value of Sni can be seen in a table of annuity. Example. If a man puts Rp 20,000 in the bank every month at interest rate 1% per month, find the total amount of annuity for 4 years. Answer. Given A=20,000, i=0.01, n=48 Answer: 48 (1 + 0.01) – 1 Sn =20000 0.01 =20000 x 61.2226078

23

=1,224,452.156

Formula Present Value of Annuity PVn =A

1- (1 + i ) i

-n

Where: PVn = Present value of annuity The formula above can be written as follows: PVn =A PVni Where: -n 1- (1 + i ) PVni = i The value of PVni can be seen in a table of annuity. Example. A debt is to be discharged by making equal payments Rp 287.680 at the end of each month for five years. If the interest rate charged is 2% per month , find the size of the debt ? Answer. Given A=287.680, i=0,02, n=60 1- (1 + 0,02 ) PVn=287680 0,02 =287680 x 34,76089 =10.000.000

-60

III Application of annuity

A father deposit in a bank Rp 100.000 every month at interest rate 1% per month for 8 years. Then, he wants his son to withdraw at equal at the end of each month during his 5 years university course. How much can he withdraw every month? Answer: For 8 years can be seen as an amount of annuity. Given A=100,000, i=0.01, n=96 Answer: 96 (1 + 0.01) – 1 Sn =100000 0.01

24

=100000 x 159.9273 =15,992,730 For the next 5 years is a process of present value of annuity. Given PVn= 15,992,730 , i=0.01, n=60, A= …..? PVn =A

1- (1 + i ) I

-n

1- (1 + 0.01 ) 0.01 15,992,730 = A x 44.95504 A = 15,992,730 / 44.95504 = 355,749. 4

-60

15,992,730=A

IV Problems Problem 1 A man deposit Rp 200,000 at the end of each month. At the end of 6 years the amount of his deposit is 21,786,930. Find the interest rate per month? Problem 2 A debt Rp 1,000,000 is to be discharged by making equal payments Rp 339,479.9 every month at interest rate 2.5% per month. Find how long time will it take to pay off the debt?

25

Interpolation Method Related to the formula Sn or PV, suppose we want to find the value of variable i if A, Sn and PV are given, we can use interpolation method : For the formula Sn

(Sni –Sni1) (Sni2 –Sni1)

I = i1 +

(i2-i1)

i1 = the smaller i i2 = the greater i Sni1 = can be found in the table or computed using the formula. Example : A man deposit at the bank Rp 250,000 at the end of each month. In 5 years the amount of his deposit will be Rp 21,784,318. Find the interest rate per month? Answer: Given : Sn

=

21,784,318

A n

= =

250,000 60

Steps in evaluating i using interpolation method. Step 1 (Evaluate Sni) Sni = Sn/A = 21,784,318 /250,000 = 87.137272 Step 2 (To Find position of Sni in the tabel ) Look at the table of Sn, for n=60,we can see i between i=1,125% and 1,25% So: I1 I2 Sni1 Sni2

= = = =

1.12500% 1.25000% 85.03512704 88.57450776

Steps 3 (Substitute to the interpolation formula )

i = 0.01125 +

(87.137272 – 85.03512704 ) (88.57450776 –85.03512704)

= 0.01199 =1.199%

26

(0.0125 – 0.01125 )

For formula PV

(PVni –PVni1) (PVni2 –PVni1)

i = i1 +

(i2-i1)

i1 = the smaller i i2 = the greater i PVni1 = can be found in the table or computed using the formula. Example : A man has a debt amounts Rp 8 millions at a bank with annuity system for 3 years. A sequence of payments should be done Rp 347,000 at the end of each month. Find the interest rate charged? Answer: Given : PV A N

= = =

8000000 347,000 36

Steps Steps 1 (Evaluate PVni) PVni = PV/A = 8 million/347,000 = 23.054755 Step 2 (To find position of PVni in the table ) (Look at table ani, for n=36, we can see i between i=2,5% dan 2,75%. Jadi: i1 i2 Pvni1 PVni2

= = = =

2.5% 2.75% 23.5562511 22.6699175

Steps 3 (Substitute to the interpolation formula )

i = 0.025 +

(23.054755 - 23.5562511) (22.6699175–23.5562511)

= 0.0264145 =2.64145%

27

(0.0275 – 0.025

)

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