anuitas umum

1. UMUM
BAB 4
ANUITAS

2. ANUITAS UMUM/General Annuities
- Anuitas dengan periode pembayaran lebih sering/lebih
sedikit dari periode bunga)
- Untuk frekuensi periode pembayaran kurang dari konversi
tingkat bunga
- Ada 2 pendekatan : pendekatan umum dan pendekatan
lanjut
- Pendekatan pertama : dua step prosedur.
- a. tentukan interest rate, samakan konversinya dengan
pembayaran yang ekivalen dengan tingkat bunga
- b. dengan hasil tingkat bunga terbaru, lakukan perhitungan
sebagaimana Bab 3

3. ANUITAS UMUM/General Annuities
- Example :
1. Find the accumulated value in 10 years if monthly due
payments of 100 are being made into a fund that credits a
nominal rate of interest at 10%, convertible semiannually.

4. 2.Tentukan nilai mendatang dari investasi selama 4 tahun
dengan deposit 100 setiap awal kuartal pada dua tahun
pertama dan 200 setiap awal kuartal pada tahun berikutnya.
Jika tingkat bunga yang berlaku adalah 12% dihitung per
bulan
ANUITAS UMUM/General Annuities

5. 3. Sebuah pinjaman sebesar 3000 akan dikembalikan dengan
mengangsur setiap akhir kuartal selama 5 tahun. Jika tingkat
bunga yang diberlakukan adalah 10% dihitung per semester,
tentukan besaran angsuran pembayaran tersebut.
ANUITAS UMUM/General Annuities

6. Annuities Payable Less Frequency
Than Interest Is Convertible
Further Analysis of Annuities Payable Less Frequency
Than Interest Is Convertible
- Let be a nominal rate of interest convertible “k”
times a year and let there be level end-of year
payments of 1
- after the first payment has been made, interest has
been converted “k” times
- after the second payment has been made, interest has
been converted “2k” times
- after the last payment has been made, interest has
been converted “n” times
- therefore, the term of the annuity (and obviously, the
number of payments) will be “n/k” years
(i.e. if “n”= 144 and is used, then the term of the
annuity is “144/12=12years”)
( )
k
i
(12)
i

7. General Annuities-immediate
n j
k j
a
PV
s


8. General Annuities-immediate
(1 )
(1 )
n
n j n j
n
k j k j
a j S
FV PV j
S S

   

9. General Annuities-due
- Let be a nominal rate of interest convertible “k”
times a year and let there be level beginning-of year
payments of 1
- after the first payment has been made, interest has
been converted “k” times
- after the second payment has been made, interest has
been converted “2k” times
- after the last payment has been made, interest has
been converted “n-k” times
- therefore, the term of the annuity (and obviously, the
number of payments) will be “1+(n-k)/k=n/k” years
(i.e. if “n”= 144 and is used, then the term of the
annuity is “144/12=12years”)
( )
k
i
(12)
i

10. General Annuities-due
n j
k j
a
PV
a


11. General Annuities-due
- Future Value
(1 ) (1 )
n j n j
n n
k j k j
a s
FV PV j j
a a
    

12. 4.Tentukan nilai mendatang dari investasi selama 4 tahun
dengan deposit 100 setiap awal kuartal pada dua tahun
pertama dan 200 setiap awal kuartal pada tahun berikutnya.
Jika tingkat bunga yang berlaku adalah 12% dihitung per
bulan

13. General Annuities-perpetuity-immediate

14. General Annuities-perpetuity-due

15. General Annuities-Continuous_Immediate
- Interest Is Convertible Continuously: i(∞) =δ
for example, the present value (at t= 0) of an annuity–immediate
where payments of 1/12 are made every month for “n” years (or
12n periods) and where the annual force of interest is δ can be
calculated as follows

16. General Annuities-Continuous_Immediate

17. Further Analysis of Annuities Payable More
Further Analysis of Annuities Payable More
FrequencyThan Interest Is Convertible
Immediate
 Payments of (1/m) are made at the end of every (1/m)-th of
year for the next “n” years
 the present value (at t=0) of an mthly annuity immediate,
where the annual effective rate of interest is “i”, shall be
denoted as and is calculated as follows:
( )
m
ni
a

18. Further Analysis of Annuities Payable More
Further Analysis of Annuities Payable More
FrequencyThan Interest Is Convertible
( )
( )
n
m
m n
Buktikan
i
a a
i


19. Further Analysis of Annuities Payable More
Further Analysis of Annuities Payable More
FrequencyThan Interest Is Convertible

20. Further Analysis of Annuities Payable More
Further Analysis of Annuities Payable More
FrequencyThan Interest Is Convertible
( )
( )
n
m
m n
Buktikan
i
s s
i


21. contoh
1.
2.

22.  Sebuah pinjaman sebesar 3000 akan dikembalikan dengan
mengangsur setiap akhir kuartal selama 5 tahun. Jika tingkat
bunga yang diberlakukan adalah 10% dihitung per semester,
tentukan besaran angsuran pembayaran tersebut.
Jawab.

24. Further Analysis of Annuities Payable More
Further Analysis of Annuities Payable More
FrequencyThan Interest Is Convertible
Relationship

25. Further Analysis of Annuities Payable More
Further Analysis of Annuities Payable More
FrequencyThan Interest Is Convertible
Due-Present Value

26. Further Analysis of Annuities Payable More
Further Analysis of Annuities Payable More
FrequencyThan Interest Is Convertible
( )
( )
n
m
m n
Buktikan
i
a a
d



27. Further Analysis of Annuities Payable More
Further Analysis of Annuities Payable More
FrequencyThan Interest Is Convertible
Due-Accumulated Value

28. Further Analysis of Annuities Payable More
Further Analysis of Annuities Payable More
FrequencyThan Interest Is Convertible
( )
( )
n
m
m n
Buktikan
i
s s
d



29. Further Analysis of Annuities Payable More
Further Analysis of Annuities Payable More
FrequencyThan Interest Is Convertible

30. Further Analysis of Annuities Payable More
Further Analysis of Annuities Payable More
FrequencyThan Interest Is Convertible
- Perpetuity Immediate
Payments of “1/m” are made at the end of every (1/m)th of year forever

31. Further Analysis of Annuities Payable More
Further Analysis of Annuities Payable More
FrequencyThan Interest Is Convertible
- Perpetuity-Due

32. Continuous Annuities
 payments are made continuously every year for
the next “n” years (i.e.m=∞)
 the present value (at t= 0) and acc. Value (at t=n)
of a continuous annuity, where the annual
effective rate of interest is “i”, shall be denoted as
as and is calculated as follows:
ni
a
ni
s
( )
( )
( )
( )
1 1 1
lim lim( )
1 1 1
lim lim( )
(1 ) 1 1
(1 )
n n n
m
m
ni ni
m m
n n n
m
m
ni ni
m m
n n
n
ni ni
v v e
a a
i
v v e
a a
d
i e
s a i



 
 
 

 

 
  
   
  
   
  
   


33. Continuous Annuities

34. contoh
Carilah force of interest untuk 20 10
3
s s


35. BasicVaryingAnnuities
payments will now vary; but the interest
conversion period will continue to coincide with
the payment frequency
3 types of varying annuities are:
i. payments varying in arithmetic progression
ii. payments varying in geometric progression
iii. other payment patterns

36. Arithmetic Progression_Immediate
Immediate: An annuity-immediate is payable over
“n” years with the first payment equal to “P” and
each subsequent payment increasing by “Q”.
Present
Future

37. Arithmetic Progression_Immediate

38. Arithmetic Progression_Immediate
Let P=1and Q = 1. In this case, the payments start
at 1 and increase by 1 every year until the final
payment of “n” is made at time n

39. Arithmetic Progression_Immediate
Let P=n and Q=−1. In this case, the payments start
at n and decrease by 1 every year until the final
payment of 1 is made at time n

40. Arithmetic Progression_Due
An annuity-due is payable over n years with the first payment
equal to P and each subsequent payment increasing by Q. The
time line diagram below illustrates the above scenario

41. Arithmetic Progression_Due
Alternative ways, present dan Accumulated value

42. Arithmetic Progression_Due
Let P=1and Q= 1. In this case, the payments start at 1 and
increase by 1 every year until the final payment of n is
made at time n−1

43. Arithmetic Progression_Due
Let P=n and Q=−1. In this case, the payments start at n
and decrease by 1 every year until the final payment of 1 is
made at time n.

44. Geometric Progression_Immediate
An annuity-immediate is payable over n years with the first
payment equal to 1 and each subsequent payment increasing by
(1 +k). The time line diagram below illustrates the above scenario:

45. Geometric Progression_Due
An annuity-due is payable over n years with the first payment
equal to 1 and each subsequent payment increasing by (1 +k). The
time line diagram below illustrates the above scenario

46. ANUITAS UMUM
1. SiToni menabung Rp.1 juta tiap akhir kuartal
di bank dengan tingkat bunga 6% dihitung
perbulan.Berapakah jumlah uangnya di bank
akhir tahun ke-3?
(12) (4)
12 4
(4)
12 4
(4)
(1 ) (1 )
12 4
0.06
(1 ) (1 )
12 4
?
4
i i
i
i
  
  

12
(4)
12
1
S 1
4
v
juta
i




47. ANUITAS UMUM
2. Si Ali menabung Rp 1 juta tiap akhir bulan
di bank dengan tingkat bunga 6% dihitung
perkuartal.Berapakah jumlah uangnya di bank
pada akhir tahun ke-3?

48. ANUITASTERTENTU
3.Hitunglah nilai tunai dari pembayaran sebesar Rp
1 juta tiap akhir kuartal selama 3 tahun disusul
dengan Rp 1,5 juta perkuartal selama 2 tahun lagi
bila tingkat bunga 12% dihitung persemester?

49. ANUITASTERTENTU
4.Harga mobil bekas Rp 120 juta.Si Ali membelinya dengan uang
muka 25% dan sisanya dia bayar dengan cicilan tiap bulan selama
5 tahun.Berapakah besar cicilan tersebut bila tingkat bunga 15 %
dihitung perkuartal?
5. Harga sebuah motor Rp 15 juta. SiTopan membelinya dengan
uang muka Rp 2,5 juta dan sisanya di bayar cicilan bulanan yang sama
besarnya selama 24 bulan. Jika tingkat bunga 6% dihitung perkuartal
maka besar cicilan yang dibayarTopan adalah
6.Tono mendepositokan uangnya Rp 10 juta pada awal tiap kuartal
untuk 2 tahun pertama,kemudian dia mendepositokan Rp 15 juta
pada awal tiap kuartal untuk 2 tahun berikutnya.Jika tingkat
bunga 6% dihitung pertahun,maka jumlah uangnya pada akhir
tahun ke-4 adalah

50. ANUITAS NAIK DANTURUN (aritmatika)
 Contoh :
7. Asrul seorang mahasiswa semester 3 yang ingin memiliki laptop baru
untuk keperluan skripsi. Ia berusaha berhemat dengan menyisihkan
sebagian uang sakunya untuk dapat membeli laptop tersebut. Jika dia
menabung sebesar 200 rb pada akhir bulan pertama, naik menjadi
300 rb bulan beriktnya dan terus naik sebesar 100 rb. Jika tingkat
bunga adalah 6% dihitung perbulan. Pada akhir semester 7 berapa
harga laptop yang mampu ia beli? Hitung pulai nilai tunainya!
8. Rani ingin membeli pakaian dan tas seharga 750rb, maka ia
menabung dari sebagian uang sakunya sebesar 50rb akhir bulan
pertama, kemudian menjadi 75 rb bulan berikutnya dan disiplin rutin
kenaikan 25 rb per bulan. Berapa waktu yang ia butuhkan untuk
mencapai uang yang ia butuhkan jika diasumsikan bunga sebesar 9%
dihitung per bulan?

51. ANUITAS NAIK DANTURUN (aritmatika)
9. Rizki ingin memberikan kejutan pada Ibunya dengan membelikan
liontin emas seharga 2.5 jt. Jika sekarang ia memiliki uang 1 jt dan
berencana bulan berikutnya akan mulai menabung 500 rb, kemudian
disusul 450 rb seterusnya berkurang 50 rb. Berapa bulan yang ia
perlukan jika tingkat suku bunga 9% dihitung per kuartal.
10. Shamila bertekad untuk tidak minta uang spp pada semester 8
nanti . Maka ia berusaha keras untuk menyisihkan uang saku dan
ikut part time bekerja. Jika SPP yang harus ia bayar sebesar 4 jt dan
saat ini ia duduk di semester 3 dan ia memulai menabung sebesar
500 rb dan bertekad semester berikutnya bisa menabung dengan
tambahan 250 rb. Jika diasumsikan tingkat bunga 6% dihitung per
kuartal, bagaimana kondisi tabungan Shamila pada akhir semester 7?
Ada lebih atau kurang untuk SPP semester 8?

52. BAB
ANUITAS UMUM

53. apendix

54. Tugas
1.
2.
3.
4.

55. Tugas
5.
6.
7.
8.
9.
10.