Ekotek - 21102011

T i m e Va l u e o f M o n e y

M . D i z a N o v i a n d i , S . T. , M . S c .

J e n i s B u n g a

Simple Interest (Bunga Biasa)

Compound Interest (Bunga Berganda)

L a j u T i n gk a t B u n g a

Nominal Interest Rate (Tingkat Bunga Nominal)

Effective Interest Rate (Tingkat Bunga Efektif)

Simple Interest

Bunga = I = P i n

P = nilai sekarang

i = tingkat bunga

n = waktu

Compound Interest

I berubah2 tiap tahunnya

Pada tahun kedua, P =modal awal + bungatahun pertama

Pada tahun ke-3, P =jumlah uang yangdidapat pada tahun ke-2

dst

BUNGA MENURUT JENISNYA (lanjutan)

Simple Interest Compound Interest

F1 = P + I1 = P + P i 1

= P (1 + i )

F2 = P + I2 = P + P i 2

= P (1 + i 2)

F3 = P + I3 = P + P i 3

= P ( 1+ i 3 )

F1 = P + P i 1

= P (1 + i)

F2 = F1 (1 + i 1)

= P (1+i)(1+i) = P(1+i)2

F3 = F2 (1 + i 1)

= P(1+i)2(1+i) = P(1+i)3

BUNGA MENURUT JENISNYA (lanjutan)

B u n g a B i a s a B u n g a B e r g a n d a

FFFFnnnn = P + P i n = P (1 + i n)= P + P i n = P (1 + i n)= P + P i n = P (1 + i n)= P + P i n = P (1 + i n)

F = Nilai yang akan datang

P = Nilai sekarang

i = tingkat bunga

n = waktu

FFFFnnnn = P ( 1 + i ) = P ( 1 + i ) = P ( 1 + i ) = P ( 1 + i ) nnnn

F = Nilai yang akan datang

P = Nilai sekarang

i = tingkat bunga

n = waktu

Contoh Soal Bunga Biasa

Contoh perhitungan 1

Seseorang meminjam uang $ 1000,-

dengan bunga i=20% pertahun. Tiga

bulan kemudian uang

dikembalikan. Berapa besarnya?

F = P (1+ 1/4 20%)

= 1000 (1 + 0.05)

= $ 1050,-


Jika pengembaliannya 6 bulan:

F = P (1+ 1/2 20%)

= 1000 (1 + 0.1 )

= $ 1100,-


Jika pengembaliannya 2 tahun:

F = P (1+ 2 20%)

= 1000 (1+ 0.40)

= $ 1400,-

Contoh Soal Bunga Berganda


Seseorang meminjam uang $ 1000,-

dengan bunga i=20% pertahun. Dua

tahun kemudian uang dikembalikan.

Berapa besarnya?

F1 = 1000 (1 + 20%)

= $ 1200,-

F2 = 1200 (1+20%)

= $ 1440,-



Maka i menjadi * 20% = 5%

F = 1000 (1 + 5%)1

= $1050,-



F = 1000 (1 + 10%)1

= $1100,-

Atau

F = 1000 (1 + 5%)2

= $1102,50

BUNGA MENURUT TINGKAT PENGEMBALIANNYA

Nominal Interest Rate

Tingkat bunga Tahunan

Dapat digandakan beberapa kali

pertahun (misal bulanan)

Misal: Bunga 8% setahun dibagi 4 kali

setahun = bunga 8% pertahun

diberlakukan secara kwartal

Maka bunga 8% Tingkat bunga

nominal tahunan

Bunga 2% Tingkat bunga

nominal kwartal

Effective Interest Rate

Perbandingan antara bunga yang

didapat dengan jumlah uang awal

pada suatu periode (misal 1 tahun)

I = F - P = P ( 1 + i )n - P

i n = (F P) / P

= [P ( 1 + i )n - P] / P

= [(1 + i)n - 1]

ieff = (1 + iannual/n)n 1

ieff = ei - 1

Example Problem

Nominal Interest Rate

The annual interest rate is 6%, andthe interest is compoundedquarterly. What is the quarterlynominal interest rate? What is theeffective annual interest rate ifcompounded quarterly?

Nominal interest rate

inominal = 6/4 = 1.5%

Effective interest rate

ieff = (1+0.015)4 1 = 6.13%

Effective Interest Rate

A local bank announces that adeposit over $1,000 will receive amonthly interest of 0.5%. If youleave $10,000 in this account, howmuch would you have at the end ofone year?

Fn = P ( 1 + i )n

= 10,000 (1+0,005)12

= 10,000 (1,0617) = $10,617

ieff = (1 + i)n - 1

= (1 + 0,005)12 1 = 0,0617

The graphic presentation of the costs and benefits over the

time is called the cash flow diagram. This is the time profile of

all the costs and benefits. It is a presentation of what costs

have to be incurred and what benefits are received at all

points in time.

The following conventions are used in the construction of

the cash flow diagram:

The horizontal axis represents time

The vertical axis represents costs and benefits

Costs are shown by downward arrows

Benefits are shown by upward arrows

Example

A car leasing company buys a car from a wholesaler for $24,000 and leases it to a customer forfour years at $5,000 per year. Since the maintenance is not included in the lease, the leasingcompany has to spend $400 per year in servicing the car. At the end of the four years, theleasing company takes back the car and sells it to a secondhand car dealer for $15,000. For themoment, in constructing the cash flow diagram, we will not consider tax, inflation, anddepreciation.

Step 1 :

Draw the horizontal axis to represent 1,2,3, and 4 years.

0 1 2 3 4

Step 2 :

At time zero, i.e., the beginning of year 1, the leasing company spends $24,000. Hence, at time zero, on the horizontal axis, a downward arrow represents this number.

0 1 2 3 4

24,000

Step 3 :

At the end of year 1, the company receives $5,000 from his customer. This is represented by an upward arrow at the end of year 1. The leasing co. also spends $400 for maintaining the car; this is represented by a downward arrow.

0 1 2 3 4

The situations at years 2 and 3 are exactly the same as year 1 and are the presentations on the cash flow diagram exactly as for the first year.

0 1 2 3 4

24,000

400

5,000

5,000

400

24,000

Step 4 :

At the end of the fourth year, in addition to the income and the expenditure as in the previous years, the leasing company receives $15,000 by selling the car. Thisadditional income is represented by an upward arrow

The project ends at this time, so we have nothing else to insert in the cash flow diagram. We have represented all the costs and benefits in the cash flow. At this point, it is a good idea to go back through the life of the project and make sure that nothing as expressed in the description of the project is left out.

15,000

Documents

Ekotek - 21102011