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PENGANTAR KEWANGAN
BDPW3103
TOPIK 3
MATEMATIK KEWANGAN
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Time Line
Refer to period of one investment.
Time 0 (t0) refer to the present time, time 1
(t1) refer to the end of the first period and
so forth.
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Compounding Interest
Types of interest
Simple Interest : Interest that will be received
based on the principal amount
Compounding Interest : Interest that will be paid
not only on the principal amount but also on any
interest payable not withdrawn throughout the
period
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Compounding Interest (Cont)
Example 4.1 : If you invested RM100 in savingaccount with the interest rate 10% per year, howmuch return will you received at the end of thefirst year.
Return (F) = Principal (P) + Interest (i)= P + P(i)
= P(1 + i)
= RM100 ( RM100 x 10%)
= RM100 + RM10= RM110
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Compounding Interest (Cont)
If the stated returns are not withdrawn from the savingaccount, and the interest rate for the second and thirdyear remained unchanged, how much return will you
receive at the end of the second and third year?F2 = P(1 + i)
2
= RM100(1 + 0.1)2
= RM121
F3 = P(1 + i)3
= RM100(1 + 0.1)3
= RM133.10
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Compounding Interest (Cont)
When the saving period I extended to tn,
the total return that will obtained in the
period (n) is
FV = PV(1 + i)n
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Calculating Future Value Using Schedule
The Future Value (FVn) equivalent to theprincipal at the point of time equal 0 or theoriginal principal amount (PV0) multiply
with the future value factor stated in theschedule of Future Value Interest Factor(FVIFi,n)
The formula of FV using the schedule is
FVn = PV0 (FVIFi,n)
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Calculating Future Value Using Schedule
(Cont)
Example 4.2 : You invested RM2,000 in the
saving account at a yearly interest rate of 5% for
the period of one year. Upon the completion of
one year, how much return will your receive?FV1 = PV0(FVIFi,n)
= RM2,000 (FVIF5% , 1)
= RM2,000 (1.0500)= RM2,100.00
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Calculating Future Value Using Schedule
(Cont)
Example 3.3 : Assume you deposited RM2,000
in the saving account at a yearly interest 5% for
the period 4 years. Upon the completion years,
how much the return will you receive?FV4 = PV0(FVIFi,n)
= RM2,000 (FVIF5% , 4)
= RM2,000 (1.216)= RM2.432.00
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Graphical Illustration of FV
There are 3 basic elements which sill
influenced the future value, these are
Principal (amount that was borrowed or
invested)
Time period (the number of frequency of
interest payment)
Interest rate payable or interest received
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Graphical Illustration of FV (Cont)
To show the interest rate influenced the FV of aninvestment, find the return for deposited RM100at Bank A, B and C that offer interest rate 8%,10% and 12% per year for 3 years.
FV for Bank A
FVA = PV0(FVIFi,n)
= RM100 (FVIF8% , 3)
= RM100 (1.2600)= RM126.00
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Graphical Illustration of FV (Cont)
FV for Bank B
FVB = PV0(FVIFi,n)
= RM100 (FVIF10% , 3)
= RM100 (1.3310)
= RM133.10
FV for Bank C
FVC = PV0(FVIFi,n)= RM100 (FVIF12% , 3)= RM100 (1.4050)
= RM140.50
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Graphical Illustration of FV (Cont)
The correlation of FV, time period and
interest rate can be shown on the graph
below
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Concept of Discounting and Present
Value (PV)
Used to ascertain the present value (PV0)
of principal value for sum of the money in
the future (FVn) that is discounted at an
interest rate (i) for the valuation period(n@t)
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Calculation of PV
There are formula to calculate PV
PV0 = FV (1 + i)n
Exmaple 3.4 : Assume you expect to receivedreturns of RM2,500 a year from now. How much
the present value if the discount rate is 8% per
year
PV0 = FV (1 + i)n
= RM2,500 (1 + 0.08)1
= RM2,314.81
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Calculation of PV (Cont)
What is the present value that you must invest ifyour expect to received RM2,500 in the period 2years and 3 years at a discount rate 8% per
year?PV0 = FV (1 + i)n
= RM2,500 (1 + 0.08)2
= RM2,143.35
PV0 = FV (1 + i)n
= RM2,500 (1 + 0.08)3= RM1,984.58
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Calculation of PV Using Schedule (Cont)
Exmaple 3.5 : Assume you expect toreceive RM3,999 in 3 years from now.How much is the PV if the discount rate is
9% per year?PV3 = FV(PVIFi,n)
= RM3,999 (PVIF9% , 3)
= RM3,999 (0.772)= RM3,087.23
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Calculation of PV Using Schedule (Cont)
Example 3.6 : You intend to accumulate saving
money at the bank for RM5,712 for the 4 years.
How much saving you must make now if the
interest rate is 10% per year?PV4 = FV(PVIFi,n)
= RM5,713 (PVIF10% , 4)
= RM5,713 (0.683)= RM3,901.98.
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Graphical Illustration of PV
Change of interest rate, time of period or
the return will changed of the present
value.
Example 3.7 : You intend to obtain return
of RM1,000 in 3 years from Bank A, B and
C that offer interest 8%, 10% and 12%.
What id the principal value that shouldmake?
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Graphical Illustration of PV (Cont)
PVA = FV(PVIFi,n)
= RM1,000 (PVIF8% , 3)
= RM1,000 (0.7938)
= RM793.80PVB = FV(PVIFi,n)
= RM1,000 (PVIF10% , 3)
= RM1,000 (0.7513)
= RM751.30PVC = FV(PVIFi,n)
= RM1,000 (PVIF12% , 3)
= RM1,000 (0.7118)
= RM711.80
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Graphical Illustration of PV (Cont)
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Single Cash Flow Money Value
Is a cash flow that only occurs once in the
period of valuation
The FV of an amount of single cash flow
invested presently will increase from time
to time with the specific interest rate
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Series Cash Flow Money Value
Is a series receiving or payments of cashthat occur throughout the valuation period.
There are several categories of series of
cash flow that isAnnuity
Derivation Cash Flow
Perpetuity
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Annuity
Series of payments @ receiving of the same
amount at the same intervals through the period
For example, Cash flow of RM5 that receive forevery month is an example of Annuity
Types of Annuity
Ordinary Annuity : Annuity occurs at the end of each
periodAnnuity Due : Annuity at the beginning of the period
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Annuity : FV of Ordinary Annuity
That occurs at the end of each period
Future Value Annuity (FVA) is the number
of annuity payments at a specific amount(n) that will increase at a specific period
based on a specific interest rate (i).
The formula of the FVA is= A[(1 + i)n 1)] i
OR = A(FVIFAi,n)
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Annuity : FV of Ordinary Annuity (Cont)
Example 3.8 : You had deposited RM100 at theend of each year for 3 years continuously in theaccount that pays a yearly interest rate 10%.
How much the FV of the said annuity?FVA = A[(1 + i)n 1)] i
= RM100[(1 + 0.1)3 -1] 0.1
= RM331
OR = A(FVIFAi,n)
= RM100 (FVIFA10%,3)
= RM100(3.310)
= RM331
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Annuity : FV of Ordinary Annuity (Cont)
Example 3.9 : Danon Company depositedRM5,000 at the end of each year for 3 yearsconsecutively in an account that pays a yearly
interest rate of 10%. What is the FVA?FVA = A[(1 + i)n 1)] i
= RM5,000[(1 + 0.1)3 -1] 0.1
= RM16,550
OR = A(FVIFAi,n)
= RM5,000 (FVIFA10%,3)
= RM100(3.310)
= RM16,550
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Annuity : FV of Annuity Due
The payment of annuity occurs at the
beginning of the period.
For example, at the beginning of each
month or each year.
The formula for FV of Annuity Due is
= [A][(1 + i)n 1)][1 + i] i
OR = A(FVIFAi,n)(1 + i)
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Annuity : PV of Ordinary Annuity
Present value of ordinary annuity can be
obtained using the below formula
PVA = A{1 [1 (1 + i)n]} I
OR = A(PVIFAi.n)
Example 3.11 : Taming Company expects to
receive RM3,000 at the end of each year for 3
consecutive years. How in the present value forthe annuity if the discount rate is 6% per year.
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Annuity : PV of Ordinary Annuity (Cont)
PVA = A{1 [1 (1 + i)n]} I
= RM3,000{1 [1 (1 + 0.06)3]}
0.06
= RM3,000[1 0.8396] 0.06= RM481.1422 0.06
= RM8,019.04
OR = A(PVIFAi.n)= RM3,000 (PVIFA6%,3)
= RM3,000 (2.673)
= RM8,019.00
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Annuity : PV of Annuity Due
The formula for PV of Annuity Due is (PVA)
= A{1 [1 (1 + i)n]} i x (1 + i)
OR = A(PVIFAi.n)(1 + i) Example 3.12 : Taming Company expects to
receive RM3,000 at the beginning of each year
for 3 consecutive years. How in the present
value for the annuity if the discount rate is 6%per year.
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Annuity : PV of Annuity Due (Cont)
PVA = A{1 [1 (1 + i)n]} i x (1 + i)
= RM3,000{1 [1 (1 + 0.06)3]}
0.06 x 1.06
= RM3,000[1 0.8396] 0.06 x 1.06= RM481.1422 0.06 x 1.06
= RM8,500.18
OR = A(PVIFAi.n) (1 + i)= RM3,000 (PVIVA6%,3) (1 + 0.06)
= RM3,000 (2.673) (1.06)
= RM8,500.14
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Non-Uniform Cash Flow
Involves a mixture of cash flow or cash
floe is irregular
The calculation for future value and
present value of an irregular cash flow is a
combination concept of determining
money value for single cash flow and
annuity cash flow
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FV of Derivation Cash Flow
Involves the determination of FV foe each of thecash flow and subsequently totaling all the FV.
The formula is
FVn = Pt(1 + i)n-1
Example 3.13 : Bikin Fulus Company made adecision to deposit RM2,000 at the end of the 1st
and 2
nd
year, withdrawing RM3,000 at the end ofthe 3rd year and depositing RM4,000 at the endof 4th year. How much is this future value cashflow at the end of the 4th year if the annualinterest rate is 10% per year?
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FV of Derivation Cash Flow (Cont)
FVn= Pt(1 + i)n-1
= (RM2,000)(1 + 0.1)4-1
+ (RM2,000)(1 + 0.1)
4-2
- (RM3,000)(1 + 0.1)4-3
+ (RM4,000)(1 + 0.1)4-4
= RM5,782
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PV of Derivation Cash Flow
Involves the determination of PV foe each of the
cash flow and subsequently totaling all the PV.
The formula is
PV0 = Pt[1 (1 + i)n]
Example 3.14 : Bikin Fulus Company expects to
receive RM1,000 at the end of 1st year and 2nd
year, RM2,000 at the end of 3rd year andRM4,000 at the end of 4th year. How much is the
present value cash flow if the yearly interest rate
is 10% per year?
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PV of Derivation Cash Flow (Cont)
PV0 = Pt[1 (1 + i)n]
= RM1,000[1 (1 + 0.1)1]
+ RM1,000[1
(1 + 0.1)
2
]+ RM2,000[1 (1 + 0.1)3]
+ RM4,000[1 (1 + 0.1)4]
= RM5,970.22
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Perpetuity
Is the annuity that have infinity period
Cannot be used in decision making
because every investment have valuation
period.
The formula is
PVp = P i
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Perpetuity (Cont)
Sukehati Company issued securities thatpromised a payment of RM100 per year atthe yearly interest rate of 8% to the
holders of that security. How much thepresent value for that cash flow?
PVp = P i
= RM100
0.08= RM1,250
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Compounding and Discounting More
Than Once A Year
Sometimes, the payment or receiving of returnoccurs more than once a year. For example,twice a year, quarterly and monthly.
For this, the period (n) must times with thenumber of payment or receiving (m) and theinterest rate (i) must be divided with the numberof payment or receiving (m) as shown below:
FV = PV x [1 + (i m)]nm
OR = PV [FVIF(im)(nm)]
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Compounding and Discounting More
Than Once A Year (Cont)
Example 3.16 : The future value of RM1 now for
6 years, using the interest rate of 10% per year
with the different compounding frequencies
Compounding Nm i/m FV
Once a year 6 x 1 = 6 0.1 1 = 0.1 RM1(1 + 0.1)6 = RM1.772
Twice a year 6 x 2 =
12
0.1 2 = 0.05 RM1(1 + 0.05)12 = RM1.796
Four times a
year
6 x 4 =
24
0.1 4 = 0.025 RM1(1 + 0.025)24 = RM1.809
Every month 6 x 12 =
72
0.1 12 =
0.0083
RM1(1 + 0.0083)72 =
RM1.817
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Compounding and Discounting More
Than Once A Year (Cont)
Example 3.17 : The present value of RM1
received in 6 years from now, discounting at the
interest rate of 10% per year with different
discounting frequencies
Discounting Nm i/m PV
Once a year 6 x 1 = 6 0.1 1 = 0.1 RM1 (1 + 0.1)6 = RM0.564
Twice a year 6 x 2 = 12 0.1 2 = 0.05 RM1 (1 + 0.05)12 =
RM0.557
Four times a
year
6 x 4 = 24 0.1 4 = 0.025 RM1 (1 + 0.025)24 =
RM0.553
Every month 6 x 12 =
72
0.1 12 =
0.0083
RM1 (1 + 0.0083)72 =
RM0.550
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Continuous Compounding and
Discounting
Some cases of the time value of money,
interest must be compounded or
discounted continuously or at each
microsecond.
The formula is
FV = PV(ein)
PV =FV (ein)
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Continuous Compounding and
Discounting (Cont)
Example 3.18 : What is the future value
for RM100 that is invested now for 6 years
with an interest rate of 8% per year and
compounded continuously?
FV = PV(ein)
= RM100(e(0.08)(6))
= RM161.61
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Continuous Compounding and
Discounting (Cont)
Example 3.19 : What is the present value
for RM161.61 that will received in 6 years
from now with an interest rate of 8% per
year and discounted continuously?
PV =FV (ein)
= RM161,61 (e(0.08)(6))
= RM100