pengantar kewangan_bab 4

7/27/2019 Pengantar Kewangan_Bab 4

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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interest payable not withdrawn throughout the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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If the stated returns are not withdrawn from the savingaccount, and the interest rate for the second and thirdyear remained unchanged, how much return will you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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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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(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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(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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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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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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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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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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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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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

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