Tài liệu ôn tập chương cuối kỳ | social pyscology | Học Viện Phụ phụ nữa Việt Nam

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Profile Kim Oanh
UNIT
TIME
VALUE
OF
MONEY
Structure
2.0 Objectives
2.1
2.2
Future of a Value Cash Flow
2.3
Future of an Annuity
2.4
Present Value
of
a Single
2.5
Value
of
Series Cash Flows of
2.5.1
Present Value
of
an
of
Uneven
Cash
Flows
2.6
Let Us
Up
2.7 Key Words
2.8
Answers to Your Progress
2.0
After studying this unit, you should
be
able to:
explain value present value concepts;
explain compound interest and discount;
value of a single and
an
annuity;
compute present value of a single amount and an annuity.
INTRODUCTION
You is must a heard that rupee today than rupee tomorroa
you why imagine, is so? Let me
tell
you an Anil's by example. grandfath
to gift him one at the end of five years; gave and a
having Rs. 75,000 today. Anil's Had you been in place choice y
made? Would you have accepted Rs. after five years or Rs. 75,00
What do you Apparently, say? Rs. 75,000 today is attractive tha
Rs. after
five
years because present is than future. You cou
Rs.
75,000 the inarket and this Rs. at the
years would have less to purchasing power due inflation, We you hav
that
a
rupee today worth rupee is more a But
are so simple. value of money concepts will unravel th
of such choices all of
us
face our daily life. in We a
of
value of constitute 90% of finance sense.
decisions involve flow occurring at different points Therefore,
of value of money very is
In
unit, you will learn about co
interest discount concepts future value of a single and a
and present of
a
single an annuity is
start value of a single for a single period
one period.
Time
of
Money
FUTURE
VALUE
CASH
of let explain the meaning of value.
By
value
the amount
of
money an investment grow to over period at
interest rate. words, In other value the is
at sometime
in
Value
of
a
Single for
If deposit Rs. you 1000
in
a account of your at
you will get year'? will You Rs.
I
is to
your amount principal Rs. Rs. 100 interest you
a year. Rs.
1
100 is the future value
o f
Rs.
1000
for
one year at per It 1000 today is worth
I
in
year given per cent is interest
Thus, invest if you one period at interest
i,
to per rupee invested.
In
tlic above is
Future Value
of
a
Single
the example, if you invest the years
will have you after two years, tlic the
You
will earn Rs. 00
+
10
+
Rs. 100
so
you
will
of Rs. (1 10). is
of
1000 at
per cent.
You can notice that Rs. 12
has First
is
1000 is
the principal second
is
Its.
in
and
part is 100 another Rs. earned
as
interest year. in
last
is Rs.
which is interest earnecl
in
second on
in
first 100
x
Rs. 10. interest So tlie earned Its. is
2
10. is
oo+ 10).
process of putting your interest on an
for more a thereby reinvesting period, interest is
Compounding tlie interest interest interest. on
We
can tlie call
result interest. The interest only on tlie
principal called is
simple interest.
value of a single flow be by tlie
value for years
cash
rate per of interest year
total
of
years
,
Foundation Finance of
The above in the table is
a
basic equation in compounding analysis
factor
is
called the compounding factor Future Interest or Value
calculations very difficult with increasing number the
called Future tables are available with different c
o f and
n.
You would see such tables attached at the end block of th
can use these tables to find value factor. you have to
at
10%
for five years, the that corresponds t o percent and the
the rows until to five years. That
is
how we found
for the example given below.
will Rs. 1000 be after five years
at
?
Year
1
2
3
n
-
The total interest on Rs.
in
five years is Rs. 61
Interest
PV
X
P
V
Amount in the beginning of
period
PV
PV(1
+i)
P
V
(1 +i)
PV
In
five years the total Rs. interest earned 500, is Rs. per year
Rs. 111 (Rs. 61 1
-
500) is compounding. Table given shows sim
interest year and and total each amount earned at end of fiv
Amount
at
the
We
have discussed the future of
a
(single) amount for num
Now
let calculate
future value of flows.
Table
2.1
Year
1
2
3
4
5
in
the
beginning
Rs. 1000
Rs.
1210
Rs. 1331
Rs. 1464.1
Simple
100
100
100
100
100
50 0
at the
0
2
33.1
46.4
110.5
121
133.1
146.4
610.5
1
1
1
1
Let
one
of
with same example. Suppose you deposit Rs. today 1000
in
a
bank
at 10%.
of
you again deposit 1000. How now you two years?
At
will have
Rs.
21 00.
1
+
deposit 1000).
you have this for deposit another year at at the end of
year you will Rs.
2
100
x
.10 Rs.
23
0.00
Let illustrate it of also called line
I
1000
2)
Future value
0
2
This is one way of out value two deposits 1000. is
another The first Rs. 1000 deposited two years is at
I
its
value
is
Rs. 1000
x x
1.2 100 Rs. 12 10
The second is deposited for so its value is Rs.
total value is 0
1
23
1
0
So there are two ways to calculate value for
the balance forward one year at
a
2)
Calculate tlie value of each and add
methods will give you same answer.
Effect
of
You may remember tlie example
of
in beginning. Suppose his great
father Rs. had 100 for years ago 60 10% rate. it would
grown till Let out today? us find value Factor.
FVIF
( 1
of
In
this case is
Rs.
600 as Rs. 29,848
compounding. Therefore, effect of is great over
to short periods
2.3
FUTURE
OF
AN
ANNUITY
annuity
is
a series (or of payments receipts) of amount p
case of life be policy and loans Annuity etc. of t
( n )
or annuity, and (b) annuity case regular
or receipt at the end of period. the each or
at the beginning o f
it
is called due.
of
Regular Annuity
The compound value is total amount
if amount at is a certain rate of and interest i
o f the period.
A
to pay Rs. 1000 a year for years is
1
if you deposit Rs. 5000 at tlie of year every
in
a
the is paying 10% interest, the value annuity
1 . 1
O r
1
=
Rs .
I
above procedure be expressed can as given below
:
I
Future
of
An
Annuity
A
Periodic cash
flow
of
the
illustration
1
FVA
=
5000
x
0.6105
0 .10
FVA
=
Rs.
30,525
the formula is called value interest factor o f
i
fin d
out the table, see yearstable for for 10% 5
5000 by 6.105 get 30525
of
Money
2:
A
person plans to contribute Rs. 2,000 retirement account
is paying 8% interest. person retires in 30 years, what is the value of
amount?
A
You can also directly find out value interest for an annuity at 8%
for 30 years from tlie value annuity table, is 13.28
value of annuity is 2,000
x
1
13.28 Rs. 2,26560
Finding interest rate
Illustration
3
:
Suppose you receive a of Rs. 94,000 at of 8 years after
paying Rs. 8,000 for 8 years. What is the implicit (i) this in
First of
all
find
96,000
=
at tlie value annuity table and row corresponding years until
we find close to 12, 12.300 it is and the is below column of 12%.
rate below is
12
per cent.
Annuity
Now, take
an
where the total
annuity
(received or
paid),
rate
of interest
and tlie
is known.
You are to find tlie amount of
annuity.
you deposit
in
a so annually you
get
Rs. at of at rate
Annual
x
=
Rs.
x
1
15.937
=
Rs. 9,412.05
So you should deposit Rs. 9,412.05 in a every year for years in order to get
Rs. at the end of years.
Note: The called a denominator.
Illustration
4:
How
a
save to accumulate Rs.
for by tlie 10 end of years, at the rate
8%.
1
Annual Annuity
=
x
Annual
x
Rs. 6,903
A
person should save Rs. 6,903 annually years to get Rs.
Future
Value
of
Due
An annuity for flows occur at of each per
annuity
due.
Lease installment are tlie of annuity due.
To annuity due. tlie methods used
in
calculating annu
wi
I
I
be applied.
Let us calculation for tlie future value of a Rs. 1,000 ordin
for
3
years at
8
percent
and
compare that it of future value of
annuity due 3 at
8
per Note that the casli flows for the o
annuity occur at the end of periods and
3,
those annu
at beginning 2, 3 and
4.
difference
value of an ordinary annuity annuity is point at which
(FV) is is calculated. For annuity. an ordinary FV calculated as of the l
flow. while for an due, annuity FV is calculated as of period t
flow.
T i e value of 3 year annuity due is equal
to
Future
3
year ordinary annuity for one period. vmore
annuity is as
future value
x
of Year
Ordinary
annuity
I
1,000
Rs.
1,000 Rs. 1,000
Future value of
an
ordinary annuity at
8%
for
3
years, is Rs. 3246
Annuity
due
Rs. 1,000 Rs. 1,000
Rs.
1,000
(Rs.
1,000) (FVIFA (Rs. 1.08) (Rs. 3,246) (1.08)
Rs.
Future value of
an
annuity due of
8%
for
3
years
(FVAD,).
=
Rs.
3,506
Progress A
1 )
What do you by value?
...............................................................................................................................
...............................................................................................................................
...............................................................................................................................
...............................................................................................................................
...............................................................................................................................
2)
What is compounding?
...............................................................................................................................
...............................................................................................................................
...............................................................................................................................
...............................................................................................................................
...............................................................................................................................
3)
What is difference between annuity annuity due? and
4)
You have deposited Rs. 10,000
in
a
fixed deposit in a
bank
at
6%
rate of
interest. How much get years? will you after 5
much Rakesh get will 12 years
if
lie deposits
in
a
fixed
disposit at
1
O%?
Foundation of
2.4
PRESENT VALUE
OF
A
SINGLE CASH
have 1 seen that tlie value of Re. for one year at 10% is Rs.
put a question
i n
a different How have way. to invest today
Re.
I
in
one year? You know tlie value here is Re. but is t
of Re. You need Re.
1
at the end of year, present the the value
that
PV
+
I
Present value of 1 Let is Re. 909. us see factor
=
FV,
(1
+
1
In
-
is the present value interest factor or discou
Suppose want you to Rs. 1500 years in three at
7%
rate interest
should you today to 1,500 get Rs.
i n
three years?
Present value is just opposite of value. future value In we do
of money.
In
present value discount present. Tlconcept we back to
reducing to their present value called is
discou
value today of the received the called present in is its value.
know PV of Rs. 500 in year
at
8%, then:
PV
x
1.08
=
500
1
PV
=
Rs. 462.5
1.08
You need not do much calculations. Value Tables help you in fin
present value of cash flow. are given These tables at end bloof this
multiply present interest value factor by amount. So,
x
0
Rs. 462.5. (See at factor
8%
for year one
i n
present table, value i
2.5
PRESENT
VALUE
OF
SERIES
OF
of
M o ne y
FLOWS
Tlie series of cash flows may be
a)
Even series of flows annuity
b)
Uneven series of flows
As tlie equation is called discount or
factor tlie rate called is discount rate. Tlie the
present value of a flow called is 'Discounted Cash Flow valuation.
2.5.1
Present Value
of
Annuity
You want Rs. to 800 at tlie of each of three years. is
What tlie present value of
There methods are two to out value. present
Under first method the present an is tlie
inflows of this annuity. can be as follows:
=
Rs. 800
x
0.9091
+
Rs. 800
x
0.8264 800
x
0.75
=
Rs. 727.28
+
66 1.12
+
60 1.04 Rs. 1989.44
Tlie above be arrived by tlie
A A
A
A
A
or PVA
=
+
-
+ +
-
(1 +;)
I
)
-
I
(
PVA
=
A
I
-
I
is present value interest for
A annuity
I
rate
of years
PVA annuity
Alternate
Method
Instead of present value for year we multiply amount by
annuity present value interest factor. See table, 2.48685 it is
at 10% for
3
years. So Rs. 800
x
2.48685
=
Rs.
1989.44 present value is an
an
Note:
If value table available is not PVIFA be calculated as
follows:
-
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UNIT TIME VALUE OF MONEY Structure 2.0 Objectives 2.1 2.2 Future Value of a Cash Flow 2.3 Future of an Annuity 2.4 Present Value of a Single 2.5 Value of Series of Cash Flows 2.5.1 Present Value of an of Uneven Cash Flows 2.6 Let Us Up 2.7 Key Words 2.8 Answers to Your Progress 2.0
After studying this unit, you should be able to: explain value present value concepts;
explain compound interest and discount; value of a single and an annuity;
compute present value of a single amount and an annuity. INTRODUCTION You must heard that a rupee today is than a rupee tomorro
you imagine, why is so? Let me tell you by a n example. Anil's grandfath to gift him one
at the end of five years; and gave a
having Rs. 75,000 today. Had you been in Anil's place choice y
made? Would you have accepted Rs. after five years or Rs. 75,00
What do you say? Apparently, Rs. 75,000 today is attractive tha Rs.
after five years because present is than future. You cou Rs. 75,000 the inarket and this Rs. at the
years would have less purchasing power due to inflation, We you hav
that a rupee today is worth more a rupee But are so simple.
value of money concepts will unravel th of such choices all of us face in our daily life. We a of value of
constitute 90% of finance sense. decisions involve
flow occurring at different points Therefore, of value of money is ve ry In unit, you will learn about co interest discount concepts future value of a single and a and present of a single an annuity is start value of a single for a single period Time of Money one period. FUTURE VALUE CASH of let explain the meaning of value. By value
the amount of money an investment grow to over period at
interest rate. In other words, value is the at sometime in Value of a Single for If you deposit Rs. 1000 in a account of your at you will get year'? You will Rs. I is to your principal amount Rs. Rs. 100 interest you a year.
Rs. 1 100 is the future value o f Rs. 1000 for one year at per It 1000 today is worth I in year given per cent is interest Thus, if you invest one period at interest i, to
per rupee invested. In tlic above is
Future Value of a Single the example, if you invest the years
will you have after two years, tlic the You will earn Rs. 00 + 10 + Rs. 100 so you will of Rs. (1 10). is of 1000 at per cent. You can notice that Rs. 12 has First is 1000 is the principal second is Its. in and
part is another Rs. 100 earned as interest in year. last is Rs. which is interest earnecl in second on in first 100 x Rs. 10. So tlie interest earned is Its. 2 10. is oo+ 10). process of putting your interest on an for more a period, thereby reinvesting interest is Compounding tlie interest
interest on interest. We can call tlie result interest. The interest only on tlie principal is c
alled simple interest. value of a single flow be by tlie value for years cash rate of interest per year total of years , Foundation of Finance Year
Amount in the beginning of Interest Amount at period the 1 PV PV X 2 PV(1 +i) 3 n- PV (1 +i) PV PV The above
in the table is a basic equation in compounding analysis
factor is called the compounding factor or Future Value Interest calculations
very difficult with increasing number the called Future tables are available with different c
o f and n. You would see such tables attached at the end block of th can use these tables to find value factor. you have to at 10% for five years, the that corresponds to percent and the the rows until
to five years. That is how we found for the example given below. will be Rs. 1000 after five years at ? The total interest on Rs. in five years is Rs. 61 In five years the total interest earned is Rs. 500, Rs. per year Rs. 111 (Rs. 61 1-500) is compounding. Table given shows sim
interest and total amount earned each year and at end of fiv Table 2.1 Year in Simple the beginning at the 1 Rs. 1000 100 0 2 Rs. 100 1 3 1210 100 2 121 1 4 Rs. 1331 100 33.1 133.1 1 5 Rs. 1464.1 100 46.4 146.4 1 500 110.5 610.5 We have discussed the future of a (single) amount for num Now let
calculate future value of flows. Let
with same example. Suppose you deposit Rs. 1000 today in a bank at 10%. of one you again deposit 1000. How now you two years? At of will have Rs. 21 00. 1 + deposit 1000). you have
this deposit for another year at at the end of year you will Rs. 2 100 x .10 Rs. 23 0.00 Let illustrate it of also called line I 1000 2) Future value 0 2 This is one way of out value two deposits 1000. is another
The first Rs. 1000 is deposited two years at I its value is Rs. 1000 x x 1.2 100 Rs. 12 10 The second is deposited for so its value is Rs. total value is 0 1 23 1 0
So there are two ways to calculate value for the balance forward one year at a 2) Calculate tlie value of each and add methods will give you same answer. Effect of
You may remember tlie example of in beginning. Suppose his great father had Rs. 100 for 60 years ago 10% rate. it would
grown till today? Let us find out value Factor. FVIF ( 1 of In this case is Rs. 600 as Rs. 29,848 compounding. Therefore, effect of is great over to short periods 2.3 FUTURE OF AN ANNUITY
annuity is a series of payments (or receipts) of amount p case of life policy and loans etc. Annuity be of t ( n ) or annuity, and (b) annuity case regular or receipt at the end of each period. the or at the beginning o f it is called due. of Regular Annuity The compound value is total amount if amount is
at a certain rate of interest and i o f the period. A
to pay Rs. 1000 a year for years is 1
if you deposit Rs. 5000 at tlie of every year in a the is paying 10% interest, the value annuity 1 . 1 O r 1 = Rs. I
above procedure can be expressed as given below : I Future of An Annuity A Periodic cash flow of the illustration 1 0.6105 FVA = 5000 x 0 . 1 0 FVA = Rs. 30,525 the formula is called value interest factor o f i find out the table, see table for 10% for 5 years 5000 by 6.105 get 30525 of Money
2: A person plans to contribute Rs. 2,000 retirement account is paying 8% interest.
person retires in 30 years, what is the value of amount? A
You can also directly find out value interest for an annuity at 8% for 30 years from tlie value annuity table, is 13.28
value of annuity is 2,000 x 1 13.28 Rs. 2,26560 Finding interest rate
Illustration 3 : Suppose you receive a of Rs. 94,000 at of 8 years after paying
Rs. 8,000 for 8 years. What is the implicit (i) in this First of all find 96,000 = at tlie value annuity table and row corresponding years until we find
close to 12, it is 12.300 and is below the column of 12%. rate is below 12 per cent. Annuity Now, take an
where the total annuity
(received or paid), rate of interest and tlie is known. You are to find tlie amount of annuity. you deposit in a annually so you get Rs. at of at rate Annual x 1 = Rs. x 15.937 = Rs. 9,412.05
So you should deposit Rs. 9,412.05 in a every year for years in order to get Rs. at the end of years. Note: The called a denominator. Illustration 4: How a save to accumulate Rs. for
by tlie end of 10 years, at the rate 8%. 1 Annual Annuity = x Annual x Rs. 6,903
A person should save Rs. 6,903 annually years to get Rs. Future Value of Due An annuity for flows occur at of each per annuity due. Lease installment are tlie of annuity due. To
annuity due. tlie methods used in calculating annu wi I I be applied. Let us
calculation for tlie future value of a Rs. 1,000 ordin
for 3 years at 8 percent and compare it that of future value of annuity due 3 at 8 per
Note that the casli flows for the o
annuity occur at the end of periods and 3, those annu at beginning 2, 3 and 4. difference value of an ordinary annuity annuity is point at which
(FV) is calculated. For an ordinary annuity. FV is calculated as of the l
flow. while for an annuity due, FV is calculated as of period t flow. T i e value of 3 year annuity due is equal to Future 3 year ordinary annuity for one more pe riod. v annuity is as future value x of Year Ordinary annuity I 1,000 Rs. 1,000 Rs. 1,000
Future value of an ordinary annuity at 8% for 3 years, is Rs. 3246 Annuity due Rs. 1,000 Rs. 1,000 Rs. 1,000 (Rs. 1,000) (FVIFA (Rs. 1.08) (Rs. 3,246) (1.08) Rs.
Future value of an annuity due of 8% for 3 years (FVAD,). = Rs. 3,506 Progress A 1 ) What do you by value?
...............................................................................................................................
...............................................................................................................................
...............................................................................................................................
...............................................................................................................................
............................................................................................................................... 2) What is compounding?
...............................................................................................................................
...............................................................................................................................
...............................................................................................................................
...............................................................................................................................
............................................................................................................................... 3) What is difference between annuity and annuity due? 4)
You have deposited Rs. 10,000 in a fixed deposit in a bank at 6% rate of
interest. How much will you get after 5 years? much Rakesh will get 12 years if lie deposits in a fixed disposit at 1 O%? Foundation of
2.4 PRESENT VALUE OF A SINGLE CASH have seen that tlie
value of Re. 1 for one year at 10% is Rs.
put a question in a different way. How have to invest today
Re. I in one year? You know tlie value here is Re. but is t of Re.
You need Re. 1 at the end of the year, the present value that PV + I Present value of 1 is Re. 909. Let us see factor = FV, (1 + 1 In -
is the present value interest factor or discou Suppose you want to
Rs. 1500 in three years at 7% rate interest should you
today to get Rs. 1,500 in three years? Present value is just opposite of value. In future value we do
of money. In present value concept we discount back to present. Tl reducing
to their present value is called discou value today of the received in the is c alled its present value. know PV of Rs. 500 in year at 8%, then: PV x 1.08 = 500 1 PV = Rs. 462.5 1.08
You need not do much calculations. Value Tables help you in fin
present value of cash flow. These tables are given at end of this blo multiply
present value interest factor by amount. So, x 0 Rs. 462.5. (See
factor at 8% for one year i n present value table, i of M o n e y 2.5
PRESENT VALUE OF SERIES OF FLOWS
Tlie series of cash flows may be a) Even series of flows annuity b) Uneven series of flows As tlie equation is called discount or factor tlie rate is called discount rate. Tlie the present value of a flow is c alled 'Discounted Cash Flow valuation. 2.5.1 Present Value of Annuity You want to Rs. 800 at tlie of each of three years. is What tlie present value of There are two methods to out present value.
Under first method the present an is tlie inflows of this annuity. can be as follows:
= Rs. 800 x 0.9091 + Rs. 800 x 0.8264 800 x 0.75 =
Rs. 727.28 + 66 1.12 + 60 1.04 Rs. 1989.44 Tlie above be arrived by tlie A A A A A or PVA = + - + + - (1 +;)I)- I ( PVA = A I-I is present value interest for A annuity I rate of years PVA annuity Alternate Method Instead of present value for year we multiply amount by
annuity present value interest factor. See table, it is 2.48685
at 10% for 3 years. So Rs. 800 x 2.48685 = Rs. 1989.44 is present value an an Note: If value table is not available PVIFA be calculated as follows:-