Transcript Module A - Basics of Business Mathematics

Accounting & Finance
for Bankers Business MathematicsModule A
SPBT College
Simple Interest
More Simple Interest …
Compound Interest:
A FV Perspective
Compounding …
Time Line: Rs78.35
Invested (5 Years, 5% Interest)
FV5 = Rs100
PV = Rs78.35
0
1
2
3
End of Year
4
5
Future Value of Rs200
(4 Years, 8% Interest )
FV4 = Rs272.10
FV3 = Rs251.94
FV2 = Rs233.28
FV1 = Rs216
PV = Rs200
0
1
2
3
4
End of Year
Compounding – the process of earning
interest in each successive year
FV of a Mixed Cash Flow
Stream (5 Years, 5.5% Interest)
Rs4,335.89
Rs4,462.12
Rs2,226.06
Rs3,165.00
Rs2,500.00
Rs3,500
0
1
Rs3,800
2
Rs2,000
3
End of Year
Rs3,000
4
Rs2,500
5
FV5 =
Rs16,689.06
Future Value Example
Power Of Compound Interest
30.00
20%
25.00
20.00
15%
15.00
10.00
5.00
1.00
10%
5%
0%
0 2 4 6 8 10 12 14 16 18 20 22 24
Periods
Format of a Future Value
Interest Factor (FVIF) Table
Period
1
2
3
4
5
6
7
1%
1.010
1.020
1.030
1.041
1.051
1.062
1.072
2%
1.020
1.040
1.061
1.082
1.104
1.126
1.149
3%
1.030
1.061
1.093
1.126
1.159
1.194
1.230
4%
1.040
1.082
1.125
1.170
1.217
1.265
1.316
5%
1.050
1.102
1.158
1.216
1.276
1.340
1.407
6%
1.060
1.124
1.191
1.262
1.338
1.419
1.504
Computing Future Values
Using Excel
You deposit Rs1,000 today at 3% interest.
How much will you have in 5 years?
PV
r
n
FV?
$
1,000
3.00%
5
$1,159.3
Excel Function
=FV (interest, periods, pmt, PV)
=FV (.03, 5, ,1000)
Present Value with
Compounding
Present Value of Rs500
(7 Years, 6% Discount Rate)
0
1
2
3
4
End of Year
PV =
Rs332.53
5
6
7
FV7 = Rs500
Present Value of Future
Amounts (4 Years, 7% Interest )
Discounting
0
1
FV1 = Rs214
2
FV2 = Rs228.98
End of Year
PV =
Rs200
3
FV3 = Rs245
4
FV4 = Rs262.16
PV of a Mixed Stream
(4 Years, 6% Interest)
0
1
2
Rs1,500,000
Rs3,000,000
End of Year
Rs1,415,100
Rs2,669,700
Rs1,679,200
Rs3,960,500
PV4 = Rs9,724,500
3
Rs2,000,000
4
Rs5,000,000
Present Value Examples
Format of a Present Value
Interest Factor (PVF) Table
Period
1
2
3
4
5
6
7
1%
0.990
0.980
0.971
0.961
0.951
0.942
0.933
2%
0.980
0.961
.942
0.924
0.906
0.888
0.871
3%
0.971
0.943
0.915
0.888
0.863
0.837
0.813
4%
0.962
0.925
0.889
0.855
0.822
0.790
0.760
5%
0.952
0.907
0.864
0.823
0.784
0.746
0.711
6%
0.943
0.890
0.840
0.792
0.747
0.705
0.665
Calculating PV Of A Single
Amount Using Excel
Example: How much must you deposit today in order
to have Rs500 in 7 years if you can earn 6% interest
on your deposit?
FV
r
n
PV?
$
500
6.00%
7
$332.5
Excel Function
=PV (interest, periods, pmt, FV)
=PV (.06, 7,,500)
FV & PV of Mixed Stream
(5 Years, 4% Interest Rate)
Compounding
Rs12,166.5
Rs3,509.6
FV
Rs4,326.4 Rs6,413.8
Rs5,624.3
Rs3,120.0
-Rs10,000
Rs3,000
0
1
Rs2,884.6
Rs5,000
2
Rs4,000
3
Rs3,000
4
Rs2,000.0
5
End of Year
Rs4,622.8
Rs3,556.0
PV
Rs5,271.7
Rs2,564.4
Rs1,643.9
W. P. Carey Executive MBA ProgramDiscounting
Slide 20
Annuity Cash Flows
FV of Ordinary Annuity
(End of 5 Years, 5.5% Interest Rate)
Rs1,238.82
Rs1,174.24
Rs1,113.02
Rs1,055.00
Rs1,000.00
Rs1,000
0
1
Rs1,000
2
Rs1,000
3
End of Year
Rs1,000
4
Rs1,000
5
(1  r )  1
FV  PMT 
 $5,581.08
r
n
FV of an Ordinary Annuity
Using Excel
How much will your deposits grow to at the end of five years if you
deposit Rs1,000 at the end of each year at 4.3% interest for 5
years?
PMT
r
n
FV?
$
1,000
4.3%
5
$5,448.8
Excel Function
=FV (interest, periods, pmt, PV)
=FV (.043, 5,1000 )
PV of Ordinary Annuity
(5 Years, 5.5% Interest)
0
1
Rs1,000
2
Rs1,000
3
Rs1,000
4
Rs1,000
5
Rs1,000
End of Year
Rs947.87
Rs898.45
Rs851.61
Rs807.22
Rs765.13
PMT 
1 
PV 
 1 
 $4,270.28
n
r
 (1  r ) 
Annuity Examples
Ordinary Annuity vs. An
Annuity Due
Annual Cash Flows
End of yeara
0
aThe
Annuity A (ordinary)
Rs
Annuity B (annuity due)
0
Rs1,000
1
1,000
1,000
2
1,000
1,000
3
1,000
1,000
4
1,000
1,000
5
1,000
0
Total
Rs5,000
Rs5,000
ends of years 0, 1,2, 3, 4 and 5 are equivalent to the beginnings of years
1, 2, 3, 4, 5, and 6 respectively
Calculating the Future Value
of an Annuity Due
•
Equation for the FV of an ordinary annuity can be converted
into an expression for the future value of an annuity due,
FVAn (annuity due), by merely multiplying by (1 + r)
n
FVAn (annuitydue)  PMT   (1  r )t 1  (1  r )
t 1
n
 PMT   (1  r )t
t 1
(1  r ) n  1
FV  PMT 
 1  r 
r
FV of an Annuity Due
Using Excel
How much will your deposits grow to at the end of five years
if you deposit Rs1,000 at the beginning of each year at 4.3%
interest for 5 years?
PMT
r
n
FV
FVA?
$1,000
4.30%
5
$5,448.89
$5,683.19
Excel Function
=FV (interest, periods, pmt, PV)
=FV (.043, 5, 1000)
=Rs5,448.89*(1.043)
Deposits Needed to
Accumulate a Future Sum



A person wishes to buy a house 5 years from now
and estimates an initial down payment of Rs35,000 will be
required at that time
She wishes to make equal annual end-of-year deposits in an
account paying annual interest of 4 percent, so she must
determine what size annuity will result in a lump sum equal to
Rs35,000 at the end of year 5
Find the annual deposit required to accumulate FVAn dollars,
given an interest rate, r, and a certain number of years, n by
solving equation PMT:
FVA5
$35,000
PMT 

 $6,461.98
FVIFA4%,5 5.4163
Loan Amortization Table
(10% interest, 4 Year Term)
Payments
End
of
year
Loan
Payment
(1)
Beginningof-year
principal
(2)
Interest
[.10 x (2)]
(3)
End-of-year
Principal
principal
[(1) – (3)]
[(2) – (4)]
(4)
(5)
1
Rs1,892.82
Rs6,000.00
Rs600.00
2
1,892.82
4,707.18
470.72
3
1,892.82
3,285.08
328.51
1,564.31
1,720.77
4
1,892.82
1,720.77
172.08
1,720.74
-a
aDue
Rs1,292.82 Rs4,707.1
8
1,422.10 3,285.08
to rounding, a slight difference (Rs.03) exists between beginning-of-year 4
principal (in column 2) and the year-4 principal payment (in column 4)