CAIIB -Financial Management

1
CAIIB -Financial Management

• Module A -Quantitative Techniques and Business
  Mathematics
• Madhav K Prabhu
• M.Tech, MIM, PMP, CISA, CAIIB, CeISB, MCTS, DCL

2
Agenda

• Time Value of Money
• Bond Valuation Theory
• Sampling
• Regression and Correlation

3
Time Value of Money
4
Objectives

• What do we mean by Time value of money
• Present Value, Discounted Value, Annuity

5
Time Value of Money

• What is Time Value of Money?
• Future Value
• Present Value
• Future Value Compounding

How would you do Compounding?
6
Compounding

• Compounding Formula
• What if compounding is done on monthly basis?

7
Compounding Exercise

• Exercise
• Prepare a table showing compounding as per
  following conditions
• Rate of Interest - 5, 12 and 15
• Compounding 2 4 times in a year
• Principal Rs.100,000/-

8
Discounting

• Present Value
• You have an option to receive Rs. 1,000/- either
  today or after one year. Which option you will
  select? Why?
• Decision will depend upon the present value of
  money which can be calculated by a process
  called Discounting (opposite of Compounding)
• Interest Rate and Time of Receipt of money decide
  Present Value
• What is the present value of Rs. 1,000/- today
  and a year later?

To compute Present Value?
9
Discounting contd

• Formula to find Present Value of Future Cash
  Receipt
• Where PV Present Value, P Principal, i Rate
  of Interest, n Number of Years after which
  money is received
• Assuming Rate of Interest is 10, value of Rs.
  1,000/- to be received after 1 year will be,
• Whereas the value of money to be received today
  will be Rs. 1,000/-

• What if you were to choose between
• Receive Rs. 1,000/- every year for 3 years, OR
• Receive Rs. 2,500/- today? (assume 10 annual
  interest rate)

10
Discounting of a Series contd

• How discounting is done for a series of cashflow?
  e.g.
• Receive Rs. 1,000/- at the end of every year for
  3 years OR
• Receive Rs. 2,500/- today
• Assume Rate of Interest _at_10

If cashflow was to occur every 6 months instead
of 1 year, what impact it will have on Present
Value?
11
Periodic Discounting

• What if the receipts are over six months
  interval ? Find Present Value of the money
  receipts
• Periodic Discounting Formula

• Receive Rs. 1,000/- at the end of every 6 months
  for 1-1/2 years OR
• Receive Rs. 2,600/- today
• Assume Rate of interest _at_10

Where, P Principal, i Rate of Interest, t
Times Payments made in a Year, n nth Period (in
this case it is half year)
12
Periodic Discounting Formula
Expressed mathematically, the equation will look
like
Generically expressed, the formula is Here, N
3
13
Charting of Cashflow

• For any financial proposition prepare a chart of
  cashflow e.g.

Interest Received 100 Sold Bond
2,050 Total
2,150
Interest Received 50
01.01.04
31.12.04
Timeline
30.06.04
30.06.05
Invested in Bonds (1,000)
Interest Received 50 New Bond
Purchased (1,020) Net
( 970)
14
Net Present Value

• Net Present Value means the difference between
  the PV of Cash Inflows Cash Outflows
• How do you compute NPV?
• Prepare Cashflow Chart
• Net off Inflow Outflow for each period
  separately
• If Inflow gt Outflow, positive cash
• If Inflow lt Outflow, negative cash
• Find present values of Inflows Outflows by
  applying Discount Factor (or Present Value
  Factor)
• NPV (PV of Inflows) LESS (PV of Outflows)
  Result can be ve OR -ve
• Continuing with our example of Bond Investment

Interest Received 100 Sold Bond
2,050 Total
2,150
Inflow
Interest Received 50
01.01.04
31.12.04
Timeline
30.06.04
30.06.05
Invested in Bonds (1,000)
Interest Received 50 New Bond
Purchased (1,020) Net
( 970)
Outflow
15
NPV contd

• If Cashflows are discounted at say 10, the sum
  of PV is 25.05, a positive number therefore the
  IRR has be higher than 10 to make Net Present
  Value to zero

What is IRR?
16
Internal Rate of Return (IRR)

• Definition The Rate at which the NPV is Zero. It
  can also be termed as Effective Rate
• If we want to find out IRR of the bond investment
  cashflow

17
IRR Contd

• To prove that at IRR of 11.38 the NPV of
  Investment Cashflow is zero, see the formula
  table

18
IRR - Additional Example

• You buy a car costing Rs. 600,000/-
• Banker is willing to finance upto Rs. 500,000/-
• The loan is repayable over 3 years, in Equated
  Monthly Installments (EMI) of Rs. 15,000/-
• Installments are payable In Arrears
• What is the IRR?
• How do you express this mathematically? What are
  the values of each component in the formula?
• What will be the impact on IRR if the EMIs are
  payable In Advance?
• Can we use IRR for computing Interest Principal
  break-up?

19
IRR - Additional Example contd

• Plot the cashflow
• EMI in Arrears

Begin
1
2
3
35
36
500,000
01.02.2006
01.03.2006
01.04.2006
01.11.2008
01.12.2008


01.01.2006
-15,000
-15,000
-15,000
-15,000
-15,000
End
Formula Expression
Values in Expression
Value of i to be determined
20
IRR - Additional Example contd

• Plot the cashflow
• EMI in Advance

Begin
1
2
3
35
36
500,000
01.02.2006
01.03.2006
01.04.2006
01.12.2008
01.01.2009


-15,000
-15,000
-15,000
-15,000
-15,000
-15,000
End
01.01.2006
Formula Expression
Values in Expression
Value of i to be determined
21
BOND VALUATION
22
Objectives

• Distinguish bonds coupon rate, current yield,
  yield to maturity
• Interest rate risk
• Bond ratings and investors demand for appropriate
  interest rates

23
Bond characteristics

• Bond - evidence of debt issued by a body
  corporate or Govt. In India, Govt predominantly
• A bond represents a loan made by investors to
  the issuer. In return for his/her money, the
  investor receives a legaI claim on future cash
  flows of the borrower.
• The issuer promises to
• Make regular coupon payments every period until
  the bond matures, and
• Pay the face/par/maturity value of the bond when
  it matures

24
How do bonds work?

• If a bond has five years to maturity, an Rs.80
  annual coupon, and a Rs.1000 face value, its cash
  flows would look like this
• Time 0 1 2 3 4 5
• Coupons Rs.80 Rs.80 Rs.80 Rs.80 Rs.80
• Face Value 1000
• Market Price Rs.____
• How much is this bond worth? It depends on the
  level of current market interest rates. If the
  going rate on bonds like this one is 10, then
  this bond has a market value of Rs.924.18. Why?

25
Coupon payments
Face value
Maturity
Lump sum component
Annuity component
26
Bond prices and Interest Rates

• Interest rate same as coupon rate
• Bond sells for face value
• Interest rate higher than coupon rate
• Bond sells at a discount
• Interest rate lower than coupon rate
• Bond sells at a premium

27
Bond terminology

• Yield to Maturity
• Discount rate that makes present value of bonds
  payments equal to its price
• Current Yield
• Annual coupon divided by the current market price
  of the bond
• Current yield 80 / 924.18 8.66

28
Rate of return

• Rate of return
• Coupon income price change
• ----------------------------------------
• Investment
• e.g. you buy 6 bond at 1010.77 and sell next
  year at 1020
• Rate of return 609.33/1010.77 6.86

29
Risks in Bonds

• Interest rate risk
• Short term v/s long term
• Default risk
• Default premium

30
Bond pricing

• The following statements about bond pricing are
  always true.
• Bond prices and market interest rates move in
  opposite directions.
• When a bonds coupon rate is (greater than /
  equal to / less than) the markets required
  return, the bonds market value will be
  (greater than / equal to / less than) its par
  value.
• Given two bonds identical but for maturity, the
  price of the longer-term bond will change more
  (in percentage terms) than that of the
  shorter-term bond, for a given change in market
  interest rates.
• Given two bonds identical but for coupon, the
  price of the lower-coupon bond will change more
  (in percentage terms) than that of the
  higher-coupon bond, for a given change in market
  interest rates.

31
SAMPLING
32
Objectives

• Distinguish sample and population
• Sampling distributions
• Sampling procedures
• Estimation data analysis and interpretation
• Testing of hypotheses one sample data
• Testing of hypotheses two sample data

33
Pouplation and Sample
Population Sample
Definition Collection of items being considered Part or portion of population chosen for study
Characteristics and Symbols Parameters Population size N Population mean m Population standard deviation s Statistics Sample size n Sample mean x Sample standard deviation S
34
Types of sampling

• Non random or judgement
• Random or probability

35
Methods of sampling

• Sampling is the fundamental method of inferring
  information about an entire population without
  going to the trouble or expense of measuring
  every member of the population. Developing the
  proper sampling technique can greatly affect the
  accuracy of your results.

36
Random sampling

• Members of the population are chosen in such a
  way that all have an equal chance to be measured.
  
• Other names for random sampling include
  representative and proportionate sampling because
  all groups should be proportionately represented.

37
Types of Random sampling

• Simple random sampling
• Systematic Sampling Every kth member of the
  population is sampled.
• Stratified Sampling The population is divided
  into two or more strata and each subpopulation is
  sampled (usually randomly).
• Cluster Sampling A population is divided into
  clusters and a few of these (often randomly
  selected) clusters are exhaustively sampled.
• Stratified v/s cluster
• Stratified when each group has small variation
  withn itself but if there is wide variation
  between groups
• Cluster when there is considerable variation
  within each group but groups are similar to each
  other

38
Sampling from Normal Populations

• Sampling Distribution of the mean
• the probability distribution of     sample
  means, with all samples     having the same
  sample size n.
• Standard error of mean for infinite populations
• sx s/n1/2
• Standard Normal probability distribution

39
Definitions

• Density Curve (or probability density
    function) the graph of a continuous probability
    distribution
• The total area under the curve must equal 1.
• Every point on the curve must have a vertical
  height that is 0 or greater.

40
Because the total area under the density curve is
equal to 1, there is a correspondence between
area and probability.
41
Definition

• Standard Normal Deviation
• a normal probability distribution that has a
• mean of 0 and a standard deviation of 1

42
Definition

• Standard Normal Deviation
• a normal probability distribution that has a
• mean of 0 and a standard deviation of 1

Area 0.3413
Area
0.4429
z 1.58
0
Score (z )
43
Table A-2 Standard Normal Distribution

• µ 0 ? 1

0 x
z
44
Table for Standard Normal (z) Distribution
z
.00 .01 .02 .03 .04
.05 .06 .07 .08 .09
.0239 .0636 .1026 .1406 .1772 .2123 .2454 .2764 .3
051 .3315 .3554 .3770 .3962 .4131 .4279 .4406 .451
5 .4608 .4686 .4750 .4803 .4846 .4881 .4909 .4931
.4948 .4961 .4971 .4979 .4985 .4989
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.
2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4
2.5 2.6 2.7 2.8 2.9 3.0
.0000 .0398 .0793 .1179 .1554 .1915 .2257 .2580 .2
881 .3159 .3413 .3643 .3849 .4032 .4192 .4332 .445
2 .4554 .4641 .4713 .4772 .4821 .4861 .4893 .4918
.4938 .4953 .4965 .4974 .4981 .4987
.0040 .0438 .0832 .1217 .1591 .1950 .2291 .2611 .2
910 .3186 .3438 .3665 .3869 .4049 .4207 .4345 .446
3 .4564 .4649 .4719 .4778 .4826 .4864 .4896 .4920
.4940 .4955 .4966 .4975 .4982 .4987
.0080 .0478 .0871 .1255 .1628 .1985 .2324 .2642 .2
939 .3212 .3461 .3686 .3888 .4066 .4222 .4357 .447
4 .4573 .4656 .4726 .4783 .4830 .4868 .4898 .4922
.4941 .4956 .4967 .4976 .4982 .4987
.0120 .0517 .0910 .1293 .1664 .2019 .2357 .2673 .2
967 .3238 .3485 .3708 .3907 .4082 .4236 .4370 .448
4 .4582 .4664 .4732 .4788 .4834 .4871 .4901 .4925
.4943 .4957 .4968 .4977 .4983 .4988
.0160 .0557 .0948 .1331 .1700 .2054 .2389 .2704 .2
995 .3264 .3508 .3729 .3925 .4099 .4251 .4382 .449
5 .4591 .4671 .4738 .4793 .4838 .4875 .4904 .4927
.4945 .4959 .4969 .4977 .4984 .4988
.0199 .0596 .0987 .1368 .1736 .2088 .2422 .2734 .3
023 .3289 .3531 .3749 .3944 .4115 .4265 .4394 .450
5 .4599 .4678 .4744 .4798 .4842 .4878 .4906 .4929
.4946 .4960 .4970 .4978 .4984 .4989
.0279 .0675 .1064 .1443 .1808 .2157 .2486 .2794 .3
078 .3340 .3577 .3790 .3980 .4147 .4292 .4418 .452
5 .4616 .4693 .4756 .4808 .4850 .4884 .4911 .4932
.4949 .4962 .4972 .4979 .4985 .4989
.0319 .0714 .1103 .1480 .1844 .2190 .2517 .2823 .3
106 .3365 .3599 .3810 .3997 .4162 .4306 .4429 .453
5 .4625 .4699 .4761 .4812 .4854 .4887 .4913 .4934
.4951 .4963 .4973 .4980 .4986 .4990
.0359 .0753 .1141 .1517 .1879 .2224 .2549 .2852 .3
133 .3389 .3621 .3830 .4015 .4177 .4319 .4441 .454
5 .4633 .4706 .4767 .4817 .4857 .4890 .4916 .4936
.4952 .4964 .4974 .4981 .4986 .4990


45
Example If a data reader has an average (mean)
reading of 0 units and a standard deviation of 1
unit and if one data reader is randomly selected,
find the probability that it gives a reading
between 0 and 1.58 units.
Area 0.4429
P ( 0 lt x lt 1.58 ) 0.4429
0 1.58

• That is 44.29 of the readings between 0 and
  1.58 degrees.

46
Central Limit Theorem

• 1. The random variable x has a distribution
  (which may or may not be normal) with mean µ and
  standard deviation ?.
• 2. Samples all of the same size n are randomly
  selected from the population of x values.

47
Central Limit Theorem
1. The distribution of sample x will, as the
sample size increases, approach a normal
distribution. 2. The mean of the sample means
will be the population mean µ. 3. The standard
deviation of the sample means will approach
??????????????
n

48
Practical Rules Commonly Used

• 1. For samples of size n larger than 30, the
  distribution of the sample means can be
  approximated reasonably well by a normal
  distribution. The approximation gets better as
  the sample size n becomes larger.
• 2. If the original population is itself normally
  distributed, then the sample means will be
  normally distributed for any sample size n (not
  just the values of n larger than 30).

49
REGRESSION - CORRELATION
50
Objectives

• Relationship between two or more variables
• Scatter diagrams
• Regression analysis
• Method of least squares

51
Regression

• Definition
• Regression Equation

52
Regression

• Definition
• Regression Equation

Given a collection of paired data, the regression
equation
algebraically describes the relationship between
the two variables

• Regression Line
• (line of best fit or least-squares line)

the graph of the regression equation
53
The Regression Equation

• x is the independent variable (predictor
  variable)

y is the dependent variable (response variable)

b0 y - intercept
y b0 b1x
b1 slope
y mx b
54
Notation for Regression Equation
Population Parameter
Sample Statistic

• y-intercept of regression equation ?0
  b0
• Slope of regression equation ?1
  b1
• Equation of the regression line y ?0
  ?1 x y b0 b1

x
55
Assumptions

• 1. We are investigating only linear
  relationships.
• 2. For each x value, y is a random variable
  having a normal (bell-shaped) distribution.
  All of these y distributions have the same
  variance. Also, for a given value of x, the
  distribution of y-values has a mean that lies
  on the regression line. (Results are not
  seriously affected if departures from normal
  distributions and equal variances are not too
  extreme.)

56
Definition

• Correlation
• exists between two variables when one of them is
  related to the other in some way

57
Assumptions

• 1. The sample of paired data (x,y) is a
  random sample.
• 2. The pairs of (x,y) data have a
  bivariate normal distribution.

58
Definition

• Scatterplot (or scatter diagram)
• is a graph in which the paired (x,y) sample data
  are plotted with a horizontal x axis and a
  vertical y axis. Each individual (x,y) pair is
  plotted as a single point.

59
Positive Linear Correlation
y
y
y
x
x
x
(a) Positive
(b) Strong positive
(c) Perfect positive
60
Negative Linear Correlation
y
y
y
x
x
x
(d) Negative
(e) Strong negative
(f) Perfect negative
61
No Linear Correlation
y
y
x
x
(h) Nonlinear Correlation
(g) No Correlation
62
TIME SERIES
63
Objectives

• Understanding four components of time series
• Compute seasonal indices
• Regression based techniques

64
Time series

• Group of data or statistical information
  accumulated at regular intervals

65
Variations in Time series

• Secular trend
• A persistent trend in a single direction. A
  market movement over the long term which does not
  reflect cyclical seasonal or technical factors.
• Cyclical fluctuation
• The term business cycle or economic cycle refers
  to the fluctuations of economic activity
  (business fluctuations) around its long-term
  growth trend. The cycle involves shifts over time
  between periods of relatively rapid growth of
  output (recovery and prosperity), and periods of
  relative stagnation or decline (contraction or
  recession).
• Seasonal variation
• Pattern of change within a year
• Irregular variation
• Unpredictable, changing in a random manner

66
Trend analysis

• To describe historical patterns
• Past trends will help us project future

67
LINEAR PROGRAMMING
68
Objectives

• Understanding Linear programming basics
• Graphic and Simplex methods

69
Linear Programming

• Problem formulation if
• All equations are linear
• Constraints are known and deterministic
• Variables should have non negative values
• Decision values are also divisible

70
Types of LP problems

• Maximisation
• Minimisation
• Transportation
• Decision making

71
Multiple Choice Questions
72

• If A invests Rs. 24 at 7 interest rate for 5
  years, total value at end of five years is
• 31.66
• 33.66
• 36.66
• 39.66

73

• If A invests Rs. 24 at 7 interest rate for 5
  years, total value at end of five years is
• 31.66
• 33.66
• 36.66
• 39.66

74

• What is the effective annual rate of 12
  compounded semiannually?
• A) 11.24
• B) 12.00
• C) 12.36
• D) 12.54

75

• What is the effective annual rate of 12
  compounded semiannually?
• A) 11.24
• B) 12.00
• C) 12.36
• D) 12.54

76

• What is the effective annual rate of 12
  compounded continuously?
• A) 11.27
• B) 12.00
• C) 12.68
• D) 12.75

77

• What is the effective annual rate of 12
  compounded continuously?
• A) 11.27
• B) 12.00
• C) 12.68
• D) 12.75

78

• A study is done to see if there is a linear
  relationship between the life expectancy of an
  individual and the year of birth. The year of
  birth is the ______________.
• A. Unable to determine
• B. dependent variable
• C. independent variable

79

• A study is done to see if there is a linear
  relationship between the life expectancy of an
  individual and the year of birth. The year of
  birth is the ______________.
• A. Unable to determine
• B. dependent variable
• C. independent variable

80

• Which of the following is an example of using
  statistical sampling? a. Statistical sampling
  will be looked upon by the courts as providing
  superior audit evidence. b. Statistical sampling
  requires the auditor to make fewer judgmental
  decisions.
• c. Statistical sampling aids the auditor in
  evaluating results. d. Statistical sampling is
  more convenient to use than nonstatistical
  sampling.

81

• Which of the following is an example of using
  statistical sampling? a. Statistical sampling
  will be looked upon by the courts as providing
  superior audit evidence. b. Statistical sampling
  requires the auditor to make fewer judgmental
  decisions.c. Statistical sampling aids the
  auditor in evaluating results. d. Statistical
  sampling is more convenient to use than
  nonstatistical sampling.

82

• Which of the following best illustrates the
  concept of sampling risk? a. An auditor may
  select audit procedures that are not appropriate
  to achieve the specific objective. b. The
  documents related to the chosen sample may not be
  available for inspection. c. A randomly chosen
  sample may not be representative of the
  population as a whole.d. An auditor may fail to
  recognize deviations in the documents examined.

83

• Which of the following best illustrates the
  concept of sampling risk? a. An auditor may
  select audit procedures that are not appropriate
  to achieve the specific objective. b. The
  documents related to the chosen sample may not be
  available for inspection. c. A randomly chosen
  sample may not be representative of the
  population as a whole. d. An auditor may fail
  to recognize deviations in the documents
  examined.

84

• The advantage of using statistical sampling
  techniques is that such techniquesa.
  Mathematically measure risk.
• b. Eliminate the need for judgmental decisions.
  c. Are easier to use than other sampling
  techniques. d. Have been established in the
  courts to be superior to nonstatistical sampling.
  

85

• The advantage of using statistical sampling
  techniques is that such techniquesa.
  Mathematically measure risk. b. Eliminate the
  need for judgmental decisions. c. Are easier to
  use than other sampling techniques. d. Have been
  established in the courts to be superior to
  nonstatistical sampling.

86

• Time series methods a. discover a pattern in
  historical data and project it into the future.
  b. include cause-effect relationships. c. are
  useful when historical information is not
  available. d. All of the alternatives are true.
  

87

• Time series methods a. discover a pattern in
  historical data and project it into the future.
  b. include cause-effect relationships. c. are
  useful when historical information is not
  available. d. All of the alternatives are true.
  

88

• Gradual shifting of a time series over a long
  period of time is called a. periodicity. b.
  cycle. c. regression. d. trend.

89

• Gradual shifting of a time series over a long
  period of time is called a. periodicity. b.
  cycle. c. regression. d. trend.

90

• Seasonal components a. cannot be predicted. b.
  are regular repeated patterns.c. are long runs
  of observations above or below the trend line.
  d. reflect a shift in the series over time.

91

• Seasonal components a. cannot be predicted. b.
  are regular repeated patterns. c. are long runs
  of observations above or below the trend line.
  d. reflect a shift in the series over time.

92

• Short-term, unanticipated, and nonrecurring
  factors in a time series provide the random
  variability known as a. uncertainty. b. the
  forecast error. c. the residuals. d. the
  irregular component.

93

• Short-term, unanticipated, and nonrecurring
  factors in a time series provide the random
  variability known as a. uncertainty. b. the
  forecast error. c. the residuals. d. the
  irregular component.

94

• The focus of smoothing methods is to smooth a.
  the irregular component.
• b. wide seasonal variations. c. significant
  trend effects. d. long range forecasts.

95

• The focus of smoothing methods is to smooth a.
  the irregular component. b. wide seasonal
  variations. c. significant trend effects. d.
  long range forecasts.

96

• . Linear trend is calculated as Tt 28.5
  .75t.  The trend projection for period 15 is a.
  11.25 b. 28.50 c. 39.75 d. 44.25

97

• . Linear trend is calculated as Tt 28.5
  .75t.  The trend projection for period 15 is a.
  11.25 b. 28.50 c. 39.75 d. 44.25

98

• The forecasting method that is appropriate when
  the time series has no significant trend,
  cyclical, or seasonal effect is a. moving
  averages
• b. mean squared error c. mean average deviation
  d. qualitative forecasting methods

99

• The forecasting method that is appropriate when
  the time series has no significant trend,
  cyclical, or seasonal effect is a. moving
  averages b. mean squared error c. mean average
  deviation d. qualitative forecasting methods

100
Thank You