Quantitative Techniques and Financial Mathematics CAIIB-Financial Management

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Quantitative Techniques andFinancial
MathematicsCAIIB-Financial Management Module A

• C.S.BALAKRISHNAN
• Faculty Member
• S.P.B.T.COLLEGE

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Concept Of Time Value Of Money,Net Present
Value,Discounted Cash Flow
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• Present value is a concept which shows that money
  has time value.
• Dealing with cash flows at different points of
  time can be made easier using a time line that
  shows both the value and timing of cash flows.
• Cash inflows are called positive cash flows and
  cash outflows are called negative negative cash
  flows.
• Discount rate is a rate at which present and
  future cash flows are traded off.

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• The process of discounting future cash flows
  converts them into cash flows in present value
  terms.
• The process of compounding converts present cash
  flows into future cash flows.
• The present value of Rs.1,00,000 a year from now
  must be less thanRs.1,00,000 today.
• Present value(pv)Discount factor x C1
• c1cash flow at time t.
• PV CFt/(1r)
• Discount factor1/(1r)

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• Suppose you have two options of investment
• Option a-Investing in a property worth Rs.4lac
  today and your investment is expected to go upto
  Rs.5lac.
• Option b-Invest in PPF to receive Rs.5 lac
  after a year say _at_9.One has to invest
  Rs.5lac/1.09 which is Rs.4,58,715.59.
• Thus one can infer that option a is better
  than option b in the above case.

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• To calculate the present value,we discount the
  expected payoff by the rate of return offered by
  equivalent investment alternatives in the capital
  or financial markets.This rate of return is often
  called as discount rate,hurdle rate or
  oppurtunity cost of capital.
• It is referred to as oppurtunity cost since it is
  the return forgone by investing in the project
  rather than investing in the securities.In our
  example the oppurtunity cost was 9.Present value
  was obtained by dividing Rs.5lac by 1.09.

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• PVDiscount Factor x C11/(1r) x C1
• 5LAC/1.09Rs.4,58,715.59
• Net Present ValuePV-Required Investment
• NPVRs.4,58,715.59-Rs.4,00,00Rs.58,715.59
• NPVC0C1/(1r).Where C0 is cash flow today
  which will be negative.
• Relation of risk to present value-We do not use
  the same discount factor while comparing
  alternative investment avenues.
• The discount rate for PPF may be 9,or 0.09 but
  discount rate for the building property may be
  11 or0.11.Only after present values are
  calculated using two different discount rates is
  the best investment avenue or project decided.

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• ReturnProfit/Investment
• (Rs.5 lac-Rs.4.30lac)/Rs.4.30 lac.
• Rs.70,000/Rs.4.30lac
• 0.163 about 16
• This cost of capital is once again the return
  foregone by not investing in securities.If the
  office building is as risky as investing in stock
  market securities where the expected return is
  14 then the return forgone is 14.Since the 16
  return on the office building exceeds the 14
  oppurtunity cost,one should go ahead with the
  project.

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• Net Present Value Rule-Accept investments that
  have positive net present values.
• Rate of Return Rule-Accept investments that offer
  rates of return in excess of their oppurtunity
  cost of capital.
• Discounting a cash flow converts it into
  present value rupees and enables the user to
  aggregate and compare.other things remaining
  equal,the present value of a cash flow will
  decrease as the discount rate increases and
  continues to decrease further into future cash
  flows.

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• Rule of 72-It is a shortcut to estimating the
  compounding effect.A cash flow growing at 6 will
  double in value in approximately 12 years,while a
  cash flow at 9 will double in value in
  approximately 8 years.
• Effective Interest Rate
• (1Stated Annual Interest Rate) -1
• N
• Where Nno.ofcompounding periods
• eg10annual int.rate,if there is semi annual
  compounding works out to an effective rate of
• (1.052-1)(1.1.025-1.0)10.25

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• Effect of compounding frequency on effective
  interest rate
• Frequency Rate() T Formula Effective
• 
  Annual Rate
• 
  ()
• Annual 10 1 10 10
• Semi-annual10 2 (110/2)2-110.25
• Monthly 10 12(110/12)12-110.47
• Daily 10 365(110/365)36510.5156
• Continous 10 continous e10-1 10.5171

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• As compounding becomes more frequent,the
  effective rate increases,and the present value of
  future cash flows decreases.
• Under rule 72,how long will it take for an
  investment to quadruple in value,if the interest
  rate is 12?
• a)10 years b)15 years c)12 years d)17 years
• In order to avoid taxes,my grandfather,starts
  giving me gifts of Rs1lac for the next 10
  years.If the interest rate is 6,how much will I
  get at the end of 10 years?
• a)10 lac b)15 lac c)12 lac d)18 lac

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• A quarterly repayments of a loan carry an
  interest rate of 8 per annum.What is the
  effective annual rate of interest?
• a)7.24 b)6.25 c)8.24 d)9.24
• Find the interest rate?Present value is
  Rs100.Future value becomes Rs.115.76 in 3 years.
• a)7 b)6 c)8 d)9
• If I take a loan of Rs8,000 and repay Rs225 per
  month,for 4 years,what is the effective annual
  rate on the loan?
• a)15.25 b)15.35 c)15.58 d)15.45

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• Sampling Methods-Presentation of data
• analysis and interpretation of data-
• Hypothesis testing

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• Sampling is the integral tool of the quantitative
  methods we use.
• To take a sample from an entire population and
  use it to describe a population.
• To make sure the samples you take are an accurate
  representation of the population they come from.
• To introduce the concepts of sampling
  distribution.
• To understand the trade offs between costs of
  larger samples and accuracy.
• To introduce experimental design-Sampling
  procedures-more information at least cost.

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• Estimation-Data analysis and interpretation.
• Testing of hypotheses- sample data.
• Four methods of sampling
• -Simple Random Sampling
• -Systematic Sampling
• -Stratified Sampling
• -Cluster Sampling
• Central Limit Theorem
• The relationship between the shape of the
• population distribution and the shape of the
• sampling distribution of the mean is called The
• Central Limit Theorem.

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• The central limit theorem is perhaps the most
  important theorem in all of statistical
  inference.It assures that the sampling
  distribution of mean approaches normal as the
  sample size increases.
• Which of the following is a method of selecting
  samples from a population?
• a)Judgement sampling b)Random sampling
  c)Probability sampling d)All of the above

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• In random sampling,we can describe mathematically
  how objective our estimates are.Why is this?
• a)We always know the chance that any
• population element will be included in the
• sample.
• b)Every sample always has an equal chance
• of being selected.
• c)All the samples are exactly the same size
• and can be counted.
• d)a and b but not c

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• Suppose you are performing stratified sampling on
  a particular population and have divided it into
  strata of different sizes.How can you now make
  your sample selection?
• a)Select at random an equal number of
• elements.
• b)Draw equal numbers of elements from each
• stratum and weigh the results.
• c)Draw numbers of elements from each
• stratum proportional to their weights in
  the
• population.
• d)(b) (c) only.

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• The dispersionamong sample means is less than the
  dispersion among the sampled items themselves
  because
• a)Each sample is smaller than the
• population from which it is drawn.
• b)Very large values are averaged down
• and very small values are averaged up.
• c) The sampled items are all drawn from
• the same population.
• d)None of these.

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• Suppose that a population with N144 has
  µ24.What is the mean of the sampling
  distribution of the mean for samples of size 25?
• a)24 b)2 c)4.8 d)Cannot be
  determined from the available information.
• The central limit theorem assures us that the
  sampling distribution of the mean is
• a)Always normal b)Always normal for
• large sample size c)Approaches normality as
  sample size increases d) Appears normal only when
  N is greater than 1,000

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• A broader patrol checkpoint that
• stops every passenger van is using
• a)Simple Random Sampling
• b)Systematic sampling
• c)Stratified sampling
• d)Complete enumeration
• A portion of the elements in a population
• chosen for direct examination or measurement
• is a --------.(sample)

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• The proportion of population contained in a
  sample is the ---------(sampling fraction)
• ------- sample should be used when each group
  considered has small variation within itself but
  there is wide variation between different
  groups.(stratified).
• --------is the degree of accuracy with which the
  sample mean can estimate the population.(precision
  )
• Within a population,groups that are similar to
  each other are called as--------(clusters)

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• Determine the sample size if standard deviation
  is 6,population mean is 25 and sample mean is
  23.The desired degree of precision is 99.
• a)60 b) 75 c) 90 d) 45
• A sample size of 90 values has a mean 55 and
  standard deviation 3.A second sample of 110
  values has mean 60 and standard deviation 2.Find
  the mean and standard deviation of the combined
  sample of 200 values.
• a)54.52.812 b)624.345 c)57.753.526
• d)61.903.89

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• REGRESSION CORRELATION-TIME SERIES

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• Regression is the measure of the average
  relationship between two or more
  variables.Regression analysis refers to the
  methods by which estimates are made of the values
  of a variable from a knowledge of the values of
  one or more variables.The study of the functional
  relationship between variables provides a
  mechanism for prediction,estimating.

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• Variable which is used to predict the variable of
  interest is called independent variableand the
  variable we are trying to predict is called
  dependent variable.Generally,independent
  variable is denoted by x and the dependent
  variable is denoted by y.
• Simple linear regression analysis -Only one
  independent variable is used.We assume linear
  relationship between variables,this is a linear
  analysis.Linear means that the equation is in a
  straight line form,like Yaxb
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  

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• Uses of regression analysis-
• -To estimate the relationship between
• economic variable like price,demand, etc.
• -Estimating errors in prediction of the
• dependent variable.
• -We can calculate the coefficient of
  coorelation.
• Coefficient of determination is the square of
  coefficient of correlation.This measures the
  degree of correlation exists between two
  variables.

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• Assume the principal of a college wants to find
  out whether there is a relationship between the
  entrance examination score to a college and final
  graduation GPA score of a student.
• Student A B C D E F G H
• Entrance
• Exam 74 69 85 63 82 60 79 91
• Cumulative
• GPA 2.6 2.2 3.4 2.3 3.1 2.1 3.2 3.8

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• When we view these points together,we see that we
  can fit a line through these scattered
  points.We try to draw the line in such a way,that
  an equal number of points lie on either side of
  line.As X increases (independent
  variable-entrance score),Y also
  increases(dependent variable-cumulative GPA
  score).We can say that there is a direct linear
  relationship.

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• Curvilinear Relationship
• In many industries we have heard of
  curvilinear relationship.The principle is that as
  the employees produce more and more of a new
  product ,the time required to produce one unit
  decreases by some fixed proportion as the total
  number of unit doubles.
• No.of planes produced 5 10 20 40
• No.of hours per plane 1000 800 640 512

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• TIME SERIES

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• The first step in making the estimates for future
  consists of gathering information from the
  past.Data is collected,and recorded at successive
  intervals of time.Such data are called time
  series.
• Time series helps in understanding past
  behaviour.It also helps in planning future
  operations as well as current operations.The
  actuals can be compared with expected results and
  cause of variation can be analysed.

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• It is customary to classify the fluctuations of a
  time series into four basic types of variations.
• a)Secular trend b)Seasonal variation
• c)Cyclical variation d)Irregular variation.
• Methods of trend analysis
• -Free hand graphic method.
• -Semi-average method.
• -Moving average method.
• -Method of least squares.

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• Light bulbs are manufactured in a factory and
  have a mean life of 540 hours and std.deviation
  of 50 hours.Calculate the fraction of bulbs that
  have less than 500 hours.
• a)0.54 b)0.2119 c)0.4234 d)0.3555
• The fraction of bulbs having life between
• 500 to 600 hours.
• a)0.6730 b)0.5478 c)0.7734 d)0.3487

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• The mean and standard deviation of marks of
  students of a class are 55 and 8
  respectively.Within what interval centred around
  the mean do at least 90 of marks lie?
• a)40 and 65 b)39 and 61 c)42 and 68
• d)30 and 72
• A binomial distribution has n20 and p0.3.Find
  the mean and the variance of the distribution.
• a)8 and 3.4 b)4 and 2.6 c)6 and 4.2 d)12
  and 4.4

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• Customer accounts at a certain cooperative bank
  have an average balance of Rs 4,800 and a
  Standard Deviation of Rs.1,600.Assume the account
  balances are normally distributed.Calculate
• a)What proportion of the accounts is over
• Rs.6,000?
• a)32 b)20 c)15.44 d)22.67
• b)What proportion of the accounts are
• between Rs4,000 and Rs.6,000?
• a)46.49 b)52.34 c)62.33 d)34.65

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• C)What proportion of the accounts is between
  Rs.2,400 and Rs.3,600 ?
• a)32.45 b)15.98 c)22.50 d)10.44.
• State whether true or false
• Secular trend refers to long term of data.
• Regular variations include only seasonal
• variations.
• Yearly data are independent of the effect
• of seasonal variations.
• The period of seasonal variations is always
• one year.

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• Series of figures arranged in a chronological
• Order are called
• a) Time series b)Trend c)Linear d)Best
• Which is an irreversible movement and continues
  in the same direction for considerable period of
  time
• a) Trend b)Time series c)Best d)Linear.
• The trend equation fitted by the method of least
  squares is known as the equation
  of-----fit.a)Linear b)Best c)Trend d)Time series

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• In the case of a -------trend,successive
  observations differ by a constant number.
• a)Time series b)Trend c)Best d)Linear
• If in a year 20,000 boats are rented
  out,average per quarter should be Rs.5,000.If the
  index for spring quarter is 142,then we estimate
  the number of boats rented out during the summer
  will be 5,000X(142/100)
• 7,100.

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• Probability distribution-confidence interval
  analysis-Estimating parameters of distribution

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• A point estimate is a single number that is used
  to estimate an unknown population
• An interval estimate is a range of values used to
  estimate a population parameter.
• Any sample statistic that is used to estimate a
  population parameter is called an estimator which
  is a sample statistic used to estimate a
  population parameter.
• An interval estimate describes a range of values
  within which a population parameter is likely to
  lie.

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• Bond Valuation

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• Bond prices vary inversely to change in interest
  rates.
• When the interest rate increases,and goes higher
  than the coupon rate,then the bond value
  decreases.This is because the present value of
  payments received decrease,leading to a fall in
  bond prices.
• When the interest rate decreases,and becomes
  lower than the coupon rate,then the present value
  of payments increases and the bonds market price
  increases.

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• When the interest rate is equal to the coupon
  rate,the market price of the bond is equal to
  face value.In this case,the bond is selling at
  par.
• When the market price of bond is greater than its
  face value,it is said that the bond is selling at
  a premium.
• RBI issues a bond with a par value of
  Rs.1000,coupon rate 10 and the maturity period
  is 10years.What will happen when interest rate
  increases?
• The price of the bond will fall.Coupon rate

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• Will not change .Yield to maturity
• increases.The interest payment received
• each year is only Rs.100 (no change).
• What is interest rate risk?
• A fall in interest rates will result in increase
  in
• bond prices.A rise in interest rates will result
  in
• decrease in bond prices.

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• A Rs.100 par value bond,bearing a coupon rate of
  12 will mature in 8 years.the required rate of
  return on this bond is 14.What is the value of
  this bond?
• a)92.87 b)90.77 c)102.76 d)78.89
• A Rs.1000 par value bond has a coupon rate of 14
  will mature after 5 years.The required rate of
  return on this bond is 13.What is the value of
  this bond?
• a)1035.4 b)1200.5 c)945.88 d)1060

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• LINEAR PROGRAMMING
• DECISION MAKING

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• Linear programming is concerned with efficient
  allocation of limited sources to known
  activities,with the objective of meeting desired
  goals such as maximizing profits or minimizing
  cost.
• Objective function is one in which we mention the
  objective quantitatively and express it as a
  linear function of the variables.It can be for
  maximizing profits or minimizing costs.
• Optimal solution satisfies all the given
• constraints.

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• Sensitivity analysis-It refers to the study of
  effect of changes in various parameters
• (constraints)on the optimal solution.
• State whether following statements are true or
  false
• a)The distinctive characteristic of linear
• programming models is that the functions
• representing the objective and the
• constraints are linear.

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• Non negativity constraints means the products are
  not produced.
• Inequality means that the capacity of each
  operation should not be exceeded.
• For finding out the optimal mix and the
  corresponding profit,linear programming method is
  not useful.

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• SIMULATION

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• Simulation is an imitation of reality.A number of
  experiments is performed on simulated models to
  determine the behaviour of the real
  system.Example
• Testing of aircraft models in wind tunnel,
• Planetarium shows represent a simulation
• of the planet system.
• Advantages of simulation
• -We can foresee difficulties and bottlenecks
  
• which may come up in real system.
• -This eliminates costly and risky trials

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• This allows experimenting with a model of the
  system without interfering with real system.
• Simulation models are comparatively flexible and
  can be modified to accommodate the changing
  environment.
• Monte Carlo method is a technique that involves
  using random numbers and probability to solve
  problems.The term Monte Carlo Method was coined
  by S.Ulam and Nicholas Metropolis in reference to
  games of chance,a popular attraction in Monte
  Carlo,Monaco.

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• State true or false
• -Simulation is a replica of real life.
• -Random numbers are not necessary for
• inputs in a simulation model.
• -Simulation is used when the problem is not
• complex,and there is a linear relationship
• between the variables.
• -Simulation involves many iterations,and
• computers are useful in solving them.

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• objective questions

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