Download Quantitative Techniques and Financial Mathematics

Quantitative Techniques
and
Financial Mathematics
C.S.BALAKRISHNAN
Faculty Member
S.P.B.T.COLLEGE
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
c1=cash flow at time t.
PV= CFt/(1+r)
Discount factor=1/(1+r)
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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 @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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PV=Discount Factor x C1=1/(1+r) x C1
=5LAC/1.09=Rs.4,58,715.59
Net Present Value=PV-Required Investment
NPV=Rs.4,58,715.59-Rs.4,00,00=Rs.58,715.59
NPV=C0+C1/(1+r).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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Return=Profit/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.
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=
(1+Stated Annual Interest Rate) -1
N
Where N=no.ofcompounding periods
eg:10%annual int.rate,if there is semi annual
compounding works out to an effective rate of
=(1.05^2-1)=(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 (1+10/2)^2-1=10.25
Monthly
10
12(1+10/12)^12-1=10.47
Daily
10
365(1+10/365)^365=10.5156
Continous 10
continous e^10-1= 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%
Sampling Methods-Presentation of data
analysis and interpretation of dataHypothesis 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.
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 N=144
has µ=24.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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broader patrol checkpoint that
stops every passenger van is using
a)Simple Random Sampling
b)Systematic sampling
c)Stratified sampling
d)Complete enumeration
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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)54.5;2.812 b)62;4.345 c)57.75;3.526
d)61.90;3.89
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 variable’and
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 Y=ax+b
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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
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 n=20 and
p=0.3.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%
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.
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.
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 bond’s 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
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
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.
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
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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.
objective questions