Quantitative Techniques Quantitative Techniques andand

Financial MathematicsFinancial Mathematics

CAIIB-Financial Management CAIIB-Financial Management –Module A–Module AC.S.BALAKRISHNANC.S.BALAKRISHNAN

Faculty MemberFaculty MemberS.P.B.T.COLLEGES.P.B.T.COLLEGE

Concept Of Time Value Of Concept Of Time Value Of Money,Money,

Net Present Value,Net Present Value,Discounted Cash FlowDiscounted Cash Flow

Present value is a concept which Present value is a concept which shows that money has time value.shows that money has time value.

Dealing with cash flows at different Dealing with cash flows at different points of time can be made easier points of time can be made easier using a time line that shows both the using a time line that shows both the value and timing of cash flows.value and timing of cash flows.

Cash inflows are called positive cash Cash inflows are called positive cash flows and cash outflows are called flows and cash outflows are called negative negative cash flows.negative negative cash flows.

Discount rate is a rate at which Discount rate is a rate at which present and future cash flows are present and future cash flows are traded off.traded off.

The process of discounting future cash The process of discounting future cash flows converts them into cash flows in flows converts them into cash flows in present value terms.present value terms.

The process of compounding converts The process of compounding converts present cash flows into future cash flows.present cash flows into future cash flows.

The present value of Rs.1,00,000 a year The present value of Rs.1,00,000 a year from now must be less thanRs.1,00,000 from now must be less thanRs.1,00,000 today.today.

Present value(pv)=Discount factor x C1Present value(pv)=Discount factor x C1

c1=cash flow at time t.c1=cash flow at time t.

PV= CFt/(1+r)PV= CFt/(1+r)

Discount factor=1/(1+r)Discount factor=1/(1+r)

Suppose you have two options of Suppose you have two options of investmentinvestment

Option a-Investing in a property worth Option a-Investing in a property worth Rs.4lac today and your investment is Rs.4lac today and your investment is expected to go upto Rs.5lac.expected to go upto Rs.5lac.

Option b-Invest in PPF to receive Rs.5 lac Option b-Invest in PPF to receive Rs.5 lac after a year say @9%.One has to invest after a year say @9%.One has to invest Rs.5lac/1.09 which is Rs.4,58,715.59.Rs.5lac/1.09 which is Rs.4,58,715.59.

Thus one can infer that option a is better Thus one can infer that option a is better than option b in the above case.than option b in the above case.

To calculate the present value,we To calculate the present value,we discount the expected payoff by the rate discount the expected payoff by the rate of return offered by equivalent of return offered by equivalent investment alternatives in the capital or investment alternatives in the capital or financial markets.financial markets.This rate of return is This rate of return is often called as discount rate,hurdle rate often called as discount rate,hurdle rate or oppurtunity cost of capital.or oppurtunity cost of capital.

It is referred to as oppurtunity cost since It is referred to as oppurtunity cost since it is the return forgone by investing in the it is the return forgone by investing in the project rather than investing in the project rather than investing in the securities.In our example the oppurtunity securities.In our example the oppurtunity cost was 9%.Present value was obtained cost was 9%.Present value was obtained by dividing Rs.5lac by 1.09. by dividing Rs.5lac by 1.09.

PV=Discount Factor x C1=1/(1+r) x C1PV=Discount Factor x C1=1/(1+r) x C1 =5LAC/1.09=Rs.4,58,715.59=5LAC/1.09=Rs.4,58,715.59 Net Present Value=PV-Required InvestmentNet Present Value=PV-Required Investment NPV=Rs.4,58,715.59-Rs.4,00,00=Rs.58,715.59 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 NPV=C0+C1/(1+r).Where C0 is cash flow today

which will be negative.which will be negative. Relation of risk to present value-We do not use Relation of risk to present value-We do not use

the same discount factor while comparing the same discount factor while comparing alternative investment avenues.alternative investment avenues.

The discount rate for PPF may be 9%,or 0.09 The discount rate for PPF may be 9%,or 0.09 but discount rate for the building property may but discount rate for the building property may be 11% or0.11.Only after present values are be 11% or0.11.Only after present values are calculated using two different discount rates is calculated using two different discount rates is the best investment avenue or project decided.the best investment avenue or project decided.

Return=Profit/InvestmentReturn=Profit/Investment

=(Rs.5 lac-Rs.4.30lac)/Rs.4.30 lac.=(Rs.5 lac-Rs.4.30lac)/Rs.4.30 lac.

=Rs.70,000/Rs.4.30lac=Rs.70,000/Rs.4.30lac

=0.163 about 16%=0.163 about 16%

This cost of capital is once again the This cost of capital is once again the return foregone by not investing in return foregone by not investing in securities. If the office building is as risky securities. If the office building is as risky as investing in stock market securities as investing in stock market securities where the expected return is 14% then the where the expected return is 14% then the return forgone is 14%.Since the 16% return forgone is 14%.Since the 16% return on the office building exceeds the return on the office building exceeds the 14% oppurtunity cost,one should go ahead 14% oppurtunity cost,one should go ahead with the project. with the project.

Net Present Value Rule-Accept Net Present Value Rule-Accept investments that have positive net investments that have positive net present values.present values.

Rate of Return Rule-Accept investments Rate of Return Rule-Accept investments that offer rates of return in excess of that offer rates of return in excess of their oppurtunity cost of capital.their oppurtunity cost of capital.

Discounting a cash flow converts it into Discounting a cash flow converts it into present value rupees and enables the present value rupees and enables the user to aggregate and compare.other user to aggregate and compare.other things remaining equal,the present value things remaining equal,the present value of a cash flow will decrease as the of a cash flow will decrease as the discount rate increases and continues to discount rate increases and continues to decrease further into future cash flows. decrease further into future cash flows.

Rule of 72-It is a shortcut to estimating the Rule of 72-It is a shortcut to estimating the compounding effect.A cash flow growing at compounding effect.A cash flow growing at 6% will double in value in approximately 6% will double in value in approximately 12 years,while a cash flow at 9% will 12 years,while a cash flow at 9% will double in value in approximately 8 years.double in value in approximately 8 years.

Effective Interest Rate=Effective Interest Rate=

(1+Stated Annual Interest Rate)(1+Stated Annual Interest Rate) -1 -1

NN

Where N=no.ofcompounding periodsWhere N=no.ofcompounding periods

eg:10%annual int.rate,if there is semi eg:10%annual int.rate,if there is semi annual compounding works out to an annual compounding works out to an effective rate ofeffective rate of

=(1.05^2-1)=(1.1.025-1.0)=10.25%=(1.05^2-1)=(1.1.025-1.0)=10.25%

Effect of compounding frequency on Effect of compounding frequency on effective interest rateeffective interest rate

Frequency Rate(%) T Formula Effective Frequency Rate(%) T Formula Effective

Annual RateAnnual Rate

(%)(%)

Annual 10 1 10 10Annual 10 1 10 10

Semi-annual10 2 (1+10/2)^2-1=10.25Semi-annual10 2 (1+10/2)^2-1=10.25

Monthly 10 12(1+10/12)^12-1=10.47Monthly 10 12(1+10/12)^12-1=10.47

Daily 10 Daily 10 365(1+10/365)^365=10.5156365(1+10/365)^365=10.5156

Continous 10 continous e^10-1= 10.5171Continous 10 continous e^10-1= 10.5171

As compounding becomes more frequent, As compounding becomes more frequent, the effective rate increases, and the the effective rate increases, and the present value of future cash flows present value of future cash flows decreases.decreases.

Under rule 72,how long will it take for an Under rule 72,how long will it take for an investment to quadruple in value ,if the investment to quadruple in value ,if the interest rate is 12%?interest rate is 12%?

a)10 years b)15 years c)12 years d)17 a)10 years b)15 years c)12 years d)17 yearsyears

In order to avoid taxes, my grandfather, In order to avoid taxes, my grandfather, starts giving me gifts of Rs1lac for the next starts giving me gifts of Rs1lac for the next 10 years. If the interest rate is 6%,how 10 years. If the interest rate is 6%,how much will I get at the end of 10 years?much will I get at the end of 10 years?

a)10 lac b)15 lac c)12 lac d)18 laca)10 lac b)15 lac c)12 lac d)18 lac

A quarterly repayments of a loan carry an A quarterly repayments of a loan carry an interest rate of 8% per annum. What is interest rate of 8% per annum. What is the effective annual rate of interest?the effective annual rate of interest?

a)7.24% b)6.25% c)8.24% d)9.24%a)7.24% b)6.25% c)8.24% d)9.24% Find the interest rate? Present value is Find the interest rate? Present value is

Rs100.Future value becomes Rs.115.76 in Rs100.Future value becomes Rs.115.76 in 3 years.3 years.

a)7% b)6% c)8% d)9%a)7% b)6% c)8% d)9% If I take a loan of Rs8,000 and repay If I take a loan of Rs8,000 and repay

Rs225 per month, for 4 years, what is the Rs225 per month, for 4 years, what is the effective annual rate on the loan?effective annual rate on the loan?

a)15.25% b)15.35% c)15.58% a)15.25% b)15.35% c)15.58% d)15.45%d)15.45%

Sampling Methods-Presentation of Sampling Methods-Presentation of datadata

analysis and interpretation of data-analysis and interpretation of data-

Hypothesis testingHypothesis testing

Sampling is the integral tool of the Sampling is the integral tool of the quantitative methods we use.quantitative methods we use.

To take a sample from an entire To take a sample from an entire population and use it to describe a population and use it to describe a population.population.

To make sure the samples you take are an To make sure the samples you take are an accurate representation of the population accurate representation of the population they come from.they come from.

To introduce the concepts of sampling To introduce the concepts of sampling distribution.distribution.

To understand the trade offs between To understand the trade offs between costs of larger samples and accuracy.costs of larger samples and accuracy.

To introduce experimental design-To introduce experimental design-Sampling procedures-more information at Sampling procedures-more information at least cost. least cost.

Estimation-Data analysis and interpretation.Estimation-Data analysis and interpretation. Testing of hypotheses- sample data.Testing of hypotheses- sample data.

Four methods of samplingFour methods of sampling

-Simple Random Sampling-Simple Random Sampling

-Systematic Sampling-Systematic Sampling

-Stratified Sampling-Stratified Sampling

-Cluster Sampling-Cluster Sampling

Central Limit TheoremCentral Limit Theorem

The relationship between the shape of the The relationship between the shape of the

population distribution and the shape of the population distribution and the shape of the

sampling distribution of the mean is called The sampling distribution of the mean is called The

Central Limit Theorem.Central Limit Theorem.

The central limit theorem is perhaps The central limit theorem is perhaps the most important theorem in all of the most important theorem in all of statistical inference.It assures that statistical inference.It assures that the sampling distribution of mean the sampling distribution of mean approaches normal as the sample approaches normal as the sample size increases.size increases.

Which of the following is a method of Which of the following is a method of selecting samples from a population?selecting samples from a population?

a)Judgement sampling b)Random a)Judgement sampling b)Random sampling c)Probability sampling d)All sampling c)Probability sampling d)All of the above of the above

In random sampling,we can describe In random sampling,we can describe mathematically how objective our estimates mathematically how objective our estimates are.Why is this?are.Why is this?

a)We always know the chance that any a)We always know the chance that any

population element will be included in the population element will be included in the

sample.sample.

b)Every sample always has an equal chance b)Every sample always has an equal chance

of being selected.of being selected.

c)All the samples are exactly the same size c)All the samples are exactly the same size

and can be counted.and can be counted.

d)a and b but not cd)a and b but not c

Suppose you are performing stratified Suppose you are performing stratified sampling on a particular population and have sampling on a particular population and have divided it into strata of different sizes. How divided it into strata of different sizes. How can you now make your sample selection?can you now make your sample selection?

a)Select at random an equal number of a)Select at random an equal number of

elements.elements.

b)Draw equal numbers of elements from each b)Draw equal numbers of elements from each

stratum and weigh the results.stratum and weigh the results.

c)Draw numbers of elements from each c)Draw numbers of elements from each

stratum proportional to their weights in the stratum proportional to their weights in the

population.population.

d)(b) & (c) only.d)(b) & (c) only.

The dispersion among sample means is The dispersion among sample means is less than the dispersion among the less than the dispersion among the sampled items themselves becausesampled items themselves because

a)Each sample is smaller than the a)Each sample is smaller than the

population from which it is drawn.population from which it is drawn.

b)Very large values are averaged down b)Very large values are averaged down

and very small values are averaged and very small values are averaged up.up.

c) The sampled items are all drawn c) The sampled items are all drawn from from

the same population. the same population.

d)None of these.d)None of these.

Suppose that a population with N=144 has Suppose that a population with N=144 has µ=24.What is the mean of the sampling µ=24.What is the mean of the sampling distribution of the mean for samples of distribution of the mean for samples of size 25?size 25?

a)24 b)2 c)4.8 d)Cannot be a)24 b)2 c)4.8 d)Cannot be determined from the available information.determined from the available information.

The central limit theorem assures us that The central limit theorem assures us that the sampling distribution of the mean is the sampling distribution of the mean is

a)Always normal b)Always normal for a)Always normal b)Always normal for

large sample size c)Approaches normality large sample size c)Approaches normality as sample size increases d) Appears as sample size increases d) Appears normal only when N is greater than 1,000normal only when N is greater than 1,000

A broader patrol checkpoint that A broader patrol checkpoint that

stops every passenger van is usingstops every passenger van is using

a)Simple Random Samplinga)Simple Random Sampling

b)Systematic samplingb)Systematic sampling

c)Stratified samplingc)Stratified sampling

d)Complete enumerationd)Complete enumeration

A portion of the elements in a population A portion of the elements in a population

chosen for direct examination or chosen for direct examination or measurement measurement

is a --------.(sample) is a --------.(sample)

The proportion of population contained The proportion of population contained in a sample is the ---------(sampling in a sample is the ---------(sampling fraction)fraction)

------- sample should be used when each ------- sample should be used when each group considered has small variation group considered has small variation within itself but there is wide variation within itself but there is wide variation between different groups.(stratified).between different groups.(stratified).

--------is the degree of accuracy with --------is the degree of accuracy with which the sample mean can estimate which the sample mean can estimate the population.(precision)the population.(precision)

Within a population,groups that are Within a population,groups that are similar to each other are called as--------similar to each other are called as--------(clusters)(clusters)

Determine the sample size if standard Determine the sample size if standard deviation is 6,population mean is 25 and deviation is 6,population mean is 25 and sample mean is 23.The desired degree of sample mean is 23.The desired degree of precision is 99%.precision is 99%.

a)60 b) 75 c) 90 d) 45a)60 b) 75 c) 90 d) 45 A sample size of 90 values has a mean 55 A sample size of 90 values has a mean 55

and standard deviation 3.A second and standard deviation 3.A second sample of 110 values has mean 60 and sample of 110 values has mean 60 and standard deviation 2.Find the mean and standard deviation 2.Find the mean and standard deviation of the combined standard deviation of the combined sample of 200 values.sample of 200 values.

a)54.5;2.812 b)62;4.345 c)57.75;3.526a)54.5;2.812 b)62;4.345 c)57.75;3.526

d)61.90;3.89d)61.90;3.89

REGRESSION – CORRELATION-TIME REGRESSION – CORRELATION-TIME SERIESSERIES

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

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 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’.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 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 line form,like Y=ax+b

Uses of regression analysis-Uses of regression analysis-

-To estimate the relationship between -To estimate the relationship between

economic variable like price,demand, etc.economic variable like price,demand, etc.

-Estimating errors in prediction of the -Estimating errors in prediction of the

dependent variable.dependent variable.

-We can calculate the coefficient of -We can calculate the coefficient of coorelation.coorelation.

Coefficient of determination is the square Coefficient of determination is the square of coefficient of correlation.This measures of coefficient of correlation.This measures the degree of correlation exists between the degree of correlation exists between two variables.two variables.

Assume the principal of a college Assume the principal of a college wants to find out whether there is a wants to find out whether there is a relationship between the entrance relationship between the entrance examination score to a college and examination score to a college and final graduation GPA score of a final graduation GPA score of a student.student.

Student A B C D E F G HStudent A B C D E F G H

EntranceEntrance

Exam 74 69 85 63 82 60 79 91Exam 74 69 85 63 82 60 79 91

CumulativeCumulative

GPA 2.6 2.2 3.4 2.3 3.1 2.1 3.2 GPA 2.6 2.2 3.4 2.3 3.1 2.1 3.2 3.83.8

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

Curvilinear RelationshipCurvilinear Relationship

In many industries we have heard of In many industries we have heard of curvilinear relationship.The principle curvilinear relationship.The principle is that as the employees produce is that as the employees produce more and more of a new product ,the more and more of a new product ,the time required to produce one unit time required to produce one unit decreases by some fixed proportion decreases by some fixed proportion as the total number of unit doubles.as the total number of unit doubles.

No.of planes produced 5 10 20 No.of planes produced 5 10 20 40 40

No.of hours per plane 1000 800 640 No.of hours per plane 1000 800 640 512512

TIME SERIESTIME SERIES

The first step in making the The first step in making the estimates for future consists of estimates for future consists of gathering information from the gathering information from the past.Data is collected,and recorded past.Data is collected,and recorded at successive intervals of time.Such at successive intervals of time.Such data are called “time series”.data are called “time series”.

Time series helps in understanding Time series helps in understanding past behaviour.It also helps in past behaviour.It also helps in planning future operations as well as planning future operations as well as current operations.The actuals can current operations.The actuals can be compared with expected results be compared with expected results and cause of variation can be and cause of variation can be analysed. analysed.

It is customary to classify the fluctuations It is customary to classify the fluctuations of a time series into four basic types of of a time series into four basic types of variations.variations.

a)Secular trend b)Seasonal variationa)Secular trend b)Seasonal variation

c)Cyclical variation d)Irregular variation.c)Cyclical variation d)Irregular variation. Methods of trend analysisMethods of trend analysis

-Free hand graphic method.-Free hand graphic method.

-Semi-average method.-Semi-average method.

-Moving average method.-Moving average method.

-Method of least squares.-Method of least squares.

Light bulbs are manufactured in a Light bulbs are manufactured in a factory and have a mean life of 540 factory and have a mean life of 540 hours and std.deviation of 50 hours and std.deviation of 50 hours.Calculate the fraction of bulbs that hours.Calculate the fraction of bulbs that have less than 500 hours.have less than 500 hours.

a)0.54 b)0.2119 c)0.4234 d)0.3555a)0.54 b)0.2119 c)0.4234 d)0.3555 The fraction of bulbs having life The fraction of bulbs having life

betweenbetween

500 to 600 hours.500 to 600 hours.

a)0.6730 b)0.5478 c)0.7734 d)0.3487a)0.6730 b)0.5478 c)0.7734 d)0.3487

The mean and standard deviation of The mean and standard deviation of marks of students of a class are 55 and marks of students of a class are 55 and 8 respectively.Within what interval 8 respectively.Within what interval centred around the mean do at least centred around the mean do at least 90% of marks lie?90% of marks lie?

a)40 and 65 b)39 and 61 c)42 and 68a)40 and 65 b)39 and 61 c)42 and 68

d)30 and 72d)30 and 72 A binomial distribution has n=20 and A binomial distribution has n=20 and

p=0.3.Find the mean and the variance p=0.3.Find the mean and the variance of the distribution.of the distribution.

a)8 and 3.4 b)4 and 2.6 c)6 and 4.2 a)8 and 3.4 b)4 and 2.6 c)6 and 4.2 d)12 and 4.4d)12 and 4.4

Customer accounts at a certain cooperative Customer accounts at a certain cooperative bank have an average balance of Rs 4,800 bank have an average balance of Rs 4,800 and a Standard Deviation of and a Standard Deviation of Rs.1,600.Assume the account balances are Rs.1,600.Assume the account balances are normally distributed.Calculatenormally distributed.Calculate

a)What proportion of the accounts is over a)What proportion of the accounts is over

Rs.6,000?Rs.6,000?

a)32% b)20% c)15.44% d)22.67%a)32% b)20% c)15.44% d)22.67%

b)What proportion of the accounts are b)What proportion of the accounts are

between Rs4,000 and Rs.6,000?between Rs4,000 and Rs.6,000?

a)46.49% b)52.34% c)62.33% d)34.65%a)46.49% b)52.34% c)62.33% d)34.65%

C)What proportion of the accounts is C)What proportion of the accounts is between Rs.2,400 and Rs.3,600 ?between Rs.2,400 and Rs.3,600 ?

a)32.45% b)15.98% c)22.50% d)10.44%.a)32.45% b)15.98% c)22.50% d)10.44%.

State whether true or falseState whether true or false

Secular trend refers to long term of data.Secular trend refers to long term of data.

Regular variations include only seasonal Regular variations include only seasonal

variations.variations.

Yearly data are independent of the effectYearly data are independent of the effect

of seasonal variations.of seasonal variations.

The period of seasonal variations is alwaysThe period of seasonal variations is always

one year.one year.

Series of figures arranged in a Series of figures arranged in a chronologicalchronological

Order are calledOrder are called a) Time series b)Trend c)Linear d)Besta) Time series b)Trend c)Linear d)Best Which is an irreversible movement and Which is an irreversible movement and

continues in the same direction for continues in the same direction for considerable period of timeconsiderable period of time

a) Trend b)Time series c)Best d)Linear.a) Trend b)Time series c)Best d)Linear. The trend equation fitted by the method The trend equation fitted by the method

of least squares is known as the of least squares is known as the equation of-----fit.a)Linear b)Best c)Trend equation of-----fit.a)Linear b)Best c)Trend d)Time seriesd)Time series

In the case of a -------trend,successive In the case of a -------trend,successive observations differ by a constant number.observations differ by a constant number.

a)Time series b)Trend c)Best d)Lineara)Time series b)Trend c)Best d)Linear

If in a year 20,000 boats are rented If in a year 20,000 boats are rented out,average per quarter should be out,average per quarter should be Rs.5,000.If the index for spring quarter is Rs.5,000.If the index for spring quarter is 142,then we estimate the number of 142,then we estimate the number of boats rented out during the summer will boats rented out during the summer will be 5,000X(142/100)be 5,000X(142/100)

=7,100.=7,100.

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

A point estimate is a single number that is A point estimate is a single number that is used to estimate an unknown populationused to estimate an unknown population

An interval estimate is a range of values An interval estimate is a range of values used to estimate a population parameter.used to estimate a population parameter.

Any sample statistic that is used to Any sample statistic that is used to estimate a population parameter is called estimate a population parameter is called an estimator which is a sample statistic an estimator which is a sample statistic used to estimate a population parameter.used to estimate a population parameter.

An interval estimate describes a range of An interval estimate describes a range of values within which a population values within which a population parameter is likely to lie.parameter is likely to lie.

Bond ValuationBond Valuation

Bond prices vary inversely to change in Bond prices vary inversely to change in interest rates.interest rates.

When the interest rate increases,and When the interest rate increases,and goes higher than the coupon rate,then goes higher than the coupon rate,then the bond value decreases.This is the bond value decreases.This is because the present value of payments because the present value of payments received decrease,leading to a fall in received decrease,leading to a fall in bond prices.bond prices.

When the interest rate decreases,and When the interest rate decreases,and becomes lower than the coupon becomes lower than the coupon rate,then the present value of rate,then the present value of payments increases and the bond’s payments increases and the bond’s market price increases.market price increases.

When the interest rate is equal to the When the interest rate is equal to the coupon rate,the market price of the coupon rate,the market price of the bond is equal to face value.In this bond is equal to face value.In this case,the bond is selling at par.case,the bond is selling at par.

When the market price of bond is When the market price of bond is greater than its face value,it is said that greater than its face value,it is said that the bond is selling at a premium.the bond is selling at a premium.

RBI issues a bond with a par value of RBI issues a bond with a par value of Rs.1000,coupon rate 10% and the Rs.1000,coupon rate 10% and the maturity period is 10years.What will maturity period is 10years.What will happen when interest rate increases?happen when interest rate increases?

The price of the bond will fall.Coupon The price of the bond will fall.Coupon rate rate

Will not change .Yield to maturity Will not change .Yield to maturity

increases.The interest payment received increases.The interest payment received

each year is only Rs.100 (no change).each year is only Rs.100 (no change).

What is interest rate risk?What is interest rate risk?

A fall in interest rates will result in A fall in interest rates will result in increase in increase in

bond prices.A rise in interest rates will bond prices.A rise in interest rates will result in result in

decrease in bond prices.decrease in bond prices.

A Rs.100 par value bond,bearing a A Rs.100 par value bond,bearing a coupon rate of 12% will mature in 8 coupon rate of 12% will mature in 8 years.the required rate of return on this years.the required rate of return on this bond is 14%.What is the value of this bond is 14%.What is the value of this bond?bond?

a)92.87 b)90.77 c)102.76 d)78.89a)92.87 b)90.77 c)102.76 d)78.89

A Rs.1000 par value bond has a coupon A Rs.1000 par value bond has a coupon rate of 14% will mature after 5 rate of 14% will mature after 5 years.The required rate of return on this years.The required rate of return on this bond is 13%.What is the value of this bond is 13%.What is the value of this bond?bond?

a)1035.4 b)1200.5 c)945.88 d)1060a)1035.4 b)1200.5 c)945.88 d)1060

LINEAR PROGRAMMINGLINEAR PROGRAMMING

DECISION MAKINGDECISION MAKING

Linear programming is concerned with Linear programming is concerned with efficient allocation of limited sources to efficient allocation of limited sources to known activities,with the objective of known activities,with the objective of meeting desired goals such as meeting desired goals such as maximizing profits or minimizing cost.maximizing profits or minimizing cost.

Objective function is one in which we Objective function is one in which we mention the objective quantitatively mention the objective quantitatively and express it as a linear function of and express it as a linear function of the variables.It can be for maximizing the variables.It can be for maximizing profits or minimizing costs.profits or minimizing costs.

Optimal solution satisfies all the given Optimal solution satisfies all the given

constraints.constraints.

Sensitivity analysis-It refers to the Sensitivity analysis-It refers to the study of effect of changes in various study of effect of changes in various parametersparameters

(constraints)on the optimal solution.(constraints)on the optimal solution.

State whether following statements are State whether following statements are true or falsetrue or false

a)The distinctive characteristic of linear a)The distinctive characteristic of linear

programming models is that the programming models is that the functions functions

representing the objective and the representing the objective and the

constraints are linear.constraints are linear.

Non negativity constraints means the Non negativity constraints means the products are not produced.products are not produced.

Inequality means that the capacity of Inequality means that the capacity of each operation should not be each operation should not be exceeded.exceeded.

For finding out the optimal mix and the For finding out the optimal mix and the corresponding profit,linear corresponding profit,linear programming method is not useful.programming method is not useful.

SIMULATIONSIMULATION

Simulation is an imitation of reality.A Simulation is an imitation of reality.A number of experiments is performed on number of experiments is performed on simulated models to determine the simulated models to determine the behaviour of the real system.Example:behaviour of the real system.Example:

Testing of aircraft models in wind tunnel,Testing of aircraft models in wind tunnel,

Planetarium shows represent a simulation Planetarium shows represent a simulation

of the planet system.of the planet system. Advantages of simulation:Advantages of simulation:

-We can foresee difficulties and -We can foresee difficulties and bottlenecks bottlenecks

which may come up in real system.which may come up in real system.

-This eliminates costly and risky trials-This eliminates costly and risky trials

This allows experimenting with a model This allows experimenting with a model of the system without interfering with of the system without interfering with real system.real system.

Simulation models are comparatively Simulation models are comparatively flexible and can be modified to flexible and can be modified to accommodate the changing environment.accommodate the changing environment.

Monte Carlo method is a technique that Monte Carlo method is a technique that involves using random numbers and involves using random numbers and probability to solve problems.The term probability to solve problems.The term Monte Carlo Method was coined by Monte Carlo Method was coined by S.Ulam and Nicholas Metropolis in S.Ulam and Nicholas Metropolis in reference to games of chance,a popular reference to games of chance,a popular attraction in Monte Carlo,Monaco.attraction in Monte Carlo,Monaco.

State true or falseState true or false

-Simulation is a replica of real life.-Simulation is a replica of real life.

-Random numbers are not necessary for -Random numbers are not necessary for

inputs in a simulation model.inputs in a simulation model.

-Simulation is used when the problem is -Simulation is used when the problem is not not

complex,and there is a linear relationship complex,and there is a linear relationship

between the variables.between the variables.

-Simulation involves many iterations,and -Simulation involves many iterations,and

computers are useful in solving them.computers are useful in solving them.

objective questions objective questions