Quantitative Techniques and Financial Mathematics

1. Quantitative Techniques andFinancial Mathematics C.S.BALAKRISHNAN Faculty Member S.P.B.T.COLLEGE

2. Concept Of Time Value Of Money,Net Present Value,Discounted Cash Flow

3. 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.

4. 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)

5. 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.

6. 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.

7. 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.

8. 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.

9. 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.

10. 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%

11. 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

12. 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

13. 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%

14. Sampling Methods-Presentation of data analysis and interpretation of data- Hypothesis testing

15. 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.

16. 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.

17. 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

18. 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

19. 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.

20. 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.

21. 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

22. 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)

23. 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)

24. 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

25. REGRESSION – CORRELATION-TIME SERIES

26. 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.

27. 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

28. 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.

29. 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

30. 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.

31. 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

32. TIME SERIES

33. 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.Theactuals can be compared with expected results and cause of variation can be analysed.

34. 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.

35. 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

36. 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

37. 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%

38. 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.

39. 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

40. 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.

41. Probability distribution-confidence interval analysis-Estimating parameters of distribution

42. 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.

43. Bond Valuation

44. 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.

45. 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

46. 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.

47. 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

48. LINEAR PROGRAMMING DECISION MAKING

49. 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.

50. 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.