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- 1. Interest rate (r) is the rate of return that an investorreceives for investing in cash or another fixed incomesecurity.Study Session 2, Reading 5
- 2. Time value of money is concerned with the equivalencerelationship between cash flows occurring on different dates. If an investor is ready to receive $550 after one year for $500given today, then the amounts may consider equivalent wherethe discount rate is 10% ($50 /$500).Study Session 2, Reading 5
- 3. Required Rate of Return Discount Rate Opportunity CostStudy Session 2, Reading 5
- 4. • Interest rate is sum of real risk-free interest rate plus fourtypes of risk premiums.• The premiums are required by investor for bearing differenttypes of risks.Study Session 2, Reading 5
- 5. • Interest Rate = Real risk-free interest rate + Inflationpremium + Default Risk Premium + Liquidity premiumStudy Session 2, Reading 5
- 6. • Financial institutions quoted rate of interest that is calledQuoted Interest or Stated Annual Interest. This rate does notconsider any compounding.• The Effective Annual Interest Rate is an interest rate that takesinto account the impact of compounding (i.e. interest on interest).Study Session 2, Reading 5
- 7. • Effective Annual Interest Rate (EAR) is calculated as:EAR = (1 + Periodic Interest Rate) m - 1where m is no of compounding periods per year• Periodic interest rate is calculated by dividing stated annualinterest rate by “m” where m is the number of compoundingperiods in one year.Study Session 2, Reading 5
- 8. • Future Value of a single cash flow is calculated by formula:FV-N = PV (1+r)N• The present value of a single cash flow is calculated by theformula:PV = FVN (1+r) – N• The future value interest factor is calculated as (1+r) N• The present Value factor is calculated as (1+r) –NStudy Session 2, Reading 5
- 9. • Future value formula in case of more than one compoundingperiod one year is:• The future value formula for continuous compounding is:Study Session 2, Reading 5
- 10. • The present value of a single cash flow is calculated by theformula PV = FVN (1+r) – N• The present value of a single cash flow in case of compoundingis calculated as:Study Session 2, Reading 5
- 11. Study Session 2, Reading 5• The future value of anordinary annuity is calculatedas:• The present value of aperpetual annuity iscalculated as:
- 12. Study Session 2, Reading 5• The present value of a perpetual annuity is calculated as:•The present value of an Annuity Due can be calculated as:
- 13. • A time line is used to index present and future cash flows.• A time line is helpful in ensuring compatibility of time unitsand interest rate per time period.• Amounts of money can be added if they indexed at the samepoint in time.Study Session 2, Reading 5
- 14. • A time line (such as the one below) can be helpful when dealingwith time value of money problems:Study Session 2, Reading 5
- 15. • The annuity cash flow that a lump sum of funds at T=0 cangenerate can be calculated as:Study Session 2, Reading 5
- 16. • The amount of money indexed at the same point in time areadditive.Study Session 2, Reading 5
- 17. • Net Present Value (NPV) of an investment can be defined aspresent value of cash inflows minus the present value of cashoutflows.Study Session 2, Reading 6• NPV is calculated using the formula:
- 18. • In mathematical terms IRR can be shown as:Study Session 2, Reading 6
- 19. • Holding period return is the return that is earned by an investorover a specified holding period (not necessarily one year).• Holding period return is used for measuring the performance ofan asset or portfolio.Study Session 2, Reading 6
- 20. • Analysing the performance of an investor involves two tasks:1) performance measurement, and 2) performance appraisal.• Holding Period Return (HPR) is the return earned over a specifiedholding period and is calculated as:HPR = (P1 – P0 + D1) / P0Study Session 2, Reading 6
- 21. • The Money Weighted Rate of Return is IRR of an investment.• The Time Weighted Rate is that return which measures thecompound rate of growth of $1 initially invested in the portfolioover the stated period of measurement.• Time Weighted Return measures the performance of a fundmanager in a better way than Money Weighted Rate, because itcontrols for fund inflows and outflows.Study Session 2, Reading 6
- 22. • In case of daily valuation of a portfolio, the time weightedreturn can be calculated by:i. Computing each day’s holding period return by using theformula:Study Session 2, Reading 6• Annual return is then calculated as:
- 23. Bank Discount Yield• T-bills are quoted on a bank discount basis:Holding Period Yield• The HPY for an instrument that makes one cash payment until itsmaturity or the end of the holding period is calculated as:Study Session 2, Reading 6
- 24. • The HPY for a discount instrument is:Effective Annual Yield (EAY)• The Effective Annual Yield is calculated as:Study Session 2, Reading 6
- 25. • Money market yield is computed based on the purchase pricebasis: rMM =(rBD)(F/P0)• The following formula is also used for calculating the MoneyMarket Yield:Study Session 2, Reading 6
- 26. • Holding Period yield can be converted back and forth to amoney market yield and an effective annual yield.• Annualizing a semi annual yield by doubling it equals the BondEquivalent Yield.• Bank discount yield is not used for determining the presentvalue of cash flows.Study Session 2, Reading 6
- 27. • Conversion of Money Market Yield (MMY) to Holding PeriodYield (HPY):HPY = (MMY) / (360/t) where t is the time to maturity in days.• Conversion of Effective Annual Yield to HPY:HPY = (1 + EAY) t/365 – 1 where t is time to maturity in days.Study Session 2, Reading 6
- 28. • Descriptive Statistics summarize data and convert it intousability information for the purpose of analysis.• Statistical inferences observe a smaller group of data (sample)and make judgement, estimates or forecasts about a largergroup of data.Study Session 2, Reading 7
- 29. • To analyse and summarize the data, four major measurementscales are used:o Nominal Scaleo Ordinal Scaleo Interval Scaleo Ration ScaleStudy Session 2, Reading 7
- 30. • A parameter is any descriptive measure of a populationcharacteristic.• A sample statistic is a quantity computed from or used todescribe a sample.• A frequency distribution can be described as the tabularpresentation of data summarized into a relatively small number ofintervals.Study Session 2, Reading 7
- 31. • Relative frequency is calculated by dividing the absolutefrequency of an interval by number of observations.• Cumulative Relative Frequency helps in ascertaining whatfraction of total observations is less than upper limit of eachreturn interval.Study Session 2, Reading 7
- 32. • A histogram is a bar chart presentation of a frequencydistribution.Study Session 2, Reading 7
- 33. • In a frequency polygon, the mid-point of each interval isplotted on the x-axis and the absolute frequency on the y-axis.These points are connected with a straight line.Study Session 2, Reading 7
- 34. • In a Cumulative Frequency Distribution line graph, the upperinterval limit of interval is plotted on the x-axis and thecumulative absolute or cumulative relative frequency on the y-axis.Study Session 2, Reading 7
- 35. • A measure of central tendency is helpful in ascertainingwhere the data is centred.• The common measures of central tendency are thearithmetic mean, the median, the mode, the weightedmean, and the geometric mean.Study Session 2, Reading 7
- 36. • The population mean (µ) can be calculated as:• The formula for calculating the Sample Mean ( ) is:Study Session 2, Reading 7
- 37. • Weighted Mean is calculated with the following formula ( ):Where the sum of the weights equals 1; that is,Study Session 2, Reading 7
- 38. • Measures of location called quantiles are used to ascertainvalues at or below which the stated proportion of the data lie.• Quartiles divide the set of observations into quarters, quintilesinto fifths, decile into tenths, and percentiles into hundredths.• Quantiles are used by investment analysts to rankperformance and investment research.Study Session 2, Reading 7
- 39. • When n entries are arranged in ascending order, theposition of a percentile can be calculated as:Study Session 2, Reading 7
- 40. • Quantiles are used to rank the performance of portfolios.• In investment research, assets are divided based on somecriteria and investment options falling below the cut off aredeclined.Study Session 2, Reading 7
- 41. • The Mean Absolute Deviation for a set of sample data can becalculated as:• Population variance ( ) can be calculated using the formula:Study Session 2, Reading 7
- 42. • The formula for calculating the Population Standard Deviation(σ) is:•The variance of the sample (s2) is calculated as:Study Session 2, Reading 7
- 43. • The Standard Deviation for a sample (S) can be calculatedas:• Semivariance is calculated as:Study Session 2, Reading 7
- 44. • Chebyshev’s Inequality states that in any distribution withfinite variance, the fraction of observations lying within kstandard deviations of arithmetic mean is at least 1-1/k2 wherek > 1.Study Session 2, Reading 7
- 45. • Coefficient of Variation (CV) calculates amount of risk per unitof mean return.• CV is a measure of relative dispersion.• Ratio of excess return (Portfolio Return – Risk Free Rate) to thestandard deviation of return of a portfolio is called the SharpeRatio.Study Session 2, Reading 7
- 46. • The coefficient of variation (CV) is calculated as:Study Session 2, Reading 7
- 47. • The Share Ratio is calculated as:Study Session 2, Reading 7
- 48. • Symmetry and skewness help us understand the distributionof returns.• A skewed distribution may be positively skewed or negativelyskewed.Study Session 2, Reading 7
- 49. Study Session 2, Reading 7
- 50. • When a set of observations are arranged in ascending ordescending order, the median is the value of the middle item.• Mode is the value that occurs most frequently in a set ofobservations.Study Session 2, Reading 7
- 51. • For positively skewed unimodal distribution:Mode ‹ Median ‹ Mean• For negatively skewed unimodal distribution:Mean ‹ Median ‹ ModeStudy Session 2, Reading 7
- 52. • Sample skewness (also called simple relative skewness), SK,is calculated as:Study Session 2, Reading 7
- 53. • As n becomes large, the formula for calculating skewnessbecomes:Study Session 2, Reading 7
- 54. • Sample Excess Kurtosis can be calculated as:•As n becomes large, the above equation can be simplified to:Study Session 2, Reading 7
- 55. • The geometric mean is appropriate for analysing pastperformance and arithmetic mean is used for estimating futureperformance.• The arithmetic mean is always greater than the geometricmean.Study Session 2, Reading 7
- 56. • Geometric mean can be calculated as:Study Session 2, Reading 7
- 57. • A random variable is a number whose outcomes areuncertain.Study Session 2, Reading 8
- 58. • An event may be defined as a specified set of outcomes.• The two key properties of probability are:1. The probability of an event can be between 0 and 1.2. The sum of probabilities of mutually exclusive and exhaustiveevents is 1.Study Session 2, Reading 8
- 59. • Mutually Exclusive Events are those events where only one canoccur at a time.• Exhaustive events mean that events cover all possibleoutcomes.Study Session 2, Reading 8
- 60. • Three types of probabilities used by investment analysts areEmpirical, Subjective and Priori.Empirical Probability• Empirical Probability is calculated based on the relativefrequency of the occurrence of historical data.Study Session 2, Reading 8
- 61. • A subjective probability in one in which empiricalprobability is adjusted to accommodate changes, there is nodata from which empirical probability can be obtained, orthe probability obtained from personal assessment.Study Session 14, Reading 51
- 62. • Probability that is based on logical analysis and not onsubjective judgement or historical data is called a prioriprobability.Study Session 2, Reading 8
- 63. • Probabilities can be stated in terms of odds.• Probability may be stated as “odds for E” or “oddsagainst E”.Study Session 2, Reading 8
- 64. • Odds for EIf probability is P (E), then:Then the odds for E = P(E)/[1-P(E)]• Odds against EIf probability is P(E), then:Odd against E = [1-P(E)]/P(E) i.e. it is reciprocal of the odds for EStudy Session 2, Reading 8
- 65. • Unconditional Probability (also called marginal probability) isthe probability of an event that is not conditional on any otherevent. It is denoted as P (A).• Conditioned probability is the probability of an event that isconditional on any other event. P (A|B) denotes the probability ofevent A, given that event B has occurred.Study Session 2, Reading 8
- 66. • The conditional probability of event A given that event B hasoccurred is:Where P(AB) is the joint probability of event A and event B.Study Session 2, Reading 8
- 67. • The multiplication rule for probability is used for ascertainingthe joint probability of two events.• The addition rule for probability helps in ascertaining theprobability of any of the two events occurring.Study Session 2, Reading 8
- 68. • The joint probability of two events occurring ( i.e. P(AB)) can becalculated as:Study Session 2, Reading 8
- 69. • Given two events, A and B, the probability of either A or B orboth occurs is calculated as:• If two event A and B are mutually exclusive then P(AB) = 0 and:P(A or B) = P(A) + P(B)Study Session 2, Reading 8
- 70. • Two events are independent if the occurrence of one of theevents does not have any effect on the probability of theoccurrence of the other event.• When the occurrence of one of the two events affects theprobability of the other event, the two events are dependent.Study Session 2, Reading 8
- 71. • Two events A and B can be said independent if:and• When two events (A and B) are independent, then the jointprobability of A and B is: P(AB) = P(A) P(B)• When there are more than two independent events (say A, B and C),then the multiplication rule becomes: P(ABC) = P(A) P(B) P(C)Study Session 2, Reading 8
- 72. • The total probability rule is used to find the unconditionalprobability of an event if probabilities are conditional uponscenarios (and the scenarios are mutually exclusive andexhaustive).Study Session 2, Reading 8
- 73. • In case of two scenarios or events, event is denoted as S andno-event is a complement of S (SC), thereby sum of the probabilitybeing P (S) + P (SC ) = 1. The total probability rule is given as:P(A) = P(AS) + P(ASC)= P(AS) P(S) + P(ASC) P(SC)Study Session 2, Reading 8
- 74. • In the case of more than two mutually exclusive and exhaustivescenarios or events (S1, S2 …….. Sn), the total probability rule isgiven as:P(A) = P(AS1) + P(AS2) +...+ P(ASn)= P(AS1) P(S1) + P(AS2) P(S2) +...+ P(ASn) P(Sn)Study Session 2, Reading 8
- 75. • The expected value of a random variable (X) conditional uponan event or scenario (S) is represented as E (X|S).• When we make adjustments to our expectations or forecaststo accommodate new information or news, we use conditionalexpected values.Study Session 2, Reading 8
- 76. • If X is the random variable and different outcomes X1, X2, X3 areconditional on S, then the conditional expected value can becalculated as:E(XS) = P(X1 S)X1 + P(X2 S)X2 +...+ P(Xn S)Xn• The total probability rule for expected values is similar to the totalprobability rule for unconditional outcomes, in terms of conditionalprobabilities. It can be calculated as:Study Session 2, Reading 8
- 77. Study Session 2, Reading 8(i) For two scenarios:E(X) = E(X | S)P(S) + E(X | SC)P(SC)(iii) For more than one scenarios ( i.e. S1, S2 …….. Sn) ofexhaustive and mutually exclusive scenarios:E(X) = E(X|S1)P(S1) + E(X|S2)P(S2) +... E(X|Sn)P(Sn)
- 78. • Tree diagrams are used in decision analysis to select frommultiple options available.• Tree diagrams help in calculating conditional probabilities.Study Session 2, Reading 8
- 79. • Covariance is used to measure co-movement betweentwo variables.• Correlation measures the linear movement betweentwo random variables.Study Session 2, Reading 8
- 80. • The covariance between two random variables (Ri andRj) can be calculated using: Cov(Ri,Rj) = E[(Ri – ERi)(Rj –ERj)]• The covariance can also be expressed as: Σ(Ri,Rj) andσijStudy Session 2, Reading 8
- 81. • The correlation between two random variables (Ri and Rj) isdenoted as or ρij and calculated as: P(Ri ,Rj) = Cov(Ri,Rj)/σ(Ri)σ(Rj)Study Session 2, Reading 8
- 82. • The expected value of random variable is used in investmentanalysis when forecasting company earnings, financial variables, ratiosetc.• Standard deviation is the positive square root of variance.Study Session 2, Reading 8
- 83. • The expected value can be calculated as:• Variance is calculated by the formula given below:e is calculated by the formula given below:Study Session 2, Reading 8
- 84. • The expected return of a portfolio can be calculated as:Study Session 2, Reading 8
- 85. • The joint probability function [P(X, Y)] provides the probability ofthe joint occurrence of the values of two random variables (X andY).• The joint probability function can be used to calculate thecovariance.Study Session 2, Reading 8
- 86. • Covariance between two random variables RA and RB is given by:• Two random variables X and Y are independent if P(X,Y) = P(X) P(Y).• The multiplication rule for the Expected Value of UncorrelatedRandom Variables is:E(XY) = E(X) E(Y) if X and Y uncorrelated.Study Session 2, Reading 8
- 87. • Bayes’ formula is based on the total probability rule inunconditional probability.What is Baye’s Formula and When Is It Used?• Bayes’ formula is used to make adjustments to probabilities whennew information is received.• The formula is used in investment and business decision making,and when assessing mutual fund performance.Study Session 2, Reading 8
- 88. • The updated probability of the event after receiving newinformation is calculated as:• The above formula can also be written as:Study Session 2, Reading 8
- 89. Multiplication Rule of Counting• The number of ways in which k task can be calculated as:(n1) (n2) (n3) ……(nK)• The number of ways in which a group of n persons can beallocated is:n! = n(n – 1) (n – 2) (n – 3) ...1Study Session 2, Reading 8
- 90. • If there are n different objects, k different labels, and n1 + n2 +………nk = n where n1 is of the first type, n2 is of second type, and soon. According to the multinomial formula, the number of ways inwhich labelling can occur is:Study Session 2, Reading 8
- 91. • Combination Formula (also called the Binomial Formula) can beshown as:• According to permutation formula:Study Session 2, Reading 8

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