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Quantitative methods Quantitative methods Presentation Transcript

  • Quantitative Methods
    Himanshu Shah
    shahhr21@gmail.com
    9970836701
  • Overview
    • Time value of Money
    • Discounted Cash Flow Applications
    • Statistical Concepts and market return
    • Probability Concept
    • Common Probability Distribution
    • Sampling and Estimation
    • Hypothesis Testing
    • Technical Analysis
  • Time Value of Money (TVM)
    • Concept is all about finding the future value of current Re.1 and finding the current value of future Re.1
    • Concept of Time Value of Money helps in arriving at comparable value of different rupee amounts at different points of time into equivalent values of particular time.
    • Investment decisions are generally taken on cash flow analysis – TVM is pre-condition
    • For TVM analysis, it is better to draw time line first denoting at what time how much cash flows have occurred
    • Decide the interest rate at which cash flows are to be discounted or compounded
  • Factors affecting Discounting Rate
  • 5
    0
    1
    2
    3
    i%
    CF0
    CF1
    CF3
    CF2
    Time lines show timing of cash flows
    Discounting Rate
    Cash Flows
    Tick marks at ends of periods.
    • Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2.
  • Present Value of Future Cash Flows
    FV1
    FV2
    FV3
    PV = FV1/(1+i)
    PV = FV2/(1+i)2
    1
    2
    3
    0
    PV = FV3/(1+i)3
    • Today’s value of Re.1 is more than tomorrows value of Re.1
    • Dividing the future cash flows by interest rate, we come at present value of future cash flows
  • Future Value of Cash Flows
    FV3 = (1+i)3PV
    FV2 = (1+i)2PV
    PV
    1
    2
    3
    0
    • Future Value of cash flow is always greater than present value of cash flow
    FV1 = (1+i)PV
    Finding FVs (moving to the right on a time line) is called compounding.
    • Compounding involves earning interest on interest for investments of more than one period.
  • A
    A
    A
    A
    A
    A
    A
    1
    2
    3
    4
    5
    6
    7
    0
    PV1 = A/(1+r)
    PV2 = A/(1+r)2
    PV3 = A/(1+r)3
    PV4 = A/(1+r)4
    etc.
    etc.
    Perpetuities
    Perpetuity is a series of constant payments, A, each period forever.
    PVperpetuity = [A/(1+i)t] = A [1/(1+i)t] = A/i
    Intuition:
    Present Value of a perpetuity is the amount that must invested today at the interest rate i to yield a payment of A each year without affecting the value of the initial investment.
  • 0
    1
    2
    3
    Annuities
    • Regular or ordinary annuity is a finite set of sequential cash flows, all with the same value A, which has a first cash flow that occurs one period from now.
    .
    Ordinary Annuity Timeline
    i%
    A
    A
    A
    9
  • 0
    1
    2
    3
    Annuities
    An annuity due is a finite set of sequential cash flows, all with the same value A, which has a first cash flow that is paid immediately.
    Ordinary Annuity Timeline
    i%
    A
    A
    A
    10
  • Ordinary Annuity
    0
    1
    2
    3
    i%
    PMT
    PMT
    PMT
    Annuity Due
    0
    1
    2
    3
    i%
    PMT
    PMT
    PMT
    PV
    FV
    Ordinary Annuity vs. Annuity Due
    11
  • TVM on the Calculator
    • Use the highlighted row of keys for solving any of the FV, PV, FVA, PVA, FVAD, and PVAD problems
    N: Number of periods
    I/Y: Interest rate per period
    PV: Present value
    PMT: Payment per period
    FV: Future value
    CLR TVM: Clears all of the inputs into the above TVM keys
  • Discounted Cash flow Applications
  • Net Present Value (NPV)
    • The difference between the cash outflows and discounted cash inflows of the project
    • How much value is created from undertaking an investment?
    • The first step is to estimate the expected future cash flows
    • The second step is to estimate the required return for projects of this risk level.
    • The third step is to find the present value of the cash flows
    • Subtract the initial investment, remainder is NPV.
  • NPV (Decision Rule)
    • If the NPV is positive, accept the project
    • A positive NPV means that the project is expected to add value to the firm and will therefore increase the wealth of the owners.
    • Since our goal is to increase owner wealth, NPV is a direct measure of how well this project will meet our goal.
  • Internal Rate of Return (IRR)
    • This is the most important alternative to NPV
    • It is based entirely on the estimated cash flows and is independent of interest rates found elsewhere
    • It is the interest rate at which PV of cash flows equates with PV cash outflows. Alternatively, at IRR, NPV of the project is zero.
    • Management, and individuals in general, often have a much better feel for percentage returns and the value that is created, than they do for dollar increases.
  • IRR – Decision Rule
    • Decision Rule:
    Accept the project if the IRR is greater than the required return
    • When there is only one project/proposal under evaluation, then NPV and IRR gives the same decision that is acceptance or rejection of the project
  • Problems Associated with IRR method
    • Some cash flow structures yield more than one IRR
    • IRR can be misleading when mutually exclusively projects are compared as this method assumes that cash flows are reinvested at IRR only which may not be possible in every situation
    • NPV is better method while comparing mutually exclusive projects
  • Calculations of NPV and IRR using calculator
    • Use the highlighted row of keys for solving any problem of NPV and IRR
  • Holding Period Return
    • The simplest measure of return is the holding period return.
    • Holding period return is independent of the passage of time.
    When comparing investments, the periods should all be of the same length.
    Ending Beginning
    value value Income
    _
    Holding
    period =
    return
    +
    Beginning value
  • Time weighted return
    • Use the basic measure of return (adjusted for cash flows) for sub-periods, and time-weighted returns for multiple periods.
    • Reflects compounding of returns
    • It is the rate at which Re.1 is compounded over a specified period of time
    • Perfect method of performance measurement as it is not affected by amount and timing of cash flow.
  • Steps of calculating Time weighted Return
    Value the portfolio immediately preceding significant addition or withdrawal
    Compute the holding period return (HPY) for each sub period
    Compute the product of (1 + HPY) for each sub period to obtain total return for the entire measurement period
    If investment period is greater than 1 year, then take geometric mean to come at annual Time Weighted Returns
  • Money weighted Return
    The Money-Weighted Rate of Return (MWRR)…same as internal rate of return of portfolio.
    Initial deposit is considered as inflows and all withdrawals and ending value is considered as outflows.
    Not good for comparing different fund managers as MWRR over 0-T is also, generally, a function of the amount and timing of net new money – which is not in the control of the fund manager.
  • Calculating Yields on T-bills
    • Since T-bills are sold on a discount basis, their returns are not directly comparable to interest bearing bonds.
    • Returns on T-bills are quoted on a “bank discount basis”:
    • Here, discount is calculated from par value and not market value of investment
    • Compounding is missing and annualized on the basis of 360 days rather than 365 days
  • Effective Annual Yield
    • Annualized value based on 365 days a year
    • Considers compounding of interest
    • It is calculated as -
  • Conversion among different rate of returns
    • Convert money market returns into holding period yields
    • Convert holding period yield into effective annual yield
    • Bond equivalent yield is just double of compounded coupon rate
  • Frequency Distribution
    When the raw data is organized into afrequency distribution, the frequency will be the number of values in a specific class of the distribution.
  • Grouped Frequency Distribution
    • Grouped Frequency Distribution -can be used when the range of values in the data set is very large. The data must be grouped into classes that are more than one unit in width.
    • Class intervals represent
    Continuous variable of X:
    • E.g. 51 is bounded by real limits of 50.5-51.5
    • If X is 8 and f is 3, does not mean they all have the same scores: they all fell somewhere between 7.5 and 8.5
  • Histogram
    • The histogram is a graph that displays the data by using
    vertical bars of various heights to represent the
    frequencies.
  • Frequency Polygon
    • A frequency polygon is a graph that displays the data by using lines that connect points plotted for frequencies at the midpoint of classes.
    • The frequencies represent the heights of the midpoints.
  • Histogram and Frequency Polygon
    Frequency Polygon
    Histogram
    6
    5
    y
    c
    4
    n
    e
    u
    q
    3
    e
    r
    F
    2
    1
    0
    2
    6
    2
    3
    2
    0
    1
    7
    1
    4
    1
    1
    8
    5
    2
    N
    u
    m
    b
    e
    r
    o
    f
    C
    i
    g
    a
    r
    e
    t
    t
    e
    s
    S
    m
    o
    k
    e
    d
    p
    e
    r
    D
    a
    y
  • Different measures of central tendency
    • Mean - 1.Arithmetic mean
    2.Harmonic mean
    3.Geometric mean
    4.Weighted mean
    • Median
    • Mode
  • Arithmetic mean
    • It is commonly used measure of central tendency.
    • It is sum all observations divided by number of observations
    • Arithmetic mean is unique
    • Sum of deviation from its mean is always zero.
    • A.M = X1+X2+…….+Xn
    n
    = sum of all observations
    Number of observations
  • Geometric Mean
    • When data contains few extremely large or small values in such case arithmetic mean is unsuitable for data
    • Geometric mean is used while calculating investment results for multiple periods or measuring compound growth rate
    • G.M. of n observation is defined as ‘n’th root of the product of n observation
    • G.M. is always less than or equal to A.M.
    • As dispersion of observation increases, difference between those two increases
  • Harmonic Mean
    • It is reciprocal of arithmetic mean of reciprocal
    observations.
    • Harmonic mean is useful for certain calculations such as
    finding average cost of shares etc.
    • HM < GM < AM
  • Weighted Mean
    • We are considering that each item in data is of equal importance. Sometimes , this is not true, some item is more important than others.
    • In such cases the usual mean is not good representative of data. Therefore we are obtaining weighted mean by assigning weights to each item according to their importance.
    • WM = sum(wx)
    sum(w)
  • Median
    • when all the observation of a variable are arranged in either ascending or descending order the middle observation is called as median.
    • It divides whole data into equal portion. In other words 50% observations will be smaller than the median and 50% will be larger than it.
    • Median is important because arithmetic mean is influenced by extreme values and median is not.
  • Mode
    • Mode: is the most frequently occurring value in the distribution. A distribution may have one, more than one, or no mode.
    • Unimodal: distribution with one most frequently occurring value
    • Bimodal: distribution with two most frequently occurring values
  • Qunatile
    • Is the general term for a value below which a stated proportion of the data in distribution lies
    • Rank ordering of data: Quartiles, Quintiles, Twentiles, Deciles, Percentiles
    • 2nd quartile is the median
    • Position of a percentile in an array with n entries sorted in ascending order
    • L(y) = (n+1) x (y/100)
  • Measures of Dispersion
    • Range
    • It is the difference between the maximum and minimum values in a dataset
    • Mathematically, Range = Maximum value – Minimum value
    • Mean Absolute Deviation
    • It is the average of the data’s absolute deviations from the mean.
    • Mathematically MAD = ∑|xi – x(bar)| / n
    • Variance
    • It is the average of the population’s squared deviations from the mean.
    • Mathematically, 2 = ∑(xi – m)2/ N
  • Measures of Dispersion (contd…)
    • Standard deviation
    • It is simply the square root of the population variance.
    • Mathematically,= square root [∑(xi – m)^2/ N ]
    • All of the above are population parameters, however in investment management quite often we need to work with subset or sample of the population
    • In the case of a sample variance, we divide by (n-1) rather than by N to obtain an unbiased estimate of the population variance
  • Chebyshev’s inequality
    • Gives the proportion of elements within k standard deviations of the mean
    • For k > 1, the proportion of observations within k standard deviations of the mean is at least (1 – 1/k^2)
    • Note: this holds regardless of the shape of the distribution and for both continuous and discrete data
  • Relative Dispersion
    • Is the amount of variability in comparison to a reference point or a benchmark
    • Allows for comparison of disparate distributions (because it would be in relation to the mean of the respective distributions)
    • Co-efficient of variation, CV = s/X(bar),
    where s is the standard deviation and
    X(bar) is the mean of the distribution
    • CV shows the amount of risk (measured by sample standard deviation s) for every % of mean return on the asset. The lower an asset’s CV, the more attractive it is in risk per unit of return.
  • Sharpe Ratio (Reward to variability Ratio)
    • A more precise measure of risk/ return is the Sharpe measure.
    • It measures the extra return an investor earns (over the risk free return) for the added risk taken
    • Sharpe Measure =
    • The higher is Sharp Measure, the better the return-risk tradeoff on the portfolio for an investor. If we assume investors dislike risk and prefer returns then large Sharpe ratios are preferable
  • Symmetry and Skewness
    • If each side of the distribution around the mean is a mirror image of the other then the distribution is said to be symmetric
    • Characteristics of a normal distribution:
    • Mean , mode and Median are equal
    • Completely described by 2 parameters
    • Mean and Variance
    • 68% within 1 standard deviation, 95% within 2 standard deviations and 99% within +- 3 standard deviations
    • Distributions that are not symmetrical are called skewed
  • Normal Distribution and probability
    Probability
    3 
    E(x)
    - 
    -2 
    2 

    -3 
    68%
    95%
    99%
  • Normal distribution
    • Bell shaped Symmetrical curve
    • Area under curve gives the total probability
    • An important symmetric distribution is the normal distribution, which is shaped like a bell curve
  • Skewed Distributions
    • A skewed distribution is not symmetrical
    • Skewness occurs because the arithmetic mean of the population is not equal to its median since the mean is influenced by some extreme outliers
    • Degree of skewness of a distribution can be measured using the coefficient of skewness, Sk
    • If Sk is positive the distribution is positively skewed
    • If Sk is negative the distribution is negatively skewed
  • Positive Skewed Distribution
    • Positive: tails to the right
    • Usually: mode < median < mean
    Example:
    Gamblers tend to like returns that are positively skewed
    because of the likelihood (however small) of a very large
    return
  • Negative Skewed Distribution
    Negative: tails to the left
    Usually: mean < median < mode
    Example
    Typically investors would not prefer negatively skewed instruments since it means that there is a likelihood (however small) of a very large loss
  • Kurtosis
    • Kurtosis is a measure of the degree to which a distribution is more or less “peaked” than a normal distribution
    • Kurtosis is an important consideration in risk management
    • Most portfolio returns exhibit some level of skewness and Kurtosis
    • Risk managers focus on the tails rather than at central tendency metrics
  • Types Of Kurtosis
    • Leptokurtic describes a distribution that is more peaked than a normal distribution
    • Fatter tails
    • More likelihood of an observed value to be either close to the mean or far from the mean
    • Can be perceived to be of increased risk due to the fatter tail
    • Platykurtic refers to a distribution that is less peaked than a normal distribution
  • Calculation of skewness and kurtosis
    • Sample Skewness is defined as the sum of the cubed deviations from the mean divided by the cubed standard deviation and normalized by the number of observations
    • Sample Skewness,
    Sk = 1/n[∑(xi – x(bar))^3}/s^3,
    where s = sample standard deviation
    • Sample Kurtosis = 1/n[∑(xi – x(bar))^4}/s^4, where s = sample standard deviation
    • Excess Kurtosis = Sample Kurtosis - 3
  • Probability
  • Probability Concepts
    • Some Definitions
    • Random Variable – uncertain or number
    • Outcome – observed value of random variable
    • Mutually exclusive events – events that cannot happen together
    • Mutually exhaustive events – includes all possible outcomes
    • Basic Properties
    • Sum of probabilities of all possible outcomes is 1.
    • Probability of any event is between 0 and 1.
  • Conditional v/s Unconditional Probability
    • Unconditional Probability :
    • Probability of an event regardless of the past/future occurrence of other events. Ex. Probability of Economic recession without considering probability of change sin interest rate and inflation
    • Conditional Probability :
    • Probability of an event after the occurrence of other events.
    Example.
    Probability of decrease in interest rates with considering the recessiona has already occurred.
    • This leads to : P(A |B) = [ P(A and B) ] / P (B) , probability of event A, after event B has already occurred.
  • Addition and Multiplication Rule
    • Compound Probabilities for Union of Events:
    P(A OR B) = P(A  B) = P(A) + P(B) - P(A  B)
    • Mutually Exclusive Events:
    P(A OR B) = P(A  B) = P(A) + P(B)
    • Compound Probabilities for Intersection of Events called Joint Events :
    P(A and B) = P(A  B) = P(A)*P(B|A) = P(B)*P(A|B)
    • For Independent Events:
    P(A and B) = P(A  B) = P(A)*P(B)
  • Dependent and Independent Events
    • Independent Events : Event Occurrence Does Not Affect Probability of Another Event
    Example :
    Two Coins are tossed and outcome from those two events are not dependent on each other. Hence, they are independent events
    Tests For Statistical Independence
    • P(A | B) = P(A)
    • P(A and B) = P(A)*P(B)
    • Dependent Events : If the above two conditions are not satisfied , events are dependent events
    Example:
    Economic recession and decrease in interest rates are dependent events.
    • Expected Value : It is the weighted average of the possible outcome of a random variable, where the weights are the probability that outcome will occur
    E(X) = P(xi)xi = P(x1)x1 + P(x2)x2 + ….. + P(xn)xn
    • Use of Conditional Expectation in Investments Application :
    • Conditional Expected values are calculated using conditional probability
    • Forecasts are made using the expected value for a stock’s return, earning etc.
    • The original forecasts are revised when new and relevant information arrives and it is revised using conditional expected values.
  • Covariance
    • Covariance between two random variables X and Y is defined as how these two variables move together
    • A negative covariance between X and Y means that when X is above its mean its is likely that Y is below its mean value.
    • If the covariance of the two random variables is zero then on average the values of the two variables are unrelated.
    Cov( Ri, Rj) = E { [Ri– E(Ri) ] [Rj– E(Rj) ] }
  • Correlation
    • Correlation is a standardized measure of how two random variables move together, i.e. correlation has no units associated with it.
    • Correlation takes on values between –1 and +1.
    • A correlation of 0 means there is no straight-line (linear) relationship between the two variables.
    • Increasingly positive (negative) correlations indicate an increasingly strong positive (negative) linear relationship between the variables.
    • When the correlation equals 1 (-1) there is a perfect positive (negative) linear relationship between the two variables.
    Corr ( Ri, Rj) = [ Cov ( Ri, Rj) ] / [(Ri) (Rj) ]
  • Portfolio Covariance & Correlation
    • Portfolio consisting of two assets A and B, wA invested in A. Asset A has expected return rA and variance s2A. Asset B has expected return rB and variance s2B. The correlation between the two returns is rAB.
    • Portfolio Expected Return:
    E(rp) = wA rA + (1-wA )rB
    • Portfolio Variance:
    Var( Rp) = (wa)22(RA) + (1- wa)22(RB) + 2wAwBCov(RA,RB)
    OR
    Var( Rp) = (wa)22(RA) + (1- wa)22(RB) + 2wAwB (RA) (RB)rAB
  • Bayes Theorem
    Probability of new info for a given event
    X (prior probability of event)
    Unconditional probability of new info
    Let us take a basket of fruits
    • Assume P(A|R) is probability that fruit is apple if it is red and round
    • P(R|A) is probability that fruit is red and round given it is an Apple
    • P(R) is the probability that a fruit is red and round (prior probability), regardless of whether it is an apple or not
    • P(A) is the probability that a fruit is an apple (prior probability) regardless of whether or not it is round and red
    • Bayes theorem allows us to infer the posterior probability P(A|R) if we know P(A), P(R) and P(R|A)
    • P(A|R) = [P(R|A)*P(A)]/P(R)
    Updated probability =
    Varied applications in business decision making including evaluation of mutual fund performance
  • Counting Problems
    • Some TIPS that can be employed to determine which counting method to use for solving counting problems :
    • Only one item to be selected from each group and two or more groups are present. Use multiplication rule.
    • When n items are to be arranged and none of them belong to any group. Use ‘n’ Factorial.
    • Each element of entire group must be a assigned a place, label in one of the three or more sub groups of predetermined size. Use the labeling formula.
    • When one needs to choose or select from two groups of predetermines size. Use Combination formula
    • When one needs to arrange and order is important. Use Permutation formula
  • Formulas for Different Methods of Counting
    • Factorial Notation :
    • n-factorial , n! = n x (n-1) x (n-2) x (n-3) x … x 2 x 1.
    • Labeling :
    • There are n items and each can receive one of k different labels
    • The total number of ways labels can be assigned :( n!) / [ n1 + n2 + … + nk ]
    • Combination :
    • Total number of ways of selecting r items from a set of n items when the order of selection does not matter
    • nCR = n! / [(n-r)!r!]
    • Permutation :
    • Total number of ways of arranging r items from a set of n items when the order of selection does matter
    • nPR = n! / [(n-r)!]
  • Common Probability Distributions
    • Probability Distribution
    • Probability Density Function
    It describes the probabilities of all the possible outcome for a random variable
    • Discreet Random Variable :
    • The number of possible outcomes can be counted and probability of each possible outcome is measurable and positive
    • P(x) = 0 when x cannot occur
    • P(x) > 0 when x can occur
    • Continuous Random Variable :
    The number of possible outcomes are infinite even within a range of values
    P(x) = 0 , even when x can occur. We can define probability P(x1 <= X <= X2 )
  • Probability Function
    • Probability function :
    • Specifies the probability of a random variable being equal to a specific value
    • Two key property of a probability function :
    • 0 <= p(x) <= 1 , probability of an event will always be between zero and 1
    • ∑p(x) = 1, The sum of probabilities for all possible outcomes x, for a random variable X equals 1
  • Probability Function
    • Probability Density Function (pdf) :
    • Function used to generate probability that outcome of a continuous distribution lie within a particular range of outcomes
    • For a continuous distribution, it is equivalent of a probability function for a discrete distribution
    • Cumulative density function (cdf):
    • It defines probability that a random variable takes on a value less or equal to a specific value.
    • It can be expressed as F(x) = P (X<=x)
    • Ex. X = { 1,2,3,4}, p(x) = x/ 10 . F(3) = 0.1 + 0.2 + 0.3 = 0.6
  • Discrete Uniform Random Variable
    • Variable for which probabilities for all possible outcomes for a discreet random variable are equal.
    • Cumulative distribution function for nth outcome of discreet random variable is n* p(x) and
    range = p(x) * k
    where k is the number of possible outcome in the range.
    • Ex. X = { 1,2,3,4,5}, p(x) = 0.2 ,
    Now p(1) = p(2) = 0.2
  • Binomial Distribution
    • Fixed number, n, of identical trials
    • Trials are independent
    • Each trial yields one of two outcomes: "Success" or "Failure“
    • P("Success") = p;
    P("Failure") = 1 - p; where 0p1 is fixed
    • For the random variable X = "the number of successes observed in n trials", X has a binomial distribution
  • Binomial Distribution
    • Expected Value of a Binomial Random Variable :
    • Expected value of X = E(x) = np
    • Variance of X = np( 1-p)
    The function below calculates the probability of x successes in n trials when the probability of success is p.
  • Continuous Uniform Distribution
    • It is defined over a range and the lower (a) and upper (b) bound of the range serve as parameters of the distribution
    • Characteristics :
    • For all a <= x1 < x2 <= b, all values of x between a & b
    • P( X < a or X > b ) = 0 , probability outside range is zero
    • P( x1 <= X <= X2 ) = (x2 – x1) / (b-a)
  • Normal Distribution
    • Similar to Binomial distribution, but for continuous interval variables and larger sample
    • Can be an approximation of Binomial distribution for count and proportion when n is large
    • Characteristics :
    • Completely defined by mean and variance
    • Skewness = 0, ie. Symmetrical about mean
    • Kutrosis = 3 Excess kutosis for any distribution is measured relative to 3, ie of Normal distribution
    • Combination of normally distributed random variables is also normally distributed
    • Probabilities of outcome above and below the mean get smaller but do not get to zero.
  • Standard Normal Distribution
    • The Standard Normal Distribution is a normal distribution with (mean),  = 0 and (std. devn.) = 1.
    • Symmetrical
    • Total area under curve = 1
    Also Note:
    area within + or - 2 standard deviations is 95.4%
    2) Area within + or - 1 standard deviation is 68.3%
    99.7%
    Wide ranging applications in Modern Portfolio Theory and Risk Management
  • Standard Normal Distribution
    • For distributions that are normal but not standard normal, we can transform them into the standard normal distribution so we can apply the same arithmetic to solve those problems
    • We can do this transformation by using the following formula for the Z score
  • Confidence Intervals
    • Confidence Intervals enable us to see if two means are the same, i.e. fall within the same interval or whether they are different
    • Hence they allow us to draw more meaningful conclusions on top of a point estimate
    • A Confidence intervals is a interval for which we can say with a certain level of probability, called the degree of confidence, that it will contain the parameter intended to estimate.
    • Three factors are relevant:
    • Standard deviation
    • Sample size
    • Precision level
    Portfolio returns
  • Determining Critical Values
    Confidence Interval can be computed as :
    Key points to note
    • X(bar) is used to estimate the mean
    • s is used to estimate the standard deviation
    • [(s)/(Square root(n)] is the standard error……not the standard deviation
    Critical Values :
    • We should use 2.15 for 99% confidence
    • We should use 1.96 for 95% confidence
    • 1.645 is used for 90% confidence
    • n is the size of the sample you have drawn
  • Confidence Interval
    Confidence Intervals :
    • 90% confidence Interval X(bar) – 1.65s to X(bar) + 1.65s
    • 95% confidence Interval X(bar) – 1.96s to X(bar) + 1.96s
    • 99% confidence Interval X(bar) – 2.58s to X(bar) + 2.58s
    Probability
    E(x)
    - 
    -2 
    2 

    68%
    95%
  • Roy’s Safety First Criteria
    • It states that the optimal portfolio minimizes the probability that the return of the portfolio falls below some minimum acceptable level.
    • The minimum acceptable level is threshold level
    • It can be calculated as :
    [ E(RP) – RL ] / P
    • Where, RP = portfolio returns
    • RL = threshold level of return
    • The portfolio with the larger SFR has the lower probability of returns below the threshold return
  • Log Normal Distribution
    • It is generated by the function ex, where x is normally distributed
    • Lognormal distribution is skewed to the right
    • Lognormal distribution is bounded by zero from left
    • Application : Pricing assets which never take negative values
    Normal Distribution
    Lognormal Distribution
  • Monte Carlo Simulation
    • It is used to draw inferences about mean, variances, etc of a security
    • Procedure :
    • It is based on the repeated generation of one or more risk factors that affect security values, in order to generate distribution of security values
    • Random values of Risk factors are generated using the computer based on the parameters of the probability distribution these risk factors follow
    • This process is repeated many times and thus the distribution obtained of simulated assets are used to arrive at various inferences about the security
    Application :
    Value complex security , estimate value at risk etc
    Limitation
    • Fairly complex
    • Simulation used is a statistical method rather than analytical
    • Provides answer which are no better than the assumptions about the distributions of the risk factors and the valuation model
  • Historical Simulation
    • Historical Simulation
    • It is based on actual changes in risk factors over some period of time
    • Simulation involves randomly selecting one of the past changes in risk factors and then calculating the value of security/ asset based on the changes in risk factors
    • Advantage
    • It uses the actual distribution of changes in risk factors which does not have to be estimated
    • Limitation
    • Past changes in risk factors may not be good indication of future
    • We can not do ‘ what if ‘ analysis using historical simulation
  • Sampling and Estimation
  • Introduction
    Simple random sampling
    • Method of selection such that each item/ person has same likely hood of being included in the sample
    Sampling Error
    • It is the difference between a sample statistic ( ex. Mean ) and its corresponding population parameter( true mean )
    • Computed as : X(bar) - 
    Sampling Distribution
    • It refers to the probability distribution of all possible values that the statistic can assume when samples of the same size are randomly drawn from the same population.
    Stratified Random Sampling
    • It is based on a classification system used to separate the population into smaller groups based on one or more distinguishing factors
    • From each group , a random sample is taken
    • The size of the sample from each group is based on the size of group( stratum) relative to the population
    • Time series data
    • It includes observations over a period of time at specific and equally spaced time intervals
    • Ex. Stock returns of Reliance Industries from January 2009 to December 2009
    • Cross sectional data
    • It includes sample of observation taken at a single point in time
    • Ex. Stock returns of all companies listed on BSE on December 31, 2009
  • Central Limit theorem
    • It states that random samples of size n and of sample mean x(bar) from a population with a mean  and finite variance 2 , approaches a normal probability distribution with mean  and variance = 2 / n as sample size becomes large ( n >= 30 )
  • Standard Error Of Sample Mean
    Standard Error
    • It is the standard deviation of the sample mean
    • Types :
    • When standard deviation is known
    •  x(bar) = / ( sq. root (n) )
    • When standard deviation is unknown but estimated
    • s x(bar) = s x(bar) / ( sq. root (n) ),
    where ‘s’ is sample standard deviation
  • Point Estimate
    • Point Estimate
    • They are single values used to estimate population parameters
    • Desirable properties of an estimator :
    • Unbiased parameter
    • Gives the expected value of the estimator which is equal to the parameter being estimated
    • Efficient
    • If the variance of the sampling distribution is smaller than all other unbiased estimators of the parameter being estimated
    • Consistent
    • The accuracy of the parameter estimate increases as the smple size increases
  • Student’s T-Distribution
    • It is a bell shaped probability distribution and is symmetrical about its mean
    • Properties
    • Symmetrical
    • Defined by a single parameter, degrees of freedom (df) are equal to the
    number of sample observations minus 1, n-1
    • Less peaked than a normal distribution, with more probability in the tails
    • As the degrees of freedom (sample size) gets larger, the shape of the t-distribution more closely approaches a standard normal distribution.
    • When compare to the normal distribution, the t-distribution is flatter with more area under the tails, as the degrees of freedom, df, for the
    t-distribution increases, its shape approaches that of the normal distribution.
  • Test Statistic Criteria
  • Various Bias
    • Sample selection bias
    • It pertains to excluding samples that may be critical in your analysis.
    • Survivorship bias
    • It is present if companies are excluded from the analysis because of problems with data availability.
    • Classic example is Mutual Funds which are closed are not taken into consideration while calculating performance
    • Look-ahead bias
    • It exists if the models uses data not available to market participants at the time the market participants act in the model.
    • Time-period bias
    • It is present if the time period used makes the result time period specific.
  • Hypothesis Testing
    • Statistical inference may be categorized into two branches:
    • Estimation: Confidence intervals around a point estimate; post event
    • Hypothesis testing: assessment of the statistical significance of a hypothesis (or statement) about a population; defined a priori
    • Estimation vs. Hypothesis testing
    • Given a sample (post event) we can define confidence intervals to state our expectation that a population parameter falls within a certain range given the observations of the sample
    • Hypothesis testing is a priori and can help define the sample size requirements for making conclusions about the differences in two populations
  • Steps in Hypothesis Testing
    • State the hypothesis
    • Identify the appropriate test statistic
    • Specify the level of significance
    • Balance risk of false positive and false negative errors
    • Formulate the decision rule regarding the acceptance or rejection of the hypothesis
    • Collect sample and calculate the sample statistics
    • Accept or reject the hypothesis (statistical decision)
    • Investment decision based on above
  • Hypothesis Testing (contd…)
    • Formulation of hypothesis
    • Null Hypothesis: it is the hypothesis to be tested
    • Alternative hypothesis: is the hypothesis to be accepted if the null hypothesis is rejected
    • If you believe the new strategy is better then the null hypothesis could be stated as - New strategy = Old strategy
    • Depending on what we are trying to accomplish we may state the hypothesis in 3 different ways:
    • H(0): Mean return = 6%; then alternative is H(A)” Mean return not equal to 6%
    • H(0): Mean return >= 6%; then alternative is Mean return < 6%
    • H(0): Mean return <= 6%; then alternative is Mean return > 6%
    • Note: The first example above calls for a two tailed test, i.e. null is rejected if the data suggests that the mean return is either smaller than 6% or larger than 6%
    • However the next two examples involve a one tailed test
  • Hypothesis Testing (contd…)
    • We need to balance the risk of false positive and false negative errors while setting the level of significance
    • Our decision on accepting or rejecting the hypothesis would be based on a comparison between the calculated test statistic and a specified set of possible values; these values are primarily a factor of the level of significance
    • There are 4 possible outcomes from a hypothesis test:
    • Accept hypothesis and it is true; correct decision
    • Reject hypothesis and it is false; correct decision
    • Reject hypothesis and it is true: type I error
    • Accept hypothesis and it is false; type II error
  • Hypothesis Testing (contd…)
  • Hypothesis Testing (contd…)
    • The probability of a type I error is denoted by the Greek letter alpha
    • It is also known as the level of significance of the test
    For Example,
    If the level of significance is 0.05, it means that there is a 5% chance of rejecting a true null
    • If we increase alpha we also increase the probability of making a mistake
    • The probability of a type II error is denoted by the Greek letter beta
    • The power of a test is the probability of correctly rejecting the null (when it is truly false)
    • 1 – beta = power of the test
  • Trade off between Type I and Type II errors
    • If we decrease the probability of type I error we increase the probability of type II error
    • It is difficult to determine beta so we typically only specify alpha
    • We need to specify alpha ahead of calculating the test statistic (else bias can creep in)
    • 3 conventional levels: 0.10, 0.05, 0.01
    • O.10 => some evidence that the null hypothesis is false
    • 0.05 => strong evidence
    • 0.01 => very strong evidence
  • Hypothesis Testing (contd…)
    • What should be the test statistic?
    • Test statistic is a quantity calculated from a sample whose value is the basis for deciding whether or not to reject the null hypothesis
    • Sample mean provides an estimate of the population mean
    • For a sample mean x(bar) calculated from a sample generated by a population with a standard deviation sigma, the standard error is given by:
    • (xbar) =  /squareroot(n); if we know sigma
    • If we do not know sigma then we need to use the sample standard deviation s to estimate it
    • S(xbar) = s/squareroot(n)
    • The test statistic then becomes [X(bar) - ]/S(xbar)
    Sample statistic – Value of population parameter under H(0)
    Test statistic =
    Standard error of the sample statistic
  • Hypothesis Testing (contd…)
    • Critical value for the test statistic
    It is the value of the test statistic for comparison with the calculated value in order to determine whether to accept or reject the hypothesis
    • The p value is the smallest level of significance at which the null hypothesis can be rejected
    • It is also an alternate way of hypothesis testing
    • Compare p value of the hypothesis test with the level of significance of the test
    • If the p value is larger than the level of significance the null hypothesis is accepted
    • If it is smaller than the level of significance the null hypotheses is not accepted
    • The p test gives the same result as the t test but it provides additional information that makes it more powerful
    • In general, the smaller the p value the stronger the evidence against the null hypothesis and in favor of the alternative hypothesis
  • Hypothesis Testing (contd…)
    • There are 4 broad categories of statistical tests of hypothesis:
    • t test: used for testing arithmetic means of populations;
    • f test: primarily for testing population variances
    • chi-square test: for hypothesis testing of independence of events
    • z test: similar to t test, where the standard normal distribution may be used
  • Hypothesis Testing (contd…)
    • For hypothesis tests concerning the population mean of a normally distributed population with unknown variance the correct tests is the t-statistic
    • We may also apply this test for small to moderate departures from the normal distribution
    • For large sample sizes (>=30 observations) we may use the Z statistic in place of the t statistic
    • The t distribution is a symmetrical distribution defined by a single parameter: degrees of freedom
    • Compared to the normal distribution the t distribution has fatter tails
    • As the sample size increases (DOF increases) the t distribution converges into a normal distribution
    • Z statistic = [X(bar) - ]/ [ /squareroot (n) ]
    • Z – test is appropriate hypothesis test of the population mean when population is normally distributed with known variance
  • Hypothesis Testing (contd…)
    • So far we looked at tests relating to a single mean, but let us say we want to know if the mean returns differ between two groups or two investment strategies, how do we set up such a test?
    • Two approaches:
    • Assumes that the population variances are equal (although unknown); samples are independent
    • If we are not able to assume the population variance are equal then we have an approximate t test
    • The null hypothesis may be formulated as (1 - 2) = 0 (or <=0 or >=0)
    With appropriate alternative hypothesis
  • Hypothesis Testing (contd…)
    • The t test is then defined as follows:
    Population variances assumed equal
    Population variances assumed unequal
    t = {[(x1(bar) – x2(bar)] - (1 - 2) }/[(sp^2/n1) + (sp^2/n2)]^(1/2)
    Where sp^2 = [(n1-1)s1^2 + (n2-1)s2^2]/(n1+n2-2)
    and DOF = (n1+n2-2)
    t = {[(x1(bar) – x2(bar)] - (1 - 2) }/[(s1^2/n1) + (s2^2/n2)]^(1/2)
    Where DOF = [(s1^2/n1) + (s2^2/n2)]^2 / {[(s1^2/n1)^2/n1] + [(s2^2/n2)^2/n2]}
    Always compute the t-statistic before computing the DF since it will sometimes be evident if it is significant or not
  • Test concerning mean diffrences
    For data consisting of paired observations from samples generated by normally distributed populations with unknown variance the t-test is as follows:
    Applications include:
    Test whether the mean returns earned by two investment strategies were equal over a particular period of time
    In this case the since both strategies are likely to have common risk factors such as market return the samples are dependent; hence we need to apply this test since by calculating the standard error based on differences we are accounting for correlation between the observations
    Pairs of before and after observations such as dividend returns before and after a change in tax laws
    t = [d(bar) – (d0)]/sd(bar)
    With n-1 degrees of freedom and
    Sd(bar) = sd/(n)^(1/2)
    Sample variance Sd^2 = {∑[(di – dbar)^2]}/(n-1)
    Sample mean difference, dbar=(1/n) ∑di
  • Test concerning variance
    • Chi Square statistic is used for tests concerning the variance of a single normally distributed population
    • Unlike normal and t distribution it is a asymmetrical distribution
    • Like the t-distribution it is a family of distributions based on the DF = (n-1)
    • Lower bound of 0 (unlike t distribution)
    • If sample is not random or underlying population is not normal then inferences can be wrong
    Test statistic, X^2 = [(n-1)(s^2)]/(0^2)
    With n-1 degrees of freedom and
    S^2 = {∑[(xi – xbar)^2]}/(n-1)
  • Test Concerning diffrence in variances
    • To test hypothesis about relative values of the variances between two normally distributed populations with means 1, 2 and 1^2, 2^2 we use the F statistic
    • For two samples the first with n1 observations and sample variance s1^2 and the second with n2 observations and sample variance of s2^2
    • And further assuming that the samples are random and independent of each other and the underlying populations are normally distributed
  • Test Concerning diffrence in variances
    • The test to be used is a ratio of the sample variances and is as follows:
    • F = S1^2/S2^2
    • DF1 = (n1 – 1) and DF2 = (n2 – 1)
    • Test is sensitive to the validity of the underlying assumptions
    • Convention is to use the larger of the two ratios (alternate one being S2^2/S1^2)
    • This ensures that the F statistic is always greater than 1
    • Hence the rejection point for any formulation of hypothesis (for this distribution and convention) is a single value in the right hand side of the relevant F distribution