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Indexes and Expectation

Indexes and Expectation. Indexes. Price weighted index Level is sum of prices divided by a “divisor”, D L E =(P 1 +P 2 +P 3 )/D Value weighted index Begin with arbitrary level (100). Call this level L0 The level next period is L1=L0(1+r vw1 )=100(1+r vw1 )

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Indexes and Expectation

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  1. Indexes and Expectation

  2. Indexes • Price weighted index • Level is sum of prices divided by a “divisor”, D • LE=(P1+P2+P3)/D • Value weighted index • Begin with arbitrary level (100). Call this level L0 • The level next period is L1=L0(1+rvw1)=100(1+rvw1) • The level next period is L2=L1(1+rvw2) • where rvwt is the return on the value-weighted portfolio of all stocks in the index at time t • Equally weighted index • Begin with arbitrary level (100). Call this level L0 • The level next period is L1=L0(1+rew1)=100(1+rew1) • The level next period is L2=L1(1+rew2) • where rvwt is the return on the equally-weighted portfolio of all stocks in the index at time t

  3. Value Weighted Index • Start with initial arbitrary value of 100 for some base year • Level next period=100(1+rvw) • Need to find return on value-weighted portfolio. • Value weighted portfolio: • Weight on asset i is wi =(Market Capi)/(Total Market Cap) • Market Cap = Price*(Shares Outstanding) • Total Market Cap = sum of individual market caps • Weight measured at beginning of period.

  4. Value Weighted Index • Total market cap at beginning (time 0)= (80 + 20 + 100 ) = $200 • w1=40%, w2=10%, w3=50% • r1=20%, r2=-20%, r3=50% • Portfolio return = .4(0.2) + 0.1(-0.2) + 0.5(0.5) = 31% • Index level next period is 100(1.31)=131

  5. Equally-Weighted Index • Start with initial arbitrary value of 100 • Level next period=100(1+rew) • Equally-weighted portfolio • Weight on asset i is wi=1/n • n is the number of assets in portfolio • Equally Weighted Return • Assume stocks pay no dividends • r1=20%, r2=-20%, r3=50%, • Portfolio return = 1/3(0.2 -0.2 + 0.5) = 16.67% • Index level next period: 100(1.167)=116.7

  6. Stock Market Indices • Dow Jones Industrial Average • Price-weighted index • Includes only 30 blue-chip companies • Standard & Poor’s Composite 500 Index • Value-weighted index • Includes 500 firms • Wilshire 5000 Index • Value-weighted index • Includes about 7,000 firms (despite its name)

  7. Investing • You’re considering whether you should buy a stock for $10 • What would you like to know? • What will be the return on my investment? • Not possible to predict with certainty • How can I make a reasonable prediction about the return? • Can I measure the uncertainty of that prediction? • What is the probability I lose 10%? 95%? • To answer these questions we first need to define • Random Variable • PDF • Expectation • White noise • Stochastic Process • Variance Pieces of a puzzle. Let’s first understand each piece before we put them together.

  8. Investing • Random Variable: • A quantity whose value is uncertain. • Example: the return on stock ABC over the next year • Event: • A specified set of outcomes • Example: all outcomes less than -10% • Probability density (distribution) function (PDF): • A function that describes the probability of each outcome for a random variable. • Examples: next slides

  9. PDF • Example • Discrete PDF • Only a finite number of outcomes • The value of the function, P(r), tells us the exact probability of getting a given outcome. • The sum of the probabilities across all possible outcomes must equal 1. • Not very realistic. Why do we use them? • Statistical intuition • Part of the CFA curriculum

  10. Continuous PDF • Example: Normal PDF • Continuous PDF • Infinite number of outcomes • The integral of the function P(r) between two points tells us the probability of getting an outcome between those two points. • The integral of the function over the range of possible outcomes must equal 1.

  11. Expected Value • Expectation or Expected Value • Notation: E[r] or m • What Expectation is: • The averagevalueof an infinite number of outcomes of a random variable. • But we never can observe an infinite number of outcomes. • If we know the PDF, we can calculate the expectation without having to observe an infinite number of outcomes.

  12. Expectation • For a discrete probability function with n outcomes • Example • E[r]=0.75*(.10)+.25*(-.15)= 3.75%

  13. Expectation • For a continuous probability function • The idea is the same as for a discrete PDF. We just integrate across all possible values rather than sum over the discrete values.

  14. Expectation • At times we are interested in the expectation of a function of a random variable. • Example • E[r]=.65*(.08)+(.35)*(-.10)=0.017 • What is E[r2]? • E[r2]=.65*(.082)+.35*(-.102)=0.008 • What is E[3r+5]? • E[3r+5]=.65*(3*.08+5)+.35*(3*(-.10)+5)=5.051

  15. Expectation • Given a discrete PDF we now know how to calculate the expected value of a random variable, and the expected value of a function of a random variable. • PDFs are like number generating machines • Sometimes, we don’t know exactly everything about the machine, but we observe the output of the machine. • Output = numbers that the machine generates • By looking at the output the machine generates, we can get some idea about what kind of machine (PDF) is generating the numbers.

  16. Expectation • Example • E[r]=.65*(.08)+(.35)*(-.10)=0.017 • Suppose the above PDF generates 100 numbers, and we take the sample average. • 100 is much less than infinity: the average of these 100 numbers is not likely to be exactly 0.017, however, it should be close. • The difference arises because of sampling error- the fact that our sample does not contain an infinite number of observations.

  17. Expectation • Given a sample of independent outcomes from the PDF, the sample average is statistically the “best” estimate of the true expectation given the information you have. • Side note: by “best” we mean • Unbiased • Consistent • Efficient • These terms are part of the CFA curriculum and we may discuss them more in depth later.

  18. Estimation • Suppose we observe the following numbers generated by a PDF: (.10, .10, .10, .10, .10,.10, -.08,-.08,-.08,-.08). • Estimate E[r] • Ê[r]=(.10+.10+ .10+.10+.10+.10-.08-.08-.08-.08)/10=0.028 • Note that it looks like these numbers may have been generated from a PDF with two possible outcomes such that • Also note that .65*(.10)+.35*(-.08)=0.037

  19. Estimation • Suppose we observe the following numbers generated by a PDF: (.10, .10, .10, .10, .10,.10, -.08,-.08,-.08,-.08). • Estimate • E[r2] • Ê[r2]=[.102+.102+ .102+.102+.102+.102+(-.08)2+(-.08)2+(-.08)2+(-.08)2]/10=0.009 • Also note that .65*(.102)+.35*(-.08)2=0.00874

  20. Expectation Summary • Given a discrete PDF we can find the true expectation by • We can also find the true expectation of a function of a random variable, g(r) as • When we don’t know anything about the PDF, but rather, observe a sample generated by the PDF, we can estimate expected value as a simple average • This works for both discrete and continuous PDFs

  21. Excel Random Number Generation • Data – Data Analysis – Random Number Generation • “Number of Variables” = number of samples to create • “Number of Random Numbers” = sample size • Choose “Discrete” for the distribution • “Value and Probability Input Range” • Here is where you specify the outcomes, and the probability of each outcome. • Example on next slide generates 1 sample of 1000 outcomes from the PDF below, and puts them in column “D”.

  22. Example

  23. Statistics Rule #1 • Rule 1: If x and y are random variables and a and b are constants, then See equation 4-13 in CFA reading, p.153

  24. Statistics Rule #1: Example • Portfolio z: 30% A, 70% B • Note the return for portfolio z is of the form we can therefore use stat rule #1

  25. Statistics Rule #1: Example • Let’s see why stat rule #1 works. • Note that the PDF for the return on portfolio z is • Expected return for portfolio z: E[r]=.8(.065)+.2(.021) = 5.62%

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