01 Properties of Returns(1)

7/27/2019 01 Properties of Returns(1)

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Gross and Net Return Log Return Returns as Random Variables Moments of Returns Applications

Basic Properties of Asset Returns

Jan Bena Jason Chen Carolin Pflueger

Sauder School of Business

University of British Columbia

January 1, 2013

1 / 29Basic Properties of Asset Returns


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Key Concept: Return on Investment

Focus for now on stock returns

Return is the payoff per $1 invested

Rt+1 =Pt+1 +Dt+1

Pt

where

Rt+1 = Gross return

Pt+1 = Price at time t+ 1

Dt+1 = Dividends received at t+ 1

Pt = Price paid at t

Historical returns are known

Future returns are uncertain because

1 Future dividends are unknown2 Future prices are unknown



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Components of Gross Return

Rt+1 =Pt+1 + Dt+1

Pt

=Pt+1

Pt+

Dt+1

Pt

Capital gain

RPt+1 =

Pt+1

Pt

Dividend yield

DYt+1 =Dt+1

Pt



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Example

Jan-4-2011 Mar-31-2011 Jan-4-2012

Pt Pt+1Dt+ 14 Rt+1 =

Pt+1+Dt+ 14

Pt

Compute gross return if

Pt= 1270.2Dt+ 1

4

= 6

Pt+1 = 1257.6?

What assumption have you made about reinvesting dividends?

What other assumptions are reasonable?



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Example with Reinvested Dividends

Pt+ 14

= 1260.1

Number of shares per dollar invested at time t: x= $1

Pt

Additional shares at time t+ 14

: x = xD

t+ 14P

t+ 14

Gross return: Rt+1 = (x+ x)Pt+1



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Gross vs. Net Return

The above is a gross return

Net return = gross return - 1

Returns relate to horizont t+ 1

1 period can be a day, a minute, a decade, etc.

Dividends dont all arrive at a point in time

-t t+16

Dividends arrive at some time between t and t+ 1



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Indices

Indices track groups of assets (e.g., stocks, bonds, commodities)

Examples of different equity indices and methods

Russell 2000 Index Russell

S&P 500 Index S&P

Compounding gross returns from the group of assets leads to ReturnIndex (e.g., the Russell 2000)

Compounding capital gains from the group of assets leads to Price Index(e.g., the S&P 500 index)

Index levels over time can be used to produce a time series of returns

http://www.russell.com/indexes/data/US_Equity/russell_US_indexes_methodology.asphttp://ca.spindices.com/indices/equity/sp-500http://ca.spindices.com/indices/equity/sp-500http://www.russell.com/indexes/data/US_Equity/russell_US_indexes_methodology.asp


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Equity Indexes in the US


G d N R L R R R d V i bl M f R A li i


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Simple vs. Log Return

Simple (i.e., gross and net) returns are convenient when aggregatingreturns of assets in portfolios.

The weighted average of simple returns is the simple return on the

portfolio. This is not true for log returns.

Log returns are convenient when aggregating returns of assets over time.

For statistics, it is more convenient to work with sums than products. Forexample, the sum of normally distributed log returns is normally

distributed, whereas the product is not.



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Compounding Simple Returns Over Time

t t+1 t+2

$1

10% 5%

Rt+1 Rt+2

Rt+1Rt+2

Two 1-period returns: Rt+1,Rt+2

The 2-period return Rtt+2 = Rt+1Rt+2 = 1.101.05

Long-horizon simple returns are products of short-horizon simple returns




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Compounding Log Returns Over Time

To generate long-horizon returns from short-horizon returns

Rtt+2 = Rt+1Rt+2= ert+1 ert+2

= ert+1+rt+2

The 2-period log return: rtt+2

rtt+2 = ln(Rtt+2) = ln(ert+1+rt+2 ) = rt+1 + rt+2

Long-horizon log returns are sums of short-horizon log returns




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Returns Summary

Gross simple returnRt+1 =

Pt+1 +Dt+1Pt

Net simple returnRt+11

Log returnrt+1 = ln(Rt+1) = ln(e

rt+1 )

2-year gross simple return

Rt+1Rt+2 = ert+1 e

rt+2

= ert+1+rt+2

2-year log returnrt+1 + rt+2




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Exercise

1 Select an index from the MSCI website MSCI

2 For December 2012, calculate:

The gross return excluding dividendsThe gross return including dividendsThe net return excluding dividendsThe net return including dividends

The log return excluding dividendsThe log return including dividends

3 For the calendar year 2012, calculate:

The gross return excluding dividendsThe gross return including dividendsThe net return excluding dividendsThe net return including dividendsThe log return excluding dividendsThe log return including dividends


http://www.msci.com/products/indices/size/standard/performance.htmlhttp://www.msci.com/products/indices/size/standard/performance.html


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g pp

Model Returns Using Probability Distributions

Question: What will be this months S&P 500 return?

Answer: Nobody knows!

BUT, if you can answer questions like the following then returns have a

distribution:What is the probability that the S&P 500 index will fall during the nextmonth?What is the probability that the S&P 500 index will increase by more than5% during the next month?

Probability distributions describe the probability of outcomes or events




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g pp

Distribution of S&P 500 Index Returns




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Assumption 1: Gross Returns are Normal




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Assumption 2: Log Returns are Normal




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Properties of Normal Distributions N(,2)




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Empirical Return Distributions are Skewed




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Empirical Return Distributions have Fat Tails




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Empirical Estimates of Moments of Returns

Random variables can be characterized by their moments

Unfortunately the moments are typically unknown

If you are willing to assume a model of the returns, then there are manyways to produce estimates of moments using historical data

Be careful here, because your model might be wrong!

Typically, estimates of the first two moments are needed

1st moment: Mean

Estimate of mean = 1TTt=1Rt

This point estimate is measured with error (i.e., it is a noisy measure of

the true return mean)2nd moment: Variance

Estimate of variance = 1T1 Tt=1(Rtmean)2

Standard deviation =

variance




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Higher-Order Moments

3rd moment: Skewness

Estimator = 1TTt=1(Rtmean)34th moment: Kurtosis

Estimator = 1TTt=1(Rtmean)4

For a Normal distribution

Skewness = 0Kurtosis = 3 (Excess Kurtosis = Kurtosis - 3 = 0)

Negative skewness

Big left tailBubbles bust

Crashes

Excess kurtosis

Fat tailsExtreme outcomes more frequent than predicted by normal




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Exercise

1 Download the S&P 500 Index data from WRDS WRDS

username: c374y13password: B3ta1sR1sk

2 Using monthly data from 1926, calculate the first four moments ofgross return excluding dividends

gross return including dividends

net return excluding dividends

net return including dividendslog return excluding dividends

log return including dividends

3 Using annual data from 1926, calculate the first four moments ofgross return excluding dividends

gross return including dividends

net return excluding dividendsnet return including dividends

log return excluding dividends

log return including dividends

4 Produce a table comparing the moments for the monthly and annual datafrequency. How do they compare?


https://wrds-web.wharton.upenn.edu/wrds/https://wrds-web.wharton.upenn.edu/wrds/


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Some Applications

Estimating long-run mean return using geometric vs. arithmetic average

Measuring risk using Value-at-Risk (VaR)

Answers questions like: x% of the time the maximum n-day loss will be

less than y (or y%)The Economist article VaR

Testing for normality

The Economist article Mandelbrot

Simulating returns, producing empirical histograms of returns


http://www.economist.com/node/15474075http://www.economist.com/node/12957753http://www.economist.com/node/12957753http://www.economist.com/node/15474075


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Geometric Average of Returns

Suppose we have 85 years of monthly index observations (1926-2010)

Level of index at end of year 85 in terms of monthly returns

P85 = R1R2 ...R1020P0, P0 = 1= er1 er2 ...er1020 = er1+r2+...+r1020

Geometric average

g = (R1R2 ...RT)1/T

Relationship to index level:

g = P1

102085

= (er1+r2+...+r1020 )1

1020

= er1+r2+...+r1020

1020

Relationship to the average of log returns: ln(g) =r1+r2+...+r1020

1020




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Expectations of Simple and Log Returns

A useful relationship

rt

N(,2)

E[Rt] = e+

2

2

Var[Rt] = e2+2

e

2 1




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Simulating Returns

The following steps can be followed to generate draws from a normaldistribution using Excel:

1 Generate a series of uniform random numbers on (0,1) using =rand()2 Use the inverse of the Standard Normal CDF (i.e., zero mean, unit variance

normal distribution) to generate a series of N(0,1) random variables,=normsinv()

3 Multiply the standard normal by the desired standard deviation and add thedesired mean to create the simulated draws from the normal distribution.

To achieve a log-normal distribution, exponentiate and subtract 1.




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Do you

Know how to access WRDS, Bloomberg

Know how to access index data with and without dividends reinvested

Know how to calculate moments of returns at a variety of horizons

Know how to produce a histogram of various returns at a variety ofhorizons

Have a general idea of the magnitude of return statistics for some commonasset classes at a variety of horizons

Know how to simulate returns from common distributions


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