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➢ Characteristics of Return Distributions

➢ Moments of Return Distribution

➢ Correlation

➢ Standard Deviation & Variance

➢ Test for Normality of Distributions

➢ Time Series Return Volatility Models

2.4 STATISTICAL FOUNDATIONS

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• Ex post Returns

➢ Historical returns also called “after the fact” returns.

• Ex post Distribution

➢ Is a function that assigns relative frequencies to ranges of values of the random variable.

• Ex ante Returns

➢ Future Returns also called “ before the fact” returns.

• Ex ante distribution

➢ Is a function that assigns probabilities to ranges of future possible values of the random

variable.

➢ Ex post distributions are used to approximate ex ante distributions, so long as

➢ i) the distribution is stationary

➢ Ii) large number of historical are available from which a proper representation of the

distribution can be arrived.

Characteristics of Return Distributions

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• Normal Distribution(Gaussian Distributions)

➢ Is the “bell shaped”, symmetrical & mesokurtic

➢ The tail of the distribution is asymptotic and continuous

➢ It can be fully explained by mean and variance

➢ Continuous compounding returns (log normal returns) follow normal distribution in

contrast to simple returns from discreet compounding.

➢ Ex: if monthly returns are normally distributed, the quarterly returns using discreet

compounding is not normally distributed, in contrast if monthly log returns are normally

distributed then quarterly log returns are also normally distributed.

Characteristics of Return Distributions

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• Lognormal Distribution

➢ It is generated by the function ex , where x is normally distributed

➢ It is positively skewed

➢ It is bounded from below by zero, hence useful for modelling asset prices which never take

negative values

➢ As per Central Limit Theorem, even if ln(1+R) does not follow a normal distribution, the sum

of the logs can be closely approximated by the normal distribution, as the number of

observations increases and uncorrelated over time.

Characteristics of Return Distributions

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Characteristics of Return Distributions

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Moments of Return Distribution

• The shape of probability distributions are described by the moments of the distribution.

• Raw Moments

➢ Are measured relative to an expected value raised to the appropriate power

➢ The first raw moment is the mean of the distribution, which is the expected value of the

returns

➢ kth raw moment:

➢ Raw moments for k>1 are not useful for return calculations.

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• Central Moments- (Variance)

➢ Are measured relative to the mean

➢ 2nd order central moment is the expected variance

➢ kth order central moment:

Moments of Return Distribution

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• Example 1:

➢Calculate the sample variance and sample standard deviation from the following set of

returns.

Period Return

1 .1

2 .06

3 -.1

Moments of Return Distribution

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• Skewness Statistic

➢ Is the standardised third central moment

➢ It refers to the extent distribution of data is not symmetric about mean

Moments of Return Distribution

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Moments of Return Distribution

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• Kurtosis Statistic

➢ Is the standardized fourth central moment

➢ It refers to the degree of peakedness in the distribution of data

➢ Normal kurtosis also called mesokurtic is equal to 3, if kurtosis greater than 3 than it is called

leptokurtic and less than 3 is called platykurtic.

Moments of Return Distribution

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Moments of Return Distribution

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Covariance

• Covariance

➢ Is an unscaled statistical measure of how two assets move together

➢ The value ranges between - ∞ to + ∞

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Correlation Coefficient

• Correlation Coefficient

➢ Also referred as the Pearson correlation coefficient is a relative measure of the strength of

relationship between two assets

➢ -1 represents perfect negative linear correlation and +1 represents perfect positive linear

correlation and zero represent no linear correlation

➢ Even for zero correlation there are possibilities of non-linear correlation

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• Correlation Coefficient

Correlation Coefficient

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➢ Covariance & Correlation

• Example 2:

➢Calculate the covariance and correlation between two assets.

Outcome Ri Rj

1 0.00 0.20

2 0.10 0.10

3 0.20 0.00

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➢ Spearman’s Rank Correlation

𝝆𝒔 = 𝟏 −𝟔σ𝒅𝒊

𝟐

𝒏(𝒏𝟐 − 𝟏)

Example: Jason Ganjaleze is examining the cross-sectional relationship between firm size and eps.

He has collected an initial sample of 10 firms.

Company EPSMarket value (in millions

of dollars)

A -0.38 705.642

B -0.62 5.0201

C -7.98 2976.3858

D 0.34 34.6617

E 0.47 1547.0867

F 2.12 3241.5314

G 1.61 1389.332

H 3.06 82853.3492

I 0.55 186.7674

J 1.19 38.408

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➢ The Correlation Coefficient and Diversification

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➢ Portfolio Diversification

• Portfolio Standard Deviation:

𝝈𝒑 = 𝒘𝒊𝟐𝝈𝒊

𝟐 +𝒘𝒋𝟐𝝈𝒋

𝟐 + 𝟐𝒘𝒊𝒘𝒋𝝆𝒊,𝒋𝝈𝒊𝝈𝒋

• Effects of Correlation on Portfolio Diversification:

Expected Return Standard Deviation

Domestic Stocks(DS) 0.20 0.30

Domestic Bonds(DB) 0.10 0.15

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➢ Beta

𝜷𝒊 =𝑪𝒐𝒗(𝑹𝒊, 𝑹𝒎)

𝑽𝒂𝒓(𝑹𝒎)=𝝆𝒊,𝒎𝝈𝒊𝝈𝒎

𝝈𝒎𝟐

= 𝝆𝒊,𝒎𝝈𝒊𝝈𝒎

Example:The covariance of returns between the RE Fund and the market portfolio equals 0.20, and the

standard deviation of returns equals 0.80 and 0.40 for the RE funds and the market portfolio,

respectively. Calculate the correlation between the RE fund and the market portfolio. Next,

calculate the beta for the RE fund.

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➢ Autocorrelation

𝒌 − 𝒐𝒓𝒅𝒆𝒓 𝒂𝒖𝒕𝒐𝒄𝒐𝒓𝒓𝒆𝒍𝒂𝒕𝒊𝒐𝒏 =𝑬[(𝑹𝒕 − 𝝁)(𝑹𝒕−𝒌 − 𝝁)

𝝈𝒕𝝈𝒕−𝒌

• First order autocorrelation(serial correlation)

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➢ Autocorrelation

• Positive & negative serial correlation

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➢ Durbin-Watson Statistic

• Hypothesis:

𝑯𝟎: 𝝆𝒕,𝒕−𝟏 = 𝟎

𝑯𝑨: 𝝆𝒕,𝒕−𝟏 ≠ 𝟎

𝑫𝑾 =σ𝒕=𝟐𝑻 (𝒆𝒕 − 𝒆𝒕−𝟏)

𝟐

σ𝒕=𝟏𝑻 𝒆𝒕

𝟐

If sample is large, approx. Durbin Watson statistic is:

𝑫𝑾 ≈ 𝟐(𝟏 − 𝝆𝒕,𝒕−𝟏)

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➢ Durbin-Watson Statistic

Example:

Assume a large sample of returns is examined for a private equity fund. The correlation od

successive returns equals 0.6. Compute and interpret the Durbin Watson statistic.

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Standard Deviation & Variance

Confidence Intervals for the Normal Distribution Using Standard Deviation

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Test of Normality

• Causes of Non-Normality1. Autocorrelation

2. Illiquidity

3. Nonlinearity

• Sample Moments

• Jarque Bera Test

where JB is the Jarque-Bera test statistic, n is the number of observations, S is the skewness of the sample, and K is the excess kurtosis of the sample.

Example:

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➢ Forecasts of Future Return Volatility

• Heteroskedastic

• Conditional Heteroskedastic

• Auto regression

• ARCH Model

• GARCH Model