4. Statistical Foundations Flashcards
practice questions
- Describe the difference between an ex ante return and an ex post return in the case of a financial asset.
Ex post returns are realized outcomes rather than anticipated outcomes. Future possible returns and their probabilities are referred to as expectational or ex ante returns.
- Contrast the kurtosis and the excess kurtosis of the normal distribution.
Kurtosis serves as an indicator of the peaks and tails of a distribution. In the case of a normally- distributed variable the kurtosis is 3. Excess kurtosis is equal to kurtosis minus 3. Thus a normally distributed variable has an excess kurtosis of 0. Excess kurtosis provides a more intuitive measure of kurtosis relative to the normal distribution since it varies around zero to indicate kurtosis that is larger (positive) or smaller (negative) than the case of the normal distribution.
- How would a large increase in the kurtosis of a return distribution affect its shape?
Kurtosis is typically viewed as capturing the fatness of the tails of a distribution, with high values of kurtosis, or positive values of excess kurtosis, indicating fatter tails (i.e., higher probabilities of extreme outcomes) than is found in the case of a normally distributed variable. Kurtosis can also be viewed as indicating the peakedness of a distribution, with a sharp narrow peak in the center being associated with high values of kurtosis, or positive values of excess kurtosis.
- Using statistical terminology, what does the volatility of a return mean?
• Volatility is often used synonymously with standard deviation in investments.
- The covariance between the returns of two financial assets is equal to the product of the standard deviations of the returns of the two assets. What is the primary statistical terminology for this relationship?
• The covariance will equal the product of the standard deviations when the correlation coefficient is equal to one.
- What is the formula for the beta of an asset using common statistical measures?
2
β(i) = Cov (R (m) ,R(i)) /Var(R ) = 𝛔(i,m) /𝛔(^2)(m) = 𝑝(i,m) 𝛔(i)/ 𝛔(m)
- What is the value of the beta of the following three investments: a fund that tracks the overall market
index, a riskless asset, and a bet at a casino table?
• +1, 0, 0 (assuming the casino bet is a traditional bet not based on market outcomes).
- In the case of a financial asset with returns that have zero autocorrelation, what is the relationship
between the variance of the asset’s daily returns and the variance of the asset’s monthly return?
• The variance of the monthly returns are T times the variance of the daily returns where T is the number of trading days in the month.
- In the case of a financial asset with returns that have autocorrelation approaching positive one, what is the relationship between the standard deviation of the asset’s monthly returns and the standard deviation of the asset’s annual return?
• In the perfectly correlated case the standard deviation of a multiperiod return is proportional to T. In this case the annual vol is 12 times the monthly vol.
- What is the general statistical issue addressed when the GARCH method is used in a time series analysis of returns?
• The tendency of an asset’s variance to change through time.
How would a normal distribution be defined as far as a graph and numerically?
bell-shaped distribution, also known as the Gaussian distribution. Symmetric, meaning that the left and right sides are mirror images of each other. With clusters or peaks near the center, with decreasing probabilities of extreme events.
Lognormal distribution?
a variable has lognormal distribution if the distribution of the logarithm of the variable is normally distributed.
What are differences between Leptokurtic, mesokurtic, and platykurtic?
Leptokurtic is the tall with shallow tails.
Platykurtic is the short and deep tails.
Then mesokurtic is in the middle of the two.
ARCH
(autoregressive conditional heteroscedasticity) is a
special case of GARCH that allows future variances to rely
only on past disturbances, whereas GARCH allows future
variances to depend on past variances as well.
Autocorrelation
time series of returns from an
investment refers to the possible correlation of the returns with
one another through time.
Autoregressive
refers to when subsequent values to a variable
are explained by past values of the same variable
Beta
defined as the covariance between
the asset’s returns and a return such as the market index,
divided by the variance of the index’s return, or, equivalently,
as the correlation coefficient multiplied by the ratio of the asset
volatility to market volatility: βi = Cov(R m,R i)∕Var(R m) =
σim∕σ2 where βi is the beta of the returns of asset i (Ri) with
respect to a market index of returns, Rm
Conditionally heteroskedastic
financial market prices have
different levels of return variation even when specified
conditions are similar (e.g., when they are viewed at similar
price levels).
Correlation coefficient
(also called the Pearson correlation
coefficient) measures the degree of association between two
variables, but unlike the covariance, the correlation coefficient
can be easily interpreted.
Covariance
return of two assets is a measure of the
degree or tendency of two variables to move in relationship
with each other.
First-order autocorrelation
refers to the correlation between
the return in time period t and the return in the immediately
previous time period, t − 1.
GARCH
(generalized autoregressive conditional
heteroskedasticity) is an example of a time-series method that
adjusts for varying volatility.
Heteroskedasticity
is when the variance of a variable changes
with respect to a variable, such as itself or time.
Homoskedasticity
is when the variance of a variable is
constant.
