4. Statistical Foundations

Question 1

1. Describe the difference between an ex ante return and an ex post return in the case of a financial asset.

Answer 1

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.

Question 2

2. Contrast the kurtosis and the excess kurtosis of the normal distribution.

Answer 2

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.

Question 3

3. How would a large increase in the kurtosis of a return distribution affect its shape?

Answer 3

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.

Question 4

4. Using statistical terminology, what does the volatility of a return mean?

Answer 4

• Volatility is often used synonymously with standard deviation in investments.

Question 5

5. 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?

Answer 5

• The covariance will equal the product of the standard deviations when the correlation coefficient is equal to one.

Question 6

6. What is the formula for the beta of an asset using common statistical measures?

Answer 6

2

β(i) = Cov (R (m) ,R(i)) /Var(R ) = 𝛔(i,m) /𝛔(^2)(m) = 𝑝(i,m) 𝛔(i)/ 𝛔(m)

Question 7

7. 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?

Answer 7

• +1, 0, 0 (assuming the casino bet is a traditional bet not based on market outcomes).

Question 8

8. 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?

Answer 8

• 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.

Question 9

9. 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?

Answer 9

• 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.

Question 10

10. What is the general statistical issue addressed when the GARCH method is used in a time series analysis of returns?

Answer 10

• The tendency of an asset’s variance to change through time.

Question 11

How would a normal distribution be defined as far as a graph and numerically?

Answer 11

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.

Question 12

Lognormal distribution?

Answer 12

a variable has lognormal distribution if the distribution of the logarithm of the variable is normally distributed.

Question 13

What are differences between Leptokurtic, mesokurtic, and platykurtic?

Answer 13

Leptokurtic is the tall with shallow tails.

Platykurtic is the short and deep tails.

Then mesokurtic is in the middle of the two.

Question 14

ARCH

Answer 14

(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.

Question 15

Autocorrelation

Answer 15

time series of returns from an
investment refers to the possible correlation of the returns with
one another through time.

Question 16

Autoregressive

Answer 16

refers to when subsequent values to a variable

are explained by past values of the same variable

Question 17

Beta

Answer 17

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

Question 18

Conditionally heteroskedastic

Answer 18

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).

Question 19

Correlation coefficient

Answer 19

(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.

Question 20

Covariance

Answer 20

return of two assets is a measure of the
degree or tendency of two variables to move in relationship
with each other.

Question 21

First-order autocorrelation

Answer 21

refers to the correlation between
the return in time period t and the return in the immediately
previous time period, t − 1.

Question 22

GARCH

Answer 22

(generalized autoregressive conditional
heteroskedasticity) is an example of a time-series method that
adjusts for varying volatility.

Question 23

Heteroskedasticity

Answer 23

is when the variance of a variable changes

with respect to a variable, such as itself or time.

Question 24

Homoskedasticity

Answer 24

is when the variance of a variable is

constant.

Question 25

Jarque-beta test

Answer 25

involves a statistic that is a function of
the skewness and excess kurtosis of the sample: JB = (n∕6)[S2
+ (K2 ∕4)] 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.

Question 26

Leptokurtosis

Answer 26

If a return distribution has positive excess kurtosis, meaning it
has more kurtosis than the normal distribution, it is said to be this or fat tailed.

Question 27

Mean

Answer 27

The most common raw moment is the first raw moment and
is known as the mean, or expected value, and is an indication
of the central tendency of the variable.

Question 28

Mesokurtosis

Answer 28

If a return distribution has no excess kurtosis, meaning it has
the same kurtosis as the normal distribution, it is said to be this or normal tailed.

Question 29

Perfect linear negative correlation

Answer 29

A correlation coefficient of −1 indicates that the two assets
move in the exact opposite direction and in the same
proportion, a result known as this.

Question 30

Perfect linear positive correlation

Answer 30

A correlation coefficient of +1 indicates that the two assets
move in the exact same direction and in the same proportion, a
result known as this.

Question 31

Platykurtosis

Answer 31

If a return distribution has negative excess kurtosis, meaning
less kurtosis than the normal distribution, it is said to be this or thin tailed.

Question 32

Skewness

Answer 32

is equal to the third central moment divided by
the standard deviation of the variable cubed and serves as a
measure of asymmetry: Skewness = E[(R − μ)3 ]∕σ3.

Question 33

Spearman rank correlation

Answer 33

is a correlation designed to
adjust for outliers by measuring the relationship between
variable ranks rather than variable values.

Question 34

SD

Answer 34

The square root of the variance is an extremely popular and
useful measure of dispersion known as the standard
deviation: Standard Deviation = √σ2 = σ

Question 35

Variance

Answer 35

is the second central moment and is the expected

value of the deviations squared,

Question 36

Volatility

Answer 36

In investment terminology, says it is a popular term that is

used synonymously with the standard deviation of returns.

Question 37

The daily returns of Fund A have a variance of
0.0001. What is the variance of the weekly returns of Fund A assuming that the
returns are uncorrelated through time?

Answer 37

The daily returns have a variance of 0.0001 and are uncorrelated through time.
The uncorrelated through time allows us to use equation 4.27. Therefore, the
variance of weekly returns of Fund A is equal to 0.0001 multiplied by 5 (number of
trading days in a week) for a product of 0.0005. Simply put, variance grows
linearly with time horizon when returns are uncorrelated.

Step One: Press 0.0001 → x → 5
Step Two: Press =
Answer: 0.0005

Question 38

The daily returns of Fund A have a standard
deviation of 1.4%. What is the standard deviation of a position that contains only
Fund A and is leveraged with $3 of assets for each $1 of equity (net worth)?

Answer 38

Utilizing equation 4.31, we multiply 1.4% (the standard deviation) by 3/1 or 3 for a
product of 4.2%, which is the standard deviation of levered returns. Simply put,
being levered with $3 of assets to $1 of equity causes the volatility of the equity to
be 3 times the volatility.

Step One: Press 3 → / → 1
Step Two: Press x → 0.014
Step Three: Press =
Answer: 0.042 or 4.2%

Question 39

The daily returns of Fund A have a standard
deviation of 1.4%. What is the standard deviation of a position that contains 40%
Fund A and 60% cash?

Answer 39

To solve this application, it is important to understand that cash has a standard
deviation of 0. Therefore, we can utilize equation 4.33 and multiply 40% (the
proportion of the fund with a standard deviation of 1.4%) by 1.4% for a product of
0.56%, which is the standard deviation of the unlevered returns.

Step One: Press 0.4 → x → 0.014
Step Two: Press =
Answer: 0.0056 or 0.56%

Question 40

The daily returns of Fund A have a standard
deviation of 1.2%. What is the standard deviation of the returns of Fund A over a
four-day period if the returns are uncorrelated through time? What is the
maximum standard deviation for other correlation assumptions?

Answer 40

With zero autocorrelation, the standard deviation of four-day returns is 2.4%
(based on the square root of the number of time periods). Simply As the
correlation approaches +1, the upper bound would be 4.8%.

Assuming uncorrelated returns through time
Step One: Press 0.012 → x → 4
Step Two: Press Square root (x) → =
Answer: 0.024 or 2.4%
Assuming perfectly correlated returns
Step One: Press 0.012 → x → 4
Step Two: Press =
Answer: 0.048 or 4.8%