Book 2. Quantitative Analysis

FRM Part 1

QA 12. Measuring Returns, Volatility and Correlation

Presented by: Sudhanshu

Module 1. Defining Returns and Volatility

Module 2. Normal and Non-normal Distributions

Module 3. Correlations and Dependence

Module 1. Defining Returns and Volatility

Topic 1. Simple and Continuously Compounded Returns

Topic 2. Volatility, Variance, and Implied Volatility

Topic 1. Simple and Continuously Compounded Returns

Definition : Returns on investments are often expressed as simple returns and continuously compounded returns.
Simple Returns Can be expressed over various periods, from an hour to a full year.
- For multiple periods, an asset's return is the product of each period's simple return.
- Formula for an asset purchased at $t - 1$ and sold at $t$ :
Continuously Compounded (Log) Returns ( $r_{t}$ ) Formula:
- For multiple periods, the total return is the sum of single period log returns:

More appropriate for shorter time horizons.
Do not accurately approximate simple returns when the simple return is large.
Conversion between Simple and Log Returns :
- The simple return always exceeds the log return.

(R_t):

R_t=\frac{P_t-P_{t-1}}{P_{t-1}}

r_t=\ln P_t-\ln P_{t-1}

r_T=\sum_{t=1}^T r_t

1+R_t=exp(r_t)

Practice Questions: Q1

Q1. Assuming a simple return of 5.00%, the log return will be closest to:

A. 4.88%.

B. 5.00%.

C. 5.05%.

D. 5.13%.

Practice Questions: Q1 Answer

Explanation: A is correct.

The equation to convert the simple return to the log return is:

Plugging in values, . Taking the natural log of each side to isolate the log return (r)
results in ln 1.05 = 0.0488 or 4.88%

1+R_t=\exp r_t

1.05=\exp r_t

Topic 2. Volatility, Variance, and Implied Volatility

Volatility ( $σ$ ) : Expressed as the standard deviation of its returns.
Variance (or Variance Rate) ( $σ^{2}$ ) : Expressed as
Return Formula ( $r_{t}$ )
- $r_{t} = μ + σ e_{t}$ (where $μ$ is mean, $e_{t}$ is a shock with zero mean and variance of one)
Annualized Volatility : Using monthly returns:
- Using daily returns (252 trading days):
Implied Volatility
- An annual volatility number backed out using option prices.
- Black-Scholes-Merton (BSM) Model : Used to price call options with inputs: current asset price, strike price, time to maturity, risk-free interest rate, and annual variance.
  - Annual variance can be derived if the option price is known, as other variables are observable.
  - Drawback: Assumes variance is constant over time.
VIX Index : Measures implied volatility for the S&P 500 for a prospective 30-calendar-day period.
- Uses option prices with future expiration dates and multiple strike prices.
- Serves as a forward-looking volatility measure.
- Requires a significant and liquid derivatives market.

\sigma^2.

\sigma_{annual}=\sqrt{12 \times \sigma^2_{monthly}}

\sigma_{annual}=\sqrt{252 \times \sigma^2_{daily}}

Practice Questions: Q2

Q2. Which of the following statements is correct in regard to using the Black-Scholes- Merton (BSM) pricing model to calculate implied volatility?

A. The option price is not needed for the calculation.

B. Variance is assumed to remain constant over time.

C. Time to maturity is not one of the components of the calculation.

D. The current asset price has to remain constant in the calculation.

Practice Questions: Q2 Answer

Explanation: B is correct.

One of the drawbacks to using the BSM pricing model to derive implied volatility is that variance must remain constant over time. The option price and time to maturity are both needed for the calculation, but there is no requirement that the current underlying asset price has to remain constant.

Module 2. Normal and Nonnormal Distributions

Topic 1. First two moments insufficient for non-normal distributions

Topic 2. Jarque-Bera Test

Topic 3. The Power Law

Topic 1. First two moments insufficient for non-normal distributions

Moments of a Probability Density Function
- First Moment: Mean
- Second Moment: Variance
- Third Moment: Skewness
- Fourth Moment: Kurtosis
Normal Distribution Properties
- Thin tails
- Symmetric
- No skewness
- No excess kurtosis (kurtosis of three)
Non-Normal Distribution Properties (Financial Returns)
- Financial returns often follow non-normal distributions.
- Exhibit skewness and excess kurtosis.
- Example: S&P 500, JPY/USD exchange rate, and gold returns show non-zero skewness and kurtosis greater than three (positive excess kurtosis).
- S&P 500 and JPY/USD had negative skewness; gold returns had positive skewness.

Practice Questions: Q1

Q1. Relative to a normal distribution, financial returns tend to have a nonnormal distribution, which will have:

A. thin tails.

B. kurtosis greater than three.

C. minimal to no skewness.

D. a symmetrical distribution.

Practice Questions: Q1 Answer

Explanation: B is correct.

A nonnormal distribution is likely to have either positive or negative skewness and a kurtosis that is different from three. A normal distribution has thin tails, kurtosis equal to three, no skewness, and a symmetrical distribution.

Topic 2. Jarque-Bera Test

Purpose: Used to test whether a distribution is normal.
- A normal distribution implies zero skewness and no excess kurtosis ( $K - 3 = 0$ ).
Hypotheses : Null Hypothesis : $S = 0$ and $K = 3$ (zero skewness and kurtosis of three)
- Alternative Hypothesis (skewness not zero and kurtosis not three)
Test Statistic
Where $T$ is the sample size, S is skewness, and K is kurtosis.
Interpretation : Smaller JB values suggest the null hypothesis is likely true (distribution is normal).
- Larger JB values suggest the null hypothesis is likely to be rejected (distribution is not normal).
- Critical values at 5% and 1% are 5.99 and 9.21, respectively. Reject null if JB is above these levels.
Impact of Time Period : Longer measurement periods often result in smaller JB statistics, approximating a normal distribution.

(H_0)

(H_A): S \neq 0 \text{ and }K \neq 3

JB=(T-1)\left(\frac{\hat{S}^2}{6}+\frac{(\hat{K}-3)^2}{24}\right)

Practice Questions: Q2

Q2. Which of the following statements regarding the Jarque-Bera (JB) test statistic is most accurate?

A. The null hypothesis states that skewness does not equal zero.

B. The alternative hypothesis states that kurtosis is equal to three.

C. The alternative hypothesis is likely to be rejected when the JB statistic is high.

D. The null hypothesis is likely to not be rejected when the JB statistic is very small.

Practice Questions: Q2 Answer

Explanation: D is correct.

When the JB test statistic is very small, the null hypothesis is likely to not be rejected. When the statistic is high, the null is likely to be rejected (with the alternative hypothesis not being rejected). The null hypothesis states that skewness is zero and kurtosis is three (with excess kurtosis therefore equal to zero). The alternative hypothesis states that skewness is not equal to zero and kurtosis is not equal to three.

Topic 3. The Power Law

Relevance to Financial Returns
- Financial returns tend to follow non-normal distributions.
- Studying the tails helps explain return distribution.
Normal Distribution Tails
- Kurtosis of three (excess kurtosis of zero).
- Thin tails.
Power Law Tails
- Found in some distributions, including Student's t-distribution.
- Probability of a return larger than $x$ : (where k and a are constants).
- "Fatter" tails than normal distributions
- Tails do not decline as quickly as normal distributions.
- Observations away from the mean are more common.

P(X>x)=k x^{-\alpha}

Practice Questions: Q3

Q3. Which of the following statements is most accurate regarding power law tails?

A. More observations tend to be closer to the mean.

B. The standard normal distribution exhibits power law tails.

C. The tails exhibit faster declines than normally distributed tails.

D. They tend to have “fatter” tails than those found in a normal distribution.

Practice Questions: Q3 Answer

Explanation: D is correct.

Power law tails tend to be “fatter” than the tails found in normal distributions. Power law tails relect more observations found farther away from the mean and they tend to exhibit slower declines than the tails in normal distributions.

Module 3. Correlation and Dependence

Topic 1. Correlation and Covariance

Topic 2. Spearman’s Rank Correlation

Topic 3. KENDALL’S τ

Topic 4. Positive Definiteness

Topic 1. Correlation and Covariance

Independence vs. Dependence : Independent Variables: Product of marginal densities equals their joint density:
- Increases diversification benefits and decreases tail risk.
- Dependent Variables: Financial assets often exhibit high linear and nonlinear dependence.
Correlation vs. Covariance : Correlation: Represents the linear relationship between two variables.
- Covariance: Represents the directional relationship between two variables.
Pearson's Correlation : Measures linear dependence.
Regression and Linear Dependence : If Y and X are standardized to unit variance, the regression slope ( $β$ ) equals the correlation.
Nonlinear Dependence : No single statistic measures it.
- Measures include Spearman's rank correlation and Kendall's $τ$ .
- Values for both lie between -1 and 1.
- Zero when returns are completely independent.
- Scale invariant.
- Positive (negative) when variables have an increasing (decreasing) relationship.

f_{x,y}(x,y)=f_x(x) f_y(y)

Practice Questions: Q1

Q1. An analyst calculates a Spearman’s rank correlation of 0.48. This output is indicative of:

A. positive linear correlation.

B. negative linear correlation.

C. positive nonlinear dependence.

D. negative nonlinear dependence.

Practice Questions: Q1 Answer

Explanation: C is correct.

Correlation will be between –1 and 1. Any number above 0 is going to represent a positive output. Because rank correlation is used to measure nonlinear dependence, an output of 0.48 indicates positive nonlinear dependence.

Practice Questions: Q2

Q2. Which of the following situations is indicative of equicorrelation in a correlation matrix?

A. Correlations which are all equal to 1.

B. Variables with correlations other than 0.

C. Variables with negative coefficients of determination.

D. Three variables with a correlation with one another of 1.25.

Practice Questions: Q2 Answer

Explanation: A is correct.

If all of the variables in a correlation matrix have correlations of 1, this is indicative of equicorrelation. They can have correlations of zero, as long as all are equal. Variables cannot have negative coeficients of determination (which are correlations squared) and correlations can never be greater than 1.

Topic 2. Spearman’s Rank Correlation

Definition : A linear correlation estimator applied to ranks of observations.
- Driven by the strength of the linear relationship between ranks, not the variables themselves.
Ranking Process : For two random variables (X, Y) with $n$ observations, and are their ranks.
- Rank 1 for the smallest value, and so on, up to rank $n$ for the largest.
Formula (Distinct Ranks):

Where (difference in ranks for the same observation).
Interpretation : Close to 1: Highly ranked values of X and Y are paired.
- Close to -1: Largest values of one variable are grouped with the smallest values of another (strong negative dependence).
Relationship to Linear Correlation : Similar if variables have a strong linear relationship.
- Large differences indicate a key nonlinear relationship.
Robustness to Outliers : Less sensitive or vulnerable to outliers than linear correlation because it uses ranks instead of actual values.

Rank_x

Rank_y

\hat{p}_s=1-\frac{6 \sum_{i=1}^n\left(d_i\right)^2}{n\left(n^2-1\right)}

d_i=Rank_{X_i}-Rank_{Y_i}

Topic 3. KENDALL’S $τ$

Purpose : Measures concordant and discordant pairs and their relative frequency.
- Represents the difference between probabilities of concordance and discordance.
Pair Classification
- Consider two pairs of random variables $(X_{i}, Y_{i})$ and $(X_{j}, Y_{j})$ .
- Concordant: If $(X_{i} < X_{j})$ and $(Y_{i} < Y_{j})$ (relative positions are in agreement).
- Discordant: If the orders are different.
- Neither: If $X_{i} = X_{j}$ and $Y_{i} = Y_{j}$ (ties).
Interpretation of Concordance/Discordance : Many concordant pairs: Strong positive dependence.
- Many discordant pairs: Strong negative relationship.
Formula :
Where:
- - $n_{c}$ = number of concordant pairs
  - $n_{d}$ = number of discordant pairs
  - $n_{t}$ = number of ties

\hat{\tau}=\frac{n_c-n_d}{n(n-1) / 2}=\frac{n_c}{n_c+n_d+n_t}-\frac{n_d}{n_c+n_d+n_t}

Topic 4. Positive Definiteness

Definition
- Every linear combination of random variables must have a non-negative variance.
- Requires that the variance of an average of components in a covariance matrix must be positive.
- When all random variables have unit variance (variance = 1), correlation matrix and covariance matrix are the same.
Ensuring Positive Definiteness in Correlation Matrices
- Typically achieved using two structured correlations:
- 1. Equicorrelation: Sets all correlations equal to the same amount.
- 2. Common Factor Exposure: Assumes correlations are due to a common factor exposure.
  - Correlation for any entries in the matrix equals $ρ_{i, j} = Y_{i} Y_{j}$ .
  - Each entry has a correlation between -1 and 1.