Estimated reading time: 41 minutes

## Introduction

We present our extensive summary for books 1 and 2 in the Level 1 FRM course.

Note: You may download this entire content on our website’s shop page, free of charge.

## FRM Level 1 Book 1 – Foundations of Risk Management

How we think about risk is the biggest determinant of whether we recognize or assess them properly

We isolate ten risk management building blocks

- The risk management process
- Identifying risk: knowns and unknowns
- Expected loss, unexpected loss, and tail loss
- Risk factor breakdown
- Structural change: from tail risk to systemic crisis
- Human agency and conflicts of interest
- Typology of risks and risk interactions 8. Risk aggregation
- Balancing risk and reward
- Enterprise risk management (ERM)

Each key risk type demands a specific set of skills and its own philosophical approach

## Market Risk

Market prices and rates continually change, creating the potential for loss

Market risk is driven by:

General market risk:

*This is the risk that an asset class will fall in value, leading to a fall in portfolio value*

Specific market risk:

*This is the risk that an individual asset will fall in value more than the general asset class*

## Credit Risk

Credit risk arises from the failure of one party to fulfill its financial obligations to another party

Examples of credit risk include:

- A debtor fails to pay interest or principal on a loan (bankruptcy risk)
- An obligor or counterparty is downgraded (downgrade risk)
- A counterparty to a market trade fails to perform (counter-party risk)

Credit risk is driven by:

- The probability of default of the obligor or counterparty
- The exposure amount at the time of default
- The amount that can be recovered in the event of a default

## Liquidity Risk

Liquidity risk is used to describe two separate kinds of risk:

- Funding liquidity risk and Market liquidity risk

Funding liquidity risk is the risk that covers the risk that a firm cannot access enough liquid cash and assets to meet its obligations

Market liquidity risk, sometimes known as trading liquidity risk, is the risk of a loss in asset value when markets temporarily seize up

## Operational Risk

Operational risk can be defined as the “risk of loss resulting from inadequate or failed internal processes, people, and systems or from external events.”

It includes legal risk, but excludes business, strategic, and reputational risk

Business risks includes all the usual worries of firms, such as customer demand, pricing decisions, supplier negotiations, and managing product innovation

## Strategic Risk

Strategic risk is distinct from business risk

Strategic risk involves making large, long-term decisions about the firm’s direction

Strategic risk is often accompanied by major investments of capital, human resources, and management reputation

## Reputation Risk

Reputation risk is the danger that a firm will suffer a sudden fall in its market standing or brand with economic consequences

## The Risk Management Process

- Identify the risk
- Analyze and Measure the risk
- Assess the impact
- Manage the risk

VaR was a popular risk aggregation measure in the years leading up to the global financial crisis

Financial risk managers have long recognized that they must overcome silo-based risk management process to build a broad picture of risk across risk types and business lines

We know this as Enterprise Risk Management (ERM)

## Managing Financial Risk

The risk management process as a road-map can be seen:

- Identify risk appetite
- Map risks, make choices
- Operationalize risk appetite
- Implement
- Re-evaluate regularly to capture changes

## Risk Hedging

Just because a risk can be hedged does not mean that it should be hedged

Hedging is simply a tool and, like any tool, it has limitations

Risk appetite describes the amount and types of risk a firm is willing to accept

This is in contrast to risk capacity, which describes the maximum amount of risk a firm can absorb

## Governance of Risk

Corporate governance is the way in which companies are run

Corporate governance describes the roles and responsibilities of a firm’s shareholders, board of directors, and senior management

Basel III designed a macro-prudential overlay intended to reduce systemic risk and lessen pro-cyclicality

Ultimately, only four basic choices need to be made in the management of corporate risk:

- The choice to undertake or not to undertake certain activities
- The choice to transfer or not transfer either all or part of a given risk to a third party
- The choice to preemptively mitigate risk through early detection and prevention
- The choice to assume or not assume risk

## The Risk Appetite Statement

Publishing a risk appetite statement (RAS) is an important component of corporate governance

The Financial Stability Board (FSB) describes an RAS as “a written articulation of the aggregate level and types of risk that a firm will accept or avoid in order to achieve its business objectives.”

The RAS includes both qualitative and quantitative statements

## Internal Auditors

Internal auditors are responsible for:

- Reviewing monitoring procedures
- Tracking the progress of risk management system upgrades
- Assessing the adequacy of application controls
- Affirming the efficacy of vetting processes

## Credit Risk Transfer Mechanisms

The core risk exposure for banks is credit risk

Banks have long had several ways to reduce their exposure to credit risk:

- Purchasing insurance from a third-party guarantor/underwriter
- Netting of exposures to counterparties
- Marking-to-market/margining
- Requiring collateral be posted
- Termination/Put options
- Reassignment of a credit exposure

## Securitization

Securitization involves the repackaging of loans and other assets into new securities that can then be sold in the securities markets

Securitization eliminates a substantial amount of risk (i.e., liquidity, interest rate, and credit risk) from the originating bank’s balance sheet when compared to the traditional buy-and-hold strategy

## Modern Portfolio Theory and Capital Asset Pricing Model

A “rational investor” is an investor who is risk averse and seeks to maximize utility

Markowitz demonstrated that a rational investor should evaluate potential portfolio allocations based upon the associated means and variances of the expected rate of return distributions

The theory also assumes:

- Capital markets are perfect
- There are no taxes or transaction costs
- All traders have costless access to all available information
- Perfect competition exists among all market participants
- Returns are normally distributed

According to Markowitz, the level of investment in a particular financial asset should be based upon that asset’s contribution to the distribution of the portfolio’s overall return (as measured by the mean and variance)

In other words, what matters is the covariability of the asset’s return with respect to the return of the overall portfolio

Along the efficient frontier, the only way to achieve a higher expected rate of return is by increasing the riskiness of the portfolio

The CAPM model shows that market equilibrium is achieved when all investors hold portfolios consisting of the riskless asset and the market portfolio

## Arbitrage Pricing Theory and Multifactor Models

The capital asset pricing model (CAPM) is a single-factor model that describes an asset’s expected rate of return as a linear function of the market’s risk premium above a risk-free rate

Beta is the coefficient (i.e., the slope) of this relationship

## Arbitrage Pricing Theory

Arbitrage Pricing Theory (APT) is based on the reasoning behind CAPM. However, it explains an asset’s expected rate of return as a linear function of several market factors

Arbitrage Pricing Theory assumes that there are no arbitrage opportunities

APT suggests that multiple factors can help explain the expected rate of return on a risky asset

These factors may include indices on stocks, bonds, and commodities, as well as macroeconomic factors

## Effective Data Aggregation and Risk Reporting

Effective risk analysis requires sufficient and high-quality data

This makes data a major asset in today’s world

A bank with a limited ability to integrate data will have difficulties in satisfying the Basel principles

A key challenge is collecting data from the various internal and external sources and feeding it into risk analytics systems

Firms need to monitor their data on an ongoing basis to ensure accuracy and integrity

Risk data should be complete, reconciled with sources, and include all material risk disclosures at a granular level

## The Key Governance Principles

Principle 1: Governance

Principle 2: Data architecture and IT infrastructure

Principle 3: Accuracy and Integrity

Principle 4: Completeness

Principle 5: Timeliness

Principle 6: Adaptability

Principle 7: Accuracy

Principle 8: Comprehensiveness

Principle 9: Clarity and usefulness

Principle 10: Frequency

Principle 11: Distribution

A study from PwC showed higher performance for compliance with Principles 7–11 (risk reporting) compared to Principles 3–6 (data aggregation)

Principles 1 (governance) and 2 (data architecture and infrastructure) have poor compliance rates

## Enterprise Risk Management and Future Trends

At the enterprise level, risks may:

- Negate each other (e.g., through netting and diversification) or
- Exacerbate each other (e.g., through risk concentrations, contagion, and cross-over risks)

Enterprise risk management (ERM) applies the perspective and resources at the top of the enterprise to manage the entire portfolio of risks and account for them in strategic decisions

Another important feature of ERM is that it supports a consistent approach

## ERM Benefits

Top ten benefits of ERM:

Helps firms define and adhere to risk appetites

Focuses oversight on most threatening risks

Identifies enterprise-scale risks generated at business line level

Manages risk concentrations across the enterprise

Manages emerging enterprise risks (e.g., cyber risk, AML (anti-money laundering) risk)

Supports regulatory compliance and stakeholder reassurance

Helps firms to understand risk-type correlations and cross-over risks

Optimizes risk transfer expenses in line with risk scale and total cost

Incorporates stress scenario capital costs into pricing and business decisions

Incorporates risk into business model selection and strategic decisions

## Risk Culture

Risk culture can be thought of as the set of goals, values, beliefs, procedures, customs, and conventions that influence how staff create, identify, manage, and think about risk within an enterprise

The Financial Stability Board (FSB) has specified four key risk culture indicators:

- Accountability
- Effective communication and challenge
- Incentives
- Tone from the top

## Learning from Financial Disasters

Over the last century, interest rate risk has caused the failure of individual firms as well as entire industries within the financial services sector

To mitigate interest rate risk, firms must manage their balance sheet structure such that the effect of any interest rate movement on assets remains highly correlated with the effect on liabilities

Funding liquidity risk can stem from external market conditions (e.g., during a financial crisis) or from structural problems within a bank’s balance sheet

Rogue trading can cause major financial institutions to collapse (as seen in the case of Barings Bank/Nick Leeson)

A main lesson from the Barings collapse is that reporting and monitoring of positions and risks (i.e., back-office operations) must be separated from trading

The case of Continental Illinois Bank is an example of how internal credit portfolio problems can precipitate a funding liquidity crisis

The 2007 failure of mortgage bank Northern Rock is a recent illustration of liquidity risk arising from structural weaknesses in a bank’s business model

In this case, a combination of an excessive use of short-term financing for long-term assets and a sudden loss of market confidence triggered a funding liquidity crisis that rapidly led to disaster

MGRM (Metallgesellschaft AG) was exposed to curve risk (i.e., the risk of shifts in the price curve between backwardation and contango)

Additionally, the firm was exposed to basis risk resulting from deviations between short-term prices and long-term prices

Long Term Capital Management (LTCM) failure reflected its inability to anticipate the dramatic increase in correlations and volatilities and the sharp drop in liquidity that can occur during an extreme crisis

LTCM also succumbed to an internal liquidity crunch brought on by large margin calls on its futures holdings

LTCM made heavy use of a Value-at-Risk (VaR) model as part of its risk control

Enron used “creative” (i.e., fraudulent) accounting practices to hide flaws in its actual financial performance

An example of Enron’s duplicity is a scheme by which the firm would build a physical asset and then immediately declare a projected mark-to-market profit on its books

## Model Risk

Model risk can stem from using an incorrect model, incorrectly specifying a model, and/or using insufficient data and incorrect estimators

Banks may mitigate funding liquidity risk by reducing the maturity of their assets

VaR is a measure of the worst-case loss for a given normal market conditions

Cyber risk has become a critically important consideration in recent years

## The Financial Crisis of 2007–2009

The cascade of events that came be known as the Great Financial Crisis of 2007–2009 (GFC) began with a downturn in the U.S. subprime mortgage market in the summer of 2007

The years preceding the crisis saw an exceptional boom in credit growth in the United States, a massive housing price bubble, and an excess of leverage in the financial system

February 2008 saw the nationalization of troubled U.K. mortgage lender Northern Rock, a victim of the first bank run that nation had experienced in 140 years

The following month, U.S. investment bank Bear Stearns was absorbed by J.P. Morgan Chase in a deal brokered by the U.S. Treasury Department and the Federal Reserve

The peak of the subprime crisis came in September 2008, which saw a cascade of events

- Lehman Brothers declared bankruptcy
- Morgan Stanley and Goldman Sachs, were converted to bank holding companies and became regulated by the Federal Reserve
- Fannie Mae and Freddie Mac were nationalized
- AIG was brought back from the brink of collapse via a USD 150 billion capital infusion by the U.S. Treasury and the Federal Reserve
- In Europe, many countries had to step in to provide massive support to their banks
- Dutch financial conglomerate Fortis was broken up and sold

Systemic risk is the risk that events at one firm, or in one market, can extend to other firms or markets

Systemic risk played a large role in exacerbating the impact of the crisis

## GARP Code of Conduct

The GARP Code of Conduct sets forth principles of professional conduct for:

- Global Association of Risk Professionals
- Financial Risk Management and Energy Risk Professional certifications
- Other GARP certification and diploma holders and candidates
- GARP’s Board of Trustees, and Regional Directors
- GARP Committee Members
- GARP’s staff

These principles promote the highest levels of ethical conduct and disclosure and provide direction and support for both the individual practitioner and the risk management profession

## Rules of Conduct

- Professional Integrity and Ethical Conduct
- Conflict of Interest
- Confidentiality
- Fundamental Responsibilities
- General Accepted Practices

## FRM Level 1 Book 2 – Quantitative Analysis

## Fundamentals of Probability

A probability measures the likelihood that some event occurs

Probability is introduced through three fundamental principles:

- The probability of any event is non-negative
- The sum of the probabilities across all outcomes is one
- The joint probability of two independent events is the product of the probability of each

## Probability Ranges

Probabilities are always between 0 and 1 (inclusive)

An event with probability 0 never occurs

An event with a probability 1 always occurs

## Conditional Probability

Conditional probability deals with computing a probability on that condition that another event occurs

## Independent Events

Two events are independent if the probability that one event occurs does not depend on whether the other event occurs

Note that two types of independence—unconditional and conditional—do not imply each other

Events can be both unconditionally dependent (i.e., not independent) and conditionally independent

Similarly, events can be independent, yet conditional on another event they may be dependent

## Random Variables

Probability can be used to describe any situation with an element of uncertainty

However, random variables restrict attention to uncertain phenomena that can be described with numeric values

This restriction allows standard mathematical tools to be applied to the analysis of random phenomena

Two functions are commonly used to describe the chance of observing various values from a random variable:

- The probability mass function (PMF) and
- The cumulative distribution function (CDF)

The PMF is particularly useful when defining the expected value of a random variable

## Quantitative Moments

Moments are used to summarize the key features of random variables

A moment is the expected value of a carefully chosen function designed to measure a characteristic of a random variable

Four moments are commonly used in finance and risk management:

- The mean (which measures the average value of the random variable)
- The variance (which measures the spread/dispersion)
- The skewness (which measures asymmetry)
- The kurtosis (which measures the chance of observing a large deviation from the mean)

## Quantile Function

The quantile function, which can be used to map a random variable’s probability to its realization, defines two moment-like measures:

- The median (which measures the central tendency of a random variable) and
- The interquartile range (which is an alternative measure of spread)

## Random Variables

A discrete random variable assigns a probability to a set of distinct values

This set can be either finite or contain a countably infinite set of values

Random variables can be described precisely using mathematical functions

The set of values that the random variable may take is called the support of the function

In most applications in finance and risk management, the assumed distributions are continuous and without regions of zero probability

## Common Univariate Random Variables

There are over two hundred named random variable distributions

Each of these distributions has been developed to explain key features of real-world phenomena

Risk managers model uncertainty in many forms, so this set includes both discrete and continuous random variables

There are three common discrete distributions:

- The Bernoulli
- The binomial
- The Poisson

The Bernoulli is a general-purpose distribution that is typically used to model binary events

The binomial distribution describes the sum of n independent Bernoulli random variables

The Poisson distribution is commonly used to model hazard rates, which count the number of events that occur in a fixed unit of time

## Mixture Distributions

Mixture distributions are built using two or more distinct component distributions

A mixture is produced by randomly sampling from each component so that the mixture distribution inherits characteristics of each component

Mixtures can be used to build distributions that match important features of financial data

## Normal Distributions

The normal distribution is the most commonly used distribution in risk management

The normal distribution is commonly referred to as a Gaussian distribution or a bell curve

A normal distribution has no skewness (because it is symmetrical) and a kurtosis of 3

## Lognormal Distributions

The lognormal distribution is a simple transformation of a normal distribution and is the distribution underlying the famous Black-Scholes Merton model

A variable Y is said to be log-normally distributed if the natural logarithm of Y is normally distributed

In other words, if X = ln Y, then Y is log-normally distributed if and only if X is normally distributed

## Chi-squared Distributions

The chi-squared distribution is frequently encountered when testing hypotheses about model parameters. It is also used when modeling variables that are always positive

## Student’s t distribution

The Student’s t distribution is closely related to the normal, but it has heavier tails. The Student’s t distribution was originally developed for testing hypotheses using small samples

A Student’s t is a one-parameter distribution

This parameter, denoted by n, is also called the degrees of freedom parameter

## F-Distribution

The F is another distribution that is commonly encountered when testing hypotheses about model parameters

The F has two parameters known as the numerator and denominator degrees of freedom

## Exponential Distributions

The exponential distribution uses a single parameter that determines both the mean and variance

The exponential distribution is closely related to the Poisson distribution

## Exponential Variables

Exponential variables are “memoryless”, meaning that their distributions are independent of their histories

## The Beta Distribution

The Beta distribution applies to continuous random variables with outcomes between 0 and 1

It is commonly used to model probabilities that naturally fall into this range

## Multivariate Random Variables

Multivariate random variables are vectors of random variables

Multivariate random variables extend the concept of a single random variable to include measures of dependence between two or more random variables

Multivariate random variables are natural extensions of univariate random variables

These are defined using PMFs (for discrete variables) or PDFs (for continuous variables), which describe the joint probability of outcome combinations

## Probability Mass Function

The probability mass function (PMF)/probability density function (PDF) for a bivariate random variable returns the probability that two random variables each take a certain value

The trinomial PMF has three parameters:

- n (i.e., the total number of experiments),
- p
_{1}, (i.e., the probability of observing outcome 1) - p
_{2}(i.e., the probability of observing outcome 2)

## Expectations

The expectation of a function of a bivariate random variable is defined analogously to that of a univariate random variable

Expectations are used to define moments of bivariate random variables in the same way that they are used to define moments for univariate random variables

## Covariance

The covariance is a measure of dispersion that captures how the variables move together

In a bivariate random variable, there are two variances and one covariance

Bivariate distributions can be transformed into either marginal or conditional distributions

A marginal distribution summarizes the information about a single variable and is simply a univariate distribution

A conditional distribution describes the probability of one random variable conditional on an outcome or a range of outcomes of another

## Correlation

Correlation measures the strength of the linear relationship between two variables and is always between -1 and 1

Correlation plays an important role in determining the benefits of portfolio diversification

When two random variables are independent, they must have zero correlation

However, if two random variables have zero correlation, they are not necessarily independent

Correlation is a measure of linear dependence

If two variables have a strong linear relationship (i.e., they produce values that lie close to a straight line), then they have a large correlation

If two random variables have no linear relationship, then their correlation is zero

## Portfolio Return

The return on a portfolio depends on:

- The distribution of the returns on the assets in the portfolio, and
- The portfolio’s weights on these assets

## Sample Moments

This segment describes how sample moments are used to estimate unknown population moments

When data are generated from independent identically distributed (iid) random variables, the mean estimator has several desirable properties:

- It is (on average) equal to the population mean
- As the number of observations grows, the sample mean becomes arbitrarily close to the population mean
- The distribution of the sample mean can be approximated using a standard normal distribution

Data can also be used to estimate higher-order moments such as variance, skewness, and kurtosis

The first four (standardized) moments (mean, variance, skewness, and kurtosis) are widely used in finance and risk management to describe the key features of data sets

## Quantiles

Quantiles provide an alternative method to describe the distribution of a data set

Quantile measures are particularly useful in applications to financial data because they are robust to extreme outliers

## Estimators

The mean estimator is a function that transforms data into an estimate of the population mean

An estimator is a mathematical procedure that calculates an estimate based on an observed data set

In contrast, an estimate is the value produced by an application of the estimator to data

The mean estimator is a function of random variables, and so it is also a random variable

The expected value of the mean estimator is the same as the population mean

The variance of the mean estimator depends on two values:

- The variance of the data
- The number of observations

The variance in the data is noise that obscures the mean

The more variable the data, the harder it is to estimate the mean of that data

The variance of the mean estimator decreases as the number of observations increases

So larger samples produce estimates of the mean that tend to be closer to the population mean

## Means and Standard Deviations

Means and standard deviations are the most widely reported statistics. Their popularity is due to several factors:

- The mean and standard deviation are often sufficient to describe the data
- These two statistics provide guidance about the likely range of values that can be observed
- The mean and standard deviation are in the same units as the data, and so can be easily compared

One challenge when using asset price data is the choice of sampling frequency

## Mean Estimator Properties

The important properties of the mean estimator include the following:

- The mean is unbiased
- When the observed data are iid, the mean has a simple expression for its standard error
- The mean estimator is BLUE
- The mean estimator is consistent, and in large samples the estimated mean is close to the population mean
- When the variance is finite, the distribution of the mean estimator can be approximated using the CLT

## Hypothesis Testing

Hypothesis testing can be reduced to one universal question:

How likely is the observed data if the hypothesis is true?

Testing a hypothesis about a population parameter starts by specifying null hypothesis and an alternative hypothesis

The null hypothesis is an assumption about the population parameter

The alternative hypothesis specifies the population parameter values (i.e., the critical values) where the null hypothesis should be rejected

The critical values are determined by:

- The distribution of the test statistic when the null hypothesis is true, and
- The size of the test, which reflects our aversion to rejecting a null hypothesis that is in fact true

Observed data are used to construct a test statistic, and the value of the test statistic is compared to the critical values to determine whether the null hypothesis should be rejected

## Components of Hypothesis Testing

A hypothesis test has six distinct components:

- The null hypothesis, which specifies a parameter value that is assumed to be true;
- The alternative hypothesis, which defines the range of values where the null should be rejected;
- The test statistic, which has a known distribution when the null is true;
- The size of the test, which captures the willingness to make a mistake and falsely reject a null hypothesis that is true;
- The critical value, which is a value that is compared to the test statistic to determine whether to reject the null hypothesis; and
- The decision rule, which combines the test statistic and critical value to determine whether to reject the null hypothesis

In some testing problems, the alternative hypothesis might not fully complement the null

The most common example of this is called a one-sided alternative, which is used when the outcome of interest is only above or only below the value assumed by the null

The test statistic is a summary of the observed data that has a known distribution when the null hypothesis is true

In an ideal world, a false (true) null would always (never) be rejected.

However, in practice there is a tradeoff between avoiding a rejection of a true null and avoiding a failure to reject a false null

## Type I Errors

Rejecting a true null hypothesis is called a Type I error

The probability of committing a Type I error is known as the test size

The test size is chosen to reflect the willingness to mistakenly reject a true null hypothesis

- The most common test size is 5%

The critical value depends on the distribution of the test statistic and defines a range of values where the null hypothesis should be rejected in favor of the alternative

- This range is known as the rejection region

When the test statistic has a standard normal distribution, the critical value depends on both the size and the type of the alternative hypothesis (i.e., whether it is one-or two-sided)

## Type II Errors

A Type II error occurs when the alternative is true, but the null is not rejected

## Confidence Intervals

A confidence interval is a range of parameters that complements the rejection region

A 95% confidence interval contains the set of parameter values where the null hypothesis cannot be rejected when using a 5% test

## P Values

A hypothesis test can also be summarized by its p-value

A p-value combines the test statistic, distribution of the test statistic, and the critical values into a single number that is always between 0 and 1

- This value can always be used to determine whether a null hypothesis should be rejected
- If the p-value is less than the size of the test, then the null is rejected

The p-value of a test statistic is equivalently defined as the smallest test size where the null is rejected

- Any test size larger than the p-value leads to rejection, whereas using a test size smaller than the p-value fails to reject the null

## Linear Regression

Linear regression is a widely applied statistical tool for modeling the relationship between random variables

Linear regression has many appealing features:

- Closed-form estimators,
- Interpretable parameters,
- A flexible specification and
- Can be adapted to a wide variety of problems

Regression analysis is the most widely used method to measure, model, and test relationships between random variables

- It is widely used in finance to measure the sensitivity of a portfolio to common risk factors, estimate optimal hedge ratios for managing specific risks, and to measure fund manager performance

The bivariate linear regression model relates a dependent variable to a single explanatory variable

Regression is surprisingly flexible and can describe a wide variety of relationships

The Ordinary Least Squares (OLS) estimators, which have a simple moment-like structure and depend on the mean, variance, and covariance of the data

## Dummy Random Variables

An important class of explanatory variable is known as a dummy

A dummy random variable is binary and only takes the value 0 or 1

Dummies are used to encode qualitative information (e.g., a bond’s country of origin)

A Dummy takes the value 1 when the observation has the quality and 0 if it does not

Dummies are also commonly constructed as binary transformations of other random variables

Example: a market direction dummy that encodes the return on the market as 1 if negative and 0 if positive

## Regression with Multiple Explanatory Variables

Linear regression with a single explanatory variable provides key insights into OLS estimators and their properties

In practice, however, models typically use multiple variables where it is possible to isolate the unique contribution of each explanatory variable

A model built with multiple variables can also distinguish the effect of a novel predictor from the set of explanatory variables known to be related to the dependent variable

## The k-Variate Regression Model

The k-variate regression model enables the coefficients to measure the distinct contribution of each explanatory variable to the variation in the dependent variable

## The Fama-French Model

The Fama-French three-factor model is a leading example of a multi-factor approach

The Fama-French three-factor model expands upon CAPM by including two additional factors:

- The size factor (which captures the propensity of small-cap firms to generate higher returns than large-cap firms) and
- The value factor (which measures the additional return that value firms earn above growth firms)

Controls are explanatory variables that are known to have a clear relationship with the dependent variable

## Model Fit

Model fit is assessed using R^{2}, which measures the ratio of the variation explained by the model to the total variation in the data

While intuitive, this measure suffers from some important limitations:

- It never decreases when an additional variable is added to a model and
- It is not interpretable when the dependent variable changes.

The adjusted R^{2} partially addresses the first of these concerns

## Regression Diagnostics

Ideally, a model should include all variables that explain the dependent variable and exclude all that do not

In practice, achieving this goal is challenging

Once a model has been selected, the specification should be checked for any obvious deficiencies

Omitting explanatory variables that affect the dependent variable creates biased coefficients

Including irrelevant variables does not bias coefficients

Determining whether a variable should be included in a model reflects a bias-variance tradeoff

Large models that include all conceivable explanatory variables are likely to have coefficients that are unbiased

## Omitted Variables

An omitted variable is one that has a non-zero coefficient but is not included in a model

Omitting a variable has two effects:

First, the remaining variables absorb the effects of the omitted variable attributable to common variation

Second, the estimated residuals are larger in magnitude than the true shocks

## Extraneous Variables

An extraneous variable is one that is included in the model but is not needed

This type of variable has a true coefficient of 0 and is consistently estimated to be 0 in large samples

The choice between omitting a relevant variable and including an irrelevant variable is ultimately a tradeoff between bias and variance

Larger models tend to have a lower bias but they also have less precise estimated parameters

Models with few explanatory variables have less estimation error but are more likely to produce biased parameter estimates

The bias-variance tradeoff is the fundamental challenge in variable selection

## Homoscedasticity

Homoscedasticity is one of the five assumptions used to determine the asymptotic distribution of an OLS estimator

It requires that the variance is constant and so does not systematically vary with any of the explanatory variables

When this is not the case, then the residuals are heteroskedastic

## Heteroskedastic Residuals

When residuals are heteroskedastic, the standard errors can be estimated using White’s estimator (also called Eiker-White in some software packages)

Parameters can be tested using t-tests by using White’s standard error in the place of standard error used for homoscedastic data

On the other hand, F-tests, are not as easy to adjust for heteroskedasticity and so caution is required when testing multiple hypotheses if the shocks are heteroskedastic

## Multicollinearity

Multicollinearity occurs when one or more explanatory variables can be substantially explained by the other(s)

Multicollinearity differs from perfect collinearity, where one of the variables is perfectly explained by the others

Multicollinearity is a common problem in finance and risk management because many regressors are simultaneously determined by and sensitive to the same news

## Residual Plots

Residual plots are standard methods used to detect deficiencies in a model specification

An ideal model would have residuals that are not systematically related to any of the included explanatory variables

## Stationary Time Series

Time-series analysis is a fundamental tool in finance and risk management

Many key time series (e.g., interest rates and spreads) have predictable components

Building accurate models allows past values to be used to forecast future changes in these series

A time series can be decomposed into three distinct components:

- The trend, which captures the changes in the level of the time series over time
- The seasonal component, which captures predictable changes in the time series according to the time of year
- The cyclical component, which captures the cycles in the data

Whereas the first two components are deterministic, the third component is determined by both the shocks to the process and the memory (i.e., persistence) of the process

A time series is covariance-stationary if its first two moments do not change across time

## Linear Processes

Any time series that is covariance-stationary can be described by a linear process

While linear processes are very general, they are also not directly applicable to modeling

Two classes of models are used to approximate general linear processes:

- Autoregressions (AR)
- Moving averages (MAs)

The ability of a model to forecast a time series depends crucially on whether the past resembles the future

## Stationarity

Stationarity is a key concept that formalizes the structure of a time series and justifies the use of historical data to build models

Covariance stationarity depends on the first two moments of a time series:

- The mean
- The autocovariances

White noise is the fundamental building block of any time-series model

White noise processes have three properties:

- Mean zero
- Constant and finite variance
- No autocorrelation or autocovariance

The lack of correlation is the essential characteristic of a white noise process and plays a key role in the estimation of time-series model parameters

Dependent white noise relaxes the iid assumption while maintaining the three properties of white noise

Autoregressive models are the most widely applied time-series models in finance and economics

## Non-Stationary Time Series

Covariance-stationary time series have means, variances, and autocovariances that do not depend on time

Any time series that is not covariance-stationary is non-stationary

This segment covers the three most pervasive sources of non-stationarity in financial and economic time series:

- Time trends
- Seasonalities
- Unit roots (more commonly known as random walks)

Time trends are the simplest deviation from stationarity

Time trend models capture the propensity of many time series to grow over time

## Seasonalities

Seasonalities induce non-stationary behavior in time series by relating the mean of the process to the month or quarter of the year

Seasonalities can be modeled in one of two ways:

- Shifts in the mean that depend on the period of the year, or
- An annual cycle where the value in the current period depends on the shock in the same period in the previous year

## Random Walks

Random walks (also called unit roots) are the most pervasive form of non-stationarity in financial and economic time series

All non-stationary time series contain trends that may be deterministic or stochastic

For deterministic trends (e.g., time trends and deterministic seasonalities), knowledge of the period is enough to measure, model, and forecast the trend

On the other hand, random walks are the most important example of a stochastic trend

A time series that follows a random walk depends equally on all past shocks

The correct approach to modeling time series with trends depends on the source of the non-stationarity

If a time series only contains deterministic trends, then directly capturing the deterministic effects is the best method to model the data

Unit roots generalize random walks by adding short-run stationary dynamics to the long-run random walk

## Measuring Returns, Volatility, and Correlation

Financial asset return volatilities are not constant, and how they change can have important implications for risk management

Capturing the dependence among assets in a portfolio is also a crucial step in portfolio construction

In portfolios with many assets, the distribution of the portfolio return is predominantly determined by the dependence between the assets held

- If the assets are weakly related, then the gains to diversification are large, and the chance of experiencing an exceptionally large loss should be small
- If the assets are highly dependent, especially in their tails, then the probability of a large loss may be surprisingly high

The volatility of a financial asset is usually measured by the standard deviation of its returns

Implied volatility is an alternative measure that is constructed using option prices

Both put and call options have payouts that are nonlinear functions of the underlying price of an asset

## The Black-Scholes-Merton Model

The most well-known expression for determining the price of an option is the Black-Scholes-Merton model

The Black-Scholes-Merton model relates the price of a call option to:

- the interest rate of a riskless asset
- the current asset price
- the strike price
- the time until maturity and
- the annual variance of the return

All values in the Black-Scholes-Merton model, including the call price, are observable except the volatility

The Black-Scholes-Merton option pricing model uses several simplifying assumptions that are not consistent with how markets actually operate

## Normal Distributions

A normal distribution is symmetric and thin-tailed, and so has no skewness or excess kurtosis

However, many return series are both skewed and fat-tailed

The Jarque-Bera (JB) test statistic is used to formally test whether sample skewness and kurtosis are compatible with an assumption that the returns are normally distributed

An alternative method to understand the non-normality of financial returns is to study the tails

Normal random variables have thin tails

The Student’s t is an example of a widely used distribution with a power law tail

Linear correlation is insufficient to capture dependence when assets have nonlinear dependence

Researchers often use two alternative dependence measures:

- Rank correlation (also known as Spearman’s correlation) and
- Kendal’s (tau)

These statistics are correlation-like: both are scale invariant, have values that always lie between -1 and 1

They are zero when the returns are independent

They are positive (negative) when there is in increasing (decreasing) relationship between the random variables

## Rank Correlation

Rank correlation is the linear correlation estimator applied to the ranks of the observations

Rank correlation has two distinct advantages over linear correlation:

- It is robust to outliers because only the ranks, not the values of X and Y, are used
- It is invariant with respect to any monotonic increasing transformation of X
_{i}and Y_{i}

Linear correlation is only invariant with respect to increasing linear transformations

## Simulation and Bootstrapping

Simulation is an important practical tool in modern risk management with a wide variety of applications

Examples of these applications include:

- Computing the expected payoff of an option
- Measuring the downside risk in a portfolio
- Assessing estimator accuracy

Monte Carlo simulation is a simple approach to approximate the expected value of a random variable using numerical methods

Another important application of simulated methods is boot-strapping

Bootstrapping uses observed data to simulate from the unknown distribution generating the observed data

This is done by combining observed data with simulated values to create a new sample that is closely related to, but different from, the observed data

The key to understanding bootstrapping lies in one simple fact:

- The unknown distribution being sampled from is the same one that produced the observed data

The bootstrap method avoids the specification of a model and instead makes the key assumption that the present resembles the past

Monte Carlo simulation and bootstrapping are closely related

- Both methods use computer-generated values to numerically approximate an expected value

## Simulation and Bootstrapping Differences

The fundamental difference between simulation and bootstrapping is the source of the simulated data:

When using simulation, the user specifies a complete data generating process (DGP) that is used to produce the simulated data

In bootstrapping, the observed data are used directly to generate the simulated data set without specifying a complete DGP

Monte Carlo experiments allow the finite sample distribution of an estimator to be tabulated and compared to its asymptotic distribution derived from the Central Limit Theorem (CLT)

Antithetic variates add a second set of random variables that are constructed to have a negative correlation with the iid variables used in the simulation

They are generated in pairs using a single uniform value

The biggest challenge when using simulation to approximate moments is the specification of the DGP

If the DGP does not adequately describe the observed data, then the approximation of the moment may be unreliable

## Bootstrap Limitations

While bootstrapping is a useful statistical technique, it has its limitations. There are two specific issues that arise when using a bootstrap:

First, bootstrapping uses the entire data set to generate a simulated sample

The second limitation arises due to structural changes in markets so that the present is significantly different from the past

Both Monte Carlo simulation and bootstrapping suffer from the “Black Swan” problem—simulations generated using either method resemble the historical data

Bootstrapping is especially sensitive to this issue, and a bootstrap sample cannot generate data that did not occur in the sample

A good statistical model, on the other hand, should allow the possibility of future losses that are larger than those that have been realized in the past

## Summary

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