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Introduction
We now turn our attention to book 4 in the FRM Level 1 course.
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FRM Level 1 Book 4 – Valuation and Risk Models
Measures of Financial Risk
Consider the two risk measures: value-at-risk (VaR) and expected shortfall (ES)
The mean and standard deviation of a portfolio’s return can be calculated using the means and standard deviations of its components’ returns
This leads to an important concept known as the efficient frontier
The efficient frontier shows the trade-offs between mean and standard deviation that are available to the holder of a well-diversified portfolio
When portfolios are described by the mean and standard deviation of their returns, it is natural to assume we are dealing with normal distributions
Most financial variables have fatter tails than the normal distribution
In other words, extreme events are more likely to occur than the normal distribution would predict
Value-at-risk (VaR)
Value-at-risk (VaR) and expected shortfall are two risk measures focusing on adverse events
Expected shortfall is less intuitive than VaR
However, expected shortfall has more desirable theoretical properties
Expected shortfall is also an example of a coherent risk measure
Investors are typically faced with a trade-off between risk and return
The greater the risks that are taken, the higher the expected return that can be achieved
Expected return does not describe the return that we expect to happen
The term is actually used by statisticians to describe the average (or mean) return
Normal Distribution
The normal (or Gaussian) distribution is one of the most well-known and widely used probability distributions
The normal (or Gaussian) distribution has two parameters: The mean and the standard deviation
Value at Risk, VaR, is an important risk measure that focuses on adverse events and their probability
The VaR for an investment opportunity is a function of two parameters: The time horizon, and the confidence level
One problem with VaR is that it does not say how bad losses might be when they exceed the VaR level
Expected shortfall is one way of overcoming this disadvantage
Expected shortfall is defined as the expected (or average) loss conditional on the loss being greater than the VaR level
Being Coherent
A risk measure that satisfies all four conditions below is termed coherent:
Monotonicity: If a portfolio always produces a worse result than another portfolio, it should have a higher risk measure
Translation Invariance: If an amount of cash K is added to a portfolio, its risk measure should decrease by K
Homogeneity: Changing the size of a portfolio by multiplying the amounts of all the components by λ results in the risk measure being multiplied by λ
Subadditivity: For any two portfolios, A and B, the risk measure for the portfolio formed by merging A and B should be no greater than the sum of the risk measures for portfolios A and B
As a risk measure, Standard Deviation is a good way of describing the overall uncertainty associated with a set of outcomes
Calculating and Applying VaR
Previously we looked at two risk measures: value-at-risk (VaR) and expected shortfall (ES). This segment discusses how they can be calculated:
One popular approach is a non-parametric method, where the future behavior of the underlying market variables is determined in a very direct way from their past behavior
This method is known as historical simulation
If we assume the returns on the underlying variables are multivariate normal, then:
- Changes in portfolio value are also normally distributed
- Calculating VaR will then be relatively straightforward
For a portfolio that is not linearly dependent on the underlying market variables (e.g., because it contains options), the delta-normal model can also be used. (However, it is less accurate in this context)
The delta-normal model works well for linear portfolios when the risk factor probability distributions are at least approximately normal
Monte Carlo Simulations
An alternative to historical simulation and the delta-normal model is provided by Monte Carlo simulations
These are like historical simulation, but their scenarios are randomly generated (rather than being determined directly from the behavior of market variables in the past)
Monte Carlo simulations generate scenarios by taking random samples from the distributions assumed for the risk factors
Historical simulation
Historical simulation is a popular method of calculating VaR and ES
Typically, the time horizon is chosen as one day
Historical simulation involves identifying the market variables on which the value of the portfolio under consideration depends
In practice, risk factors are divided into two categories:
- Those where the percentage change in the past is used to define a percentage change in the future, and
- Those where the actual change in the past is used to define an actual change in the future
Standard deviations increase during stressed market conditions (It is also true that correlations generally increase during this period)
Measuring and Monitoring Volatility
The volatility of a variable measures the extent to which its value changes through time
A constant volatility would be fairly easy to estimate using historical data
In practice, however, volatility changes through time
This leads to situations where asset returns are not normally distributed; instead, they tend to have fatter tails than a normal distribution would predict
This is important for the estimation of risk measures (such as VaR and expected shortfall) because these measures depend critically on the tails of asset return distributions
The conditionally normal model
An alternative to assuming asset returns is constantly normal is to assume they are normal conditioned on the volatility being known
When volatility is high, the daily return is normal with a high standard deviation
When the volatility is low, the daily return is normal with a low standard deviation
The conditionally normal model may not be perfect, but it is an improvement over the constant volatility model
To implement the conditionally normal model, it is necessary to monitor volatility so that a current volatility estimate is produced. We consider two ways of doing that:
- The exponentially weighted moving average (EWMA) model and
- The GARCH (1,1) model
There are three ways in which an asset’s return can deviate from normality:
- The return distribution can have fatter tails than a normal distribution
- The return distribution can be non-symmetrical
- The return distribution can be unstable with parameters that vary through time
Standard Error of an Estimate
The standard error of an estimate is the standard deviation of the difference between the estimate and the true value
The standard error of a volatility estimate calculated from m observations is approximately equal to the estimate divided by the square root of 2(m – 1)
One way of overcoming problems with estimating volatility is to use exponential smoothing
This is also referred to as the exponentially weighted moving average (EWMA)
In EWMA, the weights applied to historical data decline exponentially as we move back in time
An alternative method is the multivariate density estimation (MDE)
The GARCH model, developed by Robert Engel and Tim Bollerslev, can be regarded as an extension of EWMA
Implied volatility
Implied volatility is the volatility implied from options prices
- It is a forward-looking estimate of volatility
- It is often found to be more accurate than an estimate produced from historical data
- As volatilities increase, the value of the option increases
The correlation between two variables is their covariance divided by the product of their standard deviations
External and Internal Credit Ratings
Rating agencies are an important external source of credit risk data
The most well-known credit rating agencies are: Moody’s, Standard and Poor’s (S&P) and Fitch
The Dodd-Frank Act now requires rating agencies to make the assumptions and methodologies underlying their ratings more transparent
It has also increased the potential legal liability of rating agencies
An external credit rating is usually an attribute of an instrument issued by an entity (rather than of the entity itself)
However, bond ratings are often assumed to be attributes of the entity rather than of the bond itself
Rating levels
The highest bond rating assigned by Moody’s is Aaa
Bonds with an Aaa rating are considered to have almost no chance of defaulting
The next highest rating is Aa
Rating agencies rate publicly traded bonds and money market instruments
In addition to the ratings themselves, rating agencies provide what are termed outlooks
Rating outlooks
Outlooks are indications of the most likely direction of the rating over the medium term
- A positive outlook means that a rating may be raised
- A negative outlook means that it may be lowered
- A stable outlook means it is not likely to change
Banks and other financial institutions develop their own internal rating systems based on their assessment of potential borrowers
Banks and other financial institutions typically base their ratings on several factors: Financial ratios, Cash flow projections, An assessment of the firm’s management, etc.
Country Risk
Many large firms have business interests all over the world
For such global entities, it is important to assess the risks associated with the foreign countries they operate in:
These risks are collectively referred to as country risk
There are numerous components of country risk. One such component is political risk
Individuals and corporations can obtain diversification benefits by investing outside their domestic markets
When lending to foreign governments, it is important for lenders to consider country risk as part of their credit default risk framework
Many developing markets have economies that are growing faster than those of developed markets
However, this fast growth may be accompanied by higher economic risks and less stable political climates
There are often links between political and economic risks
Gross Domestic Product (GDP)
The growth of a country’s economy is measured its Gross Domestic Product (GDP)
GDP is the total value of goods and services produced by all the people and firms in a country
An important consideration in assessing country risk is how a country will react to economic cycles
- During economic down-turns, developing countries often see larger declines in GDP than their developed counterparts
- This is because developing economies tend to rely more heavily on commodities
Some countries are highly dependent on a single commodity
If the price of that commodity declines, the country and the value of its currency will suffer
Many African and Latin American countries fall into this category
Legal Risk
Legal risk is the risk of losses due to inadequacies or biases in a country’s legal system
A legal system that is trusted and perceived to be fair helps a country to attract foreign investment
One measure of a country’s risk is the risk it will default on its debt
There are two types of sovereign debt:
- The type issued in a foreign currency (such as the USD)
- The type issued in the country’s own currency
Debt issued in a foreign currency is attractive to global banks and other international lenders
The risk for the issuing country is that it cannot repay the debt by simply printing more money
There are several other factors that are considered when determining a rating: Social Security Commitments, The Tax Base, Political Risk, Implicit Guarantees etc.
Credit spread
The credit spread for sovereign debt in a specific currency is the excess interest paid over the risk-free rate in that currency
There is a strong correlation between credit spreads and ratings
Credit spreads can provide extra information on the ability of a country to repay its debt
Measuring Credit Risk
Economic capital is a bank’s own estimate of the capital it requires
Regulatory capital is the capital bank regulators (also known as bank supervisors) require a bank to keep
Global bank regulatory requirements are determined by the Basel Committee on Banking Supervision in Switzerland
The three different models for quantifying credit risk in this segment:
The first is a model where the mean and standard deviation of the loss from a loan portfolio is determined from the properties of the individual loans
The second model, known as the Vasicek model, is used by bank regulators to estimate an extreme percentile of the loss distribution
The third model is known as CreditMetrics and is often used by banks themselves when estimating economic capital
Typically, CreditMetrics involves time-consuming Monte Carlo simulations
A bank must keep capital for the risks it takes
When losses are incurred, they come out of the equity capital
Equity capital and Debt capital
Equity capital is sometimes referred to as “going concern capital” since as it is positive, the bank is solvent and can therefore be characterized as a going concern
Debt capital is referred to as “gone concern capital” since it only becomes an important cushion for depositors when the bank is no longer a going concern (i.e., insolvent)
Banks are subject to many risks
Credit risk has traditionally been the most important risk taken by banks
Each loan issued by a bank has some risk of default and therefore a risk of a credit loss
It is the possibility of this loss that constitutes credit risk
In 1974, the central banks of the G10 countries formed the Basel Committee to harmonize global bank regulation
By 1988, the committee had agreed on a common approach for determining the required credit risk capital for the banks under their supervision (This regulation is now known as Basel I)
Both regulatory capital and economic capital feature separate capital calculations for credit risk, market risk, and operational risk
For regulatory capital, the results are added to give the total capital requirements
For economic capital, however, correlations between the risks are often considered
The Vasicek Model
The Vasicek model is used by regulators to determine capital for loan portfolios
It uses the Gaussian copula model to define the correlation between defaults
The Vasicek model has an advantage in that the unexpected loss can be determined analytically
CreditMetrics
CreditMetrics is the model banks often use to determine economic capital
Under this model, each borrower is assigned an external or internal credit rating
A one-year transition table is used to define how ratings change
Operational Risk
Operational risk is sometimes defined very broadly as any risk that is not a market risk or a credit risk
A much narrower definition would be that it consists of risks arising from operational mistakes;
This would include the risk that a bank transaction is processed incorrectly
But it would not include the risk of fraud, cyberattacks, or damage to physical assets
Operational risk has been defined by the Basel Committee as:
The risk of loss resulting from inadequate or failed internal processes, people, and systems or from external events
Categories of operational risk
Seven categories of operational risk have been identified by the Basel Committee
- Internal fraud
- External fraud
- Employment practices and work place safety
- Clients, products, and business practices
- Damage to physical assets
- Business disruption and system failures
- Execution, delivery, and process management
Compliance risk
Compliance risk is another operational risk facing financial institutions
This is the risk that an organization will incur fines or other penalties because it knowingly or unknowingly fails to act in accordance with industry laws and regulations, internal policies, or prescribed best practices
This includes activities such as money laundering, terrorism financing, and assisting clients with tax evasion
Rogue trader risk
Rogue trader risk is the risk that an employee will take unauthorized actions resulting in large losses
One of the most notorious incidents involved Barings Bank trader Nick Leeson
Basel II rules
The final Basel II rules for banks had three approaches:
- The basic indicator approach
- The standardized approach
- The advanced measurement approach (AMA)
The key determinants of an operational risk loss distribution are:
- Average Loss frequency: the average number of times in a year that large losses occur, and
- Loss severity: the probability distribution of the size of each loss
Stress Testing
Stress testing is a risk management activity that has become increasingly important since the 2007–2008 financial crisis:
- It involves evaluating the implications of extreme scenarios that are unlikely and yet plausible
- It asks if a financial institution has enough capital / liquid assets to survive various scenarios
- Some stress tests are carried out because they are required by regulators
- Others are carried out as part of their internal risk management activities
Stress Testing Measures
Measures such as value-at-risk (VaR) and expected shortfall (ES) are often calculated and used in stress testing analysis
One disadvantage of VaR and ES is that they are usually backward-looking
They assume the future will (in some sense) be like the past
Stress testing, however, is designed to be forward-looking
Using stress tests to derive the full range of all possible outcomes is not usually possible
Risk managers have two types of analyses available to them:
- One is a backward-looking analysis where a loss distribution can be estimated
- The other is a forward-looking analysis where different scenarios are assessed
In the case of market risk, the VaR/ES approach often has a short time horizon (perhaps only one day), whereas stress testing usually looks at a much longer period
Choosing a stress-test scenario
The first step in choosing a stress-test scenario is to select a time horizon
The time horizon should be long enough for the full impact of the scenarios to be evaluated
Scenarios lasting three months to two years are more common
Very long scenarios can be necessary in some situations
Stress testing involves constructing scenarios and then evaluating their consequences
Reverse stress testing takes the opposite approach: It asks the question, “What combination of circumstances could lead to the failure of the financial institution?”
A financial institution should have written policies and procedures for stress testing and ensure that they are adhered to
Pricing Conventions, Discounting, and Arbitrage
Discount factors are numbers that allow us to relate a cash flow received in the future to its value today
Treasury bills are instruments issued by a government to finance its short-term funding needs. They last one year or less and are defined by: Their face value (or principal amount or par value), and their maturity date
While a Treasury bill lasts less than one year from the time it is issued, a Treasury bond lasts more than one year
Bonds with a maturity between one and ten years are sometimes referred to as Treasury notes
To keep the terminology simple, many refer to all coupon-bearing Treasury instruments as Treasury bonds
U.S. Treasury bonds are defined by: The face value (or principal amount or par value), The coupon rate and the maturity date
The law of one price
The law of one price states that if two portfolios provide the same future cash flows, they should sell for the same price.
If the law of one price did not hold, there would be theoretical arbitrage opportunities
STRIPS
STRIPS is an acronym for Separate Trading of Registered Interest and Principal of Securities
STRIPS are created by investment dealers when a coupon-bearing bond is delivered to the Treasury and exchanged for its principal and coupon components
Day-count conventions describe the way in which interest is earned through time
Interest Rates
Investors and traders prefer to express the time value of money in terms of interest rates rather than discount factors
To understand an interest rate: We need to understand the compounding frequency used to measure it and we also need to understand the different types of interest rates
To fully describe an interest rate, we need to specify the compounding frequency with which it is measured
The compounding frequency used for an interest rate is often the same as the frequency of payments, but this is not always the case
The spot rate is the interest rate earned when cash is received at just one future time
The spot rate is also referred to as the zero-coupon interest rate, or just the “zero”
Forward rates are the future spot rates implied by today’s spot rates.
Some Key properties of rates
If the term structure is flat (with all spot rates the same), all par rates and all forward rates equal the spot rate
If the term structure is upward-sloping, the par rate for a certain maturity is below the spot rate for that maturity
If the term structure is downward-sloping, the par rate for a certain maturity is above the spot rate for that maturity
If the term structure is upward-sloping, forward rates for a period starting at time T are greater than the spot rate for maturity T
If the term structure is downward-sloping, forward rates for a period starting at time T are less than the spot rate for maturity T
Note: The most common swap is an agreement to exchange a fixed rate for Libor
Bond Yields and Return Calculations
A bond’s realized return is calculated by comparing the initial investment’s value with its final value
A bond’s yield to maturity is the single discount rate, which if applied to all the bond’s cash flows, would make the cash flows’ present value equal to the bond’s market price
Properties of the yield-to-maturity
When the yield to maturity is equal to the coupon rate, the bond sells for its face value
When the yield to maturity is less than the coupon rate, the bond sells for more than its face value. If time passes with no change to the yield, the price of the bond declines
When the yield to maturity is greater than the coupon rate, the bond sells for less than its face value. If time passes with no change to the yield to maturity, the price of the bond increases
If the term structure is flat with all rates equal to R, the yield to maturity is equal to R for all maturities
Carry roll-down
The profit or loss from a trading strategy can be decomposed into the carry roll-down, the amount resulting from interest rate changes, and the amount resulting from spread changes
The carry roll-down is usually defined as the impact of forward rates being realized (i.e., future forward rates being equal to today’s forward rates)
However, it can also be defined as the impact of the term structure remaining unchanged, or the impact of bond yields remaining unchanged
Spread changes arise from a bond’s market price moving closer to (or further away from) its theoretical price
Applying Duration and Convexity
One-factor risk metrics are based on the assumption that interest rate term structure movements are driven by a single factor. Examples include: DV01, duration, and convexity
Yield-based measures consider what happens to a bond price when there is a small change to its yield
Effective duration and effective convexity consider what happens when all spot rates change by the same amount. The latter is referred to as a parallel shift in the term structure
One-factor term structure shifts do not need to be parallel
The term structure shape can also change completely in a one-factor model
DV01
DV01 describes the impact of a one-basis-point change in interest rates on the value of a portfolio
DV01 can be calculated for any position whose value depends on interest rates
DV01 is the decrease (increase) in the price of a bond (or other instrument) arising from a one-basis-point increase (decrease) in rates
In the case of bonds, we can define DV01, duration, and convexity in terms of small changes in yields rather than small changes in all rates
This leads to some analytic results and explains the name “duration”
The DV01 of a portfolio is the sum of the DV01s of the instruments in the portfolio
Duration
The duration (convexity) of a portfolio is the average of the durations (convexities) of the instruments in the portfolio weighted by the value of each instrument
Callable bonds
A callable bond is a bond where the issuing company has the right to buy back the bond at a pre-determined price at certain times in the future
Puttable bond
A puttable bond is a bond where the holder has the right to demand early repayment
A puttable bond should be treated like a callable bond when calculating effective duration. In this case, the probability of the put option being exercised increases as interest rates increase
Modeling Non-Parallel Term Structure Shifts and Hedging
A statistical technique known as principal components analysis can be used to understand term structure movements in historical data
This technique looks at the daily movements in rates of various maturities and identifies certain factors
DV01 can be defined as the impact of a one-basis-point shift in all spot rates on the value of a portfolio
In practice, perfectly parallel shifts are rare
Principal components analysis
Principal components analysis shows the term structure changes observed in practice consist of the following:
A component where all rates move in the same direction, but not by exactly the same amount
A component where the term structure steepens or flattens
A component where there is a bowing of the term structure
Portfolio sensitivity
The sensitivity of a portfolio to shifts in the term structure can be used to calculate the standard deviation of the daily change in the portfolio value
And can therefore provide estimates of risk measures such as VaR and expected shortfall
Binomial Trees
Binomial trees is a valuation method widely used for pricing American-style options and other derivatives
Derivatives are valued using what is called a no-arbitrage argument
This means prices are calculated on the assumption that there are no arbitrage opportunities for market participants
The law of one price states that if portfolios X and Y provide the same cash flows at the same times in the future, they should sell for the same price
Binomial trees are a convenient way of illustrating how no-arbitrage arguments apply to derivatives
We also use binomial trees to introduce risk-neutral valuation
Risk-neutral valuation is the most important principle in derivatives pricing
Risk-neutrality
A risk-neutral world is one where investors do not adjust their required expected returns for risk, so that the expected return on all assets is the risk-free rate
A risk-neutral world is one where all tradable assets have an expected return equal to the risk-free interest rate
The risk-neutral valuation principle states that if we assume we are in a risk-neutral world, we get the fair price for a derivative
Derivatives and delta
Delta is the sensitivity of a derivative’s value to the price of its underlying stock
Trees can be constructed for valuing derivatives dependent on a non-dividend paying stock
Trees can also be constructed for valuing derivatives dependent on stock indices, currencies, and futures
When stock price movements are governed by a multi-step tree, we can treat each binomial step separately and roll back through the tree to value a derivative
For American options, it is necessary to test for early exercise at each node of the tree
The Black-Scholes-Merton Model
Black and Scholes used the capital asset pricing model (CAPM) to derive the relationship between the return from a stock and the return from an option on the stock
Merton used no-arbitrage arguments like those used in connection with binomial trees:
The two papers derived the same option pricing formula
The pricing formula applies to European options on non-dividend paying stocks
It can be extended to European options on stocks paying discrete dividends and to European options on other assets (such as stock indices, currencies, and futures)
It does not apply to American options, which must be valued using the binomial tree methodology
The Black-Scholes-Merton model assumes that the return from a non-dividend paying stock over a short period of time is normally distributed
Black-Scholes-Merton assumptions
The assumptions necessary to derive the Black-Scholes-Merton options pricing model are as follows:
- There are no transaction costs or taxes
- All securities are perfectly divisible
- There are no dividends on the stock during the life of the option
- There are no riskless arbitrage opportunities
- Security trading is continuous
- Investors can borrow or lend at the same risk-free rate
- The options being considered cannot be exercised early
Normal and lognormal distribution
When the return on a stock over a short period is normally distributed, the stock price at the end of a relatively long period has a lognormal distribution
This means the logarithm of the stock price (and not the stock price itself) is normally distributed
A normal distribution is symmetrical and the variable can take any value from negative infinity to infinity
A lognormal distribution is skewed and the variable can take any positive value
Volatility
Volatility is a measure of our uncertainty about the returns provided by an investment
The implied volatility of an option is the volatility that gives the market price of the option when it is substituted into the Black-Scholes-Merton formula
There is no analytic formula for implied volatility
It must be found using an iterative trial and error procedure
Warrants
Warrants are options issued by a company on its own stock
If warrants are exercised, the company issues more shares, and the warrant holder buys the shares from the company at the strike price
Option Sensitivity Measures
The Greek letters measure different aspects of risk in derivatives portfolios:
Delta measures the sensitivity of a portfolio’s value to changes in the price of the underlying asset
Vega is the Greek letter that measures the trader’s exposure to volatility
The gamma of a stock price–dependent derivative measures the sensitivity of its delta to the stock price
The theta of an option is the rate of change in its value over time
The rho of an option measures its sensitivity to interest rates
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