Foundations of Risk Management FRM 2011 Study Notes – Vol. I By David Harper, CFA FRM CIPM www.bionicturtle.com
Foundations of Risk Management FRM 2011 Study Notes – Vol. I
By David Harper, CFA FRM CIPM
www.bionicturtle.com
FRM 2011 FOUNDATIONS OF RISK MANAGEMENT 1 www.bionicturtle.com
Table of Contents
Jorion, Chapter 1: The Need for Risk Management 2
Stulz Chapter 2: Investors & Risk Management 19
Stulz Chapter 3: Creating Value with Risk Management 27
Elton, Chapter 5: Delineating Efficient Portfolios 34
Elton, Chapter 13: The Standard Capital Asset Pricing Model 39
Elton, Chapter 14: Nonstandard Forms of Capital Asset Pricing Models 44
Elton, Chapter 16: Arbitrage Pricing Model (APT) 49
Amenc, Chapter 4: Applying CAPM to Performance Measurement 52
CAS, Overview of Enterprise Risk Management 57
Allen, Chapter 4: Financial Disasters 65
René Stulz,‖ Risk Management Failures: What are They and When Do They Happen? 71
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Jorion, Chapter 1:
The Need for Risk
Management
In this chapter…
Define risk and describe some of the major sources of risk. Differentiate between business and financial risks and give examples of each. Relate significant market events of the past several decades to the growth of
the risk management industry. Describe the functions and purposes of financial institutions as they relate to
financial risk management. Define what a derivative contract is and how it differs from a security. Describe the dual role leverage plays in derivatives and why it is relevant to a
risk manager. Define financial risk management. Define value-at-risk (VaR) and describe how it is used in risk management. Describe the advantages and disadvantages of VaR relative to other risk
management tools such as stop-loss limits, notional limits, and exposure limits. Compare and contrast valuation and risk management, using VaR as an example. Define and describe the four major types of financial risks: market, liquidity,
credit, and operational. Within market risk:
Describe and differentiate between absolute and relative market risk Describe and differentiate between directional and non-directional
market risk Describe basis risk and its sources Describe volatility risk and its sources
Within liquidity risk: Describe and differentiate between asset and funding liquidity risk
Within credit risk: Describe and differentiate between exposure and recovery rate Describe credit event and how it may relate to market risk Describe sovereign risk and its sources Describe settlement risk and its sources
Within operational risk: Describe the potential relationships between operational, market and
credit risk Describe model risk and its sources Describe people risk Describe legal risk and its sources
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Define risk and describe some of the major sources of risk.
Risk is volatility of unexpected outcomes (value of assets, equity, or earnings). Although
Jorion does not differentiate between risk and uncertainty, some authors distinguish between
risk and uncertainty:
Risk: when the outcome is random but the probability distribution is known or can be estimated or approximated. An example: a six-sided die (we know the distribution is uniform). Much of our FRM study concerns the traditional attempt to parameterize (or otherwise estimate, even if empirically) the approximately distribution of possible losses.
Uncertainty: the probability distribution is itself unknown. Example: a terrorist attack. This is when the disribution itself eludes us.
The major sources of risk include:
Human (Accident) including regulatory policy (and unintended consequence
Human (Deliberate) including terrorism and war
Natural disaster including earthquakes and hurricanes
Economic growth including the creative “disruption” caused by technological innovation
Jorion defines risk as the volatility of unexpected outcomes (change in value of assets,
equity or earnings). In doing so, he defines risk in terms of the most classic, traditional
metric (volatility) but this is not gospel. There are other definitions.
Human (Accident)
• Regulatory policy: unintended consequence
Human (Deliberate)
• Terrorism
• War
Natural disaster
• Earthquake
• Hurricane
Economic Growth
• Technological innovation
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Differentiate between business & financial risks & give examples of each.
Business risks are risks that the corporation assumes willingly. They may do this to create a competitive edge or to add shareholder value.
Financial risks are losses due to financial market activities. Examples of financial risk include losses due to interest-rate movements or defaults on financial obligations.
Deliberate, necessary
Competitive advantage
To create Shareholder value
For example
Business decisions (investments, products) & Business environment (competition & economy)
Shareholders pay for and expect firms to assume business risk!
Losses due to financial market activities
For example
Interest rate exposure
Defaults on financial obligations
Accounts receivables
To a non-financial firm, not core & firm should (probably) hedge
Banks & financial services are in the business of managing financial risk; managing
financial risk is (should be) a core strategic activity. However, industrial (non-financial)
companies typically want to hedge financial risks; i.e., the assumption of financial risk is
often non-strategic to non-financial companies.
Business Risks Financial Risks
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Relate significant market events of past several decades to growth of the
risk management industry.
Fixed exchange rate system broke down 1971 1
Oil price shocks High inflation 1973 2
Black Monday. US stocks drop 23%
10/19/87 3
Bond debacle (Fed hikes rates 6 times) 1994 4
Deflation of Japanese stock price bubble 1989 5
Asian turmoil 1997 6
Russian default Global crisis (LTCM) Aug 1998 7
Terrorist attack on New York 9/11/01 8
Visible subprime crisis Aug 2007 9
Fed takeover Fannie Mae & Freddie Mac; Merrill Lynch sold; Lehman bankruptcy; AIG Sep 2008 10
Bretton
Woods
Black
Monday
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Describe functions & purposes of financial institutions as they relate to
financial risk management.
Financial institutions (FIs) as brokers: reduce transaction and information costs between households and corporations
Financial institutions (FIs) as asset transformers: liquidity and maturity transformation
Define what a derivative contract is and how it differs from a security.
A security is a financial claim issued by a corporation to raise capital. Primary securities are
backed by real assets while secondary securities are issued by banks and backed by primary
securities.
A derivative contract is a private contract that derives its value from some underlying asset
price, reference rate, or index; e.g. the underlying may be a stock, bond, currency, or commodity.
A derivative derives its value from another security. An example would be a forward contract on
a foreign currency is a promise to buy a fixed amount at a fixed price on a future date.
Asset Transformers
Brokers
Corporations
FI
Households
Security:Financial claims issued by corporations to raise capital
Primary securities (e.g., equities, bonds) are backed by real assets
Secondary securities are issued by banks [financial institutions] and backed by primary securities
Derivative:Private contract deriving value from an underlying asset price, reference rate, or index
Forward contract on foreign currency is a promise to buy a fixed (notional) amount at a fixed price at a future date
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Balance sheet perspective:
Commercial Firms
Financial Intermediary
Assets Liabilities
Assets Liabilities
Real Assets (plant, machinery)
Primary Securities (debt, equity)
Primary securities (debt, equity)
Secondary securities
For example
Compare a corporate bond issuance (security) to a credit default swap (derivative). Both expose
investors to credit risk, but one is a security and the other is a derivative. The investor in a
corporate bond assumes default risk by purchasing the bond; this investor owns a financial
claim on the corporation’s real assets.
Describe the dual role leverage plays in derivatives and why it is relevant
to a risk manager.
Leverage is a “double-edged sword” with advantages and disadvantages:
The advantage of leverage: It makes the derivative an efficient instrument for hedging and speculation owing to very low transaction costs (Efficient)
The disadvantage of leverage: the absence of upfront cash payment makes it more difficult to assess the potential downside risk; leveraged derivative risks conseqently must be managed more carefully
XYZBond
Credit Default
Swap (CDS) on
XYZ Bond
Unfunded Short CDS (Write protection)
Funded Long Bond
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Specify confidence = 1 – significance (α)
Specify horizon
VaR is always one-tail!
Jorion says ―Leverage is a double-edged sword:‖ the derivative (unfunded) is efficient
but it is harder to assess potential downside
Define financial risk management
Financial risk management is the design and implementation of procedures for …
Identifying,
Measuring, and
Managing financial risks
Define value at risk (VaR)
VaR summarizes the worst loss over a target horizon that will not be exceeded with a
given level of confidence
“Under normal conditions, the most the portfolio can lose over a month is about $3.6 million at the 99% confidence level”
“Under normal conditions, the most the portfolio can lose over a month is $X/%X at with (1 – α) % confidence”
VaR is the worst expected (i.e., with selected confidence) loss
over a target horizon.
But better is the mathematically equivalent: ―VaR is the
minimum we expect to lose (1- confidence)% of the time over a
target horizon.‖
Efficient: Low Transaction Costs
But lack of funding makes risk
assessment difficult (What is exposure?)
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Value at risk (VaR): the equivalent but semantically better perspective:
“Under normal conditions, the portfolio should lose at least $3.6 million 1% of the time”
“Under normal conditions, we expect the portfolio to lose at least $X/%X at the selected significance (1 – confidence) level.”
Nonparametric VaR: Quantile of an EMPIRICAL distribution
Nonparametric value at risk (VaR) is the quantile of an empirical distribution. For example,
where on the distribution does the worst 5% of outcomes fall in the distribution:
Example of Nonparametric Value at Risk (VaR): Historical Simulation
Assume we observe 30 days of returns and we sort them from best to worst. The 99% VaR is
equal to PERCENTILE (array, 100% - 99%). In the sample data from the learning spreadsheet,
this worst expected loss is -1.44%, so we say “the 99% historical simulation VaR is a loss of
1.44%.”
Historical Simulation (HS) VaR (i.e., non-parametric) 1-day HS VaR:
-1.44%
1-day HS VaR:
1.44%
Period Return Sort
t - 1 -0.9%
t - 2 -0.8%
…
t - 28 0.6%
t - 29 0.4%
t - 30 -0.8%
0
20
40
60
80
100
-4 -3 -2 -1 0 1 2 3 4
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Parametric VaR: quantile of an STATISTICAL distribution
Parametric value at risk (VaR) is the quantile of a statistical distribution. Similarly, given the
statistical distribution, where is the cutoff point, such that 5% of the distribution is “to the left”
(i.e., losses in excess of):
Both (nonparametric and parametric) VaRs are statistical risk measures of potential
losses. Notice what they have in common: in both cases, we can determine the quantile
of a distribution. In order to specify VaR, we must make or produce a distributional
assumption.Then we simply select a quantile.
Example of analytical/parametric value at risk:
Given a daily volatility of 1% (and assuming zero mean; a common assumption for daily
periods), the 99% VaR is 2.33%. And, the scaled 10-day VaR is 2.33% * SQRT(10) = 7.36%.
Parametric (normal) VaR
Standard Deviation (Volatility), Daily 1.0%
Confidence Level, c 99.0%
Significance Level, α = 1-c 1%
Target Horizon (days) 10
Autocorrelation -- 1-day Value at Risk (VaR) 2.33%
Extended over Target Horizon (i.i.d.) Standard deviation (i.i.d) 3.16%
n-day VaR (i.i.d.) 7.36%
0.0%
1.0%
2.0%
3.0%
4.0%
97
.0
97
.3
97
.6
97
.9
98
.2
98
.5
98
.8
99
.1
99
.4
99
.7
10
0.0
10
0.3
10
0.6
10
0.9
10
1.2
%95%VaR ( 1.645)
-1.645
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Describe advantages & disadvantages of VaR relative to … stop-loss limits,
notional limits, and exposure limits.
Except for one challenge (i.e., VaR is not easy to calculate), Jorion finds value at risk (VaR)
superior to the other tools
Stop-loss limits: if the cumulative loss incurred by a trader exceeds some limit, the position has to be cut. The glaring problem is that the controls are applied xo post (after the fact)
Notional limit: an ex ante (forward-looking or before-the-fact) limit on the notional exposure. Although this is an ex ante control, notional is not a good proxy for exposure.
Exposures: limits placed on sensitivities; e.g., duration limit, beta limit. Jorion says, “this approach is still incomplete. It does not consider the volatility of risk factors, which could vary across markets, nor their correlations.”
Characteristic Stop Loss Notional Exposure VaR
Type Ex post Ex ante Ex ante Ex ante
Ease of Calculation
Yes Yes No No
Ease of Explanation
Yes Yes No Yes
Aggregation Yes No No Yes
Compare & contrast valuation & risk management, using VaR as example.
In a sense, VaR extends current valuation methods for derivative instruments. But where
valuation is concerned with the precise mean of the distribution, risk management is concerned
with the approximate variation in the distribution.
Derivatives Valuation Risk Management
Principle Expected Discounted Value Distribution of Future Values
Focus CENTER of distribution TAILS of distribution
Horizon Current value, discounting Future value
Precision HIGH PRECISION NEEDED for pricing purposes
LESS PRECISION needed, errors cancel out
Distribution Risk-neutral Actual (physical)
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Define and describe the four major types of financial risks: market,
liquidity, credit, and operational.
According to Jorion, the four major types of risk include:
Market risk: risk of losses owing to movements in the level or volatility of market prices. Market risk can take two forms: absolute risk (loss in dollar terms) or relative risk (loss relative to a benchmark). Market risk can be directional or nondirectional. Further, market risk includes basis risk and volatility risk.
Liquidity risk: usually treated separately from other risks discussed.
Credit risk: the risk of losses owing to counterparties unwilling/ unable to fulfill contractual obligations.
Operational risk: the risk of loss resulting from inadequate or failed internal processes, people, and systems or from external events.
Market Risk “Price level risk”
Absolute vs. Relative
Directional vs. Non
Basis
Volatility
Market Risk > Liquidity Risk
Asset-liquidity (market liquidity)
Funding-liquidity (cash-flow)
Credit Risk
Default
Credit deterioration (downgrade)
M2M loss in value
Operational Risk “Almost everything else”
Internal processes
Model risk
People risk
Legal risk
Operational
Risk
Credit
Risk
Basel II/IIICounterparty
Risk
Liquidity
Risk
Market
Risk
Investment
Risk
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Within market risk: Describe and differentiate between absolute and
relative market risk.
Absolute
Dollar (or currency) terms
Focus: Volatility of total returns
Relative
Versus a benchmark
Focus: Deviation from benchmark or tracking error
Within market risk: Describe and differentiate between directional and
non-directional market risk.
Directional
Movements in financial variables. For example:
Stock price moves down,
Interest rates drop,
Commodity prices change
Non-directional
The risks that remain, including hedged positions
Nonlinear exposures,
Basis risk
Volatilities
Within market risk: Describe basis risk and its sources.
Basis risk refers to unanticipated movements in relative prices of assets in a hedged
position, such as cash and futures or interest-rate spreads are considered basis risk. In the case
of a forward contract, the basis is the difference between the forward price (F) and the spot
price (S). In theory, the forward and spot prices should converge, such that the basis should
approach zero as the contract approaches maturity. But this is only theoretical and depends on
an exact match in the commodity, perfect timing, other key assumptions (e.g., perfect liquidity)
and the absence of other frictions. A hedge is constructed based on an anticipation of the basis
(e.g., that the basis will converge to zero). Basis risk, then, is the risk that the hedge will not
perform as expected; i.e., that the hedge will not offset the loss in the primary position.
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Basis risk is often underestimated but it is arguably one of the most important risks. All
hedges imply (at least some) basis risk: the risk the hedge will not exactly offset.
Recently in ―Lessons from the Financial Crisis,‖ Goldman’s CEO said, ―a lot of risk models
incorrectly assumed that positions could be fully hedged. After LTCM and the crisis in
emerging markets in 1998, new products like basket indices and credit default swaps
were created to help offset a number of risks. However, we didn’t, as an industry,
consider carefully enough the possibility that liquidity would dry up, making it difficult to
apply effective hedges.‖
Within market risk: Describe volatility risk & sources.
Unanticipated movements in relative prices of assets in a hedged position, such as cash and
futures or interest-rate spreads
The classic instrument that represents a trade on volatility risk is a stock option. In fact,
options are motivated as instruments of volatility. Salih Neftci writes (in Principles of
Financial Engineering, 2nd Ed), ―an option exposure, when fully put in place, is an impure
position on the way volatility is expected to change. A market maker with a net long
position in options is someone who is ―expecting‖ the volatility to increase. A market
maker who is short the option is someone who thinks that the volatility of the underlying
is going to decrease.‖
Time (T)S0
F0
ST-1
FT-1 F0ST
ST=FT
All hedges imply (at least some) basis risk: the risk the hedge will not exactly offset
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Within liquidity risk: Asset vs. funding liquidity risk.
Liquidity risk is typically divided into two different, but related, types: asset-liquidity risk (a.k.a.,
market/product-liquidity risk) and funding-liquidity risk (a.k.a., cash-flow risk). Both concern
the dilemma of not having enough time: liquidity risk implies that time can eventually solve the
problem. Funding liquidity risk refers to the inability to meet payment obligations or to fund
ongoing operations. “This is especially a problem for portfolios that are leveraged and subject to
margin calls from the lender.” Asset-liquidity risk refers to a transaction that cannot be
conducted at prevailing market prices “owing to the size of the position relative to normal
trading lots.” This is also simply when the position cannot be exited without steep discounting
(or, in the extreme case, a “fire sale”)
Asset-liquidity risk (market/product liquidity risk)
Cannot exit position at prevailing market prices due to size of the position
Varies by…
Asset class
Prevailing market conditions
Funding-liquidity risk (cash-flow risk)
Cannot meet payment obligations
Balance sheet issue, typically concern of CFO
In the context of the credit crisis, much attention has been focused on the interrelationship
between asset and funding liquidity risk. Specifically, for example, the lack of asset liquidity can
trigger margin calls that create funding liquidity risk.
Asset-liquidity risk (market/product liquidity risk)
• Transaction cannot be conducted at prevailing market prices owing to size of the position
• Varies by…
• Asset class
• Prevailing market conditions
Funding-liquidity risk (cash-flow risk)
• Inability to meet payment obligations, which may force liquidation, transforming paper losses into realized losses
Crisis Lesson: Inter-
related
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Within credit risk: Describe and differentiate between exposure and
recovery rate.
Exposure (a.k.a., amount at risk) is the potential amount that can be lost. For example, if a bond
buyer invests $10 million dollars, then the investor’s exposure is $10 million. However, upon
default, the investor expects at least some partial recovery. The recovery rate (1 – loss given
default) is the proportion recovered. Recovery is also referred to as “cents on the dollar;” e.g.,
40% recovery is “40 cents on the dollar.”
Exposure (EAD)
Amount at risk
Recovery Rate
Proportion paid back to lender (cents on the dollar)
Within credit risk: Describe credit event and how it may relate to market
risk.
A credit event occurs when there is a change in the counterparty’s ability to perform its
obligations. According to Jorion, credit risk should be defined as the potential loss in mark-to-
market value incurred owing to a credit event.
Please pay careful attention: credit risk is not limited to default. Consistent with the
broad here definition in Jorion, De Servigny includes three sub-classes of credit risk in his
overview of credit risk models: default; rating migration; and change in spread.
Therefore, credit risk include both default risk and risk of credit deterioration.
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C
M
Within credit risk: Describe sovereign risk and its sources.
Sovereign risk
Country-specific (unlike default risk which is generally company-specific)
Sources
Countries impose foreign-exchange controls that make it impossible for counterparties to honor their obligations
Within credit risk: Describe settlement risk and its sources
Settlement risk: when two payments are exchanged the same day. Risk that counterparty may
default after the institution already made its payment
Pre-settlement exposure: only netted value
On settlement day: full value of payments due
Jorion: Settlement risk is acute for foreign-exchange transactions
Within operational risk: Describe relationships between operational,
market & credit risk
Operational risks can lead to market or credit risks
A settlement fail can create market risk because cost may depend on movement in market prices
Again important is the theme of the interdependence of risks. Although operational
risks are defined separately, the point is that operational losses cannot necessarily be
analyzed in isolation of market and credit risks.
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Within operational risk: Describe model risk and its
sources
Operational risk > Model risk
Model risk is risk of losses owing to the fact that valuation models may be flawed
“Very insidious” and requires intimate knowledge of modeling process
Model risk is “the risk of error in our estimated risk measures due to inadequacies (or
deficiencies) in our models. (Kevin Dowd)” Models are by definition quantitative but
this chapter largely concerns the qualitative conundrums posed by reliance on models.
Within operational risk: Describe people risk
Operational risk > People risk
People risk includes internal or external fraud
For example:
Rogue traders
Within operational risk: Describe legal risk and its sources
Operational risk > Legal risk
Legal risk arises from exposure to fines, penalties or punitive damages resulting from supervisory actions, as well as private settlements
Examples:
Counterparties sue to invalidate credit losses [related to credit risk]
Shareholders lawsuits against corporations
What is excluded from operational risk?
Strategic risk
Business risk
Reputational risk (controversial)
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Stulz Chapter 2:
Investors & Risk
Management
In this chapter…
Explain how expected return and returns variance are used to describe the return distribution for a security or portfolio of securities.
Define and describe the significant characteristics of the efficient frontier. Differentiate between diversifiable and systematic risk and describe how
diversification can reduce risk in a portfolio. Describe the CAPM, and explain the concepts of beta and the security market
line. Calculate and interpret firm value using the CAPM. Use the CAPM to discuss the value of risk management to investors with respect
to: A firm’s diversifiable risk A firm’s systemic risk
Define and discuss the “hedging irrelevance proposition” as it relates to: Diversifiable risk Systematic risk Risks valued by investors differently from what CAPM would predict
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To reduce systemic risk requires a costly
hedge (short the market)
Reducing diversifiable risk produces no value because investors don’t care (they don’t lower
their discount rate)
Stulz Chapter 2 assumes perfect markets and concludes risk management
cannot add value.
In “perfect financial markets”
Because investors can diversity themselves, they bear unsystematic risk at no cost and will not reward its reduction
Risk management (e.g., hedging) creates no value: Reducing systematic risk is a “net zero”
The entire chapter is based on the validity of CAPM
( ) [ ( ) )]ii F M FE R R E R R
NoFriction
PerfectCompetition
PerfectInformation
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Explain how expected return & returns variance describe return
distribution for security or portfolio.
In Stulz, the working assumption is that returns are normally distributed. Under this
restrictive assumption of normality, only the mean and variance are needed. This follows the
traditional Markowitz approach (a.k.a., the mean–variance framework) which only considers
the two moments in characterizing the portfolio: mean return and return variance. This is the
same as assuming that higher order moments (skew, kurtosis) are “normal” (or null).
XYZ IBM
Expected return (mean) 26.0% 13.0%
Volatility (Standard Deviation) 60.0% 30.0%
0.0
0.5
1.0
1.5
-200% -100% 0% 100% 200%
Probability Density Function (pdf)
XYZ
IBM
0.0
0.5
1.0
1.5
-200% -100% 0% 100% 200%
Cumulative Distribution Function (CDF)
XYZ
IBM
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Define and describe the significant characteristics of the efficient
frontier.
The efficient frontier is the set of portfolio allocations that, for a given level of volatility (risk),
the returns are higher. The efficient frontier exists due to the benefits of diversification.
Differentiate between diversifiable & systematic risk & describe how
diversification can reduce risk.
Diversifiable risk is risk that disappears in a well-diversified portfolio. The risk that cannot be
eliminated through diversification is called system risk.
5.0%
7.0%
9.0%
11.0%
13.0%
15.0%
17.0%
19.0%
0.0% 10.0% 20.0%
Exp
ecte
d R
etu
rn
Standard Deviation
Risky PortfolioCMLMarket Portfolio
25.0%
30.0%
35.0%
40.0%
45.0%
50.0%
55.0%
60.0%
0%
5%
10%
15%
20%
25%
30%
-1.0 -0.5 0.0 0.5 1.0
Expected return
Volatility
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Describe the CAPM, and explain the concepts of beta and the security
market line.
The capital asset pricing model (CAPM) says that the expected excess return on a security is a
function only of its systematic risk. The security market line (SML) plots the relationship
between expected return and beta (i.e., systematic risk) that is asserted by the CAPM:
Calculate and interpret firm value using the CAPM.
The idea is:
Use the capital asset pricing model (CAPM) to determine the discount rate, then
Value the firm by discounting the future cash flow(s) at the discount rate
At the end of one year (one period), the random liquidating cash flow is (C) such that the market
value (V) of the firm is the discounted present value of E(C):
Where random liquidating cash flow (C) is market value (V) of firm at end of year
( ) 1 ( )F M FE C V R E R R
( )
1 ( )F M F
E CV
R E R R
0%
5%
10%
15%
20%
25%
0.00 1.00 2.00 3.00
Exp
ect
ed
Ret
urn
Beta
Security Market Line (SML)
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ResidualDiversifiableIdiosyncratic
“Investors can diversify themselves,
won’t pay for this”
BetaSystematic Risk
“Quantity of Risk”
Equity (Risk) Premium
“Price of Risk”
For example
If E(C) = $350 million, risk-free rate = 5%, market risk premium = 6% and beta = 0.5
( )
1 ( )F M F
E CV
R E R R
350
1 5% 0.5 6%V
Use the CAPM to discuss the value of risk management to investors with
respect to: A firm’s diversifiable risk.
Use the CAPM to discuss the value of risk management to investors with
respect to: A firm’s systemic risk.
Perfect markets view says value of firm depends on systematic risk (not total risk, not
diversifiable risk)
( ) [ ( ) )]ii F M FE R R E R R
In perfect markets, risk management to reduce …
Diversified risk does not increase value (not rewarded) because shareholders can eliminate via diversification at zero cost
Systemic risk also does not create value as two effects cancel out
Cost for risk management reduces cash flow (numerator)
Lower risk reduces discount rate
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Define and discuss the “hedging irrelevance proposition” as it relates to:
Diversifiable risk
Define and discuss the “hedging irrelevance proposition” as it relates to:
Systemic risk
To illustrate the hedging irrelevance proposition, Stulz offers the example of a firm called Pure
Gold. The riskless rate is 5% and the equity risk premium (market risk premium) is 6%. Pure
Gold can sell forward at $350 (i.e., the forward price is $350). Under the scenario where the firm
is hedged, the cash flow has no systematic risk and the expected cash flow is therefore
discounted at the riskless rate.
If, however, the beta is 0.5, the firm has systematic risk. In this scenario, investors are not
satisfied with the riskless rate. The expected future spot price [E(S)] must be greater than the
forward price (i.e., normal backwardation). Stulz’ point is: risk management (i.e., the reduction
of systematic risk) is not impacting firm value (all present values are the same: $333.3) because
higher (lower) beta corresponds to both a higher (lower) future cash flow and a higher (lower)
discount rate.
Riskless 5.0% Equity risk premium (ERP) 6.0% Gold Forward (F) $350
Systematic Risk (0 = None)
E(S) = F E(S) > F E(S) >> F
Beta 0.0 0.5 1.0
Expected Return, CAPM 5.00% 8.00% 11.00%
Expected Future Spot, E(S) $350.0 $360.0 $370.0
Present Value, S $333.3 $333.3 $333.3
All scenarios give the same present value. Systematic risk (positive beta) implies normal
backwardation!
With respect to systemic risk: Shareholders require the same risk premium for systematic risk
as all investors. Hence, eliminating it for the shareholder just means that the investors who take
it on bear it as the same cost. Stulz says this cannot create value.
With respect to diversifiable risk: it does not affect the share price, and investors do not care
about it because it gets diversified within their own portfolios. Hence, eliminating it does not
affect firm value.
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Define and discuss the “hedging irrelevance proposition” as it relates to:
Risks valued by investors differently from what CAPM would predict
Again, shareholders and other investors charge the same price for bearing such risks. The
value of the firm is the same whether the firm hedges or not. The un-hedged firm offers greater
diversifiable risk, but shareholders do not care. If they are confronted with an un-hedged firm,
either they diversify the risk away or—if they prefer the exposure to the specific risk—they can
seek the risk. If the firm could create “instant value” by hedging, then in theory, there would
exist an arbitrage opportunity that investor could also exploit. Such an arbitrage opportunity, in
theory, should not be sustainable
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Stulz Chapter 3:
Creating Value with
Risk Management
In this chapter…
Explain how risk management can create value by handling bankruptcy costs. Explain how risk management can create value moving income across time and
reducing taxes. Describe those circumstances when risk reduction benefiting a large
shareholder may increase or decrease firm value. Explain the relationship between risk management, managerial incentives, and
the structure of management compensation. Describe debt overhang, and explain how risk management can increase firm
value by reducing the probability of debt overhang. Explain how risk management can reduce the problem of information
asymmetry and increase firm value.
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Stulz Chapter 3 realizes imperfect markets and concludes risk
management can add value
Perfect financial markets risk management cannot create value
Markets are not perfect! Imperfections (and frictions) imply risk management can create value
Capital structure (financial distress)
Taxes
Agency and information asymmetries
Explain how risk management can create value by handling bankruptcy
costs.
Debt is cheaper than equity. In theory, then, firm value increases as the firm increases its financial
leverage (ratio of debt/equity or debt/total assets). However, higher leverage also increases the
probability of bankruptcy (and default) and incurs a “cost of financial distress.”
In summary:
Equity is more expensive than debt due to (i) subordinate claim and (ii) tax shield on debt
Ceteris paribus, the firm would prefer to increase leverage to lower its weighted average cost of capital (WACC)
However, the “friction” is the cost of financial distress: as leverage increases, beyond some point, the cost of equity and debt both increase due to the threat of default and bankruptcy
NoFriction
PerfectCompetition
PerfectInformation
Cost ofFinancialDistress
InformationAsymmetry
Taxest(friction!)
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Generally, interest paid by a company is tax-deductible. The tax savings achieved by the use of
debt is called the tax shield. All other things being equal, a firm that uses debt saves cash taxes
and decreases its weighted average cost of capital.
However, at a certain point, using more debt becomes counterproductive because debt carries a
fixed obligation and increases the risk of default. This is the essential trade-off: as they increase
their leverage (i.e., ratio of debt-to-equity), firms increase their tax shield but also increase the
present value of costs of financial distress. The optimal capital structure of a firm balances the
tax benefits of debt against the costs of financial distress.
A firm can reduce the present value of the costs of financial distress through risk management
by making financial distress less likely. As a result, it can take on more debt. Risk management
enables the firm to have a higher debt level, and hence a greater tax shield from debt, for any
likelihood of financial distress.
If a firm carries a risk of bankruptcy, then it incurs bankruptcy costs (e.g., as reflected in higher
interest expense). In this case, the irrelevance theorem does not hold. The present value of this
firm can be reduced by the present value of the bankruptcy costs, as follows:
Value of firm = PV(Cash flow) – PV (bankruptcy costs)
If the cost to hedge bankruptcy risk is zero (or even if it is merely less than the bankruptcy
costs), risk management creates value because the market will bear the diversifiable risk. In fact,
if the risk is diversifiable (i.e., non-systematic), the capital markets should be able to bear the
risk with zero cost. In this case, gains from risk management equal present value of the
bankruptcy costs:
Gain from risk mgmt = Value of hedged firm − Value of un-hedged firm = Present Value (bankruptcy costs)
- 2,000 4,000 6,000 8,000
10,000 12,000 14,000
0.13 0.36 0.55 0.70 0.84 0.95
Firm value(FCF/WACC)
Firm value - PV(cost of financialdistress)
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For example
In Stulz’ example, the value Pure Gold (the firm) is entirely a function of the volatile price of gold.
While the firm has present value of $333, the volatility implies there is a small chance the price
of gold causes the firm value to drop below the default threshold (i.e., the face value of debt). If
the costs of bankruptcy are $20 million and the probability of default is about 7.7%, then the
expected cost of bankruptcy are $1.53 million (notice this is an expected value which is
probability adjusted). Notice that the “threat” of bankruptcy has a cost.
Forward Price $350.0
Volatility of Gold Price 20%
Riskless rate 5%
Present Value (Cash Flow) $333.3
Debt, Face Value $250.0
Bankruptcy costs $20.0
Normal deviate (1.43)
Normal CDF 0.077
Bankruptcy costs, Expected $1.53
Bankruptcy costs, PV $1.46
Here, the present expected value of the threat of bankruptcy (the cost of financial
distress even before bankruptcy) is $1.46 million. If risk management can reduce or
eliminate that possibility, then firm value is increased by up to $1.46 million.
Explain how risk management can create value moving income across
time and reducing taxes.
Risk management creates value when it is more expensive to take a risk within the firm than to
pay the capital markets to bear that risk. Corporate taxes can increase the cost of taking risks
within the firm.
The tax argument says that if a firm can move a dollar from a high tax rate to a low tax rate, then
it reduces the present value of taxes to be paid (of course, this requires differential tax rates).
Complications include:
Carry-backs and carry-forwards: Firm with negative taxable income can offset future or past taxable income with a loss in this tax year, subject to limitations (limited number of years. No allowance for time value of money.)
Tax shields: There is a wide variety of tax shields. A critical tax shield is that on interest paid. Another is the tax shield on depreciation. Firms also have tax credits.
Personal taxes: unlikely that taxes create biases in forward contract prices.
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For example
Pure Gold (the firm) will achieve pre-tax profit of either $250 million (50% probability) or $450
million (50% probability). The simplified tax schedule is such that 50% taxes are paid only if the
pre-tax cash flow exceeds $300 million; no taxes otherwise. Alternatively, Pure Gold can sell its
gold forward at $350; note this is the same as the expected (average) value of the future spot
price. After-tax cash flow is higher for the hedged firm:
RF Rate 5%
Tax Schedule
Cash Flow < $299.0 0%
Cash Flow > $300.0 50%
Future
After-tax
Spot (S1) Tax FV PV
Probability 50% $250.0 $0.0 $250.0 Probability 50% $450.0 $75.0 $375.0 Expected
$350.0 $37.5 $312.5 $297.6
Forward
After-tax
(F0)
FV PV
$350.0 $25.0 $325.0 $309.5
If a firm can move a dollar from a high tax rate to a low tax rate, then it reduces the
present value of taxes to be paid
Describe circumstances when risk reduction benefiting a large
shareholder may increase or decrease firm value.
A large shareholder can engage in monitoring.
Evaluate management actions
Influence incentives (e.g., bonus plans, stock options)
Explain the relationship between risk management, managerial
incentives, and management compensation.
Agency problem
Management acts in their own interest instead of shareholders’ interests (e.g., “empire building”)
How to solve? Incentives (e.g., stock options) try to provide alignment between management actions (and decisions) with shareholder welfare.
But practice is more difficult
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Authors argue for:
Tying compensation to some measure of value
creation
Share ownership
Describe debt overhang and explain how risk management can increase
firm value by reducing the probability of debt overhang.
Debt overhang is too much debt: To increase shareholder value may not increase firm value!
Induces shareholders seek negative NPV projects, and/or
Avoid investing in valuable projects because they dilute
Consequently, risk management that reduces this probability increases firm value today.
Rate 5% Gold $5 Invest,
HLG Debt $400.0 Price Payoff of:
$10
Probability 50% $250.0 $260.0
Probability 50% $450.0 $460.0
Value of Debt:
50% $250.0 $260.0
50% $400.0 $400.0
FV $325.0 $330.0
PV $309.5 $314.3
Value of Equity:
50% $0.0 $0.0
50% $50.0 $60.0
FV $25.0 $30.0
PV $23.81 $28.6
Value of Firm (D+E) $333.3 $342.9
Increase in Equity Value $4.8
Existing Equity DILUTED to: $23.571
Principal-Agent (Agency) is a key friction in Subprime Crisis
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Explain how risk management can reduce the problem of information
asymmetry and increase firm value.
Information asymmetry: one party (management) knows more than the other (outside investor)
Problem for management raising funds managers know more about firms’ projects than outsiders
Ways to reduce the costs of managerial discretion (and therefore reduce the costs of the funds)
Sit large shareholder on board; e.g., private equity fund
Borrow against assets rather than against future project.
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Elton, Chapter 5:
Delineating
Efficient Portfolios
In this chapter…
Calculate the expected return and volatility of a portfolio of risky assets. Explain how covariance and correlation affect the expected return and volatility
of a portfolio of risky assets. Describe the shape of the portfolio possibilities curve. Define the minimum variance portfolio. Define the efficient frontier and describe the impact on it of various
assumptions concerning short sales and borrowing.
Calculate the expected return and volatility of a portfolio of risky assets.
Expected return
The expected return on a portfolio of two assets is given by
P A A B BR X R X R
, fraction of portfolio held in asset A, asset B
, , expected return on asset A, asset B, portfolio
A B
A B P
X X
R R R
Volatility of a portfolio of risky assets The expected return on a portfolio of two assets is given by
1 22 2 2 2(1 ) 2 (1 )P A A A B A A ABX X X X
1 22 2 2 2(1 ) 2 (1 )P A A A B A A AB A BX X X X
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A key relationship is the link between covariance and correlation. Covariance is the product of
correlation (rho) and asset volatilities:
AB AB A B
Explain how covariance & correlation affect the expected return and
volatility of a portfolio of risky assets.
Imperfect correlation reduces portfolio variance
Riskless rate 4.00%
Asset A
Expected Return 14.00%
Standard Deviation 20.00%
Variance 0.0400
Asset B
Expected Return 4.00%
Standard Deviation 10.00%
Variance 0.0100
0.0%
2.0%
4.0%
6.0%
8.0%
10.0%
12.0%
14.0%
16.0%
0.0% 5.0% 10.0% 15.0% 20.0% 25.0%
Exp
ect
ed
Ret
urn
Standard Deviation
Portfolio Possibilities Curve Concave
-1.00
-0.50
0.00
0.50
1.00
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Describe the shape of the portfolio possibilities curve.
Portfolio possibilities curve must be concave, says Elton; i.e., a straight line connecting
any two points on the curve lies entirely under the curve.
Define the minimum variance portfolio.
The minimum variance portfolio is not the ―best‖ portfolio; i.e., it is not the portfolio
with the highest Sharpe ratio.
0.0%
2.0%
4.0%
6.0%
8.0%
10.0%
12.0%
14.0%
16.0%
0.0% 10.0% 20.0% 30.0%
Exp
ect
ed
Ret
urn
Standard Deviation
Minimum variance
Most efficient
(highest excess
return/volatility)
0.0%2.0%4.0%6.0%8.0%
10.0%12.0%14.0%16.0%
0.0% 10.0% 20.0% 30.0%
Exp
ect
ed
Ret
urn
Standard Deviation
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Although not assigned, we can solve for the minimum variance portfolio. We can solve by setting
first partial derivative equal to zero (i.e., where slope of tangent is flat):
mvp
2
mvp 2 2
0
2
PA
A
B AB A BA
A B AB A B
XX
X
Define the efficient frontier & describe the impact of various assumptions
concerning short sales & borrowing.
The efficient frontier is a concave function in expected return standard deviation space that
extends from the minimum variance portfolio to the maximum return portfolio.
Prior to the inclusion of the riskless asset, the efficient frontier is the upper (concave)
region
Riskless rate 6.00%
Asset A
Expected Return 10.00%
Standard Deviation 10.00%
Variance 0.0100
Asset B
Expected Return 16.00%
Standard Deviation 20.00%
Variance 0.0400
A,B
Correlation (A,B) * 0.30
Covariance (A,B) * 0.00600
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After the addition of riskfree asset, the allocation decision is between (i) the riskless
asset and (ii) the market portfolio. The capital market line (CML) is now efficient.
With short sales, portfolios exist that give infinite expected rates of return
5%
10%
15%
20%
0.0% 5.0% 10.0% 15.0% 20.0% 25.0%
Exp
ect
ed
Ret
urn
Standard Deviation
Capital Market Line (CML)
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Elton, Chapter 13:
The Standard Capital
Asset Pricing Model
In this chapter …
Describe the CAPM and the assumptions underlying it. Derive the CAPM. Describe the capital market line. Use the CAPM to calculate the expected return on an asset.
Describe the CAPM and the assumptions underlying it.
No transaction costs.
There is no cost (no friction) to buy or sell any asset. To include transaction costs in the model
adds much complexity. Whether it is worthwhile introducing this complexity depends on the
importance of transaction costs to investors’ decisions. Given the size of transaction costs, they
are probably of minor importance.
Assets are infinitely divisible.
Investors could take any position in an investment, regardless of the size of their wealth. For
example, they can buy one dollar’s worth of IBM stock.
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Absence of personal income tax.
Implies investor (individual) is indifferent to dividends versus capital gains.
Perfect competition (individuals are “price-takers”)
An individual cannot affect the price of a stock by his/her buying or selling action. This is
analogous to the assumption of perfect competition. While no single investor can affect prices by
an individual action, investors in total determine prices by their actions.
Mean-variance framework.
Investors are make decisions solely in terms of expected values and standard deviations. of the
returns on their portfolios.
First (mean) and second (variance or standard deviation) moments only
Unlimited short sales allowed.
Individual investor can sell short any amount of any shares.
Unlimited lending and borrowing at the riskless rate.
Investor can lend or borrow any amount of funds desired at a rate of interest equal to the rate
for riskless securities.
Homogeneity of expectations: single period
All investors are assumed to define the relevant period in exactly the same manner.
Investors are concerned with the mean and variance of returns (or prices over a single period)
and all investors are assumed to define the relevant period in exactly the same manner.
Homogeneity of expectations: identical expectations.
All investors are assumed to have identical expectations with respect to the necessary inputs to
the portfolio decision: expected returns, variance of returns, and the (pairwise) correlation
matrix.
All assets are marketable.
All assets, including human capital, can be sold and bought on the market.
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Derive the CAPM [awkward: derivation will not be tested. Rather,
application of the formula is very likely be tested]
CAPM says that expected return is a linear function of systemic risk (beta)
Equivalently: Expected excess return = (price of risk) * (quantity of risk)
i F i M FR R R R
Equivalent formulation for CAPM:
Since beta is equal to covariance [security return, market return] / market return variance, an
equivalent expression is given by:
2
2
iMi
M
M F iMi F
M M
M FF iM
M
R RR R
R RR
Price of risk (MRP, ERP)
Quantity of risk
0%
5%
10%
15%
20%
25%
0.00 1.00 2.00 3.00
Exp
ect
ed
Ret
urn
Beta (quantity of systemic risk]
Security Market Line (SML)
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Describe the capital market line.
The capital market line (CML) is the set of optimal portfolios: a linear combination of (i)
the market portfolio and (ii) the risk-free asset
[Non AIM] CAPM assumes strong efficiency
CAPM’s assumption of perfect market requires (assumes) the STRONG FORM of the
Efficient Market Hypothesis (EMH)
Private Information
Strong Public Information
3
Semi-Strong
Past Prices
2
Weak 1
5%
10%
15%
20%
0.0% 5.0% 10.0% 15.0% 20.0% 25.0%
Exp
ect
ed
Ret
urn
Standard Deviation
Capital Market Line (CML)
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Use the CAPM to calculate the expected return on an asset.
Assume that the following assets are correctly priced according to the security market line.
Derive the security market line.
What is the expected return on an asset with a Beta of 2 (Elton Question 13.1)?
1 1
2 2
6% 0.5
12% 1.5
R
R
Answer:
12% 1.5
6% 0.5
6% 3%
F
F
F
R MRP
R MRP
MRP R
2.0
3% 6%
3% 6% 2 15%
R
R
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Elton, Chapter 14:
Nonstandard Forms of Capital
Asset Pricing Models
In this chapter…
Describe the impact on the CAPM of the following: Short sales disallowed Riskless lending and borrowing Personal taxes Nonmarketable assets Heterogeneous expectations Non-price-taking behavior
Describe the following multi-period versions of CAPM: Consumption-oriented CAPM CAPM including inflation Multi-beta CAPM
Describe the impact on the CAPM of the following:
In this reading, unrealistic assumptions in the CAPM (on the left, below) are variously replaced
by more realistic assumptions (on the right, below):
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Describe the impact on the CAPM of the following: Short sales disallowed The allowance of short-sales is a convenient assumption that simplifies the derivation math, but
it is not a necessary assumption: if short sales are disallowed, exactly the same CAPM result
is obtained.
Describe the impact on the CAPM of the following: No Riskless lending and
borrowing
If we assume there is neither riskless lending nor borrowing, CAPM still applies except
borrowing (lending) at the riskfree asset is replaced by shorting (going long) the zero-
beta portfolio. This is the zero-beta CAPM; also referred to as the two-factor model:
( )
expected return on zero-beta portfolio
i Z M Z i
Z
R R R R
R
No riskless lending/borrowing leads to → Zero-beta CAPM
Describe the impact on the CAPM of the following: Personal taxes
The simple form of the CAPM ignores taxes, which therefore assumes that investors are
indifferent between capital gain and dividend income; and that all investors hold the same
portfolio of risky assets.
If we introduce taxes, including the assumption that capital gains are taxed, in general, at a
lower rate than dividends, the equilibrium prices should change. Investors will evaluate after-
tax risk and return. This implies that, even with homogeneous expectations about the before-tax
return on a portfolio, the relevant (after-tax) efficient frontier faced by each investor will be
different. However, a general equilibrium relationship should still exist since, in the
aggregate, markets must clear.
( )
dividend yield of market portfolio
dividend yield for stock i
tax factor
i F i M F M F i F
M
i
E R R E R R R R
After the more realistic introduction of personal taxes: although each investor new looks
at a different efficient frontier, markets still clear to (more complex) general
equilibrium
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Describe the impact on the CAPM of the following: Nonmarketable assets
Human capital is an example of a nonmarketable asset
Adding nonmarketable assets leads to a general equilibrium relationship of the same
form as the simple model that excluded nonmarketable assets. However, the market trade-
off between return and risk is different, as is the measure of risk for any asset.
The equilibrium return for an asset can be either higher or lower than it is under the standard form of the CAPM
Describe the impact on the CAPM of the following: Heterogeneous
expectations
CAPM assumes homogenous expectations: all investors have the same inputs (expected
returns, variances [of returns], and covariance/correlation matrix)
If we relax this assumption to allow for the (certainly!) more realistic assumption that investors
have heterogeneous expectations, equilibrium can still be expressed in terms of expected
returns, covariances, and variance, but now these returns, covariances, and variances are
complex weighted averages of the estimates held by different individuals
Describe the impact on the CAPM of the following: Non-price-taking
behavior
Elton: “Lindenberg finds the price affector will hold less of the riskless asset (will be less of a risk
avoider) than [otherwise]. By doing so the price affector increases utility. “
Because the price affector still holds a combination of the riskless asset and the market portfolio,
we still get the simple form of the CAPM, but the market price of risk is lower than it
would be if all investors were price takers.
One of the CAPM assumptions is that markets are perfectly compeitive; that is, the
individuals are price-takers. Relaxing this assumption implies the idea that some investors
(e.g., large mutual or pension funds) believe their trades impact price.
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Describe the following multi-period versions of CAPM: Consumption-
oriented CAPM
A number of authors have taken a different approach to defining equilibrium in the capital
markets. They start with a set of assumptions:
Investors maximize a multiperiod utility function for lifetime consumption;
Investors have homogeneous beliefs concerning return characteristics of assets;
There is an infinitely lived fixed population;
There is a single consumption good; and
There exists a capital market that allows investors to reach a consumption pattern such that they cannot jointly fare better by additional trades.
The authors are able to show, under these assumptions, that return on assets should be
linearly related to the growth rate in aggregate consumption if the parameters of the linear
relationship can be assumed constant over time.
This model is directly analogous to the simple form of the CAPM, except the growth rate of per
capita consumption replaces the rate of return on the market portfolio.
1
1 market price of the consumption beta
expected return on portfolio with
zero consumption beta
i Z i
i
R R
Describe the following multi-period versions of CAPM: CAPM including
[uncertain] inflation
Equilibrium is similar to the simple form of the CAPM, but both the definition of the market price
of risk and the risk on an asset are modified.
As long as the correlation between the rate of return on the market and the rate of inflation is positive, the market price of risk is higher than that depicted in the standard CAPM.
Risk of any asset is not just a function of its covariance with the market; it is also a function of its
covariance with the rate of inflation.
If an asset’s rate of return is positively correlated with the rate of inflation, the standard CAPM formulation overstates the risk of the asset.
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Describe the following multi-period versions of CAPM: Multi-beta CAPM
Multi-beta CAPM says expected return is a function of (related to) several sensitivities
1 1 2 2 ...
i F
iM M F il l F il l F
R R
R R R R R R
For example:
i F
iM M F il ll F
R R
R R R R
Price of inflation risk
Sensitivity to inflation risk
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Elton, Chapter 16:
Arbitrage Pricing Model (APT)
In this chapter…
Describe the APT and the assumptions underlying it. Use the APT to calculate the expected returns on an asset. Explain the relationship between the CAPM and the APT. Describe how the APT can be used in both active and passive portfolio
management.
Describe the APT and the assumptions underlying it.
APT relaxes several of the CAPM assumptions (requirements):
APT requires that the returns on any stock be linearly related to a set of indexes:
1 1 2 2i i i i j iR a b l b l bjl e
APT is multi-index (multi-factor) model that is consistent with CAPM but less restrictive
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Use the APT to calculate the expected returns on an asset.
Elton example (S&P Index)
In the example below, we simply multiply the sensitivity by the price of risk to get the
“contribution to expected excess return” of each factor; e.g., 1.71 * 1.49% = 2.55%. Then we sum
the contributions to produce the total “Expected excess return”; e.g., 1.60% + 2.55% + 3.96% =
8.11%. Please note this is the excess return above the riskfree rate.
APT S&P Index Elton, p 380
Contribution
Price of to expected
Factor Sensitivity Risk excess return
b λ
Inflation (i) (0.37) -4.32% 1.60%
Sales growth (S) 1.71 1.49% 2.55%
Oil prices (O) - 0.00% 0.00%
Market (M) 1.00 3.96% 3.96%
Expected excess return for S&P index 8.11%
Elton example (portfolio)
In the next example, please note the prices of risk are identical as they are common factors. It
is the sensitivities that change!
APT Portfolio Elton, p 381
Same Contribution
Price of to expected
Factor Sensitivity Risk excess return
b λ
Inflation (i) (0.50) -4.32% 2.16%
Sales growth (S) 2.75 1.49% 4.10%
Oil prices (O) - 0.00% 0.00%
Market (M) 1.30 3.96% 5.15%
Expected excess return for S&P index 11.41%
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Explain the relationship between the CAPM and the APT.
The APT solution with multiple factors appropriately priced is fully consistent with the Sharpe–
Lintner–Mossin form of the CAPM.
Elton: “Employing the Roll and Ross procedure and finding that more than one risk factor (price
of risk) is significantly different from zero is not sufficient proof to reject any CAPM. If the
lambdas are not significantly different from [beta*excess market return], the empirical results
could be fully consistent with the Sharpe–Lintner–Mossin form of the CAPM. It is perfectly
possible that more than one index explains the covariance between security returns but that the
CAPM holds.”
Describe how the APT can be used in both active and passive portfolio
management.
Passive:
APT can be used to do a better job of tracking an index or to design a passive portfolio that is
appropriate for a particular client.
Simplest use is to create a portfolio of stocks that closely tracks an index.
Active:
What a multi-index model does that cannot be done with a single-index model is allow the user
to make factor bets.
“If you believe that unexpected inflation will accelerate at a rate above that anticipated by the market, then you may want to place a bet by increasing your exposure (b value) with inflation.”
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Amenc, Chapter 4:
Applying CAPM to Performance
Measurement
In this chapter…
Calculate, compare, and evaluate the Treynor measure, the Sharpe measure, and Jensen's alpha.
Compute and interpret tracking error, the information ratio, and the Sortino ratio.
Calculate, compare, and evaluate the Treynor measure, the Sharpe
measure, and Jensen's alpha.
The Treynor measure: excess return divided by portfolio beta ():
( )P FP
P
E R RT
The Sharpe measure: excess return divided by portfolio volatility (standard deviation):
( )
( )P F
PP
E R RS
R
Jensen’s alpha is the excess return equated to alpha plus expected systematic return:
( ) ( ( ) )P F P P M FE R R E R R
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Summary Comparison
The Treynor measure captures the relationship between the “excess return” on the portfolio
(i.e., the return above the risk-free rate) and the beta of the portfolio. Put another way, it is the
“excess return per unit of beta.”
The Sharpe measure has the same numerator as the Treynor measure, but it uses the total risk
of the portfolio in the denominator.
Jensen’s alpha is the “excess return” on the portfolio; i.e., the return above the risk-free rate. It is
the same as the numerator is the Treynor and Sharpe ratios.
Return Risk
Sharpe Excess Return Volatility (Total)
Treynor Excess Return Beta (Systemic)
Jensen’s Essential similar to Treynor
Information Ratio
Residual Return Tracking Error
Sortino Excess over M.A.R. Downside deviation
Portfolio Performance
The measures can be used to rank portfolios for a given period. A higher measure is better. The
Treynor and Sharpe have the same denominator and are similar, but they use different risk
definitions. Use of the Treynor should be limited to diversified portfolios.
The Treynor measure (ratio) is appropriate for evaluating the performance of a well-diversified portfolio (because it only accounts for systematic risk). Similarly, it is appropriate for evaluating the performance of a portfolio that only constitutes part of the investor’s assets.
The Sharpe ratio is good for portfolios that are not well diversified (because it accounts for total risk). Similarly, it is also suitable for evaluating a portfolio that represents an individual’s total investment.
The Jensen’s alpha can be used to rank portfolios within peer groups.
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Compute and interpret tracking error, the information ratio, and the
Sortino ratio.
Tracking Error (TE)
Tracking error (TE) is the standard deviation of the difference between the portfolio return
and the benchmark return:
( )P BTE R R
Tracking error is used to analyze benchmark funds; i.e., funds that assume a risk profile (and
construction, generally) similar to a particular profile but then deviates from the benchmark in
an attempt to add value. The ideal, of course, is to add value without assuming additional risk.
Information ratio (IR)
The information ratio (IR, aka, the appraisal ratio) is given by:
( ) ( )
( )P B
P B
E R E RIR
R R
The information ratio (IR) is also used to evaluate the manager of a benchmark fund. It helps
to answer the question, “was the manager sufficiently rewarded for the risk incurred by
deviating from the benchmark?”
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Sortino Ratio
The Sortino ratio is given by:
P
2
0
E(R )Sortino ratio =
1 T
PttR MARPt
MAR
R MART
The Sortino ratio has a similar idea to the Sharpe ratio, but the risk-free rate is replaced with
the minimum acceptable return (MAR).
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For example (from learning spreadsheet 1.a.4)
Consider the following assumptions:
Risk free rate = 7.0% Excess Market return = 5.18% Portfolio return = 16% Portfolio volatilty = 20% Portfolio beta (with respect to market) = 1.5 Tracking Error = 3.0% Years of observed performance = 5 years
Market
Riskless rate 4.00%
Exp Market Return 10.00%
Excess Market Return (ERP) 6.00%
Portfolio
Exp Return 14%
Volatility (Std Dev) 20%
Beta 1.5
Tracking Error 3.0%
Years observed (T) 5
PERFORMANCE MEASURES
Treynor 0.067
Sharpe 0.500
Jensen alpha 0.010
Information ratio (IR) 0.333
t statistic 0.745
Treynor = (14% - 4%)/1.5 = 0.67
Sharpe = (14% - 4%)/20% = 0.50
Jensen’s alpha = 14% - 4% - (1.5)*(6.0%) = 1.0%
Information ratio = alpha / tracking error = 1.0% / 3% = 0.333
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CAS,
Overview of
Enterprise Risk
Management
In this chapter…
Describe what is meant by ERM. Identify and describe risks addressed by ERM. Describe the measures, models, and tools typically used within an ERM
framework. Discuss practical considerations related to ERM implementation.
Describe what is meant by ERM.
“ERM is the discipline by which an organization in any industry assesses, controls, exploits,
finances, and monitors risks from all sources for the purpose of increasing the organization’s
short- and long-term value to its stakeholders.”
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Identify and describe risks addressed by ERM.
ERM Framework
Establish Context
Identify Risks
Analyze/Quantify Risks
Integrate Risks
Assess/Prioritize Risks
Treat/Exploit Risks
Monitor & Review
Risk Types
Hazard
Natural disaster
Theft
Liability claims
Financial Operational Strategic
Reputational
Competition
Demographic trends
Technological innovation
Regulatory & Political trends
Not operational risks!
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Hazard Risks include risks from:
Fire and other property damage,
Windstorm and other natural perils,
Theft and other crime, personal injury,
Business interruption,
Disease and disability (including work-related injuries and diseases), and
Liability claims.
Financial Risks include risks from:
price (e.g. asset value, interest rate, foreign exchange, commodity),
Liquidity (e.g. cash flow, call risk, opportunity cost),
Credit (e.g. default, downgrade)
Inflation/purchasing power, and
Hedging/basis risk
Operational Risks include risks from:
Business operations (e.g., human resources, product development, capacity, efficiency, product/service failure, channel management, supply chain management, business cyclicality),
Empowerment (e.g., leadership, change readiness),
Information technology (e.g., relevance, availability), and
Information/business reporting (e.g., budgeting and planning, accounting information, pension fund, investment evaluation, taxation).
Strategic Risks include risks from:
Reputational damage (e.g., trademark/brand erosion, fraud, unfavorable publicity)
Competition,
Customer wants,
Demographic and social/cultural trends,
Technological innovation,
Capital availability, and
Regulatory and political trends.
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Describe the measures … used within an ERM framework.
Solvency-related metrics
Concentrate on the “adverse tail” of the probability distributions
Relevant for determining economic capital (EC) requirements
Performance-related metrics
Concentrate on the mid-region of the probability distribution
Relevant to owners and their proxies
Probability of ruin:
The percentile of the probability distribution corresponding to the point at which capital is
exhausted.
Typically, a minimum acceptable probability of ruin is specified, and economic capital is derived therefrom.
Shortfall risk:
The probability that a random variable falls below some specified threshold level.
Probability of ruin is a special case of shortfall risk in which the threshold level is the point at which capital is exhausted.
Value at risk (VaR):
The maximum loss an organization can suffer, under normal market conditions, over a given
period of time at a given probability level
Technically, the inverse of the shortfall risk concept, in which the shortfall risk is specified, and the threshold level is derived therefrom.
VaR is a common measure of risk in the banking sector, where it is typically calculated daily and used to monitor trading activity.
Expected policyholder deficit (EPD) or Economic cost of ruin (ECOR):
Enhancement to probability of ruin concept (and thus shortfall risk and VaR) where severity of
ruin also reflected.
Technically, the expected value of the shortfall. (In an analogy to bond rating, comparable to considering the recovery in addition to the probability of default.)
For insurance companies, the more common term is EPD, and represents the expected shortage in the funds due to policyholders in the event of liquidation.
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Analytical
Simulation
Statistical Structural
Tail Value at Risk (Tail VaR) or Tail Conditional Expectation (TCE):
an ECOR-like measure in the sense that both the probability and the cost of “tail events” are
considered. It differs from ECOR in that it is the expected value, from first dollar, of all events
beyond the tail threshold event, not just the shortfall amount.
Performance-related metrics:
Variance: average squared difference between a random variable and its mean.
Standard deviation: square root of the variance.
Semi-variance and downside standard deviation: only unfavorable deviations from a specified target level are considered in the calculation.
Below-target-risk (BTR): expected value of unfavorable deviations of a random variable from a specified target level (such as not meeting an earnings target).
Describe the models … typically used within an ERM framework.
“As a practical matter, the choice of modeling approach is typically between statistical analytic
models and structural simulation models”
Continuum of model methods: from ―objective‖ data to experts
Rating agency models
RBC
Some option pricing models
DFA
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Describe the … tools typically used within an ERM framework.
Generic applications
Optimization: formal process by which decisions are made under conditions of uncertainty
Candidate analysis: a restricted form of optimization analysis in which only a finite number of prespecified decision options are considered
Capital management
Capital adequacy: minimum needed to satisfy economic capital (EC) constraint
Capital structure: optimal mix
Capital attribution: assignment by risk (denominator of RAROC)
Capital allocation: actual deployment to business segments
Tools typically used within an ERM framework.
Performance measurement
Investment strategy/asset allocation
Insurance/reinsurance/hedging strategy optimization
Crisis management
Contingency planning
Business expansion/contraction strategy
Distribution channel strategy
Strategic planning
Historical Data Analysis
Empirical distributions
Extreme value theory (EVT)
Regressions
Combination
System dynamics simulation
Fuzzy logic
Expert Input
Preference among bets
Judgments of Relative Likelihood
Influence diagram
Delphi Method
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Discuss practical considerations related to ERM implementation.
Designating an ERM “Champion”
Given implementation challenges, a “champion” is needed to spearhead the ERM effort
Role often fulfilled by Chief Risk Officer (CRO), who typically reports to the CEO or CFO
Organizational structure created for ERM (e.g., the CRO, the CRO’s staff, the Risk Management Committee) must have the authority to be a change agent.
Needs senior sponsorship
Making ERM part of the enterprise culture (“tearing down the silos”)
Under the historical, fragmented approach to risk management, numerous personnel are
involved in various aspects of risk management.
The successful ERM approach coordinates all these different departments, recognizes the need
for education, but allows for individual department initiative, flexibility, and autonomy.
Determining all possible risks of the organization
Many risks face every enterprise. Often the greatest risks are those not contemplated.
Some organizations have used their risk management committees to conduct and participate in
periodic, structured “disaster scenario” brainstorming exercises specifically to contemplate and,
as appropriate, plan for such “unthinkable” events.
Quantifying operational and strategic risks
Operational and strategic risk are hard to parameterize: point estimates of likelihoods
(frequency), consequences (severity), and probability distributions.
Enterprises can start with qualitative analysis of operational and strategic risk to determine
those that are material and to prioritize them. In addition, some advocate the use of causal
models, as opposed to parametric models, to quantify these risks.
Integrating risks (determining dependencies, etc.)
Building structural models in modular form, which allows enhancement in manageable
successive stages over time, is a practical approach some companies have employed.
Overcomes difficulties:
Past causal relationships are often not indicative of future relationships.
There are differences in time frames (short-term, medium-term, long-term) to consider.
Selecting correlation factors becomes cumbersome as the number of risks to review increases.
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Lack of appropriate risk transfer mechanisms
Insurance, reinsurance and capital markets markets are not complete in the sense of being able
to provide all products and services that enterprises may need.
These markets need to continue to evolve over time (such as the development of the alternative
risk market for hazard risks) in order to provide products that will meet the risk transfer needs
of enterprises. Risk transfer mechanisms for operational and strategic risks are even less
mature.
Monitoring the Process
Ideally, ERM is not a one-time “project”, but a discipline that evolves over time as risks and
opportunities within an enterprise change.
The successful ERM process will include regular progress reports and comparisons to previous
risk assessments so changes and refinements can be made as appropriate. Regularly monitoring
results can, and should, be tied to the time scales identified for the risks actively managed.
Start Slowly - Build Upon Successes
Because of the traditional, fragmented approach to risk management described earlier and the
complexity of many businesses, enterprises often find it useful to start their ERM initiative
slowly, tackling smaller projects first, so tangible results can be achieved early.
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Allen, Chapter 4:
Financial Disasters
In this chapter…
Describe the key factors that led to and the lessons learned from the following risk management case studies:
Chase Manhattan and their involvement with Drysdale Securities Kidder Peabody Barings Allied Irish Bank Long Term Capital Management (LTCM) Metallgesellschaft Bankers Trust
Describe the key factors that led to and the lessons learned from the
following risk management case studies:
Chase Manhattan & Drysdale Securities
Kidder Peabody
Barings
Allied Irish Bank
Long Term Capital Management (LTCM)
Metallgesellschaft
Banker’s Trust
Financial Disasters (Case Studies)
Misleading Reporting
• Chase/Drysdale
• Kidder Peabody
• Barings
• Allied Irish Bank
Unexpected market moves
• LTCM
• Metallgesellschaft
Conduct of Customer Business
• Banker’s Trust
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Chase Manhattan & Drysdale Securities
In 1976, Drysdale obtained $300 million in unsecured borrowing
But only had $20 million in capital
Lost money on positions.
Could not repay loans. Drysdale went bankrupt.
Reputational damage to Chase (and stock price impact)
Drysdale: Key Factors
Chase failed to detect the unauthorized positions: Chase did not believe the firm’s capital was a risk.
Inexperienced managers
Did not correctly interpret borrowing agreements that made Chase responsible for payments due.
Drysdale: Lessons Learned
More precise methods required to compute collateral value
Need process control: new products should receive prior approval “risk function”
Kidder Peabody
Between 1992 and 1994, Joseph Jett exploited an accounting-type glitch in order to book about $350 million in false profits (government bonds)
Kidder Peabody: Key Factors
System did not present value (PV) forward transactions: allowed booking of artificial profits
Management did not react to visible suspicions
Kidder Peabody: Lessons Learned
Investigate a stream of large unexpected profits
Periodically review models and systems: do assumptions need to be updated?
Barings
During 1993 to 1995, a junior trader (Leeson) took large speculative positions (Japanese stocks, interest rate futures, options) from the Singapore office
Disguised as safe transactions on behalf of fake customers!
Losses of ~ 1.25 billion forced Barings into bankruptcy
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Market risk
Leeson was short straddles on Nikkei 225. Hoped index would trade in narrow range; planned to pocket premiums. However, after Kobe earthquake (1/1995):
1. Sent index into a tailspin.
2. Earthquake increased volatility (adds value to both calls and puts) which “exploded” the
short put options
Credit risk
Management of counterparty risk & reporting of specific instrument exposures to counterparties would have been an additional signal
Barings: Key Factors
Leeson was allowed to settle his own trades
Management incompetence & poor supervision
Poor reporting
Barings: Lessons Learned
Absolute necessity of an independent trading back office
Separation of trading and settlement functions
Need to make thorough inquiries about unexpected sources of profits and/or cash movements
Allied Irish Bank
John Rusnak, a currency option trader, entered into massive unauthorized trades from 1997 to 2002, producing losses of $691 million.
Was supposed to run small arbitrage
But was disguising large naked positions
Allied Irish Bank: Key Factors
Similar to Leeson (internal deception)
Achieved by inventing imaginary trades
Allied Irish Bank: Lessons Learned
Proprietary trading is a high-risk activity
Risk management architecture is crucial
Relationship between parent and overseas units needs to be clarified
Strong and enforceable back-office controls are essential
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Long Term Capital Management (LTCM)
From 1994 to 1998, renowned quants produced spectacular returns with relative value (“arbitrage”-type) trades
In Summer 1998, series of unexpected and extreme events (e.g., Russian rouble devaluation led to flight to quality)
New York Fed coordinated a private bailout ($3.65 billion equity investment)
LTCM: Key Factors
Failure to supplement VaR with a full set of stress test scenarios
Failure to account for illiquidity of positions during stress
(Leverage too high?)
(Too much faith in models?)
LTCM: Lessons Learned
Stress scenarios including extreme stresses and interaction between market & credit risk
Incorporate liquidity
Initial margin needed if counterparty is trader
Greater counterparty disclosures
Additional LTCM lessons:
Model risk: Risk models that relied on normal distribution and extrapolation of historical
returns—did not handle once-in-a-lifetime event
Funding liquidity risk: When the firm lost nearly half its value in a sudden plunge, the
lack of equity capital created a cash flow crisis
Diversification: Risk models did not handle correlations that spiked during a crisis event
Market risk: Extreme leverage combined with concentrated market risk—LTCM had a
balance sheet leverage of 28-to-1
Transparency and disclosure
Marking to market. ―Conflict between hedging strategies and cash requirements‖
Transaction types: pairs trading, risk arbitrage, and bets on overall market volatility
Liquidity squeeze: Asian crisis → Brazil devalued its currency → Flight to quality →
Spreads increase → Value of LTCM collateral drops → LTCM liquidates to meet margin
calls
Insufficient risk management: ―underestimated the likelihood that liquidity, credit and
volatility spreads would move in a similar fashion simultaneously across markets‖
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Metallgesellschaft
MGRM wrote (sold) long-term forward contracts to sell gas/oil
Hedged with long positions in short-term futures (stack-and-roll hedge)
As spot oil prices dropped, oil futures curve shifted to contango
In 1993, creditors rescued with a $1.9 billion package
Metallgesellschaft: Key Factors
1) First factor was that the market shifted to contango (i.e., the futures price is greater
than the spot price).
Stack-and-roll hedge exposes to basis risk
Shift to contango created losses on roll return
Greatly increased the cost of the stack-and-roll hedge.
Led to cash flow (liquidity) problems
2) Second factor was German accounting methods required Metallgesellschaft to show
futures losses (i.e., from hedge) but could not recognize unrealized gains from the
forward.
Accounting standards required recognition of futures losses but not forward gains!
These reported losses triggered margin calls and a panic, which led to credit rating downgrades.
Metallgesellschaft
Basis risk
Liquidity risk
Operational risk
Shift!
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Metallgesellschaft: Lessons Learned
Short-term hedge against long-term contracts requires liquidity
Uncertainty of roll returns
Liquidity consideration may favor other than minimum variance hedge
Banker’s Trust (BT)
To reducing their funding expenses, Proctor & Gamble (P&G) and Gibson Greetings bought complex derivative products offered by BT
Due to losses (e.g., P&G lost >$100 million in 1994), customers sued BT
Claimed they were exploited because they were not sophisticated enough to understand their risks
Banker’s Trust: Key Factors
Complex derivatives
Evidence of some intent to deceive (Discovery evidence)
Banker’s Trust: Lessons Learned
Better controls for matching complexity of trade with client sophistication
Need to provision price quotes independent of the front office
Implications of internal communications that can later be made public
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René Stulz,‖
Risk Management Failures:
What are They and When Do
They Happen?
In this chapter…
Define the role of risk management and explain why a large financial loss is not necessarily a failure of risk management.
Describe how risk management can fail. Describe how risk can be mismeasured. Explain how a firm can fail to take known and unknown risks into account in
making strategic decisions. Explain the importance of communication in effective risk management. Describe how firms can fail to correctly monitor and manage risk on an ongoing
basis. Explain the role of risk metrics and discuss the shortcomings of existing risk
metrics.
Define the role of risk management and explain why a large financial loss
is not necessarily a failure of risk
To assess risks faced by firm,
Communicate risks to risk-taking decision-makers
Manage and monitor risks to ensure firm only bears risks desired by management and board of directors
But Board and Management decide to take the risks!
The articulation of risk appetite is a Board-level decision.
Why a large loss is not necessarily a failure of risk management
Stulz images a simplified example of an investor in Long-term Capital Management (LTCM)
before its collapse in September 1998. “Suppose that you stood in the shoes of the managers of
LTCM in January 1998 and had the opportunity to invest in trades that, overall, had a 99%
chance of producing a return for the fund before fees of 25% and a 1% chance of making a loss of
70% over the coming year.”
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Hypothetical payoff of strategy with expected return of 24+%
Return
Probability 99% 25%
1% -70%
Expected 24.05%
99 years out of 100 this strategy would have earned 25% before fees and the managers would
be “stars.” The occurrence of the 1% loss does not demonstrate a failure of risk management,
according to Stulz.
―the partners of LTCM knew the risks and the rewards from doing so. In the well-worn
language of financial economics, increasing leverage was a positive NPV decision
when it was made [ex ante], but obviously ex post it was a costly decision as it
meant that when assets fell in value, the fund’s equity fell in value faster than it would
have with less leverage.‖
Risk Managers try to know (and communicate) the distribution of possible outcomes
But they do not decide whether to take the risk
Assuming risk is a strategic decision based on institution’s risk appetite
Defining the risk appetite is a decision for the board and top management.
At the heart of the firm’s strategy – this is how it creates shareholder value
Strategy Shareholder
value creation
Risk appetite
• Risk Management informs
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Describe how risk management can fail
Describe how risk can be mismeasured.
Distribution
Select the wrong distribution
Specify the distribution incorrectly
Dependencies (correlations) may be mis-measured
Using and applying the data
Historical sample does not apply
No sample (“with the subprime crisis, there was no historical data of a downturn in the real estate market during which a large amount of securitized subprime mortgages was outstanding.”)
A vexing problem is the application of historical data:
Historical sample does not apply
No relevant sample may exist (“with the subprime crisis, there was no historical data of a downturn in the real estate market during which a large amount of securitized subprime mortgages was outstanding.”)
As Stulz writes, ―with the subprime crisis, there was no historical data of a downturn in
the real estate market during which a large amount of securitized subprime mortgages
was outstanding. In such a situation, risk measurement cannot be done by simply using
historical data … With such a case, statistical risk measurement reaches its limits and
risk management goes from science to art … [assessments] have a significant element of
subjectivity.
Measure risks • Mismeasurement of known risks.
• Failure to take risks into account.
Communicate • Failure in communicating the risks to top
management
Manage
• Failure in monitoring risks.
• Failure in managing risks.
• Failure to use appropriate risk metrics.
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Explain how a firm can fail to take known and unknown risks into account
in making strategic decisions.
1. A firm may ignore a risk even though that risk is known.
2. Somebody in the firm knows about a risk, but that risk is not captured by the risk
models.
3. Realization of a truly unknown risk
Risk divisions (typology) can contribute to ignored risks:
Credit vs. market vs. operational risk are “partly artificial and partly motivated by regulations.”
Trading books (market to market) versus credit book (accrual).
Explain the importance of communication in effective risk management.
Stulz key point here is that risk managers have a different job than the board and other top
management. Risk managers [risk management] need to provide timely information to the
board and top management, enabling them to make decisions (maximize shareholder value by
assuming risk). Communication, and the lack of effective communication, played a role in the
most recent crisis.
In short, because risk managers and senior management (including the board) have different
roles—risk managers analyze, quantify and assess risk while the Board determines what
orientation to assume vis-à-vis risk (how much? What type?)—the communication between the
two roles must be critical.
Risk management has to provide timely information to the board and top management
Top managers are supposed to maximize shareholder value by assuming risk
Lack of communication has played a role in the most recent crisis
Describe how firms can fail to correctly monitor and manage risk on an
ongoing basis
For a financial firm, risks can change sharply even if the firm does not take new positions.
Complex positions with derivatives
Challenging to capture these changes and adjust
Important for risk manager to identify possible solutions that can be implemented quickly
Contingency hedging plans are critical.
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Explain the role of risk metrics and discuss the shortcomings of existing
risk metrics.
Shortcomings of existing risk metrics include:
May not scale over long time horizons
Historical data samples may not predictive (history may not repeat itself)
Often, they are not designed to capture risks associated with crises (catastrophes)
In the case of Value at Risk (VaR), VaR says nothing about the magnitude of losses in excess of VaR. (“It could be that the exceedances were really small and that there were many large gains as well because volatility increased rapidly. Alternatively, there could have been very large losses and few large gains.”)
In the case of the Summer of 2007 and the abrupt withdrawal of liquidity: the risk model may not handle sudden illiquidity.
Most metrics cannot handle complicated interactions across risks and across institutions: “Statistical risk models typically take returns to be exogenous to the firm and ignore risk concentrations across institutions”