2011 bionic frm

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


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


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


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


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


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


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


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?)


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


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


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)


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


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.


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


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


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.


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.


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)


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


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


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


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


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)


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


Define and discuss the “hedging irrelevance proposition” as it relates to:

Diversifiable risk


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.



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


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.


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!)


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)


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.


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


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


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.


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


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



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


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


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



Variance 0.0100

Asset B



Variance 0.0400

A,B

Correlation (A,B) * 0.30

Covariance (A,B) * 0.00600


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)


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.


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.


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)


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)


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


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):


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


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.


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.


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


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


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%


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.”


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


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.


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?”


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).


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


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.”


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!


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.


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.


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


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


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.


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.


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


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


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


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‖


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!


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


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.”


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


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.


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.


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”

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