1a1 Foundations Slides

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2009 FRM Foundations 1a (1 of 2)

• Hosted by David Harper CFA, FRM, CIPM

• Published April 20, 2008

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Foundations 1.a Agenda (Note order of study guide)

• Jorion Value at Risk (VaR) – Chapter 1

• Amenc Portfolio Theory – Chapter 4

• Grinold Active Portfolio Management – Chapter 7

2

2009 FRM Foundations 1 (Readings 1, 2 & 3)

• 1.a.1. Introduction to VaR

• 1.a.2. CML & SML

• 1.a.3. CAPM

• 1.a.4. RAPMs

• 1.a.5. APT (trivial version, Grinold)

Related Learning Spreadsheets


Bionic Turtle FRM 2009 Legend

3

2009 FRM. Foundations. 1. Jorion Value-at-Risk. Chapter 1

• Purple plus double arrows = not really in the assignment but linked to something else in FRM

• This spreadsheet icon means there is anassociated learning spreadsheet


Chapter 1. The Need for Risk Management

4


Define risk & describe some major sources of risk

Risk is volatility of unexpected outcomes (value of assets, equity or earnings)

Regulatory policy: unintended consequence

Human (Accident)

Terrorism

Wars

Human (Deliberate)

Technological innovation

Earthquake

Hurricane

Natural disaster Economic Growth

5



Differentiate between business and financial risks and give examples of each

Business risks

• Risks the corporation assumes willingly – to create competitive

advantage and

– to add shareholder value

• Includes– Business decisions (e.g.,

investments, products, org structure)

– Business environment (competition and macro economy)

Financial risks

• Losses due to financial market activities

• For example:– Interest rate exposure

– Defaults on financial obligations

– Accounts receivables

6


For a non-financial corporation, likely to

be non-core


Relate significant market events of past several decades to growth of the risk management industry

Fixed exchange rate system broke down1971 1

Oil price shocks High inflation1973 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 bubble1989 5

7



Relate significant market events of past several decades to growth of the risk management industry

Asian turmoil1997 6

Russian default Global crisis (LTCM)Aug 1998 7

Terrorist attack on New York9/11/01 8

Visible subprime crisisAug 2007 9

Fed takeover Fannie Mae & Freddie Mac; Merrill Lynch sold; Lehman bankruptcy; AIG

Sep 2008 10

8



Describe functions & purposes of financial institutions as they relate to financial risk management

• Financial institutions (assigned Jorion):

– Function is to manage financial risks actively

– Purpose is to assume, intermediate, or advise on financial risks

9



Describe functions & purposes of financial institutions as they relate to financial risk management

• Financial institutions Functions (Saunders)

1. Brokers

2. Asset Transformers

• For example, maturity transformation

10



Define what a derivative contract is and how it differs from a security

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 its value from some underlyingasset price, reference rate, or index (e.g., stock, bond, currency, commodity)

• For example: Forward contract on foreign currency is a promise to buy a fixed (notional) amount at a fixed price at a future date

11




Common shares (security)

• Purchase $100 of MSFT

Equity option (derivative)

• Purchase call options on MSFT with face value of $100

• Principal is funded • Notional is referenced

“Notional” implies derivative!

12




• Jorion: “A position in a forward contract is equivalent to the same notional amount invested directly in the spot market, leveraged by cash so that there is zero initial investment”

• This is the essence of financial engineering: funded positions in cash market can be “replicated” with synthetically unfunded positions … for example

13




Investor funds defaultable bond (risky bond with default risk)

14



Define what a derivative contract is and how it differs from a security Investor sells credit default swap (CDS) and deposits cash into default-free money market: synthetically long the bond

15



Describe the dual role leverage plays in derivatives and why it is relevant to a risk manager

Efficient: Low Transaction Costs

But lack of funding makes risk assessment

difficult (What is exposure?)

16



Define financial risk management

• Financial risk management is the design and implementation of procedures for identifying, measuring, and managing financial risks

17



Define value at risk (VaR) and describe how it is used in risk management VaR summarizes the worst loss over a target horizon that will not be exceeded with a given level of confidence (Jorion p 17).

“Under normal market conditions, the most the portfolio can lose over a month is about $3.6 million at the 99% confidence level”

“Under normal market conditions, the most the portfolio can lose over the target horizon is $X or %Y at the selected confidence level.”

A VaR statement requires two user choices:horizon and confidence; e.g., 1 day 99%, 10-day 95%, …

18



0

10

20

30

40

50

60

70

80

90

-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4

Define value at risk (VaR) and describe how it is used in risk management

Nonparametric VaR: a quantile of an empirical distribution

19



0.0%

1.0%

2.0%

3.0%

4.0%

97

.0

97

.3

97

.6

97

.9

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.2

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.5

98

.8

99

.1

99

.4

99

.7

10

0.0

10

0.3

10

0.6

10

0.9

10

1.2

Define value at risk (VaR) and describe how it isused in risk management

Parametric VaR: a quantile of an statistical distribution

-1.645

%

95%VaR ( 1.645)

20



Define value at risk (VaR) and describe how it is used in risk management

• VaR is the worst expected (i.e., with selected confidence) loss over a target horizon

( ) 1P L VaR c

*

*1 ( ) ( )W

c f w dw P w W p

Don’t worry if you don’t follow this

formula yet…more later…

21



Describe advantages & disadvantages of VaR relative to … stop-loss limits, notional limits, and exposure limits

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

22



Compare and contrast valuation and risk management, using VaR as an example

Derivatives Valuation Risk Management

Principle Expected DiscountedValue

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)

23



Define and describe the four major types of financial risks: market, liquidity, credit, and operational

• Default

• Credit deterioration (downgrade)

• M2M loss in value

• Internal processes

• Model risk

• People risk

• Legal risk

• Asset-liquidity(market liquidity)

• Funding-liquidity(cash-flow)

• Absolute vs. Relative

• Directional vs. Non

• Basis

• Volatility Market: Price

Levels

Liquidity: Lack of

time

Credit: Losses due

to credit event

Operational:

Almost all other

24



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

25



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

26



Within market risk: Describe basis risk and its sources

Unanticipated movements in relative prices of assets in a hedged position, such as cash and futures or interest-rate spreads

Time (T)

S0

F0

ST-1

FT-1 F0ST

ST=F

T

All hedges imply (at least some) basis risk: the risk the hedge will not exactly offset

27



Within liquidity risk: Describe and differentiate between asset and 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

28



Within liquidity risk: Describe and differentiate between asset and funding liquidity risk

Funding liquidity risk

– Not enough balance sheet cashto fund ongoing operations (or precipitous drop)

– Concern of corporate Chief Financial Officer (CFO)

Asset (Market) liquidity risk

– Deterioration in asset value:

Cannot liquidate the position, and/or

Cannot sufficiently hedge the position

– Concern of market traders and market participants

29




Within liquidity risk: Describe and differentiatebetween asset and funding liquidity risk

Factors that impact liquidity, according to Culp

Market Microstructure

• Dealership structure

• Temporal aggregation

Time Horizon

Asset Type

Asset Fungibility

Purple + double

arrow icon = not in this

reading, but a useful link

30


Within credit risk: Describe and differentiate between exposure and recovery rate

Exposure

• Amount at risk

Recovery Rate

• Proportion paid back to lender (cents on the dollar)

EAD PD (EDF) LGD EL =

Exposure at DefaultLoss Given Default =

1 – Recovery rate

Probability of Default(Exp Default Freq.) Expected Loss

31



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

• Credit risk should be defined as the potential loss in mark-to-market value incurred owing to a credit event

32



Within credit risk: Describe credit event and how it may relate to market risk

33

In De Servigny, Credit Risk Portfolio Models address three types of credit event


• Default

• Ratings migration (downgrade); e.g., downgrade from Baa3/BBB-to Ba1/BB+

• Change in spread

• CreditRisk+ only models defaults

• CreditMetrics models defaults and migrations

• Portfolio Risk Tracker (PRT) models all three


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

34



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

• Settlement risk is acute for foreign-exchange transactions

35



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

36



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

37



Within operational risk: Describe people risk

• Operational risk > People risk

– People risk includes internal or external fraud

– For example:

• Rogue traders

38



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

39




• Operational risk > Legal risk

– Legal risk controlled through policies developed by the institution’s legal counsel in consult with risk managers and senior management

– Institution should ensure agreements with counterparties are legally enforceable before consummating deals

40




• What is excluded from operational risk? Depends who you ask. Jorion:

– Strategic risk

– Reputational risk (controversial)

– Business risk

41


These exclusions are often tested. Note that operational risk is not really “all other risks.” In other words: Credit + Market + Operational < Total Risk! Also, keep in mind strategic risk is not an operational risk!


Operational risk…

• Operational risk > Best Protection?

– “The best protection against operational risks consists of redundancies of systems, clear separation of responsibilities with strong internal controls, and regular contingency planning”

42



Amenc and Sound. Chapter 4. CAPM

43


Describe how covariance/correlation of returns between securities affects returns distribution of a portfolio

Variance of two-asset portfolio

2 2 2 2 2 2 cov( , )a b a a b b a bw w w w a b

cov( , )cov( , ) a b

a b

a ba b

2 2 2 2 2 2a b a a b b a b a bw w w w

44

2009 FRM. Foundations. 2. Amenc Portfolio Theory. Chapter 4 (CAPM)



Portfolio with two assets (A & B)

Riskless rate 4.00%

Asset AExp Return 14.00%Std Deviation 20.00%Variance 0.0400

Asset BExp Return 4.00%Std Deviation 10.00%Variance 0.0100

45




Imperfect correlation (< 100%) reduces portfolio variance

Riskless rate 4.00%

Asset AExp Return 14.00%Std Deviation 20.00%Variance 0.0400

Asset BExp Return 4.00%Std 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

Re

turn

Standard Deviation

-100%

-50%

0%

50%

100%

46



Describe the capital market line and constructionof the efficient frontier both with & without a risk‐free asset.

CAPITAL MARKET LINE (CML)Riskless rate 4.00%

Asset AExp Return 8.00%Variance 0.0064 Std Deviation 8.00%

Asset BExp Return 14.00%Variance 0.0256 Std Deviation 16.00%

Correlation 30.00%Covariance 0.00384

47



Describe the capital market line and the construction of the efficient frontier both with and without a risk‐free asset.

0%

2%

4%

6%

8%

10%

12%

14%

16%

18%

20%

0.0% 10.0% 20.0% 30.0%

Exp

ect

ed

Ret

urn

Standard Deviation

Portfolio





48




0%

2%

4%

6%

8%

10%

12%

14%

16%

18%

20%

0.0% 10.0% 20.0% 30.0%

Exp

ect

ed

Ret

urn

Standard Deviation

Portfolio





Efficient Frontier

49




Capital market line is the set of optimal portfolios: a linear combination of (i) the market portfolio and (ii) the risk-free asset

CAPITAL MARKET LINE (CML)

Riskless rate 4.00%

Asset A

Exp Return 8.00%

Variance 0.0064

Std Deviation 8.00%

Asset B

Exp Return 14.00%

Variance 0.0256

Std Deviation 16.00%

Correlation 30.00%

Covariance 0.00384

0%2%4%6%8%

10%12%14%16%18%20%

0.0% 10.0% 20.0% 30.0%

Exp

ect

ed

Ret

urn

Standard Deviation

Portfolio

CML

50



0%

5%

10%

15%

20%

0.0% 10.0% 20.0% 30.0%

Exp

ect

ed

Ret

urn

Standard Deviation

Portfolio

CML

Riskfree

rate

Describe the capital market line and the construction of the efficient frontier both with and without a risk‐free asset

Capital market line is the set of optimal portfolios: a linear combination of (i) the market portfolio and (ii) the risk-free asset

51



Describe Capital Asset Pricing Model (CAPM), list underlying assumptions; explain implications, contributions & limitations.

Security Market Line (SML) says the excess expected return on a security is a linear function of beta (the security’s sensitivity to the market risk premium)

0%

5%

10%

15%

20%

25%

0.00 1.00 2.00 3.00

Exp

ect

ed

Ret

urn

Beta

SECURITY MARKET LINE (SML)

Riskless rate 4.00%

Market portfolio, M PercentAsset A 56.82%Asset B 43.18%E[return] of marketportfolio M

10.59%

Std Dev of marketportfolio M

9.34%

Market excess return 6.59%

52




The security market line (SML) …

0%

5%

10%

15%

20%

25%

0.00 1.00 2.00 3.00

Exp

ect

ed

Re

turn

Beta

i F i m FE(R ) R [E(R ) R ]

53




• CAPM

• Beta (i) is asset’s sensitivity to movements in the market:

– Beta is the covariance of the return of the security with the return of the market portfolio divided by () the variance of the return of the market portfolio:

m

m

Be

( ) [ ( ) )]

tacov( , )

var( )

i F M Fi

ii

E R R E R R

R R

R

54




( ) [ ( ) )]i F M FiE R R E R R

0%5%

10%15%20%25%

0.00 1.00 2.00 3.00

Exp

ect

ed

Ret

urn

Beta

55



• Investors are risk averse and seek to maximize the expected (end of the period) utility of their wealth

• Investors only consider the expected return and the variance (the first two moments)

• Investors all consider the same one-period investment

Describe Capital Asset Pricing Model (CAPM), list underlying assumptions; explain implications, contributions & limitations

Investors risk averse & wealth seeking

Only consider return and variance

Same one period for all

( ) [ ( ) )]i F i M FE R R E R R

56



Describe Capital Asset Pricing Model (CAPM), list underlying assumptions; explain implications, contributions & limitations

• Investors can borrow and lend at the risk-free rate without limit (unlimited lending and borrowing)

• Information is simultaneously free to all All investors have the same forecast return, variance and covariance expectations for all assets (perfect, costless information)

• Perfect markets with neither taxes and nor transaction costs. All assets are infinitely divisible

Unlimited lending & borrowing

Perfect, free information

No frictions

( ) [ ( ) )]i F i M FE R R E R R

57



Define and calculate the price of risk andthe quantity of risk (beta).

( ) [ ( ) )]i F i M FE R R E R R

Mprice of risk = E(R )FR

m2

m

cov( , )

var( )i iM

iM

R Rbeta

R

58



Define market efficiency, identify the three forms of market efficiency, & discuss link between efficiency and the CAPM.

Private Information

StrongPublic Information

3

Semi-Strong

Past Prices

2

Weak1

CAPM’s assumption of perfect market requires STRONG EMH

59



Calculate, compare, and evaluate the Treynor measure, the Sharpe measure, and Jensen's alpha.

P FP

P

E(R ) RT

P FP

P

E(R ) RS

(R )

P F P P M FE(R ) R (E(R ) R )

Sharpemeasure

Jensen’smeasure

Excess Return

Excess Return

Excess return > Predicted [CAPM]

Beta

Volatility

Treynormeasure

60



Calculate, compare, and evaluate the Treynor measure, the Sharpe measure, and Jensen's alpha.

P FP

P

E(R ) RT

P FP

P

E(R ) RS

(R )

P F P P M FE(R ) R (E(R ) R )

Sharpemeasure

Jensen’smeasure

Treynormeasure Well diversified ¶ &

Portion of investor’s assets

Not diversified ¶ &Investor’s total portfolio

Rank portfolios61



Compute and interpret tracking error, the information ratio, and the Sortino ratio.

P B

P B

E(R ) E(R )IR

(R R )

Pt

P

T2

Pt

t 0R MAR

E(R ) MARSR

1R MAR

T

Tracking error (TE)

Informationratio (IR)

Sortinoratio

Residual return Residual risk

(denominator is TE)

Min. AcceptedReturn (MAR)

P BTE (R R ) Risk differential btwnportfolio & benchmark

62



Compute and interpret tracking error, the information ratio, and the Sortino ratio.

MarketRiskless rate 4.00%Exp Market Return 10.00%Excess Market Return (ERP)

6.00%

PortfolioExp Return 14%Volatility (Std Dev) 20%Beta 1.5Tracking Error 3.0%Years observed (T) 5

PERFORMANCE MEASURESTreynor 0.067 Sharpe 0.500 Jensen alpha 0.010 Information ratio (IR) 0.333 t statistic 0.745

63



Richard Grinold. Chapter 7

64


Describe limitations and deficiencies in the CAPM.

• The CAPM is based on the notion that the market portfolio is

– Mean/variance-efficient and

– Fully invested: “Even if the CAPM is true in some broader context of a worldwide portfolio, it cannot be valid in the restricted single-market world in which it is ordinarily applied.”

65

2009 FRM. Foundations. 3. Grinold Active Portfolio. Chapter 7 (APT)


Describe limitations and deficiencies in the CAPM.

• All of the other assumptions underlying the CAPM can be challenged “adding additional wounds”– Mean/ variance preferences,

– Identical expectations of mean and variance,

– No taxes or transactions costs,

– No restrictions on stock positions

• The most grievous of these is the requirement that all participants know every stock's expected excess return. – “Thus we would suspect that the CAPM can be only

approximately true. It provides a guideline that should be neither ignored nor taken as gospel.”

66


Dubious


Define and describe the components of the Arbitrage Pricing Theory (APT) model.

• The APT maintains that the expected excess return on any stock is determined by that stock's factor exposures and the factor forecasts associated with those factors.

,

1

K

n n k k n

k

r X b u

,

1

{ }K

n n n k k

k

f E r X m

67



Define and describe the components of the Arbitrage Pricing Theory (APT) model.

• The factor forecast is simply the sum of [exposure k factor k]

• The excess return contains an unexplained specific (idiosyncratic) return

,

1

K

n n k k n

k

r X b u

,

1

{ }K

n n n k k

k

f E r X m

68



Calculate a security’s expected excess returns using the APT model and interpret the results.

Factor Forecasts 2.00% 2.50% -1.50% 0.00% 6.00%

Standardized Exposures (Factor Loadings, Sensitivities or Betas) APT-

Stock Industry Forecast Growth Bond Size ROE Beta APT CAPM CAPMAmex FinServices 6.0% 0.17 -0.05 0.19 -0.28 1.16 5.93% 6.96% -1.03%AT&T Telephones 6.0% -0.16 0.74 1.47 -0.59 0.84 5.33% 5.04% 0.29%Chevron Energy 6.0% -0.53 -0.24 0.83 -0.72 0.7 3.10% 4.20% -1.11%Coca-Cola Food 6.0% -0.02 0.3 1.41 1.48 1.06 4.60% 6.36% -1.77%Disney Entertain 6.0% 0.13 -0.86 0.71 0.42 1.13 3.05% 6.78% -3.74%Dow Chemical 8.0% -0.64 -0.92 0.48 0.22 1.13 3.70% 6.78% -3.08%

69



APT

CAPM

Discuss the relationship between APT and the CAPM.

• CAPM has one factor,K = 1.

• Stock's exposure to that factor is the stock's beta

• The expected return associated with thefactor is expected excess return on the market

( ) [ ( ) )]i F i M FE R R E R R

,

1

{ }K

n n n k k

k

f E r X m

(R ) [ ( ) )]excess i M FE E R R

,1 1{ }n n nf E r X m

{ }n nf E r ERP

70


• APT is generalized CAPM


Describe the properties of a qualified model in the context of the APT.

• An APT factor model is qualified if and only if portfolio Q is diversified with respect to that factor model.

– Among all portfolios with the same factor exposures as portfolio Q, portfolio Q has minimum risk (No other portfolio with the same factor exposures is less risky than portfolio Q)

– Any factor model that is good at explaining risk should be qualified; i.e., not much specific risk

71



Describe the difficulties involved with factor forecasting.

• “Fewer is not better:” ideally, we would like the factor model to contain more (several) factors

– But our ability to forecast additional factors is challenged

– Historical factors returns are not likely to extrapolate “out of sample” into the future

• But a saving grace: we hope/trust that factors may be more stable than that asset returns!

72



Describe some of the methods typically used in factor forecasting.

• Structural factors

– Macroeconomic; e.g., interest rates, inflation, equity risk premium

– Fundamental; e.g., ROE, earnings growth

– Financial or market-related; e.g., exchange rates

• Statistical factors

– Principal component analysis (PCA)

73



Describe some of the methods typically used in factor forecasting.

• Structural [assigned Grinold]

– Model #1: Given exposures, estimate factor returns

– Model #2: Given factor returns, estimate exposures

– Model #3: Combine above

• Statistical [assigned Grinold]

– Model #1: Principal component analysis (PCA)

– Model #2: Maximum likelihood estimation (MLE)

– Model #3:The Dual of Maximum likelihood estimation

74



Describe and compare the attributes of purely statistical and structural APT models.

Structural

• Postulate some relationships between specific variables.

– Macroeconomic (unanticipated inflation, change in interest rates)

– Fundamental (growth in earnings, return on equity, market share), or

– Market-related (beta, industry membership).

• Practitioners tend to prefer: models connect factors with specific variables and link model to investment experience and intuition

Statistical

• Statistical models line up the returns data and turn a crank.

• Academics build APT models to test various hypotheses about market efficiency, the efficacy of the CAPM, etc.

• Academics tend to prefer the purely statistical models, since they can avoid putting their prejudgments into the model in that way.

75



End of 2009 Foundations 1a(1 – 3)

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1a1 Foundations Slides