Intro to Value at Risk (VaR)

Value at Risk (VaR) – Part 1(LOs 7.1 – 8.7, 12.1 – 13.9, 15.1-15.5)

Intro to VaR (Allen Chapter 1)1

VaR Mapping

4 VaR Methods

5

Cash flow at Risk (CFaR)

2 Putting VaR to Work (Allen Chapter 3)

3

Stress Testing6

LearningOutcome

Location inStudy Guide Reading

LO 7.1 to 7.6 II. Market 1.A. Intro to VaR Allen Ch. 1

LO 7.7 to 7.15 II. Market 1.B. Putting VaR to work

Allen Ch. 3

LO 8.1 to 8.7 II. Market 6.A. Firm-wide Approach to Risk

Stulz Ch. 4

LO 9.1 to 9.11 V. Investment 6.A. Portfolio Risk

Jorion Ch. 7

LO 10.1 10.7 I. Quant 3.A. Forecasting Risk and Correlation

Jorion Ch. 9

LO 11.1 to 11.10

I. Quant 1. Quantifying Volatility

Allen Ch. 2

Value at Risk (VaR) in the Readings

We are reviewinghere (Sec II)

Was reviewed in Quant (Sec I)

To be reviewedin Investments (Sec V)

LearningOutcome

Location inStudy Guide Reading

LO 12.1 to 12.6

II. Market 3.A. VaR Methods Jorion Ch. 10

LO 13.1 to 13.9

II. Market 3.B. VaR Mapping Jorion Ch. 11

LO 14.1 to 14.7

I. Quant 3.B. MCS Jorion Ch. 12

LO 15.1 to 15.5

I. Market 3.C Stress Testing Jorion Ch. 14

LO 16.1 to 16.3

I. Quant 4. EVT Kalyvas Ch. 4

LO 17.1 17.13 V. Investment 6.B. Budgeting in I/M

Jorion Ch. 17

We are reviewinghere (Sec II)

Was reviewed in Quant (Sec I)

To be reviewedin Investments (Sec V)

Value at Risk (VaR) in the Readings

Value at Risk (VaR)

Traditional metric was capital asset pricing model (CAPM) where beta is (traditional) risk metric But beta has a “tenuous connection” to actual returns CAPM is a one-factor model: too simplistic

1994, JP Morgan created an “open architecture metric” (i.e., not proprietary) called RiskMetrics TM

In 1998 Bank for International Settlements (BIS) allowed banks to use internal models (e.g., VaR) to calculate their capital requirements

VaR is relatively easy to calculate

LO 7.1 Discuss reasons for the widespread adoption of VaR as a measure of risk

Value at Risk (VaR)

%VaR z

LO 7.2 Define value at risk and calculate VaR for a single asset on both a dollar and percentage basis.

Single-Period VaR (n=1)

Value at Risk (VaR)

%VaR z


95 1 645 %%VaR ( . )


Value at Risk (VaR)

%VaR z


95 1 645 %%VaR ( . )

99 2 33 %%VaR ( . )


Value at Risk (VaR)

$$VaR W z



Value at Risk (VaR)

$$VaR W z


95 1 645 $% $VaR W ( . )


Value at Risk (VaR)

$$VaR W z


95 1 645 $% $VaR W ( . )

99 2 33 $% $VaR W ( . )


Value at Risk (VaR)

0 10001

daily return

$%%daily

W


Value at Risk (VaR) - % Basis

51 1 6451 645

%VaR( %)( . ). %

-1.645

One-Period VaR (n=1) and 95% confidence (5% significance)

Value at Risk (VaR) – Dollar Basis

5100 1 1 6451 64

%VaR V($ )( %)( . )$ .

-1.645

One-Period VaR (n=1) and 95% confidence (5% significance)

10-period VaR @ 5% significance

5 10100 1 1 645 105 20

%VaR($ )( %)( . )$ .

V

-1.64510

10-Period VaR (n=10) and 95% confidence (5% significance)

10-period (10-day) VaR5% significance, annual = +12%

5

10100 1 1 645 105 20

%

VaRVaR ($ )( %)( . )

$ .

V

$100(1+)+0.48

($95.30)

Absolute versus Relative VaR

$100(1+)

$100

$5.20

$4.73

Absolute VaR

$100(1+)

$100 510100 1 1 645 10 12 252

4 73

ABS

ABS, %

VaR ( )

VaR ($ ) ( %)( . ) ( %)

$ .

V t t

VaR Re-cap

0

0

10100 1 1

100

100 0

645 10 12

01 1 6

25

4

2

5 10

VaR ( )

VaR ( ) ( %)( . ) ( %)

$ , =1.645, =1%, t=10 days

( )

Often see this form( ) add

( .

time( )

)( . )

abs

abs

daily

relative

V t t

Given V

Var

V justV

t

t t

V

Value at Risk (VaR)Calculate the VaR for a single asset, in both dollar terms and percentage terms.

0 102

daily return

$ million%%daily

W

What is one-day VaR with 95% confidence, dollars and percentage terms?

Value at Risk (VaR)

95 1 645 2 3 29 %%VaR ( . )( %) . %

Calculate the VaR for a single asset, in both dollar terms and percentage terms.

0 102

daily return

$ million%%daily

W

What is one-day VaR with 95% confidence?

Value at Risk (VaR)

95 1 645 2 1 32 900 $%VaR ( . )( %)($ MM) $ ,

Calculate the VaR for a single asset, in both dollar terms and percentage terms.

0 102

daily return

$ million%%daily

W

What is one-day VaR with 95% confidence?

Value at Risk (VaR)LO 7.3 Convert a daily VaR measure into a weekly, monthly, or annual VaR measure.

daily

daily

daily

Weekly VaR =VaR 5Monthly VaR =VaR 20Annual VaR =VaR 250

Square-root ruleJ-day VaR = 1-day VaR Square Root of Delta Time

Assumes i.i.d.

Independent not (auto/serial) correlated

Identically distr. constant variance (homoskedastic)

Value at Risk (VaR)LO 7.3 Convert a daily VaR measure into a weekly, monthly, or annual VaR measure.

Daily VaR is (-)$10,000. What is 5-day VaR?

Value at Risk (VaR)LO 7.3 Convert a daily VaR measure into a weekly, monthly, or annual VaR measure.10-day VaR is (-)$1 million. What is annual VaR (assume 250 trading days)?

Value at Risk (VaR)

Stationarity: the (shape of the) probability distribution is constant over time

Random walk: tomorrow’s outcome is independent of today’s outcome

Non-negative: requirement: assets cannot have negative value

Time consistent: what is true for a single period is true for multiple periods; e.g., assumptions about a single week can be extended to a year

Normal: expected returns follow a normal distribution

LO 7.4 Discuss assumptions underlying VaR calculations

Value at Risk (VaR)LO 7.5 Explain why it is best to use continuously compounded rates of return when calculating VaR

June 2

June 2

$10

$11

S

S

Value at Risk (VaR)LO 7.5 Explain why it is best to use continuously compounded rates of return when calculating VaR

June 2

June 2

$10

$11

S

S

Simple = 11-10=+1

11Percentage= 10%

1011

Continuously compounded = ln 9.53%10

Value at Risk (VaR)

Absolute change (today’s price – yesterday’s price): violates stationarity requirement.

Simple change ([today price – yesterday’s price] [yesterday’s price]): satisfies stationarity requirement, but violates time consistency requirement.

Continuous compounded return is best because it satisfies the time consistency requirement: the 2-period return = sum of 1-period returns. The sum of two random variables that are jointly distributed is itself (i.e., the sum) normally distributed.

LO 7.5 Explain why it is best to use continuously compounded rates of return when calculating VaR

Except for interest rate variables: absolute

Value at Risk (VaR)LO 7.6 Calculate portfolio VaR and describe the primary factors that affect portfolio risk

2 2 2 2 21 2 1,2(1 ) 2 (1 )p w w w w

2 2 2 21 2

1,2 1 2

% (1 ) %%

2 (1 ) % %P

w VaR w VaRVaR

w w VaR VaR


2 2 2 2 21 2 1,2(1 ) 2 (1 )p w w w w

2 2 2 260% (1%) 40% (2%)

2(60%)(40%)(1%)(2%)(0.4)

1.176%

VaR ( 1.645)(1.176%) 1.94%

p

z


2 2 2 2 21 2 1,2(1 ) 2 (1 )p w w w w

2 2 2 260% (1.645%) 40% (3.29%)VaR

2(60%)(40%)(1.64%)(3.29%)(0.4)

1.94%

p


2 2 2 2

%

$

60% (1%) 40% (2%)

2(60%)(40%)(1%)(2%)(0.4)

1.176%

VaR ( 1.645)(1.176%) 1.94%

VaR W ($1 MM)( 1.645)(1.176%)

$19,351

p

z

z


$ 2 2p 1 2 1 2 1,2

2 2

VaR $VaR $VaR 2$VaR $VaR

($9,869) ($13,159)

(2)($9,869)($13,159)(0.4)

$19,351


Directional Impacts

FactorImpact on

Portfolio volatilityHigher variance Greater asset concentration More equally weighted assets Lower correlation Higher systematic risk Higher idiosyncratic risk Not relevant

Value at Risk (VaR)

Linear derivative. Price of derivative = linear function of underlying asset. For example, a futures contract on S&P 500 index is approximately linear

Non-linear derivative. Price of derivative = non-linear function of underlying asset. For example, a stock option is non-linear

LO 7.7 Differentiate between linear and non-linear derivatives.

All assets are locally linear. For example, an option: the option is convex in the value of the underlying. The delta is the slope of the tangent line. For small changes, the delta is approximately constant.

Value at Risk (VaR)LO 7.8 Describe the calculation of VAR for a linear derivative.

Linear Derivative Underlying Risk FactorVaR VaR

S&P 500 Futures ContractVaR $250 VaRIndex

Value at Risk (VaR)LO 7.9 Explain how the addition of second-order terms through the Taylor approximation improves the estimate of VAR for non-linear derivatives.

Value at Risk (VaR)LO 7.9 Explain how the addition of second-order terms through the Taylor approximation improves the estimate of VAR for non-linear derivatives.

20 0 0 0 0( ) ( ) ( )( ) 12 ( )( )f x f x f x x x f x x x

The Taylor approximation is not helpful where the derivative exhibits extreme non-linearities. Mortgage-backed securities (MBS) Fixed income securities with embedded options

Value at Risk (VaR)

Full Revaluation Every security in the portfolio is re-priced. Full revaluation is accurate but computationally burdensome.

Delta-Normal A linear approximation is created. This linear approximation is an imperfect proxy for the portfolio. This approach is computationally easy but may be less accurate. The delta-normal approach (generally) does not work for portfolios of nonlinear securities.

LO 7.11 Explain the differences between the delta-normal and full-revaluation methods for measuring the risk of non-linear derivatives.

Value at Risk (VaR)LO 7.12 Describe the structured Monte Carlo approach to measuring VAR, and identify the advantages and disadvantages of the SMC approach.

Advantage DisadvantageStructured Monte Carlo

Able to generate correlated scenarios based on a statistical distribution

By design, models multiple risk factors

Generated scenarios may not be relevant going forward

Value at Risk (VaR)

The problem with the SMC approach is that the covariance matrix is meant to be “typical”

But severe stress events wreak havoc on the correlation matrix. That’s correlation breakdown.

Scenarios can attempt to incorporate correlation breakdowns. One approach is to stress test (simulate) the correlation matrix. This is easier said than done; e.g., the variance-covariance matrix needs to be invertible.

LO 7.13 Discuss the implications of correlation breakdown for scenario analysis.

Value at Risk (VaR)LO 7.14 Describe the primary approaches to stress testing and the advantages and disadvantages of each approach.

Advantage Disadvantage

Stress Testing Can illuminate riskiness of portfolio to risk factors

Can specifically focus on the tails (extreme losses)

Complements VaR

May generate unwarranted red flags

Highly subjective (can be hard to imagine catastrophes)

Value at Risk (VaR)

The worst case scenario measure asks, what is the worst loss that can happen over a period of time? Compare this to VAR, which asks, what is the worst expected loss with 95% or 99% confidence? The probability of a “worst loss” is certain (100%); the issue is its location

LO 7.15 Describe the worst case scenario measure as an extension to VAR.

Value at Risk (VaR)

As an extension to VAR, there are three points regarding the WCS:1. The WCS assumes the firm increases its level of

investment when gains are realized; i.e., that the firm is “capital efficient.”

2. The effects of time-varying volatility are ignored3. There is still the extreme tail issue: it is still

possible to underestimate the likelihood of extreme left-tail losses

LO 7.15 Describe the worst case scenario measure as an extension to VAR.

Value at Risk (VaR)

A cash flow at risk (CAR) of $X million at the Y% percent level says: The probability is Y percent that the firm's cash flow will be lower than its expected value by at least $X million.

LO 8.1 Calculate cash flow at risk for a firm with normally distributed cash flows for any period, given the expected return and volatility of firm value, and interpret the CFAR measure.

CFAR Prob[ ( ) ] %EC C CAR p

Value at Risk (VaR)

CFAR at p percent (p%) is the cash flow shortfall (defined as expected cash flow minus [-] realized cash flow) such that there is a probability (%, percent) that the firm will have a larger cash flow shortfall.

If realized, cash flow is C and expected cash flow is E(C), we have:

LO 8.1 Calculate cash flow at risk for a firm with normally distributed cash flows for any period, given the expected return and volatility of firm value, and interpret the CFAR measure.

CFAR Prob[ ( ) ] %EC C CAR p

Value at Risk (VaR)

A firm that depends entirely on cash flows as key to its growth opportunities should measure CFAR. A firm that depends on its market value (or the market value of its assets) should measure VAR.

VAR is associated with the balance sheet: it focuses on market values. CFAR is associated with the income statement or cash flow statement: it focuses on risk to flows.

LO 8.2 Describe the characteristics of firms for which either VAR or CFAR is the more appropriate measure of risk.

Value at Risk (VaR)LO 8.2 Describe the characteristics of firms for which either VAR or CFAR is the more appropriate measure of risk.

VAR CFAR

Balance sheet (asset values)

Statement of cash flows

Banks, financial services firm, funds

Non-financial corporations

External markets for capital

Internal growth provides capital

Value at Risk (VaR)LO 8.3: Given the cost per dollar of VAR and the relevant betas, expected returns, and correlations, calculate the VAR impact and expected net gain of a project/trade that is not large relative to the firm’s portfolio of projects.

Volatility impact = ( )ip jp pw

VAR impact (at 5%)= ( ( ) ( ))

( ) 1.65i j

ip jp p

E R E R w W

w W


12

12

12

Security #1 =10%, =+10%

Security #2 =20%, =+20%

Security #3 =30%, =+30%

0.4

0.2

0.3

Porfolio = $1 MM

Impact of “small” project:Buy #1 and Sell #31% of portfolio


VAR impact (at 5%)= (10% 30%) 1% $1 MM

(0.24 0.985) 1.65 14.8% 1% $1 MM

$181


Volatility impact = (0.24 0.985) 1% $1 MM

=-$7,456


VAR impact (at 5%)= (10% 30%) 1% $1 MM

(0.24 0.985) 1.65 14.8% 1% $1 MM

$181

Return impact = 10% 30% (1%)($1 million)

$2,000


Expected gain of trade net of increase in total cost of VaR

=Expected return impact of trade Portfolio Value

- Marginal cost of VaR per unit VaR impact of trade



VAR impact (at 5%)= ( ( ) ( ))

( ) 1.65i j

ip jp p

E R E R w W

w W

Value at Risk (VaR)LO 8.4: Evaluate the impact of a project that is large relative to the firm’s portfolio of projects on CFAR, and explain how the cost of additional CFAR impacts the capital budgeting decision.

CFAR(at 5%) 1.65 CE

2 2 0.5E N E N

CFAR(at 5%, after new) 1.65

1.65[C +C +(2)COV(C ,C )]CE CN

Value at Risk (VaR)LO 8.5: Explain how to allocate CFAR and VAR to the existing activities of the firm and how to use these allocations to improve the evaluation of the economic profitability of these activities (projects, divisions, and trading).

cash flow

cash flow

Firm,project

Firm =$100, =30

Large project =$50, =10

0.3


cash flow

cash flow

Firm,project

Firm =$100, =30

Large project =$50, =10

0.3

2 2 0.5


1.65[30 10 2(30)(10)(0.3)]

34.4

CE CN



34.4CE CN

Benefit - Cost

cost of CFARNPV of Project - ($34.4)

$ of CFAR


2firm cash flowFirm cash flow variance =

2firm cash flow cov( ,Firm cash flow )iC C

cov( , )( )i

iC C

VARC

Value at Risk (VaR)

Reduce the cost of risk for a given level of VAR or CAR.

reduce risk through project choice Use derivatives

LO 8.6 Discuss how a firm can reduce the cost of VAR/CFAR.

Value at Risk (VaR)

Increase equity capital Invest is less risky projects Use derivativesHowever, at least two factors prevent the

elimination of firm-specific risk: Information asymmetries: outsiders cannot

know the same as insiders. Moral hazard: Corporate managers can take

unobserved actions that adversely affect the value of the contract.

LO 8.7 Explain the limitations on project selection and the use of derivative instruments as ways to decrease VAR/CFAR.

Value at Risk (VaR)

The local-valuation method is the use of partial derivatives as approximations. In local-valuation, value at risk (VaR) is determined by a linear relationship between the asset and the exposure.

Full-valuation is more comprehensive. Full-valuation is used when local-valuation is inadequate. Under full-valuation, the entire portfolio is re-priced at various levels. The advantage is that exposure can be modeled as non-linear.

LO 12.1: Explain the difference between local- and full-valuation methods.

Value at Risk (VaR)

Delta-normal methods rely on first-order partial derivatives. They are locally accurate due to their linearity. Second-order terms improve the accuracy by accounting for curvature. For a bond portfolio, duration is the linear sensitivity of price to yield change. The second-order term, given by C below, accounts for the curvature and is called convexity:

LO 12.2: Describe how the addition of second-order terms improves the accuracy of estimates for nonlinear relationships.

21( * ) ( )

2dV D V dy CV dy …

Value at Risk (VaR)

The best methods depend on speed required and whether the portfolio contains much optionality:

For large portfolios where optionality is not a major factor, the delta normal method is fast and efficient

For fast approximations of option values, mixed methods are efficient; e.g., delta-gamma-Monte Carlo or grid Monte Carlo

For portfolios with substantial optionality or long horizons, full valuation may be necessary

LO 12.3: Compare the delta normal, historical simulation, and Monte Carlo simulation methods, and explain their appropriate uses.

Value at Risk (VaR)LO 12.3: Compare the delta normal, historical simulation, and Monte Carlo simulation methods, and explain their appropriate uses.

Valuation Method

Risk factor Local FullAnalytical Delta-normal Not used

Delta-gamma-

delta

Simulated Delta-gamma-

Monte-Carlo

Monte Carlo

Grid Monte Carlo

Historical

Value at Risk (VaR)LO 12.4: List the advantages and disadvantages of the delta-normal model.

Advantage DisadvantageDelta-normal Easy to implement

Computationally fast

Can be run in real-time

Amenable to analysis (can run marginal and incremental VaR)

Normality assumption violated by fat-tails (compensate by increasing the confidence interval)

Inadequate for nonlinear assets

Value at Risk (VaR)LO 12.5: List the advantages and disadvantages of the historical simulation method.

Advantage DisadvantageHistorical Simulation

Simple to implement

Does not require covariance matrix

Can account for fat-tails

Robust because it does not require distributional assumption (e.g., normal)

Can do full valuation

Allows for horizon choice

Intuitive

Uses only one sample path (if history does not represent future, important tail events not captured)

High sampling variation (data in tail may be small)

Assumes stationary distribution (can be addressed with filtered simulation)

Value at Risk (VaR)LO 12.6: List the advantages and disadvantages of the Monte Carlo simulation method.

Advantage DisadvantageMonte Carlo Most powerful

Handles fat tails

Handles nonlinearities

Incorporates passage of time; e.g., including time decay of options

Computationally intensive (need lots of computer and/or time)

Can be expensive

Model risk

Sampling variation

Value at Risk (VaR)LO 13.1 Describe the two components of the typical VAR model.

Value at Risk (VaR)

First component is the data feed system:Historical data used to compute…Historical volatility and correlations, in turn used to produce…Estimated covariance matrix

LO 13.1 Describe the two components of the typical VAR model.

The second component is the “mapping system” which transforms (or maps) the portfolio positions into weights on each of the securities for which risk is measured.

Value at Risk (VaR)

1. Large underlying cash markets — demonstrates demand for the underlying asset and implies the contract can be fairly priced

2. High volatility of the underlying asset — leads to hedging needs and creates the possibility of profits for speculators

3. Lack of close substitutes—contracts are best when they hedge price risks that cannot be met with existing contracts

LO 13.2 Discuss the three qualities of successful futures contracts and why these are desired in the chosen risk factors.

Value at Risk (VaR)

Specific risk is the risk inherent to (or unique to) the company or security. It is issuer-specific as opposed to market-related. A greater number of general risk factors should create less residual risk

LO 13.3 Discuss how the residual specific risk is related to the number of risk factors.

Value at Risk (VaR)

1. Principal mapping: bond risk is associated with the maturity of the principal payment only

2. Duration mapping: the risk is associated with that of a zero-coupon bond with maturity equal to the bond duration

3. Cash flow mapping: the risk of fixed-income instruments is decomposed into the risk of each of the bond cash flows

LO 13.4 Explain the three approaches for mapping a portfolio onto the risk factors.

Value at Risk (VaR)LO 13.4 Explain the three approaches for mapping a portfolio onto the risk factors.

1. Principal

Bond risk ~with the maturity of the principal payment

Bond risk ~with a zero-coupon bond with maturity equal to the bond duration

Risk decomposed into the risk of each of the bond cash flows

2. Duration 3. Cash Flow

Value at Risk (VaR)LO 13.5 Decompose a fixed-income portfolio into positions in the standard instruments.

Value at Risk (VaR)LO 13.6 Calculate the VAR of a fixed-income portfolio using the delta-normal method, given the expected change in portfolio value and the standard deviation.

( [ ] 1.645 s.d.[ ])VAR E V V

Value at Risk (VaR)

The delta-normal method requires that a portfolio (or instrument) be expressed as a linear combination of risk factors. If a financial instrument can be “reduced” to such a linear expression—as many can—then it can be utilized by the delta-normal method.

For almost the same reason as the previous AIM: because delta-normal method relies on a linear combination of risk factors. Options and derivatives exhibit non-linear relationships.

LO 13.7: Explain why the delta-normal method can provide accurate estimates of VAR for many types of financial instruments.

Value at Risk (VaR)LO 13.8: Discuss why caution must be used in applying delta-normal VAR methods to derivatives and options, and describe when this is appropriate.

Value at Risk (VaR)LO 13.9: Explain what is meant by benchmarking a portfolio, and define tracking error.

0 0Relative VAR = ( ) ( )x x x x

Value at Risk (VaR)

Generically, simple stress testing consists of three steps:1.Create a set of extreme market scenarios (i.e.,

stressed scenarios)—often based on actual past events;

2.For each scenario, determine the price changes to individual instruments in the portfolio; sum the changes in order to determine change in portfolio value

3.Summarize the results: show estimated level of mark-to-market gains/losses for each stressed scenario; show where losses would be concentrated.

LO 15.1 Discuss the role of stress testing as a complement to the VAR measure, and describe the benefits and drawbacks of stress testing.

Value at Risk (VaR)LO 15.1 Discuss the role of stress testing as a complement to the VAR measure, and describe the benefits and drawbacks of stress testing.

VAR vs. Stress TestingVAR Stress TestingNo information on magnitude

of losses in excess of VaR

Captures the “magnitude

effect” of large market moves.

Little/no information on

direction of exposure; e.g., is

exposure due to price

increase or market decline

Simulates changes in market

rates and prices, in both

directions

Says nothing about the risk

due to omitted factors; e.g.,

due to lack of data or to

maintain simplicity

Incorporates multiple factors

and captures the effect of

nonlinear instruments.

Value at Risk (VaR)

Unidimensional scenarios focus “stressing” on key one variable at time; e.g., shift in the yield curve, change in swap spread. Scenarios consist of shocking one variable at a time. The key weakness of a unidimensional analysis is that scenarios cannot, by definition, account for correlations.The multidimensional is more realistic and attempts to “stress” multiple variables and their relationships (correlations). Multidimensional scenario analysis consists of:First, posit a state of the world (high severity event)Then, infer movements in market variables Multidimensional analysis includes:Factor push method: first, shock risk factors individually. Then, evaluate a worst-case scenario.Conditional scenario method: systematic approach

LO 15.2 Compare and contrast the use of unidimensional and multidimensional scenario analysis.

Value at Risk (VaR)

Prospective scenarios try to analyze the implications of hypothetical one-off surprises; e.g., a major bank failure, a geopolitical crisis.

Historical scenarios looks to actual past events to identify scenarios that would fall outside of the VaR window. Events that are often used include:

The one-month period in October 1987 (S&P 500 index fell by > 21%)

Exchange rate crisis (1992) and U.S. dollar interest rates changes (spring of 1994)

The 1995 Mexican crisis The East Asian crisis (summer of 1997) The Russian devaluation of August 1998 and the Brazilian

devaluation of 1999

LO 15.3 Compare and contrast the use of prospective scenarios and historical scenarios in multidimensional scenario analysis, and describe the advantages and disadvantages of each.

Value at Risk (VaR)LO 15.3 Compare and contrast the use of prospective scenarios and historical scenarios in multidimensional scenario analysis, and describe the advantages and disadvantages of each.

Advantage DisadvantageProspective Scenarios in MDA

Relies on input of managers to frame scenario and therefore may be most realistic vis-à-vis actual extreme exposures

May not be well-suited to “large, complex” portfolios

FACTOR PUSH METHOD: ignores correlations

Historical scenarios in MDA

Useful for measuring joint movements in financial variables

Typically, limited number of events to draw upon

Value at Risk (VaR)LO 15.4 Discuss an advantage and disadvantage of using the conditional scenario method as a means to generate a prospective scenario

Advantage Disadvantage

Conditional Scenario Method

More realistically incorporates correlations across variables: allows us to predict certain variables conditional on movements in key variables

Relies on correlations derived from entire sample period. Highly subjective

Value at Risk (VaR)

Although not every scenario requires a response, an institution should address relevant scenarios. The institution can:Set aside economic capital to absorb worst-case lossesPurchase protection or insuranceModify the portfolioRestructure the business or product mix to enhance diversificationDevelop a corrective or contingency plan should a scenario occurPrepare alternative funding sources in anticipation of liquidity crunches

LO 15.5 Discuss possible responses when scenario analysis reveals unacceptably large stress losses.

VaR: Question• A portfolio has an initial value (V0) of $200 and an

annual standard deviation (252 days) of 15%.

• What is the 1-day dollar VaR relative with 95%

confidence?

• What is the 10-day dollar VaR relative with 95%

confidence?

VaR: Answer1

• A portfolio has an initial value (V0) of $200 and an


• What is the 1-day VaR relative with 95% confidence?

5%VaR V t

1($200) (15%)(1.645)252$3.11

VaR: Answer2

• A portfolio has an initial value (V0) of $200 and an


• What is the 10-day VaR relative with 95% confidence?

5%VaR V t

10($200) (15%)(1.645)252$9.83

VaR: Question• A portfolio has an initial value (V0) of $100 and an

annual standard deviation (252 days) of 15%. Now

assume the annual expected return (252) of the

portfolio is +8%.

• What is the 10-day “absolute VaR” with 95%

confidence?

VaR: Answer• A portfolio has an initial value (V0) of $100 and an

annual standard deviation (252 days) of 15%. Now

assume the annual expected return (252) of the

portfolio is +8%.

• What is the 10-day “absolute VaR” with 95%

confidence?

abs

abs

VaR V( t t)

10 10VaR (200) (15%) (1.645) (8%)252 252

$9.80

VaR of nonlinear derivative

• In the case of a linear derivative, the relationship

between the derivative and the underlying asset is

linear (delta, the “transmission parameter,” is

constant)

- Example: Futures contract on the S&P 500

• Given that a futures contract is [$250 x Index], the

VaR of the futures contract is:

Ft=250 VaR(SS&P 500 Index)

European call option priceDelta

$-

$1

$2

$3

$4

$5

$6

$- $5 $10 $15 $20Stock Price

Op

tio

n P

rice


option fVaR Delta VaR

European call option priceDelta

$-

$1

$2

$3

$4

$5

$6

$- $5 $10 $15 $20Stock Price

Op

tio

n P

rice


option 0VaR ( )(S t)

VaR of nonlinear derivativePrice of Plain-vanilla Bond

Duration

$0

$50,000

$100,000

$150,000

$200,000

$250,000

0% 5% 10% 15%

Yield

Pri

ce

Bond 0 yieldVaR DV

Taylor approximationAIM: Explain how the addition of second-order terms through the Taylor approximation improves the estimate of VAR for non-linear derivatives

20 0 0 0 0f(x) f(x ) f (x )(x x ) 1 2f (x )(x x )

0 0f (x )(x x )

20 01 2f (x )(x x )

1. Constant approximation0f(x )

2. First-order (linear) approximation

3. Second-order (quadratic) approximation

Taylor approximationAIM: Explain how the addition of second-order terms through the Taylor approximation improves the estimate of VAR for non-linear derivatives

20 0 0 0 0f(x) f(x ) f (x )(x x ) 1 2f (x )(x x )

2B 1D y C( y)

B 2

Taylor approximationAIM: Discuss why the Taylor approximation is ineffective for certain types of securities

Does not perform well when the derivative shows

extreme nonlinearities. For example:

• Mortgage-backed securities (MBS)

• Fixed income securities with embedded options

When beta/duration can change rapidly, Taylor

approximation (delta-gamma approximation) is

ineffective – need more complex models.

Versus Full Re-value

0 SVaR VaR

AIM: Explain the differences between the delta-normal and full-revaluation methods for measuring the risk of non-linear derivatives

Delta-normal: linear

approximation that

assumes normality

1 0dV V(S ) V(S )

• Option = F [Delta]

• Bond = F [Duration]

Full-revaluation: linear

approximation that

assumes normality

Computationally fast but…

approximate

Accurate but…

computationally burdensome

• Re-price (simulate) the

portfolio at several price

levels

Structured Monte Carlo• Make an assumption about the behavior of the

underlying (i.e., “create artificial variables with

properties similar to portfolio risk factors”)

• Conduct several random trials (1000s, millions)

• Calculate VaR based on the worst %-ile outcome—

just like identifying the worst 1% or 5% in a historical

simulation

Simulate with one variable (e.g., GBM) or several (Cholesky)

Structured Monte CarloMonte Carlo Sims

geometric Brownian motion (10 day simulation)

$7

$8

$9

$10

$11

$12

$13

1 2 3 4 5 6 7 8 9 10

St t

S

Advantages

• Can generate

correlated scenarios

based on statistical

distribution

• Flexibility makes it

“most powerful

approach to VaR”

(Jorian)

Structured Monte Carlo

Disadvantages

• Computational time

requirements

• Simulations are not

necessarily predictive

(same problem as

extrapolation)

EvaluateCorrelation MatrixUnder Scenarios

ERMCrisis(92)

MexicanCrisis(94)

CrashOf

1987

GulfWar(90)Asian

Crisis(’97/8)

Scenario Analysis

Scenario AnalysisAIM: Discuss the implications of correlation breakdown for scenario analysis

Severe stress events wreak havoc on the covariance

matrix

21 12 13 1 12 13

221 2 13 21 2 13

231 32 3 31 32 3

var iance cov cov

cov var iance cov

cov cov var iance

Scenario AnalysisAIM: Describe the primary approaches to stress testing and the advantages and disadvantages of each approach

Provide two independent sections of the risk report:

(i) VAR-based risk report, and

(ii) Stress testing-based risk report. Either:

(a) Plugs-in historical events, or

(b) Analyzes predetermined scenarios

Historical events

+ Can inform on

portfolio weaknesses

- But could miss

weaknesses unique to

the portfolio

Scenario Analysis

Stress Scenarios

+Gives exposure to

standard risk factors

- But may generate

unwarranted red flags

- May not perform well

in regard to asset-

class-specific risk

AIM: Describe the primary approaches to stress testing and the advantages and disadvantages of each approach

Summary• VaR is the worst expected loss, with specified confidence of

a given interval

• Taylor (Series) approximation corrects for the curvature

(nonlinearity) but can only help so much – not for extreme

non-linearities

• Monte Carlo simulation – given a model for underlying

(stock) behavior, conducts many random trials

• Scenario analysis may improve by evaluating the portfolio

against “actual” historical events

Relative 0VaR V t

Probability

Random Variables

+ + +

Short-term Asset Returns

Probability distributions are models of random behavior

+ +

+

- - - -

- - -

? ? ?

Random Variables

Random variable (or stochastic variable): function that defines a point in the sample space of outcomes.

LO 1.1 Define random variable, outcome, an event, mutually exclusive events, and exhaustive events.

Random Variables


Outcome: the result of a single trial.


Random Variables


Outcome: the result of a single trial. Event: the result that reflects none, one, or

more outcomes in the sample space. Events can be simple or compound.


Random Variables


Outcome: the result of a single trial. Event: the result that reflects none, one, or more

outcomes in the sample space. Events can be simple or compound.

Mutually exclusive events: cannot simultaneously occur. Probability of (A and B) = 0. Intersection is null set.


Random Variables


Outcome: the result of a single trial. Event: the result that reflects none, one, or more

outcomes in the sample space. Events can be simple or compound.

Mutually exclusive events: cannot simultaneously occur. Probability of (A and B) = 0. Intersection is null set.

Exhaustive events: all outcomes described


Random VariablesRolling a “total of seven” (craps) with two dice is one event consisting of six outcomes

One Event:Roll a seven

One Event:Roll a seven

Six outcomesSix outcomes

ProbabilityLO 1.2 Discuss the two defining properties of probability.

x

1st condition: 0 ( ) 12nd condition: ( ) 1

f xf x

Unconditional

ConditionalLO 1.3 Compare and contrast unconditional and conditional probabilities.

# of outcomes in event A( )

Total # of outcomesP A

Conditional

ConditionalLO 1.3 Compare and contrast unconditional and conditional probabilities.

P A BP B A P A P B A P A B

P A( )

( | ) ( ) ( | ) ( )( )

# of outcomes in event A( )

Total # of outcomesP A

Joint probabilityLO 1.4 Define joint probability and interpret the joint probability of any number of independent events.

S=

$10

S=

$15

S=$2

0

Total

T=$1

5

0 2 2 4

T=$2

0

3 4 3 10

T=$3

0

3 6 3 12

Total 6 12 8 26

Joint probabilityLO 1.4 Define joint probability and interpret the joint probability of any number of independent events.

S=

$10

S=

$15

S=$2

0

Total

T=$1

5

0 2 2 4

T=$2

0

3 4 3 10

T=$3

0

3 6 3 12

Total 6 12 8 26

P S TP S T

P T( $20, $20) 3

( $20 $20)( $20) 10

TheoremsLO 1.5 Apply the three theorems on expectations for two independent variables.

E cX cE XIf c is any constant, then ( ) ( )

If X, Y are independent ( ) ( ) ( )E X Y E X E Y

If X, Y are independent ( ) ( ) ( )E XY E X EY

TheoremsLO 1.5 Apply the three theorems on expectations for two independent variables.

E cX cE XIf c is any constant, then ( ) ( )

If X, Y are independent ( ) ( ) ( )E X Y E X E Y

If X, Y are independent ( ) ( ) ( )E XY E X EY

Independent = not correlated

TheoremsLO 1.6 Apply the four theorems on variance for two independent variables.

2 2 2Variance( ) [( ) ] ( ) [ ( )]X E X E X E X 2 2 2 2Variance( ) Variance( ) cX XcX c X c

2Quantity ( ) is a minimum when ( )E X a a u E X 2 2 2If X, Y are independent X Y X Y 2 2 2If X, Y are independent X Y X Y


2 2 2Variance( ) [( ) ] ( ) [ ( )]X E X E X E X 2 2Variance( ) Variance( ) cX XcX c X c 2Quantity ( ) is a minimum when ( )E X a a u E X

2 2 2If X, Y are independent X Y X Y 2 2 2If X, Y are independent X Y X Y

What is the variance of a single six-sided die?


E X2 2 2 2 2 2 21 1 1 1 1 1 91[ ] (1) (2 ) (3 ) (4 ) (5 ) (6 )

6 6 6 6 6 6 6

2 2 291Variance( ) ( ) [ ( )] (3.5) 2.92

6X E X E X

Covariance & correlationLO 1.7 Define and calculate covariance and correlation.

XY X YX Y E X Ycov( , ) [( )( )]


XY X YX Y E X Ycov( , ) [( )( )]

X Y

3 5

2 4

4 6


XY X YX Y E X Ycov( , ) [( )( )]

X Y (X-X avg)(Y-Y

avg)

3 5 0.0

2 4 1.0

4 6 1.0

Avg = 3 Avg = 5

Avg = .67

Covariance & correlation

XY X YX Y E X Ycov( , ) [( )( )]

X Y (X-X avg)(Y-Y

avg)

3 5 0.0

2 4 1.0

4 6 1.0

Avg = 3 Avg = 5 Avg = .67s.d. =

SQRT(.67)s.d. =

SQRT(.67)Correl. = 1.0


XY

X Y

X YX Y

cov( , )StandardDev( ) StandardDev( )

X Y XY

Covariance & correlationLO 1.8 Use Bayes’ formula to determine the probability of causes for a given event.

k kk n

j jj

P A P B AP A B

P A P B A1

( ) ( | )( | )

( ) ( | )

Bayes’ formulaLO 1.8 Use Bayes’ formula to determine the probability of causes for a given event.

Bayes’ Formula

( | ) ( )( | )

( )P Info Event P Event

P Event InfoP Info

Bayes’ Formula

( | ) ( )( | )

( )P Info Event P Event

P Event InfoP Info

(90%)(60%)( | )

( )

(90%)(60%)87.1%

(90%)(60%) (20%)(40%)

P Dinner HappyP Happy

Permutations & combinationsLO 1.9 Determine the number of possible permutations of n objects taken r at a time and the number of possible combinations of n objects taken r at a time.


Given a set of seven letters: {a, b, c, d, e, f, g}

How many permutations of three letters?How many combinations of three letters?


n rn

P n n n n rn r

!( 1)( 2) ( 1)

( )!

…

n rn r

n PC

r n r r!

!( )! !

P

C 7 37 3

7!210 6 35

3!(7 3)! 3!

DistributionsLO 2.1 Distinguish between discrete random variables and continuous random variables and contrast their probability distributions

LO 2.2 Discuss a probability function, a probability density function, and a cumulative distribution function

Discrete VariablesDiscrete Random Variable

(Normal Distribution)

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

0 10 20 30 40 50 60 70 80 90 100

Continuous VariablesContinuous Random Variable

(Normal Distribution)

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

0 10 20 30 40 50 60 70 80 90 100

PDFProbability density function

f(x) = P(X ≈ 39)

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

0 10 20 30 40 50 60 70 80 90 100

Cumulative DistributionCumulative Distribution Function

F(x) = P(X ≤ 39)

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

0 10 20 30 40 50 60 70 80 90 100

Cumulative DistributionGreen is Cumulative

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

25 30 35 40 45 50 55 60 65 70 75

Normal (mean = 50, s.d = 10)

Normal (mean = 50, s.d = 10)

P (X <= 50) = 50%

P(X=50) ~ 4%

Comparison

Probability Density Function (pdf)

CumulativeDistribution

Discrete variable

Continuous variable

0%

1%

1%

2%

2%

3%

3%

4%

4%

5%

5%

25 30 35 40 45 50 55 60 65 70 75

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

25 30 35 40 45 50 55 60 65 70 75

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

25 30 35 40 45 50 55 60 65 70 75

0%

1%

1%

2%

2%

3%

3%

4%

4%

5%

5%

25 30 35 40 45 50 55 60 65 70 75

Distributions

Discrete uniform distribution Outcomes can be counted All equally likely Example: six-sided die

LO 2.3 Describe a discrete uniform random variable and a binomial random variable

Distributions

Discrete uniform distribution Outcomes can be counted All equally likely Example: six-sided dieBinomial Two outcomes (success/fail, true/false, heads/tails) Binomial experiment: binomial variables, fixed

trials, independent and identical outcomes (i.i.d.)

LO 2.3 Describe a discrete uniform random variable and a binomial random variable

Uniform Distributiona = 15, b =30

0.00.10.20.30.40.50.60.70.80.91.0

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Probability Density Function

Cumulative Distribution

Uniform Distribution

(22 18)P(18 X 22) 0.267

(30 15)

a = 15, b =30

0.00.10.20.30.40.50.60.70.80.91.0

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

Probability Density Function

Cumulative Distribution

DistributionsLO 2.3 Describe the continuous uniform distribution

f x P a x b a x bb a

1( ) ( ) for

a bE x( )

2

b ax

2( )Variance( )

12

Probability Distributions

Three Different Distributions

0%

1%

2%

3%

4%

5%

6%

0 15 30 45 60 75 90 105

120

135

150

165

180

195

NormalBinomialPoisson

Binomial

0%

5%

10%

15%

20%

25%

30%

0 1 2 3 4 5 6 7 8 9 10Number of Successes

p=0.5

Binomial

0%

5%

10%

15%

20%

25%

30%


p=0.5

x n x

x n xn x

n!P(X x) p q

x!(n x)!

C p q

Binomial

0%

5%

10%

15%

20%

25%

30%


p=0.5

4 10 4

4 10 410 4

10!P(X 4) (.5) (.5)

4!(10 4)!

C (0.5) (0.5) 0.205

Binomial – con’t

0%

5%

10%

15%

20%

25%

30%

0 1 2 3 4 5 6 7 8 9 10

Number of Successes

p=0.5

p=0.6

p=0.7

Normal distributionLO 2.5 Identify the key properties of the normal distribution

xf x e2 2( ) 21

( )2

Normal Distribution

Normal Distributionmean = 100

standard deviation = 20

0%

1%

2%

3%

4%

5%

6%

0 14 28 42 56 70 84 98 112

126

140

154

168

182

196

Also Normal

Normal Distributionboth means = 100

standard deviations = 10, 20

0%

1%

2%

3%

4%

5%

6%

0 14 28 42 56 70 84 98 112

126

140

154

168

182

196

A Big Problem with Normal

Normal Distribution

0%

1%

2%

3%

4%

5%

6%

0 15 30 45 60 75 90 105

120

135

150

165

180

195

Normal

Fat or Heavy Tails

Normal

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

6.0%

7.0%

8.0%

9.0%

10.0%

-25 -20 -15 -10 -5 0 5 10 15 20 25

1.642.33

0, 5

Normal

0.0%

1.0%

2.0%

3.0%

4.0%

5.0%

6.0%

7.0%

8.0%

9.0%

10.0%

-25 -20 -15 -10 -5 0 5 10 15 20 25

0.00135

P(X 15)

15 0 153

5 5z 3

P(X 15) 0.135%

Normal distributionLO 2.5 Identify the key properties of the normal distribution

% of all

(two-tailed)

% “to the

left” (one-

tailed) Critical values

Interval –math

(two-tailed) VaR

~ 68% ~ 34% 1

~ 90% ~ 5.0 % 1.645

(~1.65)

~ 95% ~ 2.5% 1.96

~ 98% ~ 1.0 % 2.327

(~2.33)

~ 99% ~ 0.5% 2.58

u u ˆ1.65 ˆ ˆ1.65

u ˆ1.96

u ˆ2.33 ˆ ˆ2.33

u ˆ2.58

Normal distribution

Only two parameters required: mean () and variance/standard deviation ()

Symmetrical: coefficient of skewness = 0 No excess kurtosis (“normal not fat” tails):

coefficient of kurtosis = 3

LO 2.5 Identify the key properties of the normal distribution

For Parametric VARThe normal distributionAdvantage: Large data set not required Only need two parameters, mean and

volatilityDisadvantage:Does not reflect actual distributions.

Actual returns exhibit tend to exhibit “fat tails.”

PoissonPoisson Distribution

Errors at 3%, Number of Trials at 100 (p=0.03, n=100)

0%

5%

10%

15%

20%

25%

0 1 2 3 4 5 6 7 8 9 10

Number of Errors

xe

P(X x)x!

Poisson: Question

We observe that a key operational process produces, on average, fifteen (15) errors every twenty-four hours

What is probability that exactly three (3) errors will be produced during the next eight-hour work shift?

Poisson: Answer

xe

P(X x)x!

3 555 e 125

P(X 3) e 0.143! 6

• The model is Poisson (5) and we are solving for the P(X=3):

Distributions - Poisson

Poisson distribution Discrete Occurrences over time Errors per 1,000 transaction Calls per hour Breakdowns per week

Lambda () is the mean and the variance

LO 2.7 Calculate the expected value and variance of the Poisson distribution

ComparedLO 2.8 Compare and contrast the binomial, normal, and Poisson distributions

2 2 2

Normal Binomial PoissonMean

Variance

Standard Dev. = =

np

npq

npq

In Poisson, the expected value (the mean) = variance

Variance is standard deviation2

LognormalLO 2.9 Contrast the lognormal and normal distributions

-

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0 5 10 15 20 25

Lognormal

-

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0 5 10 15 20 25

-

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0 0 0 1 10 100

Transform x-axisTo logarithmic scale

Student’s t

-

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5

Normal

Student's t

Summary Random variables Discrete or Continuous We can ask a PDF question: P(X=)

or a cumulative distribution function question P(X<=)

Distributions are attempts to model random events. For example: Normal: daily stock price change Binomial: success/failure, heads/tails, 0/1 Poisson: errors/time, exceptions/time

Key problem with normal: likely to underestimate extreme losses (“left-tail events”)

Sampling

The population: entire group, often unknowable. Denoted by a capital “N.”

The sample: subset of the population. Denoted with small “n”

A parameter is a quantity in the f(x) distribution that helps describe the distribution For example, mean () and the standard deviation

()

LO 3.1: Define a population, a parameter, and a sample.

We take a sample (from the population) in order to draw an inference about the population.

FrequenciesLO 3.2: Construct a frequency distribution, calculate relative frequencies from a frequency distribution, andillustrate the use of a histogram and a frequency polygon to the present data.

1 Year Return on NASDAQ 100 (QQQ)Monte Carlo Simulation - 100 Trials

Class FrequencyInterval (a.k.a., bin) of OccurrenceLess than -30% 0-30 to -25% 2-24.99 to -20% 3-19.99 to -15% 3-14.99 to -10% 9-9.99 to -5% 15-4.99 to 0% 190 to 4.99% 155 to 9.99% 1110 to 14.99% 915 to 19.99% 720 to 24.99% 325 to 29.99% 230 to 34.99% 135 to 39.99% 1Greater than 40% 0

1 Year Return on NASDAQ 100 (QQQ)Monte Carlo Simulation - 100 Trials

Class RelativeInterval (a.k.a., bin) FrequencyLess than -30% 0%-30 to -25% 2%-24.99 to -20% 3%-19.99 to -15% 3%-14.99 to -10% 9%-9.99 to -5% 15%-4.99 to 0% 19%0 to 4.99% 15%5 to 9.99% 11%10 to 14.99% 9%15 to 19.99% 7%20 to 24.99% 3%25 to 29.99% 2%30 to 34.99% 1%35 to 39.99% 1%Greater than 40% 0%

FrequenciesLO 3.2: Construct a frequency distribution, calculate relative frequencies from a frequency distribution, andillustrate the use of a histogram and a frequency polygon to the present data.

Histogramand Polygon Graph

0

2

4

6

8

10

12

14

16

18

20

< -3

0%-3

0 to

-25%

-24.

99 to

-20%

-19.

99 to

-15%

-14.

99 to

-10%

-9.9

9 to

-5%

-4.9

9 to

0%

0 to

4.9

9%5

to 9

.99%

10 to

14.

99%

15 to

19.

99%

20 to

24.

99%

25 to

29.

99%

30 to

34.

99%

35 to

39.

99%

> 40

%

Sampling

Population mean(arithmetic)

Sample mean(arithmetic)

LO 3.3: Calculate and interpret the following measures: population mean, sample mean, arithmetic mean, geometric mean, mode, and median.

1

N

iiX

N

1

N

iiX

Xn

Sampling

Geometric mean

Mode: most frequent Median = 50th percentile (%ile) Odd n: (n+1)/2 Even n: mean of n/2 and (n+2)/2

LO 3.3: Calculate and interpret the following measures: population mean, sample mean, arithmetic mean, geometric mean, mode, and median.

1 2 3n

nG X X X X…

Geo & Arithmetic mean

2003 5.0%2004 8.0%2005 (3.0%

)2006 9.0%

Geo.Arith.

Geo & Arithmetic mean

2003 5.0% 1.052004 8.0% 1.082005 (3.0%

)0.97

2006 9.0% 1.091.19

9Geo. 4.641

%Arith. 4.75%

SamplingLO 3.4: Discuss the properties of the sampling distribution of means, proportions, and differences and sums.

XE X( )

The sampling distribution is the probability of the sample statistic

Variance of sampling distribution of means:Infinite population or with replacement

SamplingLO 3.4: Discuss properties of the sampling distribution of means, proportions, and differences and sums.

XE Xn

22 2[( ) ]

Variance of sampling distribution of means:Finite population (size N) and without replacement

Variance of sampling distribution of means:Infinite population or with replacement

SamplingLO 3.4: Discuss properties of the sampling distribution of means, proportions, and differences and sums.

XE Xn

22 2[( ) ]

X

N nn N

22

1


Standardized Variable:“Asymptotically normal” even when population is not

normally distributed !!

XZ

n


Standardized Variable:“Asymptotically normal” even when population is not

normally distributed !!

XZ

n

Central limit theorem:Random variables are not normally distributed,

But as sample size increases →Average (and summation) tend toward normal

Sampling distribution of proportionsWhere p = probability of success


ppq p pn n

(1 )

Sampling distribution of differencesTwo populations, Two samples, difference of the

means


S S S S1 2 1 2

X X X X n n

2 22 2 1 2

1 2 1 21 2

Sampling distribution of differencesTwo populations, Two samples, sum of the means


S S S S

S S S S

1 2 1 2

2 21 2 1 2

Population Variance

VarianceLO 3.5: Calculate a sample variance, population variance, and standard deviation.

22 2

2 1 1 1( )

N N N

i i ii i i

X X X

N N N

Sample Variance


2

2 2 1

2 2 1 1( )

ˆ1 1

n

iN Ni

i ii i

XX X

nsn n

Sample Standard Deviation


2

2 2 1

1 1( )

ˆ1 1

n

iN Ni

i ii i

XX X

nsn n

Chebyshev’sLO 3.6: Determine the percentage of a distribution that lies a stated number of deviations from the mean using Chebyshev’s inequality.

P X kk2

1( )

What is the probability that random variable X (with finite mean and variance)

will differ by more than three (3) standard deviations from its mean?

Chebyshev’sLO 3.6: Determine the percentage of a distribution that lies a stated number of deviations from the mean using Chebyshev’s inequality.

P X kk2

1( )

If k = 3, then P() = 1/(32) = 1/9 = 0.1111

Skewness & KurtosisLO 3.7: Describe and interpret measures of skewness and kurtosis.

E X 3 3

3 3 3

[( ) ]Skewness =

4 4

4 4 4

[( ) ]Kurtosis =

E X

Intro to Value at Risk (VaR)

Economy & Finance

Transcript of Intro to Value at Risk (VaR)