Lecture 6

Value at Risk Models: applications, limits &

expected shortfall

Giampaolo Gabbi

Financial Investments and Risk Management

MSc in Finance

2017 - 2018

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Agenda

Main applications of VaR models

Problems of VaR models

Size of losses

Subadditivity

An alternative risk measure: expected shortfall or conditional value at risk (CVaR)

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VaR models applications

1 Creating a common risk language

2 Estimating risk-adjusted performance (RAP)

3 Setting risk limits

4 Risk-Adjusted Pricing

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VaR models applicationsA common language

VaR

Maximum potential loss ... 1. ... with a certain confidence level2. ... within a certain time horizon

Common definition of risk

This allows to:

• aggregate different types of risks;

• compare the amount of risk taken by different BUs;

• get a measure of global risk for the whole bank.

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Common language

150.4%25,033,22

74,69%25,036,733,2000.100

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BTPVaR

Price 100 EUR/USD Spot 1

Notional Value EUR 100,000 Notional USD 100,000

Maturity 10 years Market Value EUR 5.407

Coupon 6% Maturity 1 year

Modified Duration 7,36 Strike 1

Modified Convexity 69,74 Implied Volatility 10%

Yield to Maturity 6% Delta 0,5

Treasury Bond USD Call Option

Risk Profile of 2 alternative positions

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Risk-adjusted Performance (RAP)

• Risk adjusted profitability ex-ante or ex-post

VaR

LPERAROC anteex

)&(

VaR

LPRAROC postex

&

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VaR models applicationsRisk-adjusted Return on Capital (RAROC)

Bond Portfolio Equity Portfolio

Market Value € 25 mln € 25 mln

Monthly Profit € 150000 € 250000

Profitability 0.6% 1%

Volatility (IR/Stok Mkt) 0.5% 5,00%

Sensitivity (MD/Beta) 5 1

VaR 25*5*0.5%*2 = 1.25 m € 25*1*5%*2 = € 2.5 m

RAROC 150,000/1.25m = 12% 250,000/2.5 m = 10%

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VaR models applicationsRisk-control (limits)

Example of risk limits

Treasury Bonds 320.000

Treasury Bills 140.000

FRA 240.000

IRS 300.000

Total 1.000.000

• Head of trading can modify risk-capital allocation (limits)• Each desk has to respect a maximum VaR constraint• If volatility increase, the single trader has to reduce her position

(market value) in order to avoid overcoming the limit

• Assume the head of Fixed-Income Trading has a global risklimit to assign to her traders of € 1 m

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Some VaR models common critiques

VaR models do not consider exceptional events

VaR models do not consider customer relationships (short termism)

VaR models are based on unrealistic assumptions

VaR models produce different results

VaR models amplify markets instability

VaR models do not react quickly enough

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VaR models real problems

They do not consider the size of losses

VaR with confidence level 99%

the probability of loosing more than VaR is only 1%…

…however, how much could we loose if these 1% probability events occur?

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Example of Losses Relating to Two Stock Portfolios Losses (ranked starting from the worst one)

Portfolio H Portfolio K

1 150,000 60,000

2 120,000 56,000

3 100,000 55,000

4 70,000 53,000

5 60,000 51,000

6 50,000 50,000

7 48,000 45,000

8 45,000 40,000

9 42,000 35,000

10 40,000 30,000

VaR(99%) VaR99% 50,000 50,000

Maximum Loss Lmax 150,000 60,000

Maximum Excess Loss Lmax - VaR99% 100,000 10,000

Maximum Excess Loss/VaR 200% 20%

Expected Excess Loss E (L-VaR99% | L > VaR99%)

50,000 5,000

Expected Excess Loss/VaR 100% 10%

VaR models real problems

• 10 worst losses over 500 observations for two stock portfolios

Portfolio H is clearly riskier

The two portfolios have the same VaR

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VaR models real problems

Subadditivity of risk measures

Example: portfolio Z = X+Y

If we abandon the normal distribution assumption of market factors returns VaR measures could, in some particular circumstances, not respect this property

)()()()( YFXFYXFZF

)()()( YVaRXVaRYXVaR

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An alternative risk measure: expected shortfall

Expected Shortfall (ES) or Extreme Value at Risk (EVR): “the expected value of losses that the portfolio could suffer in the (1-c) worst

case during the time horizon T”

cLPEVaR )(

cc LLLEPEES /)(

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An alternative risk measure: expected shortfall

• Example: portfolio of 2 bonds both with PD = 5% e recovery rate alternatively equal to 70% (prob. 1%) or 90% (prob. 4%)

Probability Bond A Bond B

1,00% 70 70

4,00% 90 90

95,00% 100 100

E(VM) 99,3 99,3

VaR(99%) 9,3 9,3

ES(99%) 29,30 29,30

Probability distribution of 2 bonds values

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An alternative risk measure: expected shortfall

Event Probability A B A+B Cumulative Prob.

1 0,01% 70 70 140 0,01%

2 0,04% 70 90 160 0,05%

5 0,04% 90 70 160 0,09%

3 0,95% 70 100 170 1,04%

7 0,95% 100 70 170 1,99%

4 0,16% 90 90 180 2,15%

6 3,80% 90 100 190 5,95%

8 3,80% 100 90 190 9,75%

9 90,25% 100 100 200 100,00%

100,00%

Probability distribution of the bond portfolio

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An alternative risk measure: expected shortfall

Event Probability A+B Cumulative Probability

1 0,01% 140 0,01%

2 0,08% 160 0,09%

3 1,90% 170 1,99%

4 0,16% 180 2,15%

5 7,60% 190 9,75%

6 90,25% 200 100,00%

100,00%

E(VM(A+B)) 198,60

VaR(99%) 28,60

ES(99%) 40,82

Probability distribution and risk measures of the portfolio