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Unified Financial AnalysisRisk & Finance Lab

Chapter 10: Sensitivity

Willi Brammertz / Ioannis Akkizidis

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Sensitivity

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Sensitivity

> General defintiony

> Applies (in our defintion) also to higher order derivations

> β generally market value

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Where analytical sensitivities work

else

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Why bother with analytical solutions?

> Calculation efficiency

> Wide application (there are sufficient relevant cases)

> Basis for many limits

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Interest rate sensitivity

In the simplest case it is the $Duration measure: (-t*CF(t)*e^(-rt))

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Duration: Intuitive interpretation

> Average time to repricing

> Average gap

> Immunization horizon

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Hedge Investment Horizon: point in time in which the

value of a portfolio can be (almost) 100% achieved.

Value

ValueΔ

Gain/loss on

reinvestments

of CFL

Time

Duration

Duration: Immunization horizon

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Key Rate Duration

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A simple example

> A principal at maturity contract

> Date of analysis: 31.12.00

> ValueDate: 1.1.01

> MaturityDate: 1.1.03

> Interest payment frequency: 1Q, regular

> Notional: 1000 (asset)

> Interest rate: 10%, 30/360

> Rate reset cycle: 1Y, regular

> Rate spread: 0%

> Events and results

> Liquidity?

> Sensitivity?

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Intuitive explanation

Fixed cash flowsSimple discounting applies

Variable cash flowsNo spread

What is its value?

1 1+r Interest calc 1+r Discounting

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Complex example

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Derivation of Table 10.4Graphical representation

t0

1.1.00

t1

1.4.00

t2

1.7.00

t3

1.10.00

YC(t0)Rate resetƒ(t0, t1, t3)

Discountingr(t0, t1)

Discountingr(t0, t2)

Point of analysis

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Sensitivity vectorEffect of repricing mismatch

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Sensitivity vs. liquidity gap: Same example as chapter 8

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Sensitivity vs. liquidity

> Liquidity: Sum the liquidity line per time bucket

> Sensitivity: Sum the ZES line per time bucket

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Intuitive interpretation of ZES

> We decompose everything into single expected cash flows

> A single cash flow is equivalent to a zero bond

> Sensitivity of a single zero bond is -t*CF(t)*e^(-rt)

> Therefore we need to know from the contract

> t

> CF(t)

> ZES shows this information

> r comes from the market information

> ZES is additive through all financial instruments!

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Other market sensitivity

> FX

> Stocks

> Both derivable from ZES plus discounting information

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Credit exposure

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Current exposure: Example

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Current and potential exposure(taking future potential changes into account)

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Calculation of potential exposure

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Potential exposure via add-ons

> Add-ons are pre-established (canned) volatility adjustments

> Based on standard sensitivity buckets

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Basel II Haircuts

> Haircuts: Deductions from collateral value

> Similar function as Add-ons

> Language is imprecise

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Mortality sensitivitySimple example 1 year life insurance

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Limits

> Risk Limits (Chapter 11)

> Sensitivity limits> Gap limits

> First derivation limits (Interest rate, FX, Stock etc.)

> Other limit types> Volume (book value, nominal value, market value)

> Loss limits

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Gap limits

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Limit setting