BBA FN415 (18) Lect 06 VaR.pptx

Introduction to Market Risk Measurement – Value at Risk

Introduction to Market Risk Measurement – Value at Risk

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1. Overview

 _{Market risk is primarily measured with}

value at risk (VaR)

 _{VaR measures the total portfolio risk.}

 _{In theory the whole distribution of profit}

and loss should be described by the risk managers but in reality using VaR is much more practical.

 _{VaR is summarises the whole distribution}

into one number

1.1 Types of Financial Market Risks

 _{Includes Market Risk, Credit Risk, and Operational}

Risk.

 _{Market Risk: Risk of losses due to the movements}

in financial market prices or volatilities.

 _{Credit Risk: Risk of losses due to the fact that the}

counterparty may be unwilling or unable to fulfil their contractual obligations.

 _{Operational Risk: Risk of loss resulting from failed}

or inadequate internal processes, systems and people, or from external events.

 _{Oftentimes, these three categories interact with}

1.1 Example of risk interactions

 _{Suppose trader purchases 1 million worth of GBP}

spot from Bank A for settlement in two business days.

 _{The current rate is $1.5 : £1}

 _{This simple transaction involves a series of risks}

– _{Market Risk} – _{Credit Risk}

1.2 Risk Management Tools

 _{In the past, risks were measured with}

notional amounts, sensitivity measures and scenario analysis.

 _{Some intuition provided but it failed to}

capture the downside risk of the total portfolio.

 _{They failed to capture}

– _{Differences in volatilities across markets} – _{Correlations between risk factors}

1.2 Risk Management Tools; Example

 _{Consider a five year inverse floater, which pays}

coupons 16%-2xLIBOR, if positive.

 Notional Principal of $100 million

 Initial market value of note is $100 million

 _{Extremely sensitive to movement in the interest rates}  _{Question is how much could the investor lose on this}

investment over a specified horizon?

 _{Notional only provides an indication of the potential}

loss.

1.2 Risk Management Tools; Example (ii)

 _{A sensitivity measure such as duration is more helpful}

 _{Suppose that this bond has the modified duration of 13.5}

years.

 _{This sensitivity measures reveals the extreme sensitivity}

of the bond to interest rates movement but it doesn’t tell us anything about the probability of occurrence.

 _{Scenario analysis can help but it still doesn’t associate the}

loss with a probability.

 _{The great beauty about VaR is that it provides a neat}

answer to all these questions.

 _{VaR gives one number which aggregates the risks across}

1.2 Risk Management Tools; Example (iii)

 _{If the worst increase in yield at the 95% level is 1.65%, we can}

compute the VaR as

VaR = Market Value x Modified Duration x Worst Yield Increase.

 This gives VaR = $100million X 13.5 X 0.0165 = $22 million

 _{The investor can now make the following statement:}

1.2 Risk Management Tools; VaR Revolution

 _{This started in 1993 when the Group of Thirty} (G30) endorsed VaR as part of best practices for dealing with derivatives.

 _{However, the methodology behind VaR is not} new

 _{It came from the merging of financial theory,} which focuses on the pricing of financial

instruments, and statistics, which studies the behaviour of risk factors.

 _{The idea of VaR can be traced back to the}

2.1 VaR: Definition



_{Value at Risk is a summary measure}

of downside risk expressed in dollars,

or in the reference currency.



_{A general definition is}

“VaR is the maximum loss over a

target horizon such that there is a

2.1 VaR: Example (i)



_{Consider for instance a position of $4}

billion short the Yen, Long the dollar.



_{This position corresponds to a}

well-known hedge fund that took a bet

that the yen would fall in value

against the dollar.



_{How much could this position lose}

2.1 VaR: Example (ii)

 _{To answer this question, we could use historical}

data

 _{The simulated daily return in dollars in then}

R_t($) = Q₀($) [S_t-S_t-1]/S_t-1

where Q₀ is the current dollar value of the position and S is the spot rate in Yen per dollar measured over two consecutive days.

 So if S₁=112.0 and S₂=111.8. The simulated return is

2.1 VaR: Example (iii)

 _{Repeat this operation over the whole sample}

2.1 VaR: Example (iv)

 _{Next we construct a frequency distribution of daily}

returns.

 _{This is ordered from the worst return to the best}

2.1 VaR: Example (v)

 _{Now we wish to summarize the}

distribution by one number.

 _{We could describe the quantile, the level of}

loss that will not be exceeded at some high confidence level.

 _{Select for instance the confidence level as c}

= 95% which correspond to the right tail probability.

 _{VaR is defined in terms of the left tail}

2.1 VaR: Example (vi)

 _{Now we wish to summarize the}

distribution by one number.

 _{We could describe the quantile, the level of}

loss that will not be exceeded at some high confidence level.

 _{Select for instance the confidence level as c}

= 95% which correspond to the right tail probability.

 _{VaR is defined in terms of the left tail}

2.1 VaR: Example (vii)

 _{We want to find the cutoff value R}* _such

that the probability of loss worse than - R*

is p = 1 – c = 5%

 _{With a total of T = 2,527 observations, this}

corresponds to a total of pT = 0.05*2527 = 126.

 _{We pick from the ordered distribution the}

cutoff value, which is R*_{=$47.1 million.}

 _{Now we can make the following VaR}

2.1 VaR: Example (viii)

 _{The maximum loss over one day is about}

$47 million at the 95% confidence level.

 _{Note that this describes risk in a way that}

notional amounts or exposures cannot convey.

 _{We can also calculate the number of}

expected exceedences n over a period of N days:

2.2 VaR: Caveats



_{VaR does not describe the}

worst loss.



_{VaR does not describe the}

losses in the left tail.



_{VaR is measured with some}

2.3 Alternative Measures of Risk



_{The Entire distribution}



_{The Conditional VaR}



_{The Standard Deviation}



_{The Semi-standard Deviation}

The Entire Distribution



_{In our example VAR is simply one}

quantile in the distribution.



_{The risk manager however has access}

to the whole distribution and could

report a range of VAR numbers for

increasing confidence levels



_{The entire distribution will describe}

Conditional VaR

 A related concept is the expected value of the loss when it exceeds VaR.

 _{This measures the average of the loss conditional}

on the fact that it is greater than VAR.

 Formally the conditional VAR (CVAR) is the negative of

 This ratio is also called expected shortfall, tail conditional expectation, conditional loss, or expected tail loss.

 _{This is similar to the standard deviation but}

consider only data points that represent a loss.

 _{Advantage: it takes into account the}

Drawdown

 _{Drawdown is the decline from peak over a fixed}

time interval.

 _{Define x}MAX _{as the local maximum over this period}

t of [0, T], which occurs at time t_MAX of [0, T]

 _{Relative to this value, the drawdown at time t is}

 _{The maximum drawdown is the largest such value}

Maximum Drawdown 1 8 /0 8 /1 99 5 1 3 /1 1/ 1 9 9 5 0 9 /0 2 /1 99 6 1 4 /0 5 /1 99 6 1 3 /0 8 /1 99 6 0 6 /1 1/ 1 9 9 6 0 4 /0 2 /1 9 9 7 0 7 /0 5/ 1 9 9 7 0 4 /0 8 /1 99 7 2 9 /1 0/ 1 9 9 7 2 7 /0 1 /1 9 9 8 2 8 /0 4/ 1 9 9 8 2 8 /0 7 /1 9 9 8 2 1 /1 0 /1 9 9 8 2 0 /0 1 /1 9 9 9 2 1 /0 4/ 1 9 9 9 2 0 /0 7 /1 9 9 9 1 4 /1 0 /1 9 9 9 1 3 /0 1 /2 00 0 1 0 /0 4 /2 0 0 0 1 0 /0 7 /2 0 0 0 0 4 /1 0 /2 0 0 0 0 3 /0 1 /2 00 1 2 9 /0 3 /2 0 0 1 2 9 /0 6 /2 00 1 2 6 /0 9 /2 00 1 2 4 /1 2 /2 00 1 2 1 /0 3 /2 00 2 2 1 /0 6 /2 00 2 1 8 /0 9 /2 00 2 1 6 /1 2 /2 00 2 1 4 /0 3/ 2 0 0 3 1 6 /0 6 /2 00 3 1 1 /0 9 /2 00 3 0 8 /1 2 /2 00 3 0 4 /0 3/ 2 0 0 4 0 8 /0 6 /2 00 4 0 3 /0 9/ 2 0 0 4 2 9 /1 1/ 2 0 0 4 2 8 /0 2/ 2 0 0 5 0 1 /0 6 /2 0 0 5 2 9 /0 8/ 2 0 0 5 2 2 /1 1 /2 0 0 5 2 0 /0 2 /2 0 0 6 2 4 /0 5 /2 0 0 6 2 2 /0 8/ 2 0 0 6 1 6 /1 1 /2 0 0 6 1 4 /0 2 /2 0 0 7 1 7 /0 5 /2 00 7 1 4 /0 8 /2 0 0 7 0 200 400 600 800 1000 1200 0% 10% 20% 30% 40% 50% 60% 70% 80%

SET50 Max. Down

521.71

238.80

-54.23%

433.69

136.97

Maximum Down = -68.42% Avg. Down = -33.29%

-68.42%

3. VaR Parameters

To measure VAR, we first

need to define two

quantitative parameters the

confidence level and the

3.1 VaR Parameters; Confidence Level

 _{The higher the confidence level c, the greater}

the VAR measure.

 _{Extreme losses. It is not clear, however,}

whether one should stop at 99%, 99.9%, 99.99% and so on. Each of these values will create an increasingly larger loss, but with less likelihood.

 _{Another problem is that as c increases the}

3.2 VaR Parameters; Horizon

 _{The longer the horizon T, the greater the VAR}

measure

 _{To extrapolate from a one-day horizon to a longer}

horizon, we need to assume that returns are

independent and identically distributed. If so, the daily volatility can be transformed into to a

multiple day volatility by multiplication by the square root of time.

 _{We also need to assume that the distribution of}

5. Stress Testing

 VaR does not purport to account for extreme

losses. This is why VAR should be complemented by stress-testing.

 _{Stress testing is a key risk management process}

which includes

– _{(1) scenario analysis}

– _{(2) stressing models, volatilities, and correlations} – _{(3) developing policy response}

 Scenario Analysis

– _{Moving key variables one at a time} – _{Using historical scenario}

6. Liquidity Risk



_{Liquidity risk is usually viewed as}

a component of market risk.



_{Lack of liquidity can cause the}

failure of an institution even

when it is technically liquid.



_{Liquidity risk can be separated}

into

–

_{Asset Liquidity risk}

Desirable Properties of Risk Measures r  Monotonicity:

 _{Translation invariance:}

 _Homogeneity:

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