© K.Cuthbertson, D.Nitzsche 1 LECTURE Market Risk/Value at Risk: Basic Concepts Version 1/9/2001

© K.Cuthbertson, D.Nitzsche 1

LECTURE

Market Risk/Value at Risk: Basic Concepts

Version 1/9/2001

FINANCIAL ENGINEERING:DERIVATIVES AND RISK MANAGEMENT(J. Wiley, 2001)

K. Cuthbertson and D. Nitzsche


• Value at Risk (VaR)

• Forecasting and Backtesting

• Validation of Risk Measures

• Basle Capital Adequacy for Market Risk

and Other Approaches/ Uses of VaR

TOPICS


Value at Risk (VaR)


VALUE AT RISK:RiskMetrics™, J.P. MORGAN

• Common methodology

• Handles market risk of (linear instruments):

stocks, bonds, foreign assets -using the ‘parametric’ or “variance-covariance” approach (also ‘delta-normal’ approach!)

• Options (non-linear) - using MCS


VaR: CONCEPT

If at 4.15pm the reported daily VaR is $10m then:

Maximum amount I expect to lose in 19 out of next 20

days, is $10m

OR

I expect to lose more than $10m only 1 day in every 20

days (ie. 5% of the time)

The VaR of $10m assumes my portfolio of assets fixed

Exactly how much will I lose on any one day?

Unknown !!!


VALUE AT RISK

TOPICS

VaR for stocks of – single asset – portfolio of (domestic) assets

© K.Cuthbertson, D.Nitzsche

Fig. 22.1 : Standard Normal Distribution, N(0,1)

Probability

-1.65 0.0 +1.65

5% of the area

5% of the area

Return

Mean = 0, = 1

Single Asset


VaR: Single Asset

Normal Distribution (5% lower tail) implies:

Only 5% of the time will the actual % return be below:

“ - 1.65 1”

= Mean Return

When then:5% lower tail, cut off point = -1.65 1

Only 5% of the time will the % loss be more than “ 1.65 1

This is a PARAMETRIC approach since it requires

estimation of volatility

R

R0R


VaR: Single Asset

Example: Mean return = 0 %

Let return) = 0.02 (per.day) (equivalent to 2%)

Only 5% of the time will the loss be more than 3.3% (=1.65 x 2%)

VaR of a single asset (Initial Position V0 =$200m in equities)

VaR = V0 (1.65 ) = 200 ( 0.033) = $6.6m

That is “(dollar) VaR is 3.3% of $200m”

VaR is reported as a positive number (even though it’s a loss)


VaR of a Single Asset

Are Daily Returns Normally Distributed?-not quite but close

• fat tails (excess kurtosis)• peak is higher and narrower• negative skewness• small (positive) autocorrelations• squared returns have strong autocorrelation,

ARCH• But niid is a (good ) approx for equities, long

term bonds, spot FX , and futures (but not for short term interest rates or options)


VaR : Portfolio of Assets : Diversification

1) Text Book Approach

p =

VaRp = Vp (1.65 p )

Vp = total $’s held in whole portfolio

Can we express the above formula in terms of VaR of each individual asset? (VaR1 , VaR2 etc)

Yes!

wi i

wiwj ij i j

2 21

2


VaR : Portfolio of Assets

• “Street/City” uses $ or £’s held in each asset, Vi

• Note that wi = Vi / Vp (substitute in above equation)

then VaRp =

• where VaRi = Vi (1.65 i) - for single asset-i

2/121

22

21 ]2[ VaRVaRVaRVaR



• Can we put VaR in matrix form? - Yes

• Given VaRp =

where VaRi = Vi (1.65 i) - for single asset-i

Let

Z = [ VaR1, VaR2 ] (2 x 1 vector)

C = [ 1 ; 1 ] = correlation matrix (2x2)

THEN: VaRp = [Z C Z’]1/2

2/121

22

21 ]2[ VaRVaRVaRVaR


• An arithmetical nuance

• If asset-1 is held long and asset-2 has been ‘short sold’ then for example V1 = +$100 and V2 = -$50

• So when constructing the “Z” vector then:

VaR1 = ($100)1.65 1 and VaR2 = (-$50)1.65 2

• we still have Z = [ VaR1, VaR2 ] and VaRp = [Z C Z’]1/2

• Above is known as the “Variance-Covariance” approach



“Worst Case VaR”

• Assume all correlations are +1 and all assets are held “long” then

VaRp =

which with all =+1 gives

VaRp = { VaR1 + VaR2 }

-i.e. no “diversification effect”

2/121

22

21 ]2[ VaRVaRVaRVaR


Forecasting and

Backtesting


Forecasting Volatility

Simple Moving Average ( Assume Mean Return = 0)

t+1|t = (1/n) R2t-i

Exponentially Weighted Moving Average EWMA

t+1|t = R2t-i wi = (1-)

It can be shown that this may be re-written:

t+1|t =

t| t-1 + (1- Rt2

Longer Horizons :” -rule” - for returns iid.

T

T


EWMA (Recursive)

t+1|t =

t| t-1 + (1- Rt2

is found to Min. forecast error (Rt+12- t+1|t)

Cut off point : Start recursion in the computer using about 74 days of historical data, so that

0 = (1/n) R2t-i

Covariance xy,t+1|t = xy,t|t-1 +(1- ) Rx,t Ry,t

Correlation = xy/xy

Non-synchronous trading ?

Forecasting Volatility


Validation of Risk Measures

“Backtesting”



1.Individual Returns Series

How many?

Are about 5% of actual (individual) returns Rt+1

‘greater than’ the forecast of 1.65 t+1|t ? Yes !

How big?

Are the actual returns in lower 5th percentile the

same size as those for the normal distribution?

• “Yes” for equity, bond and spot FX - returns


2. Portfolio of Assets

• Portfolio Returns• Take equal wtd portfolio of 200 assets.

• Forecast all the individual VaRi’s = Vi1.65 t+1|t ,

• calculate portfolio VaR for each day:

VaRp = [Z C Z’]1/2

• then see if actual portfolio losses exceed this only 5% of the time (over some historic period, eg. 252 days).


© K.Cuthbertson, D.Nitzsche

Figure 22.2: Backtesting

Days

Daily $m profit/loss

0

2.5

5.0

-2.5

-5.0

= forecast = actual


Basle Capital Adequacy for Market Risk

and

Other approaches/ uses of VaR


Capital Adequacy : Basle

Basle Internal Models Approach (VaR)

Calc VaR for worst 1% of losses over 10 days

Use at least 1-year of daily data to forecast t+1|t

VaRi = 2.33

left tail critical value )

Capital Charge KC

KC = Max ( Av.of prev 60-days VaR x M,

or, previous day’s VaR) + SR

M = multiplier (min = 3)

SR = specific risk (equity and fixed income)

10


Pre-commitment Approach

Capital charge = announced VaR

If losses exceed VaR, then impose a penalty

Penalties

Go public Financial penalty

Greater regulatory surveillance

� Transparent and simple� Reduces compliance costs� Minimises portfolio distortions� But does not avert ‘go for broke strategy’


Assessing Performance of Investment Managers

Whose got the biggest Sharpe Ratio ?

S = (Rav - r ) /

Rav = actual monthly returns averaged over 1 year (say)

= s.d. of portfolio of assets (calculated as in VaR framework (but with returns measured over a 1-day horizon and grossed up with the “root T “ rule to 1-month).


LECTURE ENDS


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© K.Cuthbertson, D.Nitzsche 1 LECTURE Market Risk/Value at Risk: Basic Concepts Version 1/9/2001