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

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

© K.Cuthbertson, D.Nitzsche 2 Value at Risk (VaR) Forecasting and Backtesting Validation of Risk Measures Basle Capital Adequacy for Market Risk and Other Approaches/ Uses of VaR TOPICS

© K.Cuthbertson, D.Nitzsche 3 Value at Risk (VaR)

© K.Cuthbertson, D.Nitzsche 4 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

© K.Cuthbertson, D.Nitzsche 5 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 !!!

© K.Cuthbertson, D.Nitzsche 6 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 Return Mean = 0,  = 1 Single Asset

© K.Cuthbertson, D.Nitzsche 8 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

© K.Cuthbertson, D.Nitzsche 9 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 V 0 =\$200m in equities) VaR = V 0 (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)

© K.Cuthbertson, D.Nitzsche 10 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)

© K.Cuthbertson, D.Nitzsche 11 VaR : Portfolio of Assets : Diversification 1) Text Book Approach  p = VaR p = V p (1.65  p ) V p = total \$’s held in whole portfolio Can we express the above formula in terms of VaR of each individual asset? (VaR 1, VaR 2 etc) Yes!

© K.Cuthbertson, D.Nitzsche 12 VaR : Portfolio of Assets “Street/City” uses \$ or £’s held in each asset, V i Note that w i = V i / V p (substitute in above equation) then VaR p = where VaR i = V i (1.65  i ) - for single asset-i

© K.Cuthbertson, D.Nitzsche 13 VaR : Portfolio of Assets Can we put VaR in matrix form? - Yes Given VaR p = where VaR i = V i (1.65  i ) - for single asset-i Let Z = [ VaR 1, VaR 2 ] (2 x 1 vector) C = [ 1  ;  1 ] = correlation matrix (2x2) THEN: VaR p = [Z C Z’] 1/2

© K.Cuthbertson, D.Nitzsche 14 An arithmetical nuance If asset-1 is held long and asset-2 has been ‘short sold’ then for example V 1 = +\$100 and V 2 = -\$50 So when constructing the “Z” vector then: VaR 1 = (\$100)1.65  1 and VaR 2 = (-\$50)1.65  2 we still have Z = [ VaR 1, VaR 2 ] and VaR p = [Z C Z’] 1/2 Above is known as the “Variance-Covariance” approach VaR : Portfolio of Assets

© K.Cuthbertson, D.Nitzsche 15 “Worst Case VaR” Assume all correlations are +1 and all assets are held “long” then VaR p = which with all  =+1 gives VaR p = { VaR 1 + VaR 2 } -i.e. no “diversification effect”

© K.Cuthbertson, D.Nitzsche 16 Forecasting and Backtesting

© K.Cuthbertson, D.Nitzsche 17 Forecasting Volatility Simple Moving Average ( Assume Mean Return = 0)    t+1|t = (1/n)  R 2 t-i Exponentially Weighted Moving Average EWMA    t+1|t =    R 2 t-i w i = (1- )   It can be shown that this may be re-written:   t+1|t =   t| t-1 + (1-  R t 2 Longer Horizons :” -rule” - for returns iid.   

© K.Cuthbertson, D.Nitzsche 18 EWMA (Recursive)   t+1|t =   t| t-1 + (1-  R t 2  is found to  Min. forecast error (R t+1 2 -  t+1|t ) Cut off point : Start recursion in the computer using about 74 days of historical data, so that   0 = (1/n)  R 2 t-i Covariance  xy,t+1|t =  xy,t|t-1 +(1- ) R x,t R y,t Correlation  =  xy /  x  y Non-synchronous trading ? Forecasting Volatility

© K.Cuthbertson, D.Nitzsche 19 Validation of Risk Measures “Backtesting”

© K.Cuthbertson, D.Nitzsche 20 Validation of Risk Measures 1.Individual Returns Series How many? Are about 5% of actual (individual) returns R t+1 ‘greater than’ the forecast of 1.65  t+1|t ? Yes ! How big? Are the actual returns in lower 5 th percentile the same size as those for the normal distribution? “Yes” for equity, bond and spot FX - returns

© K.Cuthbertson, D.Nitzsche 21 2. Portfolio of Assets Portfolio Returns Take equal wtd portfolio of 200 assets. Forecast all the individual VaR i ’s = V i 1.65  t+1|t, calculate portfolio VaR for each day: VaR p = [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). Validation of Risk Measures

© K.Cuthbertson, D.Nitzsche Figure 22.2: Backtesting Days Daily \$m profit/loss 0 2.5 5.0 -2.5 -5.0 = forecast = actual

© K.Cuthbertson, D.Nitzsche 23 Basle Capital Adequacy for Market Risk and Other approaches/ uses of VaR

© K.Cuthbertson, D.Nitzsche 24 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 VaR i = 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)

© K.Cuthbertson, D.Nitzsche 25 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’

© K.Cuthbertson, D.Nitzsche 26 Assessing Performance of Investment Managers Whose got the biggest Sharpe Ratio ? S = (R av - r ) /  R av = 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).

© K.Cuthbertson, D.Nitzsche 27 LECTURE ENDS

© K.Cuthbertson, D.Nitzsche 28 TAKE A BREAK