Copyright K.Cuthbertson, D. Nitzsche 1 FINANCIAL ENGINEERING: DERIVATIVES AND RISK MANAGEMENT (J. Wiley, 2001) K. Cuthbertson and D. Nitzsche Lecture Value at Risk: Mapping Version 1/9/2001

Copyright K.Cuthbertson, D. Nitzsche 2 VaR for Different Assets (Mapping) Stocks Foreign Assets Coupon Paying Bonds Other Assets (All of above have portfolio returns that are approximately ‘linear’ in individual returns ~ hence use VCV method) Topics

Copyright K.Cuthbertson, D. Nitzsche 3 Your Cash : Is it Safe in Their Hands ? *

Copyright K.Cuthbertson, D. Nitzsche 4 VaR for Different Assets (Mapping)

Copyright K.Cuthbertson, D. Nitzsche 5 VaR for Different Assets: Practical Issues PROBLEMS STOCKS : Too many covariances [= n(n-1)/2 ] FOREIGN ASSETS : Need VaR in “home currency” BONDS: Many different coupons paid at different times DERIVATIVES: Options payoffs can be highly non- linear (ie. NOT normally distributed)

Copyright K.Cuthbertson, D. Nitzsche 6 VaR for Different Assets: Practical Issues SOLUTIONS = “Mapping” STOCKS : Within each country use “single index model” SIM FOREIGN ASSETS : Treat asset in foreign country = “local currency risk”+ spot FX risk BONDS: treat each bond as a series of “zeros” OTHER ASSETS: Forward-FX, FRA’s Swaps: decompose into ‘constituent parts’. DERIVATIVES: ~ next lecture

Copyright K.Cuthbertson, D. Nitzsche 7 Stocks/Equities and SIM

Copyright K.Cuthbertson, D. Nitzsche 8 “Mapping” Equities using SIM Problem : Too many covariances to estimate Soln. All n(n-1)/2 covariances “collapse or mapped” into  m and the asset betas (n-of them) Single Index Model: R i = a i + b i R m + e i R k = a k + b k R m + e k assume E  i  k = 0 and cov (R m, e) = 0 All the systematic variation in R i AND R k is due to R m ‘p’ = portfolio of stocks held in one country (R m,  m ) for eg. S&P500 in US

Copyright K.Cuthbertson, D. Nitzsche 9 Intuition 1) In a diversified portfolio     ) =0 2) Then each return-i depends only on the market return (and beta). Hence ALL variances and covariances also only depend on these 2 factors. We then only require( n-betas +   m )to calculate ALL our inputs for VaR. 3)  = 1 because (in a well diversified portfolio) each return moves only with R m 4) We end up with [1]VaR p = V p  p (1.65  m ) or equivalentlyVaR p = (Z C Z ’ ) 1/2 where C is the unit matrix

Copyright K.Cuthbertson, D. Nitzsche 10 THE MATHS OF SIM AND VaR - OPTIONAL ! SIM implies:  i,k =cov(R i,R k )=  i  k   m  i, 2 = var(R i ) =   i   m +     ) BUT in diversified portfolio     ) = 0 and  = 1

Copyright K.Cuthbertson, D. Nitzsche 11 We have(linear): R p = w 1 R 1 + w 2 R 2 + …. From the SIM we can deduce that for a PORTFOLIO of equities in one country Standard Deviation of the PORTFOLIO is given by :  p =  p  m where:  p =  w i  i (ie. portfolio beta requires, only n-beta’s) Hence: [1]VaR p = V p  p (1.65  m ) THE MATHS OF SIM AND VaR - OPTIONAL !

Copyright K.Cuthbertson, D. Nitzsche 12 Alternative Representation Eqn [1] above can be written: VaR p = (Z C Z ’ ) 1/2 where: VaR 1 = V (  1  m ) and VaR 2 =V (  2  m ) Z = [ VaR 1, VaR 2 ] C is the identity matrix C = ( 1 1 ; 1 1 ), since  = 1 for the SIM and a well diversified portfolio.

Copyright K.Cuthbertson, D. Nitzsche 13 SIM: Checking the Formulae [1] VaR p = V p  p (1.65  m ) = V p (  w i  i ) (1.65  m ) = (V 1  1 + V 2  2 ) (1.65  m ) The above can easily be shown to be the same as VaR p = (Z C Z ’ ) 1/2 = (1.65  m ) [ (V 1  1 ) 2 + (V 2  2 ) V 1  1 V 2  2 ] 1/2 where VaR 1 = V (  1  m ) VaR 2 =V (  2  m ) and Z = [ VaR 1, VaR 2 ] and C = ( 1 1 ; 1 1 )

Copyright K.Cuthbertson, D. Nitzsche 14 “Mapping” Foreign Assets into Domestic Currency

Copyright K.Cuthbertson, D. Nitzsche 15 “Mapping” Foreign Assets into Domestic Currency US based investor with DM140m in DAX Two sources of risk a) variance of the DAX b) variance of $-DM exchange rate c) one covariance/correlation coefficient  (between FX-rate and DAX) eg. Suppose when DAX falls then the DM also falls - ‘double whammy’ from this positive correlation

Copyright K.Cuthbertson, D. Nitzsche 16 “ Mapping” Foreign Assets into Domestic Currency US based investor with DM140m in DAX FX rate : S = $/DM ( 1.4 DM/$ ) Dollar initial value V o$ = 140/1.4 = $100m Linear R $ = R DAX + R $/DM above implies w i = V i / V 0$ = 1 and  p = Dollar-VaR p = V o$ 1.65  p  = correlation between return on DAX and FX rate

Copyright K.Cuthbertson, D. Nitzsche 17 “Mapping” Foreign Assets into Domestic Currency Alternative Representation Dollar VaR Let Z = [ V 0,$ 1.65  DAX, V 0,$ 1.65  S ] = [ VaR 1, VaR 2 ] V 0,$ = $100m for both entries in the Z-vector Then VaR p = (Z C Z ’ ) 1/2

Copyright K.Cuthbertson, D. Nitzsche 18 “Mapping”Coupon Paying Bonds

Copyright K.Cuthbertson, D. Nitzsche 19 “Mapping”Coupon Paying Bonds Coupons paid at t=5 and t=7 Treat each coupon as a separate zero coupon bond P is linear in the ‘price’ of the zeros, V 5 and V 7 We require two variances of “prices” V 5 and V 7 and covariance between these prices.  5 (dV / V) = D  (dy 5 )

Copyright K.Cuthbertson, D. Nitzsche 20 Coupon Paying Bonds Treat each coupon as a zero Calculate PV of coupon = price of zero, V 5 = 100 / (1+y 5 ) 5 VaR 5 = V 5 (1.65  5 ) VaR 7 = V 7 (1.65  7 ) VaR (both coupon payments) =  = correlation: bond prices at t=5 and t=7 (approx )

Copyright K.Cuthbertson, D. Nitzsche Actual Cash Flow $ 100 m 57 RM Cash Flow Weights , 1- , are chosen to ensure weighted average volatility based on RM values of  at 5 and 7 equals the interpolated volatility at “6” Mapping on to “standard” RMetrics Vertices

Copyright K.Cuthbertson, D. Nitzsche 22 “MAPPING” OTHER ASSETS SWAP: =LONG FIXED RATE BOND AND SHORT AN FRN FRA (6m x 12m) = BORROW AT 6 MONTHS SPOT RATE +LEND AT 12M SPOT RATE FORWARD FX: =BORROW AND LEND IN DOMESTIC AND FOREIGN SPOT INTEREST RATES AND CONVERT PROCEEDS AT CURRENT SPOT-FX

Copyright K.Cuthbertson, D. Nitzsche 23 End of Slides