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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:
Ri = ai + bi Rm + ei
Rk = ak + bk Rm + ek
assume Ei k = 0 and cov (Rm , e) = 0
All the systematic variation in Ri AND Rk is due to Rm
‘p’ = portfolio of stocks held in one country (Rm , 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 Rm
4) We end up with [1] VaRp = Vp p (1.65 m )or equivalently VaRp = (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(Ri,Rk )= i k
m
i,2 = var(Ri) =
i m + )
BUT in diversified portfolio ) = 0 and = 1
Copyright K.Cuthbertson, D. Nitzsche 11
We have(linear): Rp = w1R1 + w2 R2 + ….
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 = wi i
(ie. portfolio beta requires, only n-beta’s)
Hence: [1] VaRp = Vp 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:
VaRp = (Z C Z’ )1/2
where:
VaR1 = V1 1.65 ( 1 m ) and VaR2 =V2 1.65 ( 2 m)
Z = [ VaR1 , VaR2 ]
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] VaRp = Vp p (1.65 m ) = Vp ( wi i) (1.65 m)
= (V1 1 + V2 2 ) (1.65 m)
The above can easily be shown to be the same as
VaRp = (Z C Z’ )1/2
= (1.65 m) [ (V1 1) 2 + (V2 2)
2 + 2 V1 1 V2 2 ]1/2
where VaR1= V1 1.65( 1 m)
VaR2 =V2 1.65 ( 2 m )
and Z = [ VaR1 , VaR2 ] 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
2s
2DAX
Copyright K.Cuthbertson, D. Nitzsche 16
“Mapping” Foreign Assets into Domestic Currency
US based investor with DM140m in DAX
FX rate : S = 0.714 $/DM ( 1.4 DM/$ )
Dollar initial value Vo$ = 140/1.4 = $100m
Linear R$ = RDAX + R$/DM
above implies wi = Vi / V0$ = 1 and
p =
Dollar-VaRp = Vo$ 1.65 p
= correlation between return on DAX and FX rate
21
222sDAXsDAX
Copyright K.Cuthbertson, D. Nitzsche 17
“Mapping” Foreign Assets into Domestic Currency
Alternative Representation
Dollar VaR
Let Z = [ V0,$ 1.65DAX , V0,$ 1.65S ]
= [ VaR1 , VaR2 ]
V0,$ = $100m for both entries in the Z-vector
Then
VaRp = (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, V5 and V7
We require two variances of “prices” V5 and V7 and
covariance between these prices.
5(dV / V) = D (dy5)
77
55 )1(
100
)1(
100
yyP
75 VVP
Copyright K.Cuthbertson, D. Nitzsche 20
Coupon Paying Bonds
Treat each coupon as a zero
Calculate PV of coupon = price of zero, V5 = 100 / (1+y5)5
VaR5 = V5 (1.65 5)
VaR7 = V7 (1.65 7)
VaR (both coupon payments)
=
= correlation: bond prices at t=5 and t=7
(approx 0.95 - 0.99 )
[ ] /VaR VaR VaRVaR52
72
5 71 22
Copyright K.Cuthbertson, D. Nitzsche 21
5 6 7Actual Cash Flow
$ 100 m
5 7RM 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
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