VaR by example
Zvi Wiener
02-588-3049
http://pluto.mscc.huji.ac.il/~mswiener/zvi.html
FRM
VaR
Deposit 1yr. 6%
Bonds 10yr. 5%
Credit
3yr. 15%
Assets
NIS TSAMUD $
Yen
4,000
2,000
8,000
Today L=6%
Liabilities
Saving 2yr. 4%
Deposit 1mo. 11%
Deposit 3mo. L-2%
Total:
Zvi Wiener
NIS TSAMUD $
1,800
8,200
Yen
3,000
(200)
VaR example
200
4,000 (3,000)
slide 2
Risk Factors
• USD/NIS exchange rate
• Yen/NIS exchange rate
• Inflation
• Real NIS interest rates (IR, 10 yr., 2 yr.)
• Nominal NIS IR (1mo., 10 yr.)
• USD IR, (1 yr.)
• Yen IR, (Libor 3 mo.)
Zvi Wiener
VaR example
slide 3
Fair Value
8000  2000  4000  8200 1800  3000
For risk measurement we need not only the
fair value, but the fair value as a function of
risk factors in order to estimate the potential
profit/loss.
Zvi Wiener
VaR example
slide 4
Fair Value Function
M (1  r
8000 1
M (1  r
0
3
NIS , 3 y
1
3
NIS , 3 y
0
2000
)
)
(1  r
)
(1  r
)
0
10
real ,10 y
1
10
real ,10 y
d M 1 r
4000 0 1
d M 1 r
1
Zvi Wiener
0
VaR example
0
$,1 y
1
$,1 y
slide 5
Fair Value Function
 r

1 


0 
12
M 

8200 1
1
M  rNIS ,1mo 
1 



12 

0
NIS ,1mo
Zvi Wiener
VaR example
slide 6
Fair Value Function
Y M 1  0.25( L
3000 0 1
Y M 1  0.25( L
1
1800
Zvi Wiener
0
Y ,1 y
1
Y ,1 y
0
(1  r
)
(1  r
)
 0.02)
 0.02)
0
2
real, 2 y
1
2
real, 2 y
VaR example
slide 7
Sensitivity
CPI
USD
Yen
rnominal1mo
rnominal3yr
rreal2yr
rreal10yr
rUSD1yr
rYen3mo
Zvi Wiener
0.1%
1%
2%
0.5%
0.5%
0.5%
0.5%
0.25%
0.25%
-8
40
-60
3
-103
17
-93
-10
2
VaR example
Significant risk
Significant risk
Biggest market risk
slide 8
Risky Scenario
Real r
2yr
Zvi Wiener
10 yr
VaR example
T
slide 9
Sensitivity
CPI
USD
Yen
rnominal1mo
rnominal3yr
rreal2yr
rreal10yr
rUSD1yr
rYen3mo
Zvi Wiener
0.1%
1%
2%
0.5%
0.5%
0.5%
0.5%
0.25%
0.25%
-8
40
-60
3
-103
17
-93
-10
2
VaR example
Are not included
into BoI
requirements
slide 10
Gradient Vector
Direction of fastest decay (loss).
Take the sensitivity vector and divide it by the
assumed changes in the risk factors.
f ' ( x)  lim
f ( x   )  f ( x)

 0
V ' ( x) 
Zvi Wiener
V ( x   )  V ( x)

VaR example
slide 11
What if ...
The sensitivity vector allows to estimate
quickly an impact of a certain market move on
the value of the portfolio.
Scalar multiplication of the gradient vector
and the hypothetical market change vector
gives the predicted loss/gain.
Zvi Wiener
VaR example
slide 12
Risk Measurement
• The gradient vector describes my exposure
to risk factors
• The distribution of risk factors allows me to
estimate the potential loss together with
probability of such an event.
• The stress test will describe the response to
specific (the most interesting) scenarios.
Zvi Wiener
VaR example
slide 13
Risk Management
• Swap Dollar Yen
• Two forward contracts
• Quanto option
• FRA (?)
• Fixed - floating swap
Zvi Wiener
VaR example
slide 14
Duration and IR sensitivity
Zvi Wiener
VaR example
slide 15
The Yield to Maturity
The yield to maturity of a fixed coupon
bond y is given by
n
p (t )   ci e
 (Ti  t ) y
i 1
Zvi Wiener
VaR example
slide 16
Macaulay Duration
Definition of duration, assuming t=0.
n
D
Zvi Wiener
T c e
i 1
Ti y
i i
p
VaR example
slide 17
Macaulay Duration
T
T
CFt
1
D   t wt 
t

t
Bond Pr ice t 1 (1  y)
t 1
A weighted sum of times to maturities of each
coupon.
What is the duration of a zero coupon bond?
Zvi Wiener
VaR example
slide 18
Meaning of Duration
dp d 
Ti y 

 ci e    Dp
dy dy  i 1

n
$
r
Zvi Wiener
VaR example
slide 19
Proposition 15.12 TS of IR
With a term structure of IR (note yi), the
duration can be expressed as:
n
D
T c e
i 1
Ti yi
i i
p
d 
Ti ( yi  s ) 
 ci e
   Dp
ds  i 1
 s 0
n
Zvi Wiener
VaR example
slide 20
Convexity
 p
C 2
y
2
$
r
Zvi Wiener
VaR example
slide 21
FRA Forward Rate Agreement
A contract entered at t=0, where the parties (a
lender and a borrower) agree to let a certain
interest rate R*, act on a prespecified principal,
K, over some future time period [S,T].
Assuming continuous compounding we have
at time S: -K
at time T: KeR*(T-S)
Calculate the FRA rate R* which makes PV=0
hint: it is equal to forward rate
Zvi Wiener
VaR example
slide 22
Exercise 15.7
Consider a consol bond, i.e. a bond which
will forever pay one unit of cash at t=1,2,…
Suppose that the market yield is y - flat.
Calculate the price of consol.
Find its duration.
Find an analytical formula for duration.
Compute the convexity of the consol.
Zvi Wiener
VaR example
slide 23
ALM Duration
1 L
DL  
L r
1 A
DA  
A r
D A L
1  ( A  L)

A L
r
• Does NOT work!
• Wrong units of measurement
• Division by a small number
Zvi Wiener
VaR example
slide 24
ALM Duration
1 P
VaRP r
A similar problem with measuring yield
Zvi Wiener
VaR example
slide 25