VaR

1
VaR by example

• Zvi Wiener
• 02-588-3049
• http//pluto.mscc.huji.ac.il/mswiener/zvi.html

2
Assets

• NIS TSAMUD Yen
• Deposit 1yr. 6 4,000
• Bonds 10yr. 5 2,000
• Credit 3yr. 15 8,000

Liabilities
Today L6
NIS TSAMUD Yen Saving 2yr. 4
1,800 Deposit 1mo. 11 8,200 Deposit 3mo.
L-2 3,000
Total (200) 200 4,000 (3,000)
3
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.)

4
Fair Value

• 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.

5
Fair Value Function
6
Fair Value Function
7
Fair Value Function
8
Sensitivity

• CPI
• USD
• Yen
• rnominal1mo
• rnominal3yr
• rreal2yr
• rreal10yr
• rUSD1yr
• rYen3mo

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
9
Risky Scenario
10
Sensitivity

• CPI
• USD
• Yen
• rnominal1mo
• rnominal3yr
• rreal2yr
• rreal10yr
• rUSD1yr
• rYen3mo

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
11
Gradient Vector

• Direction of fastest decay (loss).
• Take the sensitivity vector and divide it by the
  assumed changes in the risk factors.

12
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.

13
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.

14
Risk Management

• Swap Dollar Yen
• Two forward contracts
• Quanto option
• FRA (?)
• Fixed - floating swap

15
Duration and IR sensitivity
16
The Yield to Maturity

• The yield to maturity of a fixed coupon bond y is
  given by

17
Macaulay Duration

• Definition of duration, assuming t0.

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Macaulay Duration
A weighted sum of times to maturities of each
coupon.

• What is the duration of a zero coupon bond?

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Meaning of Duration
20
Proposition 15.12 TS of IR

• With a term structure of IR (note yi), the
  duration can be expressed as

21
Convexity
22
FRA Forward Rate Agreement

• A contract entered at t0, 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 PV0
• hint it is equal to forward rate

23
Exercise 15.7

• Consider a consol bond, i.e. a bond which will
  forever pay one unit of cash at t1,2,
• 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.

24
ALM Duration

• Does NOT work!
• Wrong units of measurement
• Division by a small number

25
ALM Duration

• A similar problem with measuring yield