2-1. Three Mapping System
i. Principal mapping: average portfolio maturity
ii. Duration mapping: portfolio duration
iii. Cash-flow mapping: maturity buckets, term-structure vertices
2-2. Stress Test
- assumption: all zeros are perfectly correlated
- decrease all zeroes' values by their VaR -> generate a new distribution of P.V. factors -> undiversified VaR
2-3. Benchmarking a Portfolio
- compute VaR in relative terms, that is, relative to a performance benchmark
- Tracking error VaR: 
where:
x = vector of position for the portfolio
x0 = vector of positions for the index
- tracking error can be measured in terms of variance reduction
e.g.) Absolute risk of the index (Absolute VaR)= $1.99, TE-VaR=$0.43
=> 1 - (0.43/1.99)^2 = 95.4%
- as correlations decrease for more distance maturities, we should expect that a duration-matched portfolio should have the lowest absolute risk for the combination of most distance maturities (babell portfolio)
**Babell Portfolio: A bond portfolio that has high concentrations of bonds in both short-term and long-term fixed-income instruments with only a few in intermediate-term bonds. A barbell portfolio implements a trading strategy that concentrates on investments in the short- and long-term end of bond maturities. A barbell strategy is useful when short-term and long-term interest rates are higher relative to intermediate interest rates. This strategy allows investors to earn higher overall yields while still retaining the desired time frame for the bond portfolio.
3. Mapping Linear Derivatives
3-1. Forward Contracts 
where:
St = spot price of one unit of the underlying cash asset
K = contracted forward price
r = domestic risk-free rate
y = income flow on the asset
tau = time to maturity
Ft = current forward rate
- appropriate for application of the delta-normal method; liner combinations of normally distributed risk factors
e.g.) a forward contract to purchase pounds for dallars one year from now
risk positions:
. a short position in a U.S. T-bill
. a long position in a one-year U.K. bond
. a long position in the British pound spot market
- the current value of the forward contract is the present value of the difference between the current forward rate and the locked-in delivery rate. -> the initial value of the contract is zero
- the value of the contract may change, creating market risk
3-2. Commodity Forwards
- more complex than for financial assets such as currencies, bonds, or stock indices
- most products consumable -> creating implied benefit (convenience yield)
*convenience yield is not tied to another financial variable -> highly variable, creating it own source of risk
- main driver of the value of the contract: current forward price for commodities
- commodities are much more volatile than typical financial assets
- volatilities decrease with maturity; the effect is strongest for less storable products (energy products).
**financial assets: volatilites driven primarily by spot prices (constant volatilities across contract maturities)
3-3. Forward Rate Agreements
- the buyer of an FRA locks in a borrowing rate; the seller locks in a lending rate
e.g.) Long 6 x 12 FRA = long 6-month bill + short 12-month bill
(borrowing for 6 months & investing the proceeds for 12 months)
360-day spot rate = 5.8125%, 180-day rate = 5.6250%
=> (1 + F1,2 / 2) = (1 + 5.8125%) / (1 + 5.6250%/2) => F = 5.836%
3-4. Interest Rate Swaps
Interest-rate swaps viewed in two different ways:
- a combined position in a fixed-rate bond and in a floating-rate bond or
- a portrolio of forward contracts
4. Mapping Options
- mapping proccess for nonlinear derivatives
- delta is not a constant, which may make linear methods inappropriate for measuring the risk of options. Delta increases with the underlying spot price. The relationship becomes more nolinear for short-term options.
- with a small delta change, the linear effect will dominate the nonlinear effect -> linear approximations may be acceptable for options with long maturities when the risk horizon is short
Long option = long delta asset + short(delta asset - c) bill
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