P2.T5.208. VaR Mapping

[Suzanne Evans]

Questions:

208.1. Each of the following is true about value at risk (VaR) mapping EXCEPT which is false?

a. VaR mapping is compatible with all three basic approaches (delta-normal, historical simulation, and Monte Carlo simulation)
b. VaR mapping is capable of estimating diversified portfolio VaR; i.e., incorporating risk factor correlations
c. For the same position, risk measurement via VaR mapping may justifiably rely on fewer factors than the pricing (valuation) model
d. VaR mapping is NOT well-suited to complex portfolios with non-linearities

208.2. To map the risk of his fund's bond portfolio, risk manager Joe Smith uses zero-coupon U.S. Treasury bonds, at annual maturities, as the primitive (or elementary) risk factors. With respect to this primitives:

• The yield VaR for all T-bond maturities is 2.0%; yield VaR is monthly yield volatility scaled (multiplied by) the confidence deviate.
• Modified durations for the U.S. T-bond zero-coupon curve are given by: duration of 0.97 at 1.0 year vertex, 1.95 at 2.0 years, 2.93 at 3.0 years, 3.90 at 4.0 years, and duration of 4.85 at 5.0 year vertex.
• This implies, for example, that the returns VaR at the 1.0 year vertex is given by 2% yield VaR * 0.97 duration = 1.940% returns VaR.

Joe wants to duration map a two-bond $200.0 million portfolio, where the market value of each bond is $100.0 million. The first bond is a 7-year coupon bond with modified duration of 5.6 years and the second bond is a 3-year coupon bond with modified duration of 2.8 years. If Joe employs linear interpolation between vertices, and maps duration by replacing the portfolio with a zero-coupon bond with maturity equal to the duration of the portfolio, which is nearest to the duration-based value at risk (VaR) of the bond portfolio?

a. $13.3 million
b. $16.4 million
c. $18.7 million
d. $19.5 million

208.3. According to Jorion, the delta-normal method assumes that portfolio exposures are linear and that the risk factors are jointly normally distributed. Each of the following is TRUE about the delta-normal method EXCEPT for which of the following?

a. The method is the simplest of the three basic methods, is also called the variance-covariance matrix method, and unlike the other two, is a local valuation method
b. As this method assumes the portfolio return is a linear combination of normal variables, the portfolio return itself is normally distributed
c. This method allows for the precise analytical expression for the incremental VaR of a portfolio of options
d. This method allows for the precise analytical expression of component and marginal VaR of a portfolio of futures contracts

Answers:

• Here in forum

 

[ashanks]

Think 208.2 is a well-framed question that actually demonstrates duration based VaR mapping in action. It's easy to get lost in too much descriptive text when there is no example. Attempting this question feels like I've improved my understanding. Thanks!

I'd be checking option b], with the calculation as follows:
The portfolio's single, aggregated measure of portfolio duration would be 4.2. i.e. the portfolio behaves (approximately) as if it was one bond of duration Interpolate the given graph to arrive at 4.09 as corresponding T-bond maturity. VaR is then 2%*4.09*200 ~16.4 MM.