The methodology that we described in this chapter agrees quite closely with that
presented in RiskMetrics, although we deviate from it on some points of details. In
this appendix, we discuss the motivations for some of these deviations.
Mapping of non-linear instruments
In the previous appendix we described a general approach to treating instruments
whose value depends non-linearly on the primary assets. This approach involved
computing a first-order Taylor series expansion in what we termed PAVs. The coefficients of this expansion gave positions in primary assets that comprised a firstorder
proxy portfolio to the instrument in question. The RiskMetrics Technical
Document (Morgan and Reuters, 1996) describes a similar approach to mapping
non-linear FX and/or interest-rate derivatives. The main difference is that the firstorder
Taylor series expansion of the PV function is taken with respect to FX rates
and zero-coupon discount factors (ZCDF). Using this expansion, it is possible to
construct a portfolio of proxy instruments whose risk is, to first order, equivalent.
This approach, termed the delta approximation, is illustrated by a number of relatively
simple examples in Morgan and Reuters (1996).
While the approach described in Morgan and Reuters (1996) is workable, it has a
number of significant drawbacks relative to the DEAF approach:
1 As will be seen below, the proxy instrument constructed to reflect risk with
respect to ZCDFs in foreign currencies does not correspond to a commonly traded
asset; it is effectively a zero-coupon bond in a foreign currency whose FX risk has
been removed. In contrast, the proxy instrument for the DEAF approach, a fixed
cashflow, is both simple and natural.
2 Different proxy types are used for FX and interest-rate risk. In contrast, the DEAF
approach captures both FX and interest-rate risk with a single proxy type.
3 The value of some primary instruments is non-linear in the base variables, with
the proxy-position depending on current market data. This means they need to
be recomputed whenever the market data changes. In contrast, primary instruments
are linear in the DEAF approach, so the proxy positions are independent
of current market data.
4 The expansion is base-currency dependent, and thus needs to be recomputed
every time the base currency is changed. In contrast, proxies in the DEAF
approach are base-currency independent.
In general, when an example of first-order deal proxies is given in Morgan and
Reuters (1996), the basic variable is taken to be the market price. This is stated in
Morgan and Reuters (1996, table 6.3), where the underlying market variables are
stated to be FX rates, bond prices, and stock prices. For example, in Morgan and
Reuters (1996, §1.2.2.1), the return on a DEM put is written as dDEM/USD, where
rDEM/USD is the return on the DEM/USD exchange rate and d is the delta for the
option.4
Consider the case of a GBP-denominated zero-coupon bond with a maturity of q
years. For a USD-based investor, the PV of this bond is given by
vt0óUSDNXGBP/USD(t0)DGBP
q (t0)
where XGBP/USD is the (unitless) GBP/USD exchange rate, DGBP
q (t0) denotes the q-year
ZCDF for GBP, and N is the principal amount of the bond. We see that the PV is a
non-linear function of the GBP/USD exchange rate and the ZCDF for GBP. (While
the valuation function is a linear function of the risk factors individually, it is
quadratic in the set of risk factors as a whole.) Expanding the valuation function in
a first-order Taylor series in these variables, we get
This says that the delta-equivalent position is a spot GBP position of NDGBP
T and a
q-year GBP N ZCDF position with an exchange rate locked at XGBP/USD. Note that
these positions are precisely what one would obtain by following the standard
mapping procedure for foreign cashflows in RiskMetrics (see example 2 in Morgan
and Reuters, 1996). This non-linearity contrasts with the DEAF formulation, where
we have seen that the value of this instrument is a linear function of the PAV GBPT(T)
and the delta-equivalent position is just a fixed cashflow of GBP N at time T.
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