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Interpolation of Interest Rate from Selected Risk Factors

Since the 1.25-year interest rate is not usable directly, we need to allocate the position to the time points retained as risk factors. How much value should we allocate to the 1-year time point and how much to the 2-year time point?

The first step is finding the value of the 1.25-year position, from interpolation of the 1.25-year interest rate from the 1-year and the 2-year interest rates. In order to find the appropriate interest rate applicable to the 1.25 years position, a linear interpolation provides a proxy of the 1.25 years interest rate:

With this rate, the value of the initial position is calculated.

Value Preservation

The value for the position using the proxy of the 1.25-year interest rate (/) is €1000,000/(1 + Ï)125, where / is the above interpolated rate from the two rates. The value is:

1000,000/(1 25 =938,052

Next we need to define the volatility of this position.

Volatility Preservation

The mapping process should preserve volatility. We know only the volatilities for the 1-year and 2-year bonds as percentages of their values. Using available 1-year and 2-year volatilities, we find the interpolated 1.25-year volatility.

The variance of the 1.25-year bond is 8.5%2 — 0.723%.

Mapping and Allocation

The final step is to determine the allocations of the initial position to the two equivalent 1-year and 2-year positions.

We could use the distance to the time points as the allocation rule, or allocate 0.25 x (1.25 -1) to 1 year and 0.75 x (2 - 1.25) to 2 years. But the variance of such a portfolio will not match the interpolated volatility of the 1.25-year position. The correct process consists of finding the allocation percentages that make the variance of the replicating portfolio of two positions identical to the interpolated volatility. If we allocate w to the 1 -year bond and 1 - w to the 2-year bond, the resulting variance is that of a portfolio of these two positions. The formula of portfolio variance is:

TABLE 35.4 Allocating weights and values to each reference position

Allocating weights and values to each reference position

Matching this variance with the interpolated variance implies that the coefficient w, with the volatilities of the bonds and the correlation coefficient of 90%, is such that this variance matches exactly the interpolated variance 0.723% (Table 35.4).

The allocation is w — 64.1% for the 1-year bond and 1 - w — 35.9% for the 2-year bond. Such an allocation applied to the interpolated value of the 1.25-year position, 93 8,042, provides the values of each of the two 1-year and 2-year positions.

The initial position is equivalent to these two equivalent positions. In the VaR calculation, the initial position will be replaced by the two equivalent positions, which depend on retained risk factors only, not on the 1.25-year interest rate which is not selected. Such allocation preserves value (interest rate interpolation) and volatility (volatility interpolation). The replicating portfolio is:

Since we have the percentage allocations, we can proceed with risk factors 1-year and 2-year. The sensitivities with respect to each interest rate are the percentage allocations multiplied by the sensitivities of the two elementary positions. The volatilities of each position are expressed as percentage allocations multiplied by the volatilities of the two elementary positions.

Next we proceed to a more general example of delta-VaR using as an example a forward contract on foreign exchange. The example shows the methodology for decomposition of a contract that depends on a non-linear relation with several risk factors.

 
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