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The PCA allows simulating the values of interest rates. Assume that there is a small variation of the factors AP and AP2. We calculate the variations of all interest rates as linear functions of the variations of the two first factors, ignoring residuals since they have very small influences on interest rates. For example, considering the Euribor 3-month, E_m3, its standardized value is a linear function of the factors:

Generating random variations of P and P2 generates random variations of the standardized interest rate. Applying similar relationships between each interest rate and the same common two factors, all standardized interest rates are linear functions of P and P , with coefficients given in Table 37.2. If we generate random values of those two factors, we generate values of all 11 interest rates. They are dependent because they derive from the same factors, with different sensitivities. This principle allows generating entire term structures of the 11 interest rates by varying the two factors randomly.

Generating random values of P and P2 is extremely simple since those two factors are independent and standardized principal components. All we need to do is generate two series of random values of a uniform standard function U(0, 1) and take the normal standard inverse function of these random values. The two uniform standard values of Ul and U2, and the random values of Pj andP2 are:

For simulation k, the values of Pl and P2 are plk = <&~l{ulk) andp2k = <&~l{u2k)

For moving back from factor values to interest rates, we have to revert from the standardized value of interest rates to non-standardized values. The statistics of the interest rates provide the historical estimates of mean and standard deviation of each rate. The standardized variable derived from the non-standard variable Xis Xs — (X-[i)/c since Xs has mean zero and O equal to 1. Reversing the equation, we obtain the non-standard variable X— Xsc + [l. Since the Euribor 3-month is non-standard, and since we simulate its standardized value, the simulated values are obtained by the above transformation. The historical mean and standard deviations of each rate are required for converting standardized values to non-standardized values. Both the mean and the standard deviation of each interest rate are in Table 37.3. For the principal components, there is no need to transform them, since they serve for modeling standardized values of interest rates.

The algorithm uses the following steps, illustrated for the first simulations of the Euribor 3-month. The first step generates random values of the independent P and P , for example 0.340 and -1.313 in the first simulation run. Table 37.4 reproduces the first five simulations[1]. The standardized value of Euribor 3-month results from the linear function with principal components, where the second subscript"/" of the coefficients refers to the Euribor 3-month:

TABLE 37.3 Mean and standard deviation of interest rates

Mean and standard deviation of interest rates

TABLE 37.4 Simulations of the principal components

Simulations of the principal components

The value obtained with the first couple of simulated values of P and P2 is a standardized value of the Euribor 3-month. The non-standardized value of any one of the interest rates is / — isc(i) + where c(i) and |J.(0 are the standard deviation and mean of interest rates, and i is the standardized value of/. The first simulated value of the Euribor 3-month is:

The numerical values are:

The simulation of P and P2 are in Table 37.5.

Replacing Pj and P2 by the simulated values, respectively, 0.340 and -1.313, we find a simulated value of the Euribor 3-month in percentage:

The five first simulations of all eleven rates are in rows in Table 37.5. The simulated factor values are in the first two columns of the table and the 3 -month interest rate is the column labeled "EU_m3." The equations ensure that the interest rates are equal to their long-term mean values

TABLE 37.5 Simulations of interest rates from PCA

Simulations of interest rates from PCA

Simulation of zero interest rates

FIGURE 37.1 Simulation of zero interest rates

plus a random element obtained as the linear combination of the random factor values, using factor loadings as coefficients, multiplied by the standard deviation of the 3-month rate.

Using the same algorithm, for example 100 times, we obtain 100 randomly generated term structures of rates. Figure 37.1 shows the first 20 random term structures of interest rates. The graph provides a partial image of interest rate scenarios matching the fitting period used for determining the factor loadings.

  • [1] As usual, two independent uniform standard variables are generated and converted into standard normal variables using the normal standard inverse function.
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