The Kriging Weights

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The Kriging weights w_i1,w_i2,w_i3,…w_ik can be used to estimate the fair value of VA contracts x ⃑_i by the following formula, y ̂_i=∑_(j=1)^k▒〖w_ij∙y_j 〗.
The kriging weights can be calculated by the following linear equations, Where the is a control variable, which is used to make sure that ∑_(j=1)^k▒w_ij =1.

V_rs=α+exp⁡(-3/β D(z ⃑_r,z ⃑_s,λ)),r,s=1,2,3,…,k,
D_ij=α+exp⁡(-3/β D(x ⃑_i,z ⃑_j,λ)),j=1,2,3,…,k,

The D(.) function is the distance function mentioned in the clustering section,α≥0 and β≥0 are two parameters. The above linear equation system has a unique solution because D(z ⃑_r,z ⃑_s,λ)>0 for all r,s=1,2,3,…,k,.

Then the fair value of whole portfolio X=(x ⃑_1,x ⃑_2,x ⃑_3,…,x ⃑_n)can be estimated by the following formula,
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Also, this linear equation system is the sum of both sides of the previous linear equation system fromi=1,2,3,…,n. It is more efficient because we only need to solve one linear equation system instead of solve n linear equation systems and then sum the results.
3 Proposed Alternatives to Step 1

In this section, some alternatives to step1 are introduced and tested. Clustering is a data mining technique that is used for observing interesting objects, and it is an unsupervised learning algorithms comparing with classification (Ahmad & Dey, 2007). More similar objectives are partitioned into same cluster, and interesting objectives may be discovered. The k-prototypes algorithm generalizes the k-means and k-modes algorithms, which is efficient in clustering large data sets with mixed numerical and categorical values.

The clustering part is time consuming in the three steps method for valuation of large portfolio of VAs at most of the time. Speed and accuracy are two important performance measures of the pricing method for a large portfolio. Different data clustering methods are
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The only difference is how the representative contracts are selected. The k-centroids μ ⃑_1,μ ⃑_2,μ ⃑_3,…,μ ⃑_kare obtained by k-prototype algorithm. Then, for the MAS, the representative contracts are selected from the whole portfolio. For the MWS method, the representative contracts are selected from each subset of the portfolio X=(x ⃑_1,x ⃑_2,x ⃑_3,…,x ⃑_n). For example, 100000 contracts are divided into 20 subsets, each subset has 5000 contracts. For the MAS method, we obtain 5 centroids in the subset, then choose 5 closest contracts from the whole portfolio X=(x ⃑_1,x ⃑_2,x ⃑_3,…,x ⃑_n ) that contains 100000 contracts. For the MWS method, we obtain 5 centroids in the subset, then choose 5 closest contracts from the 5000 contracts in the same subset as representative contracts. The MWS method can save a lot of time comparing with the MAS method by avoiding a lot of distance calculations in the nearest neighbor mapping closest contract step. However, the closest contracts in the MAS method usually have shorter distance to the centroids. The 5000 contracts in each subset are also included in the whole portfolio X=(x ⃑_1,x ⃑_2,x ⃑_3,…,x ⃑_n), and it is possible that exists contracts with shorter distance to the centroids from other subsets. This impact on accuracy is tested in Table 4 and Table 5. For example, in Figure 2, if the centroid is denoted as C and the closest contract in the same subset is B. The contract

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