Time Consistent Strategies for Mean-Variance Asset-Liability Management Problems

Table 1

Model notations of Section 3.

For 𝑡=𝑇−1:	For 𝑡=𝑇−2,…,0:
𝜆^1𝑇−1=𝜆𝑇−1	𝜆^1𝑡=𝜆𝑡+𝜆^1𝑡+1𝑞𝑡+1
Σ^1𝑇−1=Σ𝑇−1	Σ^1𝑡=Σ𝑡+𝐸(𝑅𝑡𝑅^′𝑡)/𝑎𝑡+1
𝑎𝑇−1=𝐼^′Σ^1𝑇−1^−1𝐼	𝑎𝑡=𝐼^′Σ^1𝑡^−1𝐼
𝑏𝑇−1=𝐼^′Σ^1𝑇−1^−1𝜇𝑇−1	𝑏𝑡=𝐼^′Σ^1𝑡^−1𝜇𝑡
𝑐𝑇−1=𝜇^′𝑇−1Σ^1𝑇−1^−1𝜇𝑇−1	𝑐𝑡=𝜇^′𝑡Σ^1𝑡^−1𝜇𝑡
𝑑𝑇−1=𝐼^′Σ^1𝑇−1^−1Σ^0𝑇−1	𝑑𝑡=𝐼^′Σ^1𝑡^−1Σ^0𝑡
𝑞𝑇−1=𝑏𝑇−1/𝑎𝑇−1	𝑞𝑡=𝑏𝑡/𝑎𝑡
𝛼𝑇−1=𝑐𝑇−1−𝑞𝑇−1𝑏𝑇−1	𝛼𝑡=𝑐𝑡−𝑞𝑡𝑏𝑡
𝛾𝑇−1=𝑑𝑇−1/𝑎𝑇−1	𝛾𝑡=𝑑𝑡/𝑎𝑡
𝛾𝑇−1=𝛾𝑇−1	𝛾𝑡=𝛾𝑡+𝛾𝑡+1𝐸(𝑄𝑡𝑅^′𝑡)Σ^1𝑡^−1𝐼/𝑎𝑡
𝛽1,𝑇−1=𝜆𝑇−1𝐸(𝑄𝑇−1)−𝜆𝑇−1Σ^0𝑇−1^′Σ^1𝑇−1^−1𝜇𝑇−1	𝛽1,𝑡=−𝜆^1𝑡𝜇^′𝑡Σ^1𝑡^−1(Σ^0𝑡−𝛾𝑡𝐼)−𝜆^1𝑡𝛾𝑡+1𝜇^′𝑡Σ^1𝑡^−1𝐸(𝑄𝑡𝑅𝑡)
+𝜆𝑇−1𝑏𝑇−1𝛾𝑇−1	+(𝜆𝑡+𝛽1,𝑡+1)𝐸(𝑄𝑡)
𝛽2,𝑇−1=Var(𝑄𝑇−1)−Σ^0𝑇−1^′Σ^1𝑇−1^−1Σ^0𝑇−1+𝛾𝑇−1𝑑𝑇−1	𝛽2,𝑡=Var(𝑄𝑡)+𝛽2,𝑡+1𝐸(𝑄^2𝑡)+𝑎𝑡𝛾^2𝑡−Σ^0𝑡+𝛾𝑡+1𝐸(𝑄𝑡𝑅𝑡)^′
	⋅(Σ^1𝑡)^−1[Σ^0𝑡+𝛾𝑡+1𝐸(𝑄𝑡𝑅𝑡)]