臺大管理論叢 NTU Management Review VOL.29 NO.2

Effect of Inflation on the Cost of Inflation-Linked Annuities Considering Stochastic Interest Rate and Inflation Rate Models 48 Effect of Inflation on the Cost of Inflation-Linked Annuities Considering Stochastic Interest Rate and Inflation Rate Models 1. Objective To protect the real income from retirement pensions against the risk of inflation, developing an annuity model that can be linked to the inflation index is necessary. Although inflation-linked annuities increase the cost of retirement pensions for governments, the risk posed by inflation to retirement annuities must be understood. In this paper, an inflation-linked annuity with a cap and floor model is developed. The model is a general form that can be reduced to special forms such as ordinary annuities, inflation- linked annuities without barriers, inflation-linked annuities with a cap model, and inflation-linked annuities with a floor model. Interest rate risk and inflation risk are considered within the model. Furthermore, we derive a closed-form solution for the model. The valuation framework can benefit annuity providers to understand the cost of inflation-linked annuities. 2. Methodology Consider an inflation-linked annuity: the annuity has annual floating rate payments at time t, which are linked to an inflation index and a survival probability. Therefore, if investors are still alive at time t, then they will obtain A×S(t, x 0 )×Y t , where A is a constant and S(t, x 0 ) is the survival probability in which a person aged x 0 at time t 0 is still alive at time t. Let Y t = (1 + β) l I t > 1+β + I t l 1+α ≤ I t ≤ 1+β + (1 + α) l I t < 1+α with constants α, β, and α < β. I t is the inflation index at time t. Thus, payments of inflation-linked annuities at time t can be written as Equation (1). , (1) for all j = 1, 2, ..., m, t 0 = 0, and t m = T (maturity). The payments can also be demonstrated as shown in Figure 1. Fen-Ying Chen , Department of Finance, Shih Hsin University Sharon S. Yang , Department of Finance, National Central University Hong-Chih Huang , Department of Risk Management and Insurance, National Chengchi University