Hattendorff's Theorem, attributed to K. Hattendorff (1868), is a theorem in actuarial science that describes the allocation of the variance or risk of the loss random variable over the lifetime of an actuarial reserve. In other words, Hattendorff's theorem demonstrates that the variation in the present value of the loss of an issued insurance policy can be allocated to the future years during which the insured is still alive. This, in turn, facilitates the management of risk prevalent in such insurance contracts over short periods of time.
Hattendorff's Theorem The main result of the theorem has three equivalent formulations:
where:
In its above formulation, and in particular the first result, Hattendorff's theorem states that the variance of L h {\displaystyle L_{h}} , the insurer's total loss over the remaining life of the policy at time h, can be calculated by discounting the variances of the yearly net losses (cash losses plus changes in net liabilities) Λ k {\displaystyle \Lambda _{k}} in future years.
Background Source: In the most general stochastic setting in which the analysis of reserves is carried out, consider an insurance policy written at time zero, over which the insured pays yearly premiums π 0 , π 1 … π K ( x ) {\displaystyle \pi _{0},\pi _{1}\dots \pi _{K(x)}} at the beginning of each year starting today until the year of death of the insured. Furthermore, the insured receives a benefit of K ( x ) + 1 {\displaystyle K(x)+1} , at the end of the year of death, equal to b K ( x ) + 1 {\displaystyle b_{K(x)+1}} . No other payments are received nor paid over the lifetime of the policy. Suppose an insurance company is interested to know the cash loss from this policy over the year (h, h+1). Of course, if the death of the insured happens prior to time h, or when K ( x ) < h {\displaystyle K(x)<h} , then there is no remaining loss and C h = 0 {\displaystyle C_{h}=0} . If the death of the insured occurs exactly at time h, or when K ( x ) = h {\displaystyle K(x)=h} , then the loss on the policy is equal to the present value of the benefit paid in the following year, v b h + 1 {\displaystyle vb_{h+1}} , less the premium paid at time h. Hence in this case C h = v b h + 1 − π h . {\displaystyle C_{h}=vb_{h+1}-\pi _{h}.} Lastly, if the death of the insured occurs after time h, or when K ( x ) > h {\displaystyle K(x)>h} , then the cash loss in the year (h, h+1) is just the negative of the premium received at time h (cash inflows are treated as negative losses). Hence we summarize this result as
C h = { 0 if K ( x ) = 0 , 1 … h − 1 v b h + 1 − π h if K ( x ) = h − π h if K ( x ) = h + 1 , h + 2 … {\displaystyle C_{h}={\begin{cases}0&{\mbox{if }}K(x)=0,1\dots h-1\\vb_{h+1}-\pi _{h}&{\mbox{if }}K(x)=h\\-\pi _{h}&{\mbox{if }}K(x)=h+1,h+2\dots \\\end{cases}}}
Furthermore, the actuarial present value of the future cash losses in each year has the explicit formula
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