Life Assurance Contracts PDF

# Whole Life Assurance Contracts | Product | PV | EPV | Variance | |---|---|---|---| | Benefits payable at the end of death year (Discrete) | $V_Kx_t$ for all $K_x$ | $\sum_{k=0}^\infty V_{k+1} k_{12}^*x$ | $2A_x - (A_x)^2$ | | $A_x$ | | | | | Benefits payable immediately on death (Continuous)...

# Whole Life Assurance Contracts | Product | PV | EPV | Variance | |---|---|---|---| | Benefits payable at the end of death year (Discrete) | $V_Kx_t$ for all $K_x$ | $\sum_{k=0}^\infty V_{k+1} k_{12}^*x$ | $2A_x - (A_x)^2$ | | $A_x$ | | | | | Benefits payable immediately on death (Continuous) | $V^tx$ for all $T_x$ | $\int_0^\infty v^t {}_tp_x \mu_{x+t} dt$ | $2A_x - (A_x)^2$ | | $A_x$ | | | | | Deferred Whole (Benefits at the end) (Discrete) | $S_0$ if $k \le n$ \ $V_k{x + k}_t$ if $k > n$ | $\sum_{k=n}^\infty V_{k+1}k_{12}^*x$ | $n^2A_x - (nA_x)^2$ | | $n|A_x$ | | $V^n {}n p_x A_{x+n}$ | | | Deferred Whole (Benefits payable immediately (Continuous) | $S_0$ if $T_x \le n$ $V^t_x$ if $T_x > n$| $\int_n^\infty v^t {}_tp_x \mu_{x+t} dt$ | $n^2A_x - (nA_x)^2$ | | $n|A_x$ | | | | # Term Assurance Contract | Product | PV | EPV | Variance | |---|---|---|---| | Benefits payable at the end of death year (Discrete) | $S_0$ if $k \le n$ \ $V_k{x + k}_t$ if $k > n$ | $\sum_{k=0}^n V_{k+1}k_{12}^*x$ | $2A_{x:n} - (A_{x:n})^2$ | | $A_{x:n}$ | | | | | Benefits payable immediately on death (Continuous) | $S_0$ if $T_x \le n$ \ $V^t_x$ if $T_x > n$| $\int_0^n v^t {}_tp_x \mu_{x+t} dt$ | $2A_{x:n} - (A_{x:n})^2$ | | $A_{x:n}$ | | | | | 'Deferred' benefits payable end of death year (Discrete) | $S_0$ if $0 \le k \le m$ $V_k{x + k}_t$ if $m \le k \le m+n$ \ $0$ if $k \ge m+n$ | $\sum_{k=m}^{m+n-1} V_{k+1}k_{12}^*x$ | $m^2A_{x:m}^2 - (mA_{x:m})^2$ | | $m|A_{x:m}$ | | $V^m {}mp_x A_{x+m:n}$ | | | 'Deferred' benefits payable immediately (Continuous) | $S_0$ if $T_x \le m$ $V^t_x$ if $m \le T_x \le m+n$ \ $0$ if $T_x \ge m+n$ | $\int_m^{m+n} v^t {}_tp_x \mu_{x+t} dt$ | $2A_{x:m} - (A_{x:m})^2$ | | $m|A_{x:m}$ | | $V^m {}mp_x A_{x+m:n}$| | # Endowment Life Assurance Contracts | Product | PV | EPV | Variance | |---|---|---|---| | Benefits payable at the end of death year (Discrete) | $S_0$ if $k \le n$ \ $V_k{x + k}_t$ if $k > n$ OR \ $V_n(min(k_x t_n))$ | $\sum_{k=0}^n V_{k+1}k_{12}^*x$ OR \ $V_nx + V_n{n}p_x(A_{x+n})$ | $2A_{x:n} - (A_{x:n})^2$ | | $A_{x:n}$ | $\overline{A}_x + \frac{1}{\delta}A_{x:n}$ | | | | Benefits payable immediately on death (Continuous) | $S_0$ if $T_x \le n$ \ $V^t_x$ if $T_x > n$ OR \ $V_n(min(T_x,n))$ | $\int_0^n v^t {}_tp_x \mu_{x+t} dt$ OR \ $V^nA_x + V^n{n}p_x A_{x+n\bar{n}}$ | $2A_{x:n} - (A_{x:n})^2$ | | $A_{x:n}$ | $\overline{A}_x + \frac{1}{\delta}A_{x:n}$ | | | | 'Deferred' benefits payable at the end of death year (Discrete) | $S_0$ if $k \le m$ \ $V_k{x + k}_t$ if $m \le k \le m+n$ $V_n$ if $k > m+n$ | $\sum_{k=m}^{m+n-1} V_{k+1}k_{12}^*x$ + $V^m{m}p_xA_{x+m:n}$ OR \ $(m^2A_{x:m}) + (A_{x+m:n})$ | $m^2A_{x:m} - (mA_{x:m})^2$ | | $m|A_{x:n}$ | | | | | 'Deferred' benefits payable immediately (Continuous) | $S_0$ if $T_x \le m$ \ $V^t_x$ if $m\le T_x \le m+n$ \ $V^n$ if $T_x \ge m+n$ OR \ $(m^2A_{x:m}) + (A_{x+m:n})$ | $\int_m^{m+n} v^t {}_tp_x \mu_{x+t} dt$ + ${n+m}p_x A_{x+m:n}$ OR \ $(m^2A_{x:m}) + (A_{x+m:n})$ | $m^2A_{x:m} - (mA_{x:m})^2$ | | $m|A_{x:n}$ | | | | # Pure Endowment Assurance Contract | Product | PV | EPV | Variance | |---|---|---|---| | Benefits payable at the end of death year (Discrete) | $S_0$ if $k \le n$ \ $V_n$ if $k> n$ | $V^n{n}p_x$ | $2A_{x:n} - (A_{x:n})^2$ | | $A_{x:n}$ | | | | **Note**: The benefits under pure endowment assurance is paid at fixed time period, it doesn't depend on the survival of life. So, the benefits can't be paid earlier than the maturity date. So, there is no need for a random variable of 'benefits paid immediately'. Also, there is no need to derive the 'deferred pure endowment assurance' separately as it would be the same as 'Pure Endowment'.