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Give step-by-step solution with explanation and final answer: Solution: This problem requires the calculation of policy reserves at different durations for a fully discrete 10-pay 20-year term life insurance policy. The policy has a death benefit of 10,000, issued to an individual aged 30. The calculations will use the Illustrative Life Table (ILT) with an interest rate of $i = 5\%$. The reserve calculations will be performed using the prospective method. The general formula for the reserve at time $t$, denoted as $_tV$, is: $$_tV = (\text{APV of future benefits at } t) - (\text{APV of future premiums at } t)$$ where APV stands for Actuarial Present Value. First, we must determine the annual net premium, $P$, using the equivalence principle at the time of issue ($t=0$). The APV of premiums must equal the APV of benefits. ### 1. Calculation of the Annual Premium (P) The equivalence principle states that the APV of premiums equals the APV of benefits at issue. $$P \cdot \ddot{a}_{30:\overline{10}|} = 10,000 \cdot A_{30:\overline{20}|}^{1}$$ The annual premium $P$ is therefore: $$P = 10,000 \cdot \frac{A_{30:\overline{20}|}^{1}}{\ddot{a}_{30:\overline{10}|}}$$ Using commutation functions from the ILT at $i=5\%$: - $A_{30:\overline{20}|}^{1} = \frac{M_{30} - M_{50}}{D_{30}}$ - $\ddot{a}_{30:\overline{10}|} = \frac{N_{30} - N_{40}}{D_{30}}$ Thus, the premium is calculated as: $$P = 10,000 \cdot \frac{M_{30} - M_{50}}{N_{30} - N_{40}}$$ The required values from the ILT at 5% are: - $M_{30} = 3593.75$ - $M_{50} = 3213.00$ - $N_{30} = 561842$ - $N_{40} = 370524$ $\begin{aligned} P &= 10,000 \cdot \frac{3593.75 - 3213.00}{561842 - 370524} \\ &= 10,000 \cdot \frac{380.75}{191318} \\ &\approx 19.90 \end{aligned}$ ### 2. Calculation of the Reserve at t=1 ($_{1}V$) At $t=1$, the insured is age 31. There are 9 remaining premium payments and 19 years left on the term of the insurance. The reserve is: $$_{1}V = 10,000 \cdot A_{31:\overline{19}|}^{1} - P \cdot \ddot{a}_{31:\overline{9}|}$$ Using commutation functions: $$_{1}V = 10,000 \cdot \frac{M_{31} - M_{50}}{D_{31}} - P \cdot \frac{N_{31} - N_{40}}{D_{31}}$$ The required values from the ILT at 5% are: - $M_{31} = 3578.96$ - $N_{31} = 530408$ - $D_{31} = 30045.6$ - $M_{50} = 3213.00$ - $N_{40} = 370524$ $\begin{aligned} _{1}V &= 10,000 \cdot \frac{3578.96 - 3213.00}{30045.6} - 19.9014 \cdot \frac{530408 - 370524}{30045.6} \\ &= 10,000 \cdot \frac{365.96}{30045.6} - 19.9014 \cdot \frac{159884}{30045.6} \\ &= 121.80 - 105.90 \\ &\approx 15.90 \end{aligned}$ ### 3. Calculation of the Reserve at t=10 ($_{10}V$) At $t=10$, the insured is age 40. The premium payment period has just ended. There are 10 years remaining on the term of the insurance. The reserve is: $$_{10}V = 10,000 \cdot A_{40:\overline{10}|}^{1}$$ Using commutation functions: $$_{10}V = 10,000 \cdot \frac{M_{40} - M_{50}}{D_{40}}$$ The required values from the ILT at 5% are: - $M_{40} = 3443.08$ - $D_{40} = 19133.3$ - $M_{50} = 3213.00$ $\begin{aligned} _{10}V &= 10,000 \cdot \frac{3443.08 - 3213.00}{19133.3} \\ &= 10,000 \cdot \frac{230.08}{19133.3} \\ &\approx 120.25 \end{aligned}$ ### 4. Calculation of the Reserve at t=18 ($_{18}V$) At $t=18$, the insured is age 48. The premium payment period has ended. There are 2 years remaining on the term of the insurance. The reserve is: $$_{18}V = 10,000 \cdot A_{48:\overline{2}|}^{1}$$ Using commutation functions: $$_{18}V = 10,000 \cdot \frac{M_{48} - M_{50}}{D_{48}}$$ The required values from the ILT at 5% are: - $M_{48} = 3280.40$ - $D_{48} = 12533.1$ - $M_{50} = 3213.00$ $\begin{aligned} _{18}V &= 10,000 \cdot \frac{3280.40 - 3213.00}{12533.1} \\ &= 10,000 \cdot \frac{67.40}{12533.1} \\ &\approx 53.78 \end{aligned}$ The calculated reserves are: - Reserve at t=1: **15.90** - Reserve at t=10: **120.25** - Reserve at t=18: **53.78** $\boxed{_{1}V \approx 15.90, \quad _{10}V \approx 120.25, \quad _{18}V \approx 53.78}$ can you rewite this as it is act like a chegg tutor

Question:

Give step-by-step solution with explanation and final answer: Solution: This problem requires the calculation of policy reserves at different durations for a fully discrete 10-pay 20-year term life insurance policy. The policy has a death benefit of 10,000, issued to an individual aged 30. The calculations will use the Illustrative Life Table (ILT) with an interest rate of $i = 5\%$. The reserve calculations will be performed using the prospective method. The general formula for the reserve at time $t$, denoted as $_tV$, is: $$_tV = (\text{APV of future benefits at } t) - (\text{APV of future premiums at } t)$$ where APV stands for Actuarial Present Value. First, we must determine the annual net premium, $P$, using the equivalence principle at the time of issue ($t=0$). The APV of premiums must equal the APV of benefits. ### 1. Calculation of the Annual Premium (P) The equivalence principle states that the APV of premiums equals the APV of benefits at issue. $$P \cdot \ddot{a}_{30:\overline{10}|} = 10,000 \cdot A_{30:\overline{20}|}^{1}$$ The annual premium $P$ is therefore: $$P = 10,000 \cdot \frac{A_{30:\overline{20}|}^{1}}{\ddot{a}_{30:\overline{10}|}}$$ Using commutation functions from the ILT at $i=5\%$: - $A_{30:\overline{20}|}^{1} = \frac{M_{30} - M_{50}}{D_{30}}$ - $\ddot{a}_{30:\overline{10}|} = \frac{N_{30} - N_{40}}{D_{30}}$ Thus, the premium is calculated as: $$P = 10,000 \cdot \frac{M_{30} - M_{50}}{N_{30} - N_{40}}$$ The required values from the ILT at 5% are: - $M_{30} = 3593.75$ - $M_{50} = 3213.00$ - $N_{30} = 561842$ - $N_{40} = 370524$ $\begin{aligned} P &= 10,000 \cdot \frac{3593.75 - 3213.00}{561842 - 370524} \\ &= 10,000 \cdot \frac{380.75}{191318} \\ &\approx 19.90 \end{aligned}$ ### 2. Calculation of the Reserve at t=1 ($_{1}V$) At $t=1$, the insured is age 31. There are 9 remaining premium payments and 19 years left on the term of the insurance. The reserve is: $$_{1}V = 10,000 \cdot A_{31:\overline{19}|}^{1} - P \cdot \ddot{a}_{31:\overline{9}|}$$ Using commutation functions: $$_{1}V = 10,000 \cdot \frac{M_{31} - M_{50}}{D_{31}} - P \cdot \frac{N_{31} - N_{40}}{D_{31}}$$ The required values from the ILT at 5% are: - $M_{31} = 3578.96$ - $N_{31} = 530408$ - $D_{31} = 30045.6$ - $M_{50} = 3213.00$ - $N_{40} = 370524$ $\begin{aligned} _{1}V &= 10,000 \cdot \frac{3578.96 - 3213.00}{30045.6} - 19.9014 \cdot \frac{530408 - 370524}{30045.6} \\ &= 10,000 \cdot \frac{365.96}{30045.6} - 19.9014 \cdot \frac{159884}{30045.6} \\ &= 121.80 - 105.90 \\ &\approx 15.90 \end{aligned}$ ### 3. Calculation of the Reserve at t=10 ($_{10}V$) At $t=10$, the insured is age 40. The premium payment period has just ended. There are 10 years remaining on the term of the insurance. The reserve is: $$_{10}V = 10,000 \cdot A_{40:\overline{10}|}^{1}$$ Using commutation functions: $$_{10}V = 10,000 \cdot \frac{M_{40} - M_{50}}{D_{40}}$$ The required values from the ILT at 5% are: - $M_{40} = 3443.08$ - $D_{40} = 19133.3$ - $M_{50} = 3213.00$ $\begin{aligned} _{10}V &= 10,000 \cdot \frac{3443.08 - 3213.00}{19133.3} \\ &= 10,000 \cdot \frac{230.08}{19133.3} \\ &\approx 120.25 \end{aligned}$ ### 4. Calculation of the Reserve at t=18 ($_{18}V$) At $t=18$, the insured is age 48. The premium payment period has ended. There are 2 years remaining on the term of the insurance. The reserve is: $$_{18}V = 10,000 \cdot A_{48:\overline{2}|}^{1}$$ Using commutation functions: $$_{18}V = 10,000 \cdot \frac{M_{48} - M_{50}}{D_{48}}$$ The required values from the ILT at 5% are: - $M_{48} = 3280.40$ - $D_{48} = 12533.1$ - $M_{50} = 3213.00$ $\begin{aligned} _{18}V &= 10,000 \cdot \frac{3280.40 - 3213.00}{12533.1} \\ &= 10,000 \cdot \frac{67.40}{12533.1} \\ &\approx 53.78 \end{aligned}$ The calculated reserves are: - Reserve at t=1: **15.90** - Reserve at t=10: **120.25** - Reserve at t=18: **53.78** $\boxed{_{1}V \approx 15.90, \quad _{10}V \approx 120.25, \quad _{18}V \approx 53.78}$ can you rewite this as it is act like a chegg tutor

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Asked by: Nani
Created at: 2025-08-25 18:38:52

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Absolutely! Here’s a rewrite in the style of a Chegg tutor, step-by-step, friendly, and clear. --- **Problem Restatement:** We have a fully discrete 10-pay, 20-year term insurance policy with a death benefit of \$10,000, issued at age 30. We are to calculate the policy reserves at different durations using the prospective method and the Illustrative Life Table (ILT) at 5% interest. Specifically, we need the reserves at times \( t = 1, 10, 18 \). --- ## Step 1: Find the Annual Net Premium (P) First, let's find the annual net premium using the **equivalence principle**. The present value of premiums paid over 10 years equals the present value of benefits payable over the 20-year term. **Formula:** \[ P \cdot \ddot{a}_{30:\overline{10}|} = 10{,}000 \cdot A_{30:\overline{20}|}^{1} \] So, \[ P = 10{,}000 \cdot \frac{A_{30:\overline{20}|}^{1}}{\ddot{a}_{30:\overline{10}|}} \] We use **commutation functions** to express these actuarial present values: - \( A_{x:\overline{n}|}^1 = \frac{M_x - M_{x+n}}{D_x} \) - \( \ddot{a}_{x:\overline{m}|} = \frac{N_x - N_{x+m}}{D_x} \) For this problem: - \( x = 30 \), \( n = 20 \), \( m = 10 \) **Given ILT Values:** - \( M_{30} = 3593.75 \) - \( M_{50} = 3213.00 \) - \( N_{30} = 561842 \) - \( N_{40} = 370524 \) - \( D_{30} = 29540.7 \) (not needed for premium, as it cancels out) Plugging in: \[ A_{30:\overline{20}|}^{1} = \frac{3593.75 - 3213.00}{D_{30}} \] \[ \ddot{a}_{30:\overline{10}|} = \frac{561842 - 370524}{D_{30}} \] So, \[ P = 10,000 \cdot \frac{3593.75 - 3213.00}{561842 - 370524} = 10,000 \cdot \frac{380.75}{191318} \approx 19.90 \] **So the annual net premium is:** \[ \boxed{P \approx 19.90} \] --- ## Step 2: Reserve at \( t = 1 \) (\( _1V \)) At \( t = 1 \), the insured is now age 31. There are 19 years of coverage left, and 9 premiums left to pay. **Reserve formula:** \[ _{1}V = (\text{APV of future benefits at 31 over 19 years}) - (\text{APV of future premiums at 31 over 9 years}) \] So, \[ _{1}V = 10,000 \cdot A_{31:\overline{19}|}^{1} - P \cdot \ddot{a}_{31:\overline{9}|} \] In commutation function form: \[ _{1}V = 10,000 \cdot \frac{M_{31} - M_{50}}{D_{31}} - P \cdot \frac{N_{31} - N_{40}}{D_{31}} \] **ILT values:** - \( M_{31} = 3578.96 \) - \( M_{50} = 3213.00 \) - \( N_{31} = 530408 \) - \( N_{40} = 370524 \) - \( D_{31} = 30045.6 \) Plug in: \[ _{1}V = 10,000 \cdot \frac{3578.96 - 3213.00}{30045.6} - 19.90 \cdot \frac{530408 - 370524}{30045.6} \] \[ = 10,000 \cdot \frac{365.96}{30045.6} - 19.90 \cdot \frac{159884}{30045.6} \] \[ = 10,000 \cdot 0.01218 - 19.90 \cdot 5.324 \] \[ = 121.80 - 105.90 \] \[ \approx 15.90 \] **So, the reserve at \( t = 1 \) is:** \[ \boxed{_{1}V \approx 15.90} \] --- ## Step 3: Reserve at \( t = 10 \) (\( _{10}V \)) At \( t = 10 \), the insured is now age 40. The premium payment period has ended. There are 10 coverage years left. **Reserve formula:** \[ _{10}V = 10,000 \cdot A_{40:\overline{10}|}^{1} = 10,000 \cdot \frac{M_{40} - M_{50}}{D_{40}} \] **ILT values:** - \( M_{40} = 3443.08 \) - \( M_{50} = 3213.00 \) - \( D_{40} = 19133.3 \) Plug in: \[ _{10}V = 10,000 \cdot \frac{3443.08 - 3213.00}{19133.3} = 10,000 \cdot \frac{230.08}{19133.3} \] \[ = 10,000 \cdot 0.01203 \] \[ = 120.25 \] **So, the reserve at \( t = 10 \) is:** \[ \boxed{_{10}V \approx 120.25} \] --- ## Step 4: Reserve at \( t = 18 \) (\( _{18}V \)) At \( t = 18 \), the insured is now age 48. No premiums are left to pay, and only 2 coverage years remain. **Reserve formula:** \[ _{18}V = 10,000 \cdot A_{48:\overline{2}|}^{1} = 10,000 \cdot \frac{M_{48} - M_{50}}{D_{48}} \] **ILT Values:** - \( M_{48} = 3280.40 \) - \( M_{50} = 3213.00 \) - \( D_{48} = 12533.1 \) Plug in: \[ _{18}V = 10,000 \cdot \frac{3280.40 - 3213.00}{12533.1} = 10,000 \cdot \frac{67.40}{12533.1} \] \[ = 10,000 \cdot 0.00538 \] \[ = 53.78 \] **So, the reserve at \( t = 18 \) is:** \[ \boxed{_{18}V \approx 53.78} \] --- ## **Final Answers (Summary Table):** | Time (\( t \)) | Reserve (\( _tV \)) | |:--------------:|:-------------------:| | 1 | 15.90 | | 10 | 120.25 | | 18 | 53.78 | **Boxed Answers:** \[ \boxed{ _{1}V \approx 15.90,\quad _{10}V \approx 120.25,\quad _{18}V \approx 53.78 } \] --- **Let me know if you need more explanation or if you want to see the retrospective method as well!**

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