A Husband and wife are both aged 65 exact. Upon wife’s death, husband will receive Rs.
20000 per annum payable annually in advance for the rest of his life starting from the end
of the year of wife’s death, provided her death occurs in the next 10 years.
Husband’s mortality follows PMA92C20 and wife’s mortality follows PFA92C20. Interest
rate is 4% for all future years. Calculate the EPV of benefit to husband.

To calculate the Expected Present Value (EPV) of the benefit to the husband, we need to consider the contingent nature of the benefit, which is paid upon the death of the wife within the next 10 years, provided the husband is still alive.

### Step-by-Step Calculation:

1. **Interest Rate Factor \( v \)**:
   \[
   v = \frac{1}{1 + i} = \frac{1}{1 + 0.04} = 0.9615
   \]

2. **Mortality Data**:
   - The husband's mortality follows the table \( PMA92C20 \).
   - The wife's mortality follows the table \( PFA92C20 \).

3. **Define the Benefit**:
   - Benefit of Rs. 20,000 per annum, payable annually in advance, starting at the end of the year in which the wife dies, for the remainder of the husband's life.
   - The wife must die within the next 10 years for the benefit to be payable.

4. **EPV Formula**:
   The EPV of the benefit can be written as:
   \[
   \text{EPV} = 20000 \times \sum_{k=1}^{10} \, v^k \, \, q_{65}^{\text{wife}}(k) \, \times \, \, \ddot{a}_{65+k}^{\text{husband}}
   \]
   where:
   - \( q_{65}^{\text{wife}}(k) \) is the probability that the wife dies in the \(k\)-th year.
   - \( \ddot{a}_{65+k}^{\text{husband}} \) is the present value of an annuity-due for the husband aged \( 65+k \) years at the time of the wife's death.

5. **Calculate the Individual Components**:
   - **Mortality of Wife**: Look up \( q_{65+k} \) from \( PFA92C20 \) table for each year \( k = 1 \) to \( 10 \).
   - **Present Value of Annuity-Due for Husband**: Calculate \( \ddot{a}_{65+k} \) using \( PMA92C20 \) for each year \( k = 1 \) to \( 10 \).
   - **Discount Factor**: Apply the discount factor \( v^k \) for each year \( k \).

6. **Sum the Contributions**:
   - For each year \( k \), multiply the probability of the wife dying in that year \( q_{65+k} \) by the present value of the annuity for the husband \( \ddot{a}_{65+k} \) and by the discount factor \( v^k \).
   - Sum these products over all 10 years.

### Example Calculation Framework (Without Mortality Tables):
Given that I don't have the specific mortality rates from the tables, the steps would be:

\[
\text{EPV} = 20000 \times \left( v^1 \times q_{65}^{\text{wife}}(1) \times \ddot{a}_{66}^{\text{husband}} + v^2 \times q_{65}^{\text{wife}}(2) \times \ddot{a}_{67}^{\text{husband}} + \dots + v^{10} \times q_{65}^{\text{wife}}(10) \times \ddot{a}_{75}^{\text{husband}} \right)
\]

### Summary:
To get the exact EPV, you would need the values of:
- \( q_{65+k} \) from \( PFA92C20 \) for the wife.
- \( \ddot{a}_{65+k} \) from \( PMA92C20 \) for the husband.

With these values, you can plug them into the EPV formula to get the final value.

Would you like to explore any of these steps in detail or ask additional questions?

### 5 Related Questions:
1. How do mortality rates change with age according to different mortality tables?
2. How would the EPV change if the annuity payments were made at the end of each year instead of in advance?
3. How does the selection of a different interest rate affect the EPV of the benefit?
4. Can the EPV of a benefit be calculated if mortality improvements are considered?
5. How do you compute the annuity value if the husband's mortality follows a different table?

**Tip:** When calculating EPVs, ensure that you are consistently using mortality rates and interest rates that align with the specific period and conditions of the benefit in question.

Learn how to calculate the Expected Present Value (EPV) of a benefit to the husband based on actuarial science principles. This problem involves using mortality tables PMA92C20 and PFA92C20, an interest rate of 4%, and an annuity benefit of Rs. 20,000 per annum. Suitable for actuarial professionals and students of actuarial science.

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