calculate the value of S55:10| with mortality PFA92C20 and interest rate of 4% per annum

To calculate the value of the life annuity \(\ddot{S}_{55:\overline{10}|}\), which is a 10-year temporary annuity-due for a life aged 55, we need to use the given mortality table \(PFA92C20\) and an annual interest rate of 4%.

### Steps to calculate \(\ddot{S}_{55:\overline{10}|}\):

1. **Determine the interest rate factor \(v\)**:
   \[
   v = \frac{1}{1 + i} = \frac{1}{1 + 0.04} = \frac{1}{1.04} \approx 0.9615
   \]

2. **Determine the discount factors**:
   We use \(v^k\) for \(k = 1, 2, \ldots, 10\) where \(v^k = (0.9615)^k\).

3. **Obtain the survival probabilities** \(p_{55}, p_{56}, \ldots, p_{64}\) from the mortality table \(PFA92C20\).

4. **Calculate the present value of each annuity payment**:
   The present value of each payment is given by:
   \[
   \text{PV} = v^k \times p_{55} \times p_{56} \times \ldots \times p_{54+k}
   \]
   for \(k = 0, 1, \ldots, 9\).

5. **Sum up the present values** to get \(\ddot{S}_{55:\overline{10}|}\):
   \[
   \ddot{S}_{55:\overline{10}|} = \sum_{k=0}^{9} v^k \times p_{55+k}
   \]

However, without the actual survival probabilities from the \(PFA92C20\) table, the exact numerical calculation cannot be done here. You will need the specific values from the mortality table to perform this computation.

### Summary
1. **Interest rate factor**: \(v = 0.9615\)
2. **Mortality table**: \(PFA92C20\) (you would need to look up survival probabilities \(p_{55+k}\))
3. **Present value calculation**: \(v^k \times p_{55+k}\) for each year
4. **Final step**: Sum these present values to find \(\ddot{S}_{55:\overline{10}|}\).

Do you want details on any of these steps or have further questions? 

### 5 Related Questions:
1. How do you calculate the present value of a life annuity with varying payment amounts?
2. What is the impact of increasing the interest rate on the value of an annuity?
3. How do mortality improvements impact the value of an annuity?
4. Can you calculate the annuity value using a different mortality table?
5. How is the force of mortality \(\mu_x\) related to the survival probability \(p_x\)?

**Tip:** When working with actuarial calculations, it's essential to carefully select the correct mortality table and interest rate, as these have a significant impact on the annuity value.

Learn how to calculate the value of the life annuity \(\ddot{S}_{55:\overline{10}|}\), a 10-year temporary annuity-due for a life aged 55, using the mortality table PFA92C20 and an annual interest rate of 4%. Understand the steps involving discount factors, survival probabilities, and present value calculations. Suitable for actuarial science students and professionals.

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