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

To calculate the value of the temporary annuity \( S_{55:10|} \), which represents the present value of an annuity certain payable annually in arrears for a maximum of 10 years, assuming the annuitant is currently aged 55, we need to use the following steps:

### Steps to Calculate \( S_{55:10|} \):

1. **Identify the Present Value Factor for an Annuity:**
   - \( S_{55:10|} \) is the present value of an annuity of 1 unit per year payable for 10 years, starting at age 55, discounted at a 4% per annum interest rate, and considering mortality based on the PFA92C20 table.

2. **Mortality Adjusted Annuity Calculation:**
   - The calculation is based on the probability of survival and the interest rate.

3. **Formula:**
   - \( S_{55:10|} \) is calculated as:
     \[
     S_{55:10|} = \sum_{k=1}^{10} v^k \cdot \,_{k}p_{55}
     \]
     Where:
     - \( v = \frac{1}{1 + i} \) is the discount factor, with \( i = 0.04 \) (4% per annum).
     - \( _{k}p_{55} \) is the probability that a person aged 55 survives for \( k \) years.

4. **Look Up \( _{k}p_{55} \) from PFA92C20:**
   - Using the PFA92C20 mortality table, find the values of \( _{k}p_{55} \) for \( k = 1 \) to 10.

5. **Compute the Sum of the Present Values:**
   - For each year \( k \), calculate \( v^k \cdot \,_{k}p_{55} \) and sum these values to get \( S_{55:10|} \).

### Calculation:

1. **Discount Factor:**
   \[
   v = \frac{1}{1.04} = 0.96154
   \]

2. **Survival Probabilities \( _{k}p_{55} \) from PFA92C20:**
   - Let's assume the following approximate values (since I don't have the exact table):
   \[
   \begin{aligned}
   _{1}p_{55} &\approx 0.995 \\
   _{2}p_{55} &\approx 0.990 \\
   _{3}p_{55} &\approx 0.985 \\
   _{4}p_{55} &\approx 0.980 \\
   _{5}p_{55} &\approx 0.975 \\
   _{6}p_{55} &\approx 0.970 \\
   _{7}p_{55} &\approx 0.965 \\
   _{8}p_{55} &\approx 0.960 \\
   _{9}p_{55} &\approx 0.955 \\
   _{10}p_{55} &\approx 0.950 \\
   \end{aligned}
   \]

3. **Calculate Each Term:**

\[
\begin{aligned}
S_{55:10|} & = 0.96154^1 \times 0.995 + 0.96154^2 \times 0.990 + 0.96154^3 \times 0.985 + \dots + 0.96154^{10} \times 0.950 \\
& = 0.95673 + 0.92013 + 0.88379 + 0.84768 + 0.81180 + 0.77613 + 0.74067 + 0.70540 + 0.67031 + 0.63540
\end{aligned}
\]

4. **Sum the Values:**
\[
S_{55:10|} \approx 7.94704
\]

### Conclusion:

The value of \( S_{55:10|} \) is approximately **7.94704**.

### Next Steps:

Would you like to verify this value using exact probabilities from the PFA92C20 table, or do you have any questions about the calculation?

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**Further Questions:**

1. How would the value of \( S_{55:10|} \) change with a different interest rate?
2. What is the difference between \( S_{55:10|} \) and a whole life annuity?
3. How does the mortality table impact the value of annuities?
4. What other factors influence the pricing of life annuities?
5. How would you calculate the present value of an annuity certain without considering mortality?

**Tip:** Always ensure to use the correct mortality table and interest rate when calculating annuities to obtain accurate values.

Learn how to calculate the present value of annuity S55:10|, which represents an annuity certain payable annually for up to 10 years starting at age 55, using the PFA92C20 mortality table and a 4% per annum interest rate. Includes step-by-step calculation with survival probabilities.

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