. An annuity is payable annually in arrears for the lifetime of a woman currently aged exactly 68.
The first annuity payment, due in one year’s time, will be £2,000, and thereafter the payments will
increase at the rate of 2.5% per annum compound.
Calculate the expected present value of this annuity, assuming that mortality follows the PFA92C20
table and that the effective annual interest rate is 6.6%.

To calculate the expected present value (EPV) of the annuity, we need to take into account both the increasing annuity payments and the mortality of the annuitant. Here's how we can approach this problem step by step:

### 1. **Define the Variables and Formula:**

- Let \( a(t) \) be the annuity payment at time \( t \) (years from now).
- The first payment is \( a(1) = 2000 \).
- The payments increase at a compound rate of 2.5% per annum, so \( a(t) = 2000 \times (1.025)^{t-1} \) for \( t = 1, 2, 3, \dots \).
- Let \( v = \frac{1}{1 + i} \) be the discount factor, where \( i = 0.066 \) (6.6% effective annual interest rate).
- \( v = \frac{1}{1.066} \approx 0.938967 \).

The expected present value (EPV) of the annuity is given by:
\[
\text{EPV} = \sum_{t=1}^{\infty} a(t) \times v^t \times \, _tp_{68}
\]
where \( \, _tp_{68} \) is the probability that a woman aged 68 survives to age \( 68 + t \).

### 2. **Express the Payments and Discount Factor:**

\[
a(t) = 2000 \times (1.025)^{t-1}
\]
\[
v^t = (0.938967)^t
\]
So the EPV formula becomes:
\[
\text{EPV} = 2000 \times \sum_{t=1}^{\infty} (1.025)^{t-1} \times (0.938967)^t \times \, _tp_{68}
\]

### 3. **Simplify the EPV Formula:**

Factor out the constant:
\[
\text{EPV} = 2000 \times \sum_{t=1}^{\infty} \left(\frac{1.025}{1.066}\right)^{t-1} \times \, _tp_{68}
\]
Here, \( \frac{1.025}{1.066} \approx 0.96154 \). So the formula reduces to:
\[
\text{EPV} = 2000 \times \sum_{t=1}^{\infty} 0.96154^{t-1} \times v^t \times \, _tp_{68}
\]

### 4. **Use the PFA92C20 Mortality Table:**

The survival probabilities \( \, _tp_{68} \) are derived from the PFA92C20 table.

### 5. **Calculate the EPV:**

Using the formula:
\[
\text{EPV} = 2000 \times \sum_{t=1}^{\infty} 0.96154^{t-1} \times (0.938967)^t \times \, _tp_{68}
\]
This requires the use of life tables to get \( \, _tp_{68} \) values and then calculating the sum.

Given the mortality table and the survival probabilities \( \, _tp_{68} \) from the PFA92C20, we can plug in the values to compute the sum.

### 6. **Approximate EPV Calculation:**

Typically, this calculation would be performed using actuarial software or spreadsheet functions, but here's an outline of how to proceed:

- Look up the \( \, _tp_{68} \) values for each \( t \) using the PFA92C20 table.
- Compute \( 0.96154^{t-1} \times (0.938967)^t \times \, _tp_{68} \) for each \( t \).
- Sum these terms multiplied by 2000 to get the EPV.

**However, the exact value depends on the specific values from the PFA92C20 table, which I would need to perform the full calculation. If you have access to these values, you can sum the series using a tool like Excel or an actuarial calculator.**

Would you like to proceed with further steps, or do you have specific data from the mortality table? 

---

### 5 Related Questions:
1. How do survival probabilities change as age increases?
2. What is the significance of the PFA92C20 mortality table in actuarial calculations?
3. How does the interest rate affect the present value of future payments?
4. What is the impact of different annuity payment increase rates on the EPV?
5. How can this approach be generalized for annuities payable monthly instead of annually?

### Tip:
When working with increasing annuities and life tables, using actuarial software or a spreadsheet significantly simplifies the complex summation and multiplication involved.

Learn how to calculate the expected present value (EPV) of an increasing annuity with annual payments, starting at £2,000 and increasing at 2.5% compound interest per annum. This problem uses the PFA92C20 mortality table and an effective annual interest rate of 6.6% to determine the EPV. Suitable for advanced mathematics and actuarial science.

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