Welcome to Expected Accumulations!
In your actuarial journey so far, you have spent a lot of time "discounting" future cashflows to find their Expected Present Value (EPV). However, sometimes we need to look in the opposite direction. Instead of asking "What is it worth today?", we ask "How much will have built up by the end of the term?"
In this chapter, we explore Expected Accumulations. This is a vital concept for understanding how funds grow under insurance and annuity contracts, especially when we account for both interest and the "bonus" that survivors get when others in the pool pass away. Don't worry if it feels like a bit of a mental flip—we will take it step-by-step!
1. The Logic of Actuarial Accumulation
When we accumulate money in a standard bank account, we only care about the interest rate \( i \). But in actuarial science, we are dealing with a group of lives. If a group of people pays into a fund, the total amount at the end is shared among the survivors.
The Core Principle:
The Expected Accumulation at the end of \( n \) years for a contract is the Expected Present Value (EPV) rolled forward to time \( n \) with interest AND divided by the probability of survival.
Mathematically, we move the EPV forward by dividing it by the Pure Endowment factor, \( {}_n E_x \):
\( \text{Expected Accumulation} = \frac{\text{EPV}}{{}_n E_x} \)
Since \( {}_n E_x = v^n \cdot {}_n p_x \), the formula effectively does two things:
1. The \( v^n \) in the denominator moves the value forward by \( (1+i)^n \) (the interest part).
2. The \( {}_n p_x \) in the denominator increases the share for each survivor (the mortality part).
Why do we divide by the survival probability?
Analogy: Imagine 100 people each put \$100 into a pot. After a year, the pot has grown with interest. If only 90 people are still alive, those 90 people share the entire pot. Each survivor gets a bigger slice than if all 100 had survived! This "extra" gain is often called the mortality profit or survival benefit.
2. Accumulations for Annuities
The most common application of this is the Accumulated Life Annuity. This represents the total value at age \( x+n \) of a series of payments made over the last \( n \) years, provided the life survived to make/receive those payments.
Accumulated Annuities-Due
For an annuity-due (payments at the start of the year), the symbol is \( \ddot{s}_{x:\overline{n}|} \). This is the value at age \( x+n \) of payments of 1 made at ages \( x, x+1, \dots, x+n-1 \), contingent on survival.
\( \ddot{s}_{x:\overline{n}|} = \frac{\ddot{a}_{x:\overline{n}|}}{{}_n E_x} = \frac{\ddot{a}_{x:\overline{n}|}}{v^n \cdot {}_n p_x} \)
Accumulated Annuities-Arrear
For an annuity-in-arrear (payments at the end of the year), the symbol is \( s_{x:\overline{n}|} \). This covers payments made at ages \( x+1, x+2, \dots, x+n \).
\( s_{x:\overline{n}|} = \frac{a_{x:\overline{n}|}}{{}_n E_x} = \frac{a_{x:\overline{n}|}}{v^n \cdot {}_n p_x} \)
Quick Review Box:
Always remember: Accumulation = EPV / Pure Endowment. If you know your EPV formulas, you are already 90% of the way there!
3. Accumulations for Assurance Contracts
While less common in exam questions than annuities, we can also calculate the expected accumulation of an assurance benefit. This represents the "expected value" of the insurance cover provided over the term, valued at the end of that term.
For a Term Assurance \( A^1_{x:\overline{n}|} \), the expected accumulation would be:
\( \text{Accumulated Term Assurance} = \frac{A^1_{x:\overline{n}|}}{{}_n E_x} \)
Did you know?
This concept is rarely used for simple death benefits in the real world, but it is a vital mathematical building block for calculating Retrospective Reserves (which you will study in the Pricing and Reserving section). It helps the insurer understand if the premiums accumulated so far are enough to cover the risks that have already passed.
4. Working with Different Frequencies
The syllabus requires you to handle payments that occur more frequently than annual (pthly) or continuously. The logic remains exactly the same—just swap the annual EPV for the pthly or continuous EPV.
- For pthly payments: \( \ddot{s}^{(p)}_{x:\overline{n}|} = \frac{\ddot{a}^{(p)}_{x:\overline{n}|}}{{}_n E_x} \)
- For continuous payments: \( \bar{s}_{x:\overline{n}|} = \frac{\bar{a}_{x:\overline{n}|}}{{}_n E_x} \)
Common Mistake to Avoid:
When calculating pthly accumulations, students sometimes try to adjust the \( {}_n E_x \) factor. Don't do this! The denominator \( {}_n E_x \) always refers to the survival and discounting over the entire term \( n \), regardless of how often the payments happened during that term.
5. Step-by-Step Calculation Guide
If you are asked to calculate an expected accumulation in the exam, follow these steps:
- Identify the EPV: Calculate the Expected Present Value of the contract as if it were a normal annuity or assurance problem.
- Calculate the Pure Endowment: Find \( {}_n E_x \). Remember that \( {}_n E_x = \frac{l_{x+n}}{l_x} \cdot v^n \).
- Divide: Divide the result of Step 1 by the result of Step 2.
- Check for Select Mortality: If the question mentions "select" lives (e.g., life recently took out a policy), ensure you use \( {}_n E_{[x]} = \frac{l_{[x]+n}}{l_{[x]}} \cdot v^n \).
Key Takeaways
1. Definition: Expected accumulation is the value at the end of a term for a survivor, representing the accumulated cashflows adjusted for interest and the benefit of being a survivor.
2. The Golden Rule: Every accumulation formula is just \( \frac{\text{EPV}}{{}_n E_x} \).
3. Survivors' Bonus: The accumulation is higher than a "certain" accumulation (like \( s_{\overline{n}|} \)) because survivors inherit the interest and principal from those who died during the term.
4. Consistency: Ensure the term \( n \) used in the EPV matches the term \( n \) used in the denominator \( {}_n E_x \).
Ready for the next step? This chapter links directly to the "Pricing and Reserving" section, where you will use these accumulations to see how much money an insurance company actually has on its books after a policy has been running for a few years!