Premium and benefit payment patterns in EPV formulae

Introduction: Getting the Timing Right

Welcome! In our previous study of Expected Present Values (EPVs), we looked at the basic building blocks of life insurance and annuities. However, in the real world, cashflows don't always happen neatly once a year. Some people pay their premiums monthly; some insurance claims are paid the moment someone passes away, rather than waiting until the end of the year.

In this chapter, we explore the payment patterns of these cashflows. We will learn how to adjust our formulas to account for different frequencies (like monthly or quarterly payments) and different timings (like payments made at the start of a period versus the end). Think of this as fine-tuning our actuarial "clock" to ensure our valuations are as accurate as possible.

1. Annuity Payment Patterns: Advance, Arrear, and Frequency

Annuities are series of payments contingent on survival. The timing of these payments significantly impacts their present value.

Advance vs. Arrear

The most basic distinction is when the payment occurs within the year:

• In Advance (Annuity-due): Payments are made at the beginning of the period. We use the "double dot" notation: \( \ddot{a}_x \).
• In Arrear (Annuity-immediate): Payments are made at the end of the period. We use the notation: \( a_x \).

The Relationship: Since an annuity in advance pays one extra payment at time 0 and ends one period earlier than an annuity in arrear, we can relate them simply: \( \ddot{a}_x = 1 + a_x \). However, for life annuities, the standard relationship is \( \ddot{a}_x = 1 + a_x \) only if we assume the person survives the first year. More formally, for a whole life annuity: \( \ddot{a}_x = 1 + a_x \), because the "in arrear" version is just the "in advance" version minus the very first payment (which is guaranteed at \( t=0 \) if the life is currently alive).

Payments more frequent than annual (\(p\)-thly)

What if someone pays monthly (\( p=12 \)) or quarterly (\( p=4 \))? We use the notation \( \ddot{a}_x^{(p)} \). This represents an annuity paying \( 1/p \) at the start of each \( 1/p \)-th of a year.

Handy Approximation: You don't always need complex tables for \( p \)-thly annuities. A very common approximation used in exams (assuming a uniform distribution of deaths) is: \( \ddot{a}_x^{(p)} \approx \ddot{a}_x - \frac{p-1}{2p} \)

Quick Tip: If \( p \) is very large (like daily payments), the fraction \( \frac{p-1}{2p} \) gets closer and closer to \( 0.5 \). This makes sense intuitively—on average, a monthly payment starts about half a year "closer" to today than an annual payment.

2. Assurance Benefit Patterns: When is the Claim Paid?

Assurance benefits are paid upon death. But when exactly does the check get sent out?

End of the Year of Death (\( A_x \))

This is the simplest model. We assume the life dies at some point during the year, but the insurance company waits until the end of that year to pay the benefit. This is the standard "discrete" case.

Immediately on Death (\( \bar{A}_x \))

In reality, most policies pay out as soon as the claim is processed. We represent this "continuous" payment with a bar over the \( A \). Since the payment happens, on average, halfway through the year of death (under the assumption of uniform distribution of deaths), it is worth more today than a payment at the end of the year.

The Relationship: \( \bar{A}_x \approx (1+i)^{1/2} A_x \)

Essentially, we are "un-discounting" the end-of-year payment by half a year to bring it back to the middle of the year.

3. Increasing and Decreasing Benefits

Sometimes the benefit amount isn't level. It might increase to keep up with inflation or decrease as a loan is paid off.

Increasing Benefits (\( IA \))

An Increasing Whole Life Assurance pays 1 if death occurs in year 1, 2 if in year 2, and so on. The EPV is denoted as \( (IA)_x \). If the benefit increases continuously and is paid immediately, we use the "double bar" notation: \( (\bar{I}\bar{A})_x \).

Increasing Annuities (\( I\ddot{a} \))

Similarly, an annuity might pay 1 in the first year, 2 in the second, etc. \( (I\ddot{a})_x = \sum_{t=0}^{\infty} (t+1) v^t {}_t p_x \)

Common Pitfall: Students often forget that for an increasing annuity, the "1" starts at time 0. For a deferred increasing annuity, you must be very careful with the timing of when the "stepping up" begins!

4. Key Relationships and the Equation of Value

One of the most important skills in CM1 is moving between these different patterns using the Equation of Value. The syllabus highlights the relationship between assurance and annuity factors. This is a life-saver in the exam!

The Golden Formula: \( A_x = 1 - d \ddot{a}_x \)

This works because of a clever piece of logic: If you have \$1 today, you can either keep it (the "1") OR you can give it away in exchange for the interest it earns every year while you are alive (the annuity \( d \ddot{a}_x \)) plus getting the \$1 back when you die (the assurance \( A_x \)).

This relationship also holds for continuous functions: \( \bar{A}_x = 1 - \delta \bar{a}_x \)

5. Summary of Timing Effects

When you are solving problems, use this "Rule of Thumb" to check if your answers make sense:

1. For Annuities: The more frequent the payment, the smaller the EPV (because you are getting your money slightly later in the year compared to a single lump sum at the start of the year).
   Example: \( \ddot{a}_x > \ddot{a}_x^{(12)} > \bar{a}_x > a_x \)
2. For Assurances: The sooner the benefit is paid after death, the larger the EPV.
   Example: \( \bar{A}_x > A_x \)

Step-by-Step for Exam Questions:
1. Identify if the contract is Assurance (on death) or Annuity (while alive).
2. Check the timing: Is it advance, arrear, or continuous?
3. Check the frequency: Is it annual or \( p \)-thly?
4. Look for variations: Is it increasing, decreasing, or deferred?
5. Use the relationships (like \( A_x = 1 - d \ddot{a}_x \)) to convert into forms you can find in your Formulae and Tables book.

Don't worry if the notation feels like a "alphabet soup" at first! With practice, you'll start to see the logic in the symbols. The bar always means continuous, the dots always mean advance, and the (p) always means frequency.