Interest and discount terminology

Welcome to the Foundation of Financial Math!

Welcome, future Actuary! You are starting your journey into Exam FM (Financial Mathematics). This first chapter is all about the Time Value of Money. Think of this as learning the "language" of finance. Before we can solve complex problems about annuities and bonds, we need to understand how money grows and how we describe that growth.

Why is this important? Because a dollar today is worth more than a dollar tomorrow. In these notes, we will explore exactly how to measure that difference using interest and discount. Don't worry if the formulas look a bit intimidating at first—we’ll break them down step-by-step!

1. The Accumulation and Amount Functions

To understand growth, we need a way to track it. We use two main functions:

The Accumulation Function \( a(t) \): This tells us what $1 invested at time 0 will grow to at time \( t \). By definition, \( a(0) = 1 \).
\nThe Amount Function \( A(t) \): This tells us what an initial investment of \( k \) dollars (the principal) will grow to at time \( t \).
\nThe relationship is simple: \( A(t) = k \cdot a(t) \).

Key Differences:

1. \( a(t) \) always starts at 1.
\n2. \( A(t) \) starts at whatever amount you actually invested (\( k \)).

Analogy: Think of \( a(t) \) like a recipe for a single cupcake. If you want to know how many ingredients you need for 100 cupcakes (\( A(t) \)), you just multiply the single recipe by 100 (\( k \)).

2. Simple Interest vs. Compound Interest

There are two primary ways interest is calculated. Understanding the difference is vital for Exam FM.

Simple Interest

In Simple Interest, you only earn interest on your original principal. The interest earned each year is constant.

The Formula: \( a(t) = 1 + i \cdot t \)

Example: If you invest $100 at 5% simple interest, you earn $5 every year. After 10 years, you have your original $100 plus $50 in interest.

Compound Interest

In Compound Interest, you earn interest on your principal and on the interest you've already earned. This is the "snowball effect."

The Formula: \( a(t) = (1 + i)^t \)

Did you know? Compound interest is the "standard" for Exam FM. If a problem doesn't specify which type to use, always assume it is compound interest!

Common Mistake to Avoid: When using simple interest, \( t \) can be a fraction, but the growth is linear. When using compound interest, the growth is exponential. Over long periods, compound interest will always result in much more money!

3. The Effective Rate of Interest (\( i \))

The Effective Rate of Interest is the amount of money earned by $1 invested at the beginning of a period, earned during that specific period.

For any period \( n \), the effective rate \( i_n \) is calculated as:
\( i_n = \frac{A(n) - A(n-1)}{A(n-1)} \)

Quick Review:
- In Compound Interest, the effective rate \( i \) is constant for every year.
- In Simple Interest, the effective rate \( i_n \) decreases over time because you are earning the same dollar amount on a larger and larger total balance.

4. The Effective Rate of Discount (\( d \))

This is often where students get tripped up. While interest (\( i \)) is paid at the end of a period, discount (\( d \)) is like interest paid at the beginning of a period.

The Formula: \( d = \frac{A(n) - A(n-1)}{A(n)} \)

Analogy: Imagine you borrow $100.
\nIf the lender charges 10% interest, they give you $100 now, and you pay back $110 later.
\nIf the lender charges a 10% discount, they take the $10 "fee" out immediately. They give you $90 now, and you pay back $100 later.

The Relationship between \( i \) and \( d \):

You will use these conversions constantly! It is worth memorizing them:
\( d = \frac{i}{1+i} \)
\( i = \frac{d}{1-d} \)
\( (1+i)(1-d) = 1 \)

Key Takeaway: Discount is always a smaller numerical value than its equivalent interest rate. For example, a 10% interest rate is equivalent to about a 9.09% discount rate.

5. Present Value and the Discount Factor (\( v \))

If we want to know what a future payment is worth today, we are finding its Present Value (PV). To do this, we use the Discount Factor \( v \).

The Formula: \( v = \frac{1}{1+i} \)

To move a value backward in time by \( t \) years, we multiply by \( v^t \).
\( PV = \text{Future Value} \cdot v^t \)

Memory Aid:
- To move money forward (Accumulate): Multiply by \( (1+i) \).
- To move money backward (Discount): Multiply by \( v \).

6. Nominal Rates of Interest and Discount

Sometimes, interest isn't calculated once a year. It might be calculated monthly, quarterly, or daily. These are called Nominal Rates.

\( i^{(m)} \): Nominal interest rate compounded \( m \) times per year.
\( d^{(m)} \): Nominal discount rate compounded \( m \) times per year.

The actual rate applied in each tiny sub-period is \( \frac{i^{(m)}}{m} \). To find the annual effective rate, we use:
\( 1 + i = (1 + \frac{i^{(m)}}{m})^m \)

Similarly for discount:
\( 1 - d = (1 - \frac{d^{(m)}}{m})^m \)

Example: If a bank says "12% compounded monthly," then \( i^{(12)} = 0.12 \). The monthly interest rate is \( 0.12 / 12 = 0.01 \) (or 1%).

7. The Force of Interest (\( \delta \))

What if we compound interest every single second? What if we compound it infinitely often? This is the Force of Interest, denoted by the Greek letter delta (\( \delta \)).

The Relationship: \( e^\delta = 1 + i \)
This means \( \delta = \ln(1+i) \).

The accumulation function using the force of interest is: \( a(t) = e^{\int_0^t \delta_s ds} \).
If \( \delta \) is constant, it simplifies to: \( a(t) = e^{\delta t} \).

Don't worry if this seems tricky at first! Just remember that \( \delta \) is simply the "limit" of nominal interest as the number of compounding periods goes to infinity.

Chapter Summary Checklist

Before moving to the next chapter, make sure you can:
1. Distinguish between \( a(t) \) and \( A(t) \).
2. Calculate simple vs. compound interest growth.
3. Convert between \( i, d, v, \) and \( \delta \).
4. Calculate an effective annual rate from a nominal rate \( i^{(m)} \).
5. Move money back and forth across a timeline using \( (1+i)^t \) and \( v^t \).

You've got this! These definitions are the building blocks for everything else in Exam FM. Practice converting between these rates until it feels like second nature!