Welcome to the foundational concepts of deterministic interest rates in financial mathematics for actuaries. Understanding how money grows or shrinks over time under a fixed rate is crucial for pricing insurance products, valuing financial instruments, and managing financial risk. This module will equip you with the core principles and calculations essential for actuarial exams.

The fundamental idea behind interest rates is the time value of money. A dollar today is worth more than a dollar in the future due to its potential earning capacity. We explore two key concepts:

• Future Value (FV): The value of an investment at a specific future date, given a certain interest rate.
• Present Value (PV): The current worth of a future sum of money or stream of cash flows, given a specified rate of return.

Compound interest is the standard in most financial applications, including actuarial work, due to its realistic representation of how investments grow. The power of compounding is often referred to as the 'eighth wonder of the world'.

Interest rates can be quoted in different ways. It's crucial to distinguish between nominal and effective rates to ensure accurate calculations.

• Nominal Interest Rate: An interest rate quoted without considering the effect of compounding. It is usually expressed as an annual rate, but interest may be compounded more frequently (e.g., semi-annually, quarterly, monthly).
• Effective Interest Rate: The actual rate of interest earned or paid over a period, taking into account the effect of compounding. The effective annual interest rate (EAR) is the most common form.

Annuities are a series of equal payments made at regular intervals. They are fundamental to many financial products, such as mortgages, pensions, and life insurance payouts. We distinguish between:

• Ordinary Annuity: Payments are made at the end of each period.
• Annuity Due: Payments are made at the beginning of each period.

The present and future values of annuities are calculated using specific formulas that sum up the present or future values of each individual payment. These formulas are derived from geometric series.

Let $P$ be the periodic payment, $i$ be the effective interest rate per period, and $n$ be the number of periods.

• Future Value of an Ordinary Annuity ($FV_o$): $FV_o = P \times \frac{(1+i)^n - 1}{i}$
• Present Value of an Ordinary Annuity ($PV_o$): $PV_o = P \times \frac{1 - (1+i)^{-n}}{i}$
• Future Value of an Annuity Due ($FV_d$): $FV_d = P \times \frac{(1+i)^n - 1}{i} \times (1+i) = FV_o \times (1+i)$
• Present Value of an Annuity Due ($PV_d$): $PV_d = P \times \frac{1 - (1+i)^{-n}}{i} \times (1+i) = PV_o \times (1+i)$

A perpetuity is a special type of annuity that continues forever. The present value of a perpetuity-immediate (payments at the end of each period) is calculated as:

$PV = \frac{P}{i}$

For a perpetuity-due (payments at the beginning of each period), the present value is:

$PV = \frac{P}{i} \times (1+i)$

Deterministic interest rates are the bedrock for many actuarial calculations, including:

• Pricing Life Insurance: Determining premiums based on expected future payouts.
• Valuing Pension Obligations: Estimating the present value of future pension payments to retirees.
• Loan Amortization: Calculating payment schedules for loans.
• Investment Analysis: Evaluating the potential returns of various financial instruments.