Compound Interest

TL;DR

Compound interest is when you earn interest not only on your initial money but also on the interest that money has already earned. It's often called "interest on interest" and makes your money grow much faster over time. Understanding it is crucial for making smart financial decisions whether you're saving or borrowing.

1. The Mental Model

Think of compound interest like a snowball rolling down a hill: it starts small, but as it collects more snow (interest), it gets bigger faster and faster. Each time it rolls, the new "interest" snow adds to its size, allowing it to collect even more snow on the next roll.

2. The Core Material

Compound interest means your money grows because you're earning interest on your initial investment (the principal) PLUS on all the accumulated interest from previous periods. This is different from simple interest, where you only earn interest on the original principal.

The key factors are:
* Principal (P): Your starting amount of money.
* Annual Interest Rate (r): The percentage your money grows each year (expressed as a decimal, e.g., 5% is 0.05).
* Number of Times Compounded Per Year (n): How often the interest is calculated and added to the principal within a year (e.g., annually n=1, semi-annually n=2, quarterly n=4, monthly n=12).
* Time (t): The number of years your money is invested or borrowed.

The formula for calculating the future value (A) of an investment with compound interest is:

A = P (1 + r/n)^(nt)

Let's break down how compounding frequency impacts growth:
* Annually (n=1): Interest is added once a year.
* Semi-annually (n=2): Interest is added twice a year (every 6 months).
* Quarterly (n=4): Interest is added four times a year (every 3 months).
* Monthly (n=12): Interest is added twelve times a year (every month).
* Daily (n=365): Interest is added every day.

The more frequently interest is compounded, the faster your money grows, because you start earning interest on that interest more quickly.

```mermaid
graph TD
A["Initial Principal (P)"] --> B{"First Compounding Period"};
B --> C["Interest Earned (P * r/n)"];
B --> D["New Principal = P + Interest Earned"];
D --> E{"Second Compounding Period"};
E --> F["Interest Earned (New Principal * r/n)"];
E --> G["Even Newer Principal = New Principal + Interest Earned"];
G --> H{"Subsequent Compounding Periods"};
H --> I["Future Value

Introduction to Financial Mathematics and Basic Concepts

TL;DR

Financial mathematics helps you understand how money changes over time and how to value investments. It uses tools like interest rates and present/future values to make smart financial decisions. Focusing on key concepts will build foundational skills for more complex topics.

1. The Mental Model

Think of financial mathematics like a time machine for money. You're learning to move money forward (future value) or backward (present value) in time, always accounting for its growth or decay due to interest.

2. The Core Material

Financial mathematics is all about understanding the time value of money. A dollar today isn't worth the same as a dollar next year because you could invest it and earn interest.

Simple vs. Compound Interest

• Simple Interest: This is interest calculated only on the initial amount (the principal). It's straightforward: you earn the same amount of interest every period.

  • Formula: $I = P \times r \times t$
    • $I$ = Interest earned
    • $P$ = Principal (initial amount)
    • $r$ = Annual interest rate (as a decimal)
    • $t$ = Time period (in years)

• Compound Interest: This is where interest earns interest. After each period, the interest earned is added to the principal, and the next period's interest is calculated on this new, larger amount. This is how most real-world investments work, and it's incredibly powerful over time.

  • Formula for Future Value: $FV = P (1 + r)^t$
    • $FV$ = Future Value
    • $P$ = Principal
    • $r$ = Annual interest rate (as a decimal)
    • $t$ = Number of compounding periods (usually years, but can be months/quarters if rate adjusted)

Let's visualize the difference using a simple flow:

graph TD
    A["Initial Principal (P)"] --> B{"Choose Interest Type?"}
    B -- "Simple" --> C["Interest = P * r * t"]
    C --> D["Final Amount (P + I)"]
    B -- "Compound" --> E["Calculate Interest on P"]
    E --> F["Add Interest to P (New Principal)"]
    F --> G{{"Repeat for 't' Periods?"}}
    G -- "Yes" --> E
    G -- "No" --> H["Final Amount (FV = P(1+r)^t)"]

Present Value and Future Value

• Future Value (FV): This is the value of an investment or a sum of money at a specified date in the future, assuming a certain interest rate. You're asking, "What will my money be worth later?"
  • The compound interest formula abo

Annuities: Future and Present Values

TL;DR

Annuities are a series of equal payments made at regular intervals. You'll learn how to calculate their future value (what they'll be worth later) and their present value (what they're worth today). Understanding these concepts helps you make smart financial decisions about savings and loans.

1. The Mental Model

Think of an annuity as a financial conveyor belt where you consistently put in or take out the same amount of money. The future value is how much you've accumulated at the end, considering interest, while the present value is how much that whole series of payments would be worth if you received it all right now.

2. The Core Material

An annuity is a fixed series of payments or receipts over a specified period. When these payments occur at the end of each period, it's called an ordinary annuity. If they happen at the beginning, it's an annuity due. We'll focus on ordinary annuities for now, as they're more common in many applications like loan repayments or regular savings.

Future Value of an Ordinary Annuity (FVA)

The future value (FV) tells you how much a series of regular payments will be worth at a specific point in the future, assuming a certain interest rate. Imagine saving the same amount every month; FVA calculates your total savings plus all the interest earned on those savings.

The formula for the Future Value of an Ordinary Annuity is:

FVA = P * [((1 + i)^n - 1) / i]

Where:
* P = Payment amount per period
* i = Interest rate per period (e.g., annual rate / number of periods per year)
* n = Total number of periods

Let's break down how the future value accumulates:

graph TD
    A["Start Saving"] --> B{First Payment (P)}
    B --> C["Earns Interest for n-1 periods"]
    D{Second Payment (P)} --> E["Earns Interest for n-2 periods"]
    F{...}
    G{Last Payment (P)} --> H["Earns Interest for 0 periods (ends up as P)"]
    C --> I["Accumulated Value 1"]
    E --> I
    H --> I
    I --> J["Total FVA (Sum of all payments + earned interest)"]

Present Value of an Ordinary Annuity (PVA)

The present value (PV) tells you what a series of future payments is worth today. This is super useful for decisions like how much you need to save now to fund future withdrawals, or figuring out the true principal amount of a loan based on its regular payments.

The formula for the Present Value of an Ordinary Annuity is:

`PVA = P * [(1 - (