Financial Mathematics: Loans and Investments

TL;DR

You'll learn about how money grows or shrinks over time with loans, investments, and annuities. This involves understanding interest, regular payments, and the final value of your money. We'll also briefly touch on how graphs and networks can help you make smart decisions with this financial info.

1. The Mental Model

Think of money as a seed. With investments, it grows; with loans, you pay for someone else's seed to grow for you. Annuities are like a steady harvest from your investment.

2. The Core Material

You noted loans, investments, and annuities as key areas. These concepts are all about how money changes value over time, usually because of interest.

Loans

A loan is money borrowed that you pay back, usually with interest. Interest is the extra cost of borrowing money. For example, if you borrow \$100 at 5% interest per year, you'll pay back more than \$100.
* Simple Interest: Interest is only calculated on the original amount borrowed (the principal). It's less common for loans but good to understand.
* Compound Interest: Interest is calculated on the principal and any accumulated interest. This is how most loans (and investments) work, and it makes your money grow (or your debt increase) much faster.

Investments

An investment is money you put away to grow over time, often earning interest.
* Simple Interest: Same as above. Your interest earnings are fixed based on the initial investment.
* Compound Interest: The magic of investing! Interest earned is added to your principal, and then the next interest calculation is based on this new, larger amount. This leads to exponential growth.
* Formula for Compound Interest: While not in your notes, it's good to know the basic idea. If you invest P dollars at an annual interest rate r (as a decimal) compounded n times per year for t years, the future value (FV) is roughly: FV = P * (1 + r/n)^(nt).

Annuities

An annuity is a series of equal payments made at regular intervals. They can be for savings (where you make regular payments into an investment) or for payout (where you receive regular payments from an investment).
* Savings Annuities: You make regular payments into an account, and the money grows with compound interest. Think of saving for retirement with regular contributions.
* Payout Annuities: You receive regular payments from a lump sum of money. Think of a pension where you get a fixed amount every month after retirement.

Graphs and Networks

You mentioned graphs and networks and Networks and Decision Making. In the context of financial maths, these usually apply to:
* Decision Making: You can use network diagrams (like flow charts) to map out different financial choices and their potential outcomes. For example, comparing loan options or investment strategies.
* Representing Data: Graphs can visually show how loans grow, investments compound, or annuity balances change over time. This helps you quickly understand trends and make decisions. For instance, a graph could compare how fast two different investments grow.

3. Worked Example

Let's look at a simple investment with compound interest, as that's often where the biggest gains are.

You invest \$1,000 for 3 years at an annual interest rate of 5%, compounded annually.

• Year 1:
  • Starting principal: \$1,000
  • Interest earned: \$1,000 * 0.05 = \$50
  • Ending balance: \$1,000 + \$50 = \$1,050
• Year 2:
  • Starting principal: \$1,050 (your previous ending balance)
  • Interest earned: \$1,050 * 0.05 = \$52.50
  • Ending balance: \$1,050 + \$52.50 = \$1,102.50
• Year 3:
  • Starting principal: \$1,102.50
  • Interest earned: \$1,102.50 * 0.05 = \$55.125 (round to \$55.13)
  • Ending balance: \$1,102.50 + \$55.13 = \$1,157.63

After 3 years, your \$1,000 investment has grown to \$1,157.63 due to compound interest. Notice how the interest earned increases each year because it's calculated on a larger balance.

4. Key Takeaways

• Loans cost you money (interest) for borrowing, while investments earn you money (interest) for saving.
• Compound interest is super powerful for both loans and investments, making money grow much faster than simple interest.
• Annuities involve a series of regular payments, either going into your account (savings) or coming out (payout).
• Visual tools like graphs help you see trends in financial data, while networks aid in decision-making by mapping choices.
• Understanding interest rates and compounding frequency is crucial for comparing financial products.
• Always distinguish between simple and compound interest calculations.
• Financial decisions often involve trade-offs that networks can help illuminate.

Common Mistakes to Avoid:
* Confusing simple interest with compound interest, especially when calculating future values.
* Forgetting to convert annual interest rates to the correct period (e.g., monthly) if compounding isn't annual.
* Not reading the fine print on fees or charges associated with loans and investments.
* Ignoring the impact of inflation over long-term investments.

5. Now Try It

Imagine you're offered two investment options for \$500 over 2 years:
1. Option A: 10% simple interest per year.
2. Option B: 9.5% compound interest per year, compounded annually.

Calculate the final value of each investment after 2 years. Which option should you choose and why?

Success will look like: You've correctly calculated the final value for both options and can clearly explain which one is better and why, referencing the concepts of simple vs. compound interest.