THE TIME VALUE OF MONEY
Simple Interest: The Foundation of Earning on Your Money
What's the Big Idea?
Imagine your Gogo gives you $100 USD. If you just hide that money, in one year, you'll still have just $100. But what if you could make that money grow? The extra money you earn is called interest. Simple Interest is the most straightforward way to calculate this earning. It means the interest is calculated only on the initial amount of money you started with.
Key Vocabulary
- Principal (P): The initial, original amount of money.
- Interest (I): The extra money paid for borrowing or earned from investing.
- Rate (R): The percentage at which interest is charged, usually per year.
- Time (T): The duration for which the money is borrowed or invested.
- Simple Interest: Interest that is calculated only on the original principal amount.
- Amount (A): The total sum at the end, which is the principal plus the interest (A = P + I).
The Simple Interest Formula: I = P × R × T
To calculate simple interest, we use this formula. It is crucial to get the Rate and Time in the correct format.
- Rate (R): You must convert the percentage to a decimal by dividing by 100 (e.g., 8% becomes 0.08).
- Time (T): The time period must match the rate period. If the rate is per year, time must be in years. To convert months to years, divide by 12 (e.g., 6 months = 6/12 = 0.5 years).
Worked Example: "Putting it to Work in Zimbabwe"
Scenario: Tendai borrows $500 USD for his poultry business. The loan has a simple interest rate of 15% per annum, and he will repay it after 9 months.
- Step 1: Identify and convert values.
- Principal (P) = $500
- Rate (R) = 15% ÷ 100 = 0.15
- Time (T) = 9 months / 12 = 0.75 years
- Step 2: Calculate the Simple Interest (I).
- I = P × R × T
- I = 500 × 0.15 × 0.75 = $56.25
- Step 3: Calculate the Total Amount (A).
- A = P + I
- A = $500 + $56.25 = $556.25
Answer: Tendai will pay $56.25 in interest, and the total amount he must repay is $556.25.
Common Mistakes to Avoid
- Forgetting to convert the Time to years.
- Forgetting to convert the Rate from a percentage to a decimal.
- Giving only the interest (I) as the answer when the question asks for the total amount (A).
Compound Interest: The Power of Growth on Growth
What's the Big Idea?
Compound interest is the interest you earn not just on your original principal, but also on the accumulated interest from previous periods. In other words, your interest starts earning its own interest. This snowball effect can turn a small amount of savings into a very large amount over time. It is the single most important concept for long-term saving and investing.
Key Vocabulary
- Compound Interest: Interest calculated on the initial principal and also on the accumulated interest.
- Future Value (FV): The value of a current asset at a specified date in the future.
- Present Value (PV): The current value of a future sum of money.
- Compounding Period: The frequency with which interest is calculated (e.g., annually, semi-annually).
Future and Present Value of a Lump Sum
Future Value Formula: FV = PV (1 + r)ⁿ
- Answers: "If I invest a lump sum today, what will it be worth in the future?"
- r is the interest rate per compounding period.
- n is the total number of compounding periods.
Present Value Formula: PV = FV / (1 + r)ⁿ
- Answers: "To have a certain amount in the future, how much do I need to invest today?"
Worked Example
Scenario: Rumbidzai invests a lump sum of $2,000 in an account earning 8% per annum, compounded annually, for 3 years.
Future Value Calculation:
- FV = 2000 (1 + 0.08)³
- FV = 2000 (1.259712) = $2,519.42
Present Value Calculation (to have $2,000 in 3 years):
- PV = 2000 / (1 + 0.08)³
- PV = 2000 / 1.259712 = $1,587.66
Present Value of Annuities: What Is a Stream of Future Payments Worth Today?
What's the Big Idea?
An annuity is a series of equal, regular payments. The Present Value of an Annuity (PVA) calculates the single lump-sum value, right now, of all those future payments combined. It's essential for comparing a lump-sum offer (like a pension payout) with a stream of payments, allowing you to make a fair financial decision.
Key Vocabulary
- Annuity: A series of equal payments made at fixed, regular intervals.
- Payment (PMT): The amount of the regular, fixed payment in an annuity.
- Present Value of an Annuity (PVA): The total value today of a series of future payments.
The Present Value of an Annuity Formula
PVA = PMT [ (1 - (1 + r)⁻ⁿ) / r ]
- PMT is the amount of each regular payment.
- r is the interest rate per period.
- n is the total number of payment periods.
Future Value of Annuities: Building Wealth Through Regular Savings
What's the Big Idea?
The Future Value of an Annuity (FVA) calculates the final total amount you will have if you make a series of regular deposits into an interest-earning account. It shows the power of disciplined saving and how small, consistent contributions can grow into a substantial sum over time.
Key Vocabulary
- Future Value of an Annuity (FVA): The total value of a series of regular payments at a future date, including all payments plus all accumulated interest.
- Sinking Fund: A savings fund built by making regular annuity payments to meet a specific future financial goal.
The Future Value of an Annuity Formula
FVA = PMT [ ((1 + r)ⁿ - 1) / r ]
- PMT is the amount of your regular payment.
- r is the interest rate per period.
- n is the total number of payment periods.
Worked Example
Scenario: Tariro wants to save for her business. She deposits $70 every month for 4 years into an account that earns 12% per year, compounded monthly.
- Convert r and n:
- r = 12% / 12 = 1% per month (0.01)
- n = 4 years × 12 = 48 months
- Apply FVA Formula:
- FVA = 70 [ ((1 + 0.01)⁴⁸ - 1) / 0.01 ]
- FVA = 70 [ (1.612226 - 1) / 0.01 ]
- FVA = 70 [ 61.2226 ] = $4,285.58
Answer: After 4 years, Tariro will have $4,285.58. Her total contribution was $70 x 48 = $3,360, so she earned $925.58 in interest.