IB Mathematics AA SL Compound interest Study Notes
IB Mathematics AA SL Compound interest Study Notes Offer a clear explanation of Compound interest , including various formula, rules, exam style questions as example to explain the topics. Worked Out examples and common problem types provided here will be sufficient to cover for topic Compound interest
Concept of Simple Interest
Concept of Simple Interest
Simple interest is a method of calculating interest where the interest is earned only on the original principal amount. It does not compound over time.
The formula for calculating simple interest is:
\( I = P \cdot r \cdot t \)
- \( I \) is the interest earned
- \( P \) is the principal (initial amount of money)
- \( r \) is the annual interest rate (in decimal)
- \( t \) is the time in years
The total amount after \( t \) years is given by:
\( A = P + I = P(1 + rt) \)
Example
Calculate the simple interest and total amount on a loan of $5000 at an annual interest rate of 6% over 4 years.
▶️ Answer/Explanation
Given: \( P = 5000 \), \( r = 0.06 \), \( t = 4 \)
Use the formula \( I = Prt \):
\( I = 5000 \cdot 0.06 \cdot 4 = 1200 \)
Total amount: \( A = P + I = 5000 + 1200 = 6200 \)
Answer: Interest = $1200, Total Amount = $6200
Concept of Compound Interest
Concept of Compound Interest
Compound interest is interest calculated on the initial principal, which also includes all of the accumulated interest from previous periods.
The formula for compound interest is:
\( A = P \left(1 + \frac{r}{n} \right)^{nt} \)
- \( A \) is the total amount after time \( t \)
- \( P \) is the principal (initial amount)
- \( r \) is the annual interest rate (in decimal)
- \( n \) is the number of times interest is compounded per year
- \( t \) is the time in years
If interest is compounded annually, then \( n = 1 \), and the formula simplifies to:
\( A = P(1 + r)^t \)
Example
Calculate the amount after 5 years if $3000 is invested at a compound interest rate of 4% per annum, compounded annually.
▶️ Answer/Explanation
Given: \( P = 3000 \), \( r = 0.04 \), \( t = 5 \), \( n = 1 \)
Use the formula: \( A = 3000(1 + 0.04)^5 \)
Compute: \( A = 3000(1.04)^5 = 3000 \cdot 1.21665 = 3649.95 \)
Answer: Total Amount = $3649.95
Annual Depreciation
Annual Depreciation
Depreciation refers to the decrease in the value of an asset over time due to wear and tear, age, or obsolescence. When depreciation is applied annually, the value of the asset reduces by a fixed percentage each year.
Formula for Depreciation (using compound decrease):
\( V = P(1 – r)^t \)
- \( V \): Value of the asset after \( t \) years
- \( P \): Original value (initial cost of asset)
- \( r \): Annual depreciation rate (as a decimal)
- \( t \): Number of years
This formula assumes the depreciation is a fixed percentage of the current value each year.
Example
A car was bought for $25,000 and depreciates at a rate of 12% per year. What is its value after 4 years?
▶️ Answer/Explanation
Given: \( P = 25000 \), \( r = 0.12 \), \( t = 4 \)
Use the formula: \( V = 25000(1 – 0.12)^4 \)
\( V = 25000(0.88)^4 \)
\( V = 25000 \times 0.5997 = 14992.5 \)
Answer: The car’s value after 4 years is approximately $14,992.50
Real Value of an Investment
The real value of an investment adjusts the nominal (or stated) interest rate to account for the effect of inflation. Inflation reduces the purchasing power of money over time, so the real return reflects the actual increase in value after removing inflation’s impact.
Formula for Real Interest Rate (approximate):
\( r_{real} \approx r_{nominal} – i \)
- \( r_{real} \): Real interest rate (adjusted for inflation)
- \( r_{nominal} \): Nominal interest rate (stated rate without adjustment)
- \( i \): Inflation rate (as a decimal)
This is a simple approximation, valid when rates are small.
More accurate formula (Fisher Equation):
\( 1 + r_{real} = \frac{1 + r_{nominal}}{1 + i} \)
Using this, you can calculate the real value of your investment after \( t \) years by applying the real interest rate:
\( V = P (1 + r_{real})^t \)
- \( V \): Real value of the investment after \( t \) years
- \( P \): Initial investment amount
- \( r_{real} \): Real interest rate (from the Fisher Equation)
- \( t \): Number of years
Example
An investment of $10,000 has a nominal interest rate of 8% per year. The inflation rate is 3% per year. What is the real value of the investment after 5 years?
▶️ Answer/Explanation
Given: \( P = 10000 \), \( r_{nominal} = 0.08 \), \( i = 0.03 \), \( t = 5 \)
Calculate real interest rate using Fisher Equation: \[ 1 + r_{real} = \frac{1 + 0.08}{1 + 0.03} = \frac{1.08}{1.03} \approx 1.0485 \] So, \( r_{real} \approx 0.0485 \) or 4.85%
Calculate real value after 5 years: \[ V = 10000 \times (1.0485)^5 = 10000 \times 1.267 = 12670 \]
Answer: The real value of the investment after 5 years is approximately $12,670
Using Technology for Financial Calculations
Using Technology for Financial Calculations
Examinations may allow or require the use of financial calculators, spreadsheet software (e.g., Excel), or financial packages in programming languages. These tools can quickly compute values such as compound interest, depreciation, and real values, reducing calculation errors.
Common Built-in Functions:
- Excel:
FV(rate, nper, pmt, [pv], [type])to calculate future value. - Financial Calculators: Keys for PV (present value), FV (future value), I/Y (interest rate), N (number of periods).
- Programming (Python, R, etc.): Functions for compound interest, e.g.,
FV = P * (1 + r)^t.
Example
Calculate the future value of a $15,000 investment that earns 6% annual interest for 8 years using Excel.
▶️ Answer/Explanation
Use Excel’s FV function: =FV(rate, nper, pmt, [pv], [type])
Here, rate = 0.06, nper = 8, pmt = 0 (no periodic payments), pv = -15000 (initial investment as a negative number)
Formula: =FV(0.06, 8, 0, -15000)
Result: $23,918.95
Answer: The future value of the investment after 8 years is approximately $23,918.95.
Example
Using a GDC, find the future value of a $12,000 investment that earns 5% interest per year, compounded annually, after 10 years.
▶️ Answer/Explanation
Turn on the calculator and open the Finance or TVM Solver menu.
Input the values:
\( N = 10 \) (number of years)
\( I\% = 5 \) (interest rate per year)
\( PV = -12000 \) (present value, negative because it’s an outflow)
\( PMT = 0 \) (no additional payments)
Leave \( FV \) blank to solve for it.
Calculate/solve for \( FV \).
Result: \( FV \approx 19506.64 \)
Answer: The future value after 10 years is approximately $19,506.64.