IB Mathematics AI – Topic 1

Compound Interest Formula:

\[ FV = PV(1 + r)^n \]

where:

• FV: Future Value (final amount)
• PV: Present Value (initial principal)
• r: Interest rate per compounding period (as decimal)
• n: Number of compounding periods

Annual Compounding with Different Frequencies:

\[ FV = PV\left(1 + \frac{r}{k}\right)^{nk} \]

where k = number of times interest is compounded per year

• Annually: k = 1
• Semi-annually: k = 2
• Quarterly: k = 4
• Monthly: k = 12
• Daily: k = 365

⚠️ Common Pitfalls & Tips:

• Convert percentage to decimal: 5% = 0.05
• Match time units: if rate is annual, n should be in years
• For monthly compounding with annual rate: divide rate by 12
• Use GDC Finance Solver for complex calculations

Solution:

Identify variables:

PV = $5,000 (initial investment)

Annual rate = 4% = 0.04

Time = 3 years

Compounded monthly: k = 12

Calculate:

Number of compounding periods: n × k = 3 × 12 = 36

Rate per period: r/k = 0.04/12 = 0.003333...

Using formula:

\[ FV = 5000\left(1 + \frac{0.04}{12}\right)^{36} \]

\[ = 5000(1.003333...)^{36} \]

\[ = 5000(1.127328...) \]

\[ = 5636.64 \]

Using GDC Finance Solver:

N = 36, I% = 4, PV = -5000, PMT = 0, FV = solve

Answer: $5,636.64

Depreciation Formula:

\[ FV = PV(1 - r)^n \]

Note: (1 - r) because value is decreasing

Key Difference from Compound Interest:

Depreciation uses (1 - r) instead of (1 + r) because the value is decreasing

⚠️ Common Pitfalls & Tips:

• Depreciation uses subtraction: (1 - r), not (1 + r)
• Same formula structure as compound interest, just negative growth
• Depreciation rate of 15% means asset retains 85% of value each year

Solution:

Identify variables:

PV = $28,000

Depreciation rate: r = 18% = 0.18

Time: n = 5 years

Calculate:

\[ FV = 28000(1 - 0.18)^5 \]

\[ = 28000(0.82)^5 \]

\[ = 28000(0.3707) \]

\[ = 10,379.30 \]

Answer: $10,379.30 (or $10,379 to nearest dollar)

Loan Payment Formula:

\[ PMT = PV \times \frac{r(1+r)^n}{(1+r)^n - 1} \]

where:

• PMT: Regular payment amount
• PV: Loan amount (principal)
• r: Interest rate per payment period
• n: Total number of payments

Key Concepts:

• Amortization: Process of paying off loan over time with regular payments
• Principal: Amount of each payment that reduces the loan balance
• Interest: Amount of each payment that goes to the lender
• Early payments: more interest, less principal
• Later payments: less interest, more principal

⚠️ Common Pitfalls & Tips:

• Always use GDC Finance Solver for loans – formula is complex
• For monthly payments with annual rate: divide rate by 12
• PV is positive (loan received), PMT is negative (payment made)
• FV = 0 for fully amortized loans

Solution:

Identify variables:

PV = $250,000

Annual interest rate: 5.4%

Monthly rate: r = 5.4%/12 = 0.45% = 0.0045

Number of payments: n = 25 × 12 = 300 months

FV = 0 (loan fully paid off)

(a) Using GDC Finance Solver (TVM Solver):

N = 300

I% = 5.4

PV = 250000

PMT = solve

FV = 0

P/Y = 12 (payments per year)

C/Y = 12 (compounding per year)

Result: PMT = -$1,538.46

(Negative because it's money paid out)

Answer (a): Monthly payment = $1,538.46

(b) Total amount paid:

\[ \text{Total} = PMT \times n = 1538.46 \times 300 = 461,538 \]

Answer (b): $461,538

Interest paid = $461,538 - $250,000 = $211,538

Future Value of Annuity Formula:

\[ FV = PMT \times \frac{(1+r)^n - 1}{r} \]

Present Value of Annuity Formula:

\[ PV = PMT \times \frac{1 - (1+r)^{-n}}{r} \]

Types of Annuities:

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

Common Applications:

• Retirement savings plans
• Regular investment contributions
• Pension payments
• Savings account deposits

⚠️ Common Pitfalls & Tips:

• Use GDC Finance Solver – annuity formulas are complex
• For savings: PV = 0, solve for FV
• PMT is negative when you're making deposits
• Most IB problems assume ordinary annuity (end of period)

Solution:

Identify variables:

PMT = -$500 (monthly deposit, negative because money out)

Annual rate: 6%

Monthly rate: r = 6%/12 = 0.5% = 0.005

Number of payments: n = 30 × 12 = 360 months

PV = 0 (no initial lump sum)

FV = ? (what we want to find)

Using GDC Finance Solver:

N = 360

I% = 6

PV = 0

PMT = -500

FV = solve

P/Y = 12, C/Y = 12

Result: FV = $502,257.65

Verification using formula:

\[ FV = 500 \times \frac{(1.005)^{360} - 1}{0.005} \]

\[ = 500 \times \frac{6.023 - 1}{0.005} = 500 \times 1004.515 \]

\[ = 502,257.65 \]

Answer: $502,258 (rounded)

Total deposited = $500 × 360 = $180,000

Interest earned = $502,258 - $180,000 = $322,258

TVM Variables (Time Value of Money):

• N: Total number of payment periods
• I%: Annual interest rate (as percentage, not decimal)
• PV: Present Value (initial amount)
• PMT: Regular payment amount
• FV: Future Value (final amount)
• P/Y: Payments per year
• C/Y: Compounding periods per year

Accessing TVM Solver:

TI-84 series: APPS → Finance → TVM Solver

Casio: MENU → TVM

Sign Conventions:

• Positive: Money received (inflow)
• Negative: Money paid out (outflow)
• Example: Deposit $1000 → PV = -1000; Receive $1500 → FV = 1500

Common Problem Types:

⚠️ Common Pitfalls & Tips:

• Set P/Y and C/Y correctly! Usually both equal to 12 for monthly
• Enter I% as percentage (6 not 0.06)
• Use correct signs: outflows negative, inflows positive
• Always check P/Y and C/Y before solving
• For annual compounding: P/Y = C/Y = 1

Compound Interest

• \(FV = PV(1+r)^n\)
• Money grows exponentially
• PMT = 0 in calculator

Depreciation

• \(FV = PV(1-r)^n\)
• Value decreases
• Use (1-r) not (1+r)

Loans

• Solve for PMT
• FV = 0 (paid off)
• Use Finance Solver

Annuities

• Regular payments
• PV = 0 for savings
• Solve for FV

→ Compound Interest:

"Invest", "deposit", "savings account", "compound", "grows"

→ Depreciation:

"Decreases in value", "depreciate", "car value", "loses value"

→ Loan:

"Borrow", "mortgage", "loan", "monthly payment", "repay"

→ Annuity:

"Regular deposits", "monthly contributions", "retirement savings", "equal payments"