Financial mathematics and models

Financial mathematics involves applying mathematical models to real-world financial situations, particularly compound interest, loans, investments, and annuities.

Simple Interest (SI) is calculated only on the principal amount: $SI = \frac{Prt}{100}$ or $SI = Prt$ (when r is decimal) where P = principal, r = rate per period, t = time

Compound Interest grows exponentially as interest is earned on interest: $FV = PV(1 + r)^n$ where FV = future value, PV = present value, r = interest rate per compounding period (as decimal), n = number of compounding periods

Annual Percentage Rate (APR) vs nominal rate: When interest compounds more frequently than annually, use: $r = \frac{\text{nominal rate}}{k}$ and $n = kt$ where k = compounding frequency per year, t = years

Annuities involve regular payments. For a present value annuity: $PV = \frac{R[1-(1+r)^{-n}]}{r}$ For future value annuity: $FV = \frac{R[(1+r)^n-1]}{r}$ where R = regular payment amount

Depreciation models asset value decline. Straight-line: constant decrease. Reducing balance: $V = V_0(1-r)^n$ where r = depreciation rate

Amortization involves loan repayment schedules showing principal vs interest portions. The loan formula connects monthly payment, principal, and interest rate.

Memory aid: FV grows UP (multiply), PV shrinks DOWN (divide). Compound interest = exponential growth; annuities = geometric series.

Financial mathematics powers every mortgage, student loan, retirement fund, and credit card on Earth. Understanding these models empowers informed financial decisions.

Compound Interest in Action: Imagine depositing £5,000 in a savings account with 4% annual interest, compounded quarterly. Each quarter, you earn interest on your growing balance. After 10 years:

• Compounding periods: n = 10 × 4 = 40
• Rate per period: r = 0.04/4 = 0.01
• FV = 5000(1.01)^40 ≈ £7,449.23

Compare this to simple interest: £7,000. The extra £449.23 comes from interest earning interest—the power of compounding Einstein allegedly called "the eighth wonder of the world."

Mortgages as Annuities: When you take a £200,000 mortgage at 3.5% annually over 25 years with monthly payments, you're solving an annuity equation. The bank wants £200,000 back now (PV), you'll make 300 monthly payments (n = 25 × 12), and monthly rate is r = 0.035/12. Using the PV annuity formula solved for R gives monthly payment ≈ £1,001.

Depreciation Reality: A new car costing £30,000 might depreciate 20% annually (reducing balance). After 5 years: V = 30000(0.8)^5 ≈ £9,830—losing two-thirds of its value!

Inflation Connection: £100 today won't buy the same in 10 years. With 2% inflation, purchasing power follows: PV = 100(1.02)^(-10) ≈ £82—your money loses 18% real value.

Analogy: Compound interest is like a snowball rolling downhill—it gathers more snow (interest) as it grows larger, accelerating exponentially.

Example 1: Sarah invests $8,000 at 6% per annum compounded monthly. How much after 8 years?

Solution:

1. Identify variables: PV = 8000, r = 0.06/12 = 0.005, n = 8 × 12 = 96
2. Apply formula: FV = 8000(1.005)^96
3. Calculate: FV = 8000(1.6141...) ≈ $12,912.83

Examiner note: Always show the formula substitution explicitly for method marks.

Example 2: A loan of £15,000 at 7.2% annual interest (monthly compounding) requires monthly payments of £300. How many payments to clear the loan?

Solution:

1. Set up PV annuity: 15000 = \frac{300[1-(1+r)^{-n}]}{r} where r = 0.072/12 = 0.006
2. Rearrange: 15000(0.006) = 300[1-(1.006)^{-n}]
3. Simplify: 90/300 = 1-(1.006)^{-n}, so 0.3 = 1-(1.006)^{-n}
4. Therefore: (1.006)^{-n} = 0.7
5. Take logs: -n·ln(1.006) = ln(0.7)
6. Solve: n = -ln(0.7)/ln(1.006) ≈ 59.66 months

Answer: 60 payments (must round UP—you can't make partial payments)

Example 3: A machine costs $25,000 and depreciates at 15% annually (reducing balance). Find value after 6 years.

Solution:

1. V = V₀(1-r)^n = 25000(1-0.15)^6
2. V = 25000(0.85)^6
3. V ≈ $9,392.76

Key insight: Depreciation uses (1-r); appreciation/growth uses (1+r).