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Unit 7: Financial Mathematics

Financial Mathematics applies quantitative methods to financial problems. It is essential for banking, investing, taxation, and economic modeling.

1. Interest and Interest Rates

Interest is the cost of borrowing money or the return on an investment.

Simple Interest (SI)

Calculated only on the principal amount. $$ \text{SI} = \frac{P \times R \times T}{100} $$ Where $P$ = Principal, $R$ = Rate of Interest per annum, $T$ = Time in years.

Compound Interest (CI)

Calculated on the principal and the accumulated interest of previous periods. $$ A = P \left(1 + \frac{r}{100}\right)^n $$ $$ \text{CI} = A - P $$ Where $A$ = Amount, $r$ = Rate per compounding period, $n$ = Total number of periods.

Effective Rate of Interest

The true annual interest rate considering the effect of compounding. $$ E = \left(1 + \frac{i}{n}\right)^n - 1 $$ Where $i$ = Nominal interest rate, $n$ = Number of compounding periods per year.

2. Annuities

An annuity is a series of equal payments made at regular intervals.

Regular Annuity

Payments are made at the end of each period. Future Value of a Regular Annuity ($FV$): $$ FV = C \times \left[ \frac{(1 + i)^n - 1}{i} \right] $$ Where $C$ = Cash flow per period, $i$ = Interest rate per period, $n$ = Number of periods.

(Note: Simple applications up to 3 periods only are expected).

3. Taxation

Taxes are mandatory contributions levied on individuals or corporations by the government.

Goods and Services Tax (GST)

An indirect tax used in India on the supply of goods and services.

  • CGST: Central GST (varies by state/center split).
  • SGST: State GST.
  • IGST: Integrated GST (inter-state transactions).

$$ \text{Tax Amount} = \text{Base Price} \times \left( \frac{\text{Rate}}{100} \right) $$

Income Tax

Computed by adding income from various sources (salary, house property, business, capital gain, etc.) and deducting allowances defined under the Income Tax Act (PF, PPF, LIC, Housing loan, etc.). $$ \text{Taxable Income} = \text{Gross Income} - \text{Deductions} $$ Income tax is calculated based on predefined tax slabs.

4. Utility Bills

The calculation of utility bills involves evaluating fixed charges and variable usage.

Electricity and Water Bills

  • Tariff Rates: Depending on user brackets (residential, commercial).
  • Fixed Charge: A mandatory stationary fee regardless of usage.
  • Service Charge/Surcharge: Extra percentage added to the main cost or specific service fees.

$$ \text{Total Bill} = \text{Fixed Charge} + (\text{Units Consumed} \times \text{Rate per Unit}) + \text{Taxes/Surcharges} $$

Competency-Based Questions

  1. (Simple vs. Compound Trade-off): A business wants to borrow ₹500,000 for 3 years. Bank A offers 6.5% Simple Interest. Bank B offers 6.2% Compound Interest compounded annually. By calculating the total interest for both loans, which bank offers the lower cost of borrowing?

  2. (Annuity Application): A software engineer decides to invest ₹10,000 at the end of every year into a tech fund that guarantees a return of 5% compounded annually. Using the Regular Annuity formula, calculate the accumulated future value of this investment at the end of exactly 3 years.

  3. (GST Computation in E-commerce): A customer buys an electronic CPU online. The base price is ₹15,000. If the GST on electronics is divided equally into 9% CGST and 9% SGST, calculate the total amount the customer pays during checkout.

  4. (Income Tax Deduction Logic): An individual has a gross yearly income of ₹1,200,000. They have total deductions of ₹250,000 under sections PF, LIC, and Medical. If the tax slab imposes a 10% rate on taxable income above ₹500,000 (and 0% below), calculate their final income tax liability.

  5. (Tariff and Utility Modeling): Model an electricity bill using an algebraic expression where the fixed meter rent is $F$, the first 100 units cost $R_1$ per unit, and units above 100 cost $R_2$ per unit. Write the unified total cost function $C(x)$ for total units consumed $x$, assuming $x > 100$. Evaluate $C(250)$ if $F = ₹50, R_1 = ₹3, R_2 = ₹5$.


Answers to Competency-Based Questions

1. Simple vs. Compound Trade-off: Bank A (Simple Interest): $$ \text{SI} = \frac{500,000 \times 6.5 \times 3}{100} = ₹97,500 $$ Bank B (Compound Interest): $$ A = 500,000 \left(1 + \frac{6.2}{100}\right)^3 = 500,000 (1.062)^3 \approx 500,000(1.19777) = ₹598,885.16 $$ $$ \text{CI} = 598,885.16 - 500,000 = ₹98,885.16 $$ Since $₹97,500 < ₹98,885.16$, Bank A (Simple Interest) offers the lower cost of borrowing.

2. Annuity Application: Using the Future Value of a Regular Annuity formula: $$ FV = 10,000 \times \left[ \frac{(1 + 0.05)^3 - 1}{0.05} \right] $$ $$ FV = 10,000 \times \left[ \frac{(1.157625) - 1}{0.05} \right] = 10,000 \times \left[ \frac{0.157625}{0.05} \right] = 10,000 \times 3.1525 $$ $$ FV = ₹31,525 $$

3. GST Computation in E-commerce: Base Price = ₹15,000. $$ \text{CGST} (9%) = 15,000 \times 0.09 = ₹1,350 $$ $$ \text{SGST} (9%) = 15,000 \times 0.09 = ₹1,350 $$ $$ \text{Total Bill} = 15,000 + 1,350 + 1,350 = ₹17,700 $$

4. Income Tax Deduction Logic: $$ \text{Gross Income} = ₹1,200,000 $$ $$ \text{Taxable Income} = \text{Gross} - \text{Deductions} = 1,200,000 - 250,000 = ₹950,000 $$ Tax is applied only on the amount above ₹500,000: $$ \text{Amount subject to Tax} = 950,000 - 500,000 = ₹450,000 $$ $$ \text{Final Tax Liability} = 10% \text{ of } 450,000 = ₹45,000 $$

5. Tariff and Utility Modeling: The cost function $C(x)$ for an electricity bill when consumption $x > 100$ is modeled algebraically as: $$ C(x) = F + 100 R_1 + (x - 100) R_2 $$ Evaluating $C(250)$ given $F = 50, R_1 = 3, R_2 = 5$: $$ C(250) = 50 + 100(3) + (250 - 100)(5) $$ $$ C(250) = 50 + 300 + 150(5) = 350 + 750 = ₹1,100 $$