Business Mathematics IUnit 78 min read

Maths of Finance: Interest, Annuities & NPV

Unit 7 of Business Mathematics I covers simple and compound interest, annuities (ordinary and annuity due), present value, net present value (NPV), and loan amortization—essential tools for loans, investments, and financial decision-making in business.

Padhai · Based on the official syllabus and past papers
Updated 7 Oct 2026

TAKEAWAYS:

Simple vs. compound interest: Understand how time value of money works and why compounding grows wealth faster.
Annuities: Master the formulas for ordinary annuities and annuities due to calculate future or present value of regular payments.
NPV: Learn how to evaluate investment projects by discounting cash flows to present value.
Loan amortization: Break down how monthly payments cover interest and principal over time.
Real-world applications: Apply these concepts to loans (e.g., bank loans), savings (e.g., fixed deposits), and business investments (e.g., NEPSE stocks).

1. Simple Interest

Definition: Interest calculated only on the original principal amount for a given period. Formula: $I = P \times r \times t$ where:

$I$ = Interest
$P$ = Principal (initial amount)
$r$ = Annual interest rate (in decimal)
$t$ = Time in years

Worked Example 1: A business takes a ₹50,000 loan from a bank at 8% simple interest for 3 years. Calculate the total interest and maturity amount. Answer: Total interest = ₹12,000; Maturity amount = ₹62,000.

Comparison Table: Simple vs. Compound Interest

Feature	Simple Interest	Compound Interest
Calculation	Only on principal	On principal + accumulated interest
Growth	Linear (fixed amount per period)	Exponential (accelerates over time)
Formula	$I = P \times r \times t$	$A = P (1 + r)^{t}$
Use Case	Short-term loans (e.g., overdrafts)	Long-term investments (e.g., fixed deposits)

2. Compound Interest

Definition: Interest calculated on the initial principal and the accumulated interest of previous periods. Formula: $A = P (1 + \frac{r}{n})^{n t}$ where:

$A$ = Amount after time $t$
$n$ = Number of times interest is compounded per year (e.g., $n = 12$ for monthly)

Worked Example 2: A company invests ₹100,000 in a fixed deposit at 6% annual interest, compounded quarterly for 5 years. Calculate the maturity amount. Answer: Maturity amount ≈ ₹134,690.

Graph: Growth of ₹100,000 at 6% Compound Interest (Annually vs. Quarterly)

3. Annuities: Ordinary vs. Annuity Due

Definitions:

Ordinary Annuity: Payments made at the end of each period (e.g., rent, loan EMIs).
Annuity Due: Payments made at the beginning of each period (e.g., lease payments, insurance premiums).

Formulas:

Type	Future Value (FV)	Present Value (PV)
Ordinary Annuity	$F V = P M T \times \frac{( 1 + r ) ^{n} - 1}{r}$	$P V = P M T \times \frac{1 - ( 1 + r ) ^{- n}}{r}$
Annuity Due	$F V = P M T \times \frac{( 1 + r ) ^{n} - 1}{r} \times (1 + r)$	$P V = P M T \times \frac{1 - ( 1 + r ) ^{- n}}{r} \times (1 + r)$

Worked Example 3: A business deposits ₹5,000 at the end of each year in a savings account offering 5% annual interest for 10 years. Calculate the future value. Answer: Future value ≈ ₹62,889.

Mermaid Diagram: Ordinary Annuity vs. Annuity Due Cash Flows

flowchart TD
    A["Ordinary Annuity"] --> B["Payments at End of Period"]
    C["Annuity Due"] --> D["Payments at Beginning of Period"]
    B --> E["Example: Loan EMIs"]
    D --> F["Example: Rent Paid in Advance"]

4. Present Value and Net Present Value (NPV)

Present Value (PV): The current worth of future cash flows, discounted at a specific rate. Formula: $P V = \frac{F V}{( 1 + r ) ^{n}}$

Net Present Value (NPV): Used to evaluate investment projects by comparing present value of cash inflows and outflows. Formula: $N P V = t = 0 \sum n \frac{C F _{t}}{( 1 + r ) ^{t}} - I ni t ia l I n v es t m e n t$ where $C F_{t}$ = Cash flow at time $t$ .

Worked Example 4: A company is considering an investment with:

Initial cost = ₹50,000
Cash inflows: ₹15,000/year for 4 years
Discount rate = 10% Calculate NPV. Answer: NPV = ₹7,546 (Project is acceptable since NPV > 0).

Graph: NPV Profile for Different Discount Rates

5. Loan Amortization

Definition: The process of paying off a loan in equal installments, where each payment covers interest and principal. Formula for Monthly Payment (M): $M = P \times \frac{r ( 1 + r ) ^{n}}{( 1 + r ) ^{n} - 1}$ where:

$P$ = Loan principal
$r$ = Monthly interest rate ( $r = \frac{ann u a l r a t e}{12}$ )
$n$ = Total number of payments

Worked Example 5: A business takes a ₹200,000 loan at 8% annual interest for 5 years. Calculate the monthly payment and amortization schedule for the first 3 months. Answer: Monthly payment = ₹4,264.

Mermaid Diagram: Loan Amortization Process

flowchart TD
    A["Loan Amount"] --> B["Calculate Monthly Payment"]
    B --> C["Allocate Payment to Interest & Principal"]
    C --> D["Update Remaining Balance"]
    D --> E["Repeat Until Loan is Paid Off"]
    E --> F["Final Payment (if any)"]

In the Real World

Khalti & eSewa (Digital Payments):
- Concept Used: Compound Interest
- How: When users save money in Khalti’s "Khalti Save" or eSewa’s fixed deposit-like features, the interest is compounded annually, growing their savings faster than simple interest would.
Ncell & NTC (Mobile Loans):
- Concept Used: Loan Amortization
- How: When customers take a ₹50,000 loan from Ncell at 12% annual interest for 2 years, their monthly payments are calculated using amortization. Each payment covers part of the principal and part of the interest, reducing the loan balance over time.
NEPSE Stock Investments:
- Concept Used: Net Present Value (NPV)
- How: Investors use NPV to decide whether to buy stocks. For example, if a company’s future dividends discounted at 10% yield an NPV of ₹50,000 (above the current stock price), it’s a good investment.
Daraz & Pathao (Business Loans):
- Concept Used: Annuities (Ordinary)
- How: Many small businesses take working capital loans from banks or fintech companies (e.g., Pathao’s merchant loans). They repay fixed monthly installments (ordinary annuity), where each payment is calculated to cover both interest and principal over the loan term.
Bank Fixed Deposits (e.g., NMB, Global IME):
- Concept Used: Compound Interest
- How: When a customer deposits ₹100,000 for 3 years at 7% compounded annually, the bank calculates the maturity amount using compound interest, ensuring higher returns than simple interest.

Exam Tip

Memorize Key Formulas:
- Simple/Compound Interest, Annuity FV/PV, NPV, and Amortization.
- Shortcut: Write them on a flashcard and practice deriving them from first principles.
Unit Consistency:
- Always ensure time ( $t$ ) is in years if the rate ( $r$ ) is annual. For monthly compounding, adjust $r$ and $n$ accordingly.
NPV Decision Rule:
- If NPV > 0, accept the project.
- If NPV < 0, reject it.
- Common Mistake: Forgetting to subtract the initial investment from the sum of discounted cash flows.
Amortization Tables:
- Examiners often ask for partial amortization schedules. Practice calculating the first 2–3 rows manually to show understanding.
Real-World Scenarios:
- Questions may tie concepts to loans, investments, or business decisions. For example:
  - "A company wants to buy machinery costing ₹500,000 with cash flows of ₹150,000/year for 5 years. Calculate NPV at 12% and advise."
- Always label your answer clearly (e.g., "NPV = ₹X → Recommendation: [Accept/Reject]").
Graphs in Exams:
- If asked to plot (e.g., NPV vs. discount rate), sketch a rough graph showing trends. Label axes and key points.
Common Pitfalls:
- Mixing ordinary vs. annuity due: Remember, annuity due payments are at the start of the period.
- Ignoring compounding frequency: If interest is compounded monthly, $n = 12$ in the formula.
- Sign errors in NPV: Cash inflows are +, outflows are -.

Final Note: Mathematics of Finance is 80% application, 20% theory. Practice 5–10 numerical problems per topic, especially those involving time value of money. Use a financial calculator or Excel (=FV, =PV, =NPV functions) to verify your answers.

Based on the PU BBA (PU) syllabus for Business Mathematics I, unit 7.

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