Time Value of Money & Financial Mathematics: Concepts, Formulas & Applications

TAKEAWAYS:

• Time value of money (TVM) means money today is worth more than the same amount in the future due to its earning potential (interest, inflation, or investment returns).
• Key formulas include:
  • Future Value (FV) = PV × (1 + r)^n
  • Present Value (PV) = FV / (1 + r)^n
  • Annuity formulas for periodic payments (ordinary vs. annuity due).
• Applications include loan repayments (e.g., bank loans), investment evaluations (e.g., NEPSE stocks), and break-even analysis (e.g., Daraz’s order fulfillment costs).
• Amortization schedules show how loans (e.g., Ncell’s financing plans) are repaid over time, splitting interest and principal.
• Break-even point determines the sales volume where total revenue equals total costs (critical for businesses like Kathmandu Manufacturing Company).
• Discounting cash flows helps compare projects (e.g., NTC’s infrastructure investments) by converting future cash flows to present value.

1. Introduction to Time Value of Money (TVM)

Why Does Money Have a Time Value?

Money loses purchasing power over time due to:

• Inflation: Rs 1,000 today buys less in 5 years (e.g., a cup of coffee in Kathmandu cost Rs 120 in 2019; it’s Rs 180 in 2024).
• Opportunity cost: Investing Rs 1,000 today could earn interest or returns (e.g., NEPSE’s average return is ~12% annually).
• Risk: Future cash flows are uncertain (e.g., a Daraz seller’s future sales depend on market trends).

Key Principle:

"A rupee today is worth more than a rupee tomorrow."

Visual: The Power of Compounding

graph LR
    A["Rs 1,000 Today"] -->|"Invested at 10%"| B["Rs 1,100 Year 1"]
    B -->|"Reinvested"| C["Rs 1,210 Year 2"]
    C -->|"Reinvested"| D["Rs 1,331 Year 3"]
    D -->|"Reinvested"| E["Rs 1,464 Year 4"]

Example: If you invest Rs 1,000 at 10% annual interest, it grows to Rs 1,464 in 4 years. Without compounding (simple interest), it would only be Rs 1,400.

2. Core TVM Formulas

A. Future Value (FV)

Converts today’s money to its future value. Formula: $$FV=PV\times (1+r{)}^n$$

• $PV$ = Present Value (today’s money)
• $r$ = Interest rate per period (e.g., 8% = 0.08)
• $n$ = Number of periods (years, months, etc.)

Example: You deposit Rs 50,000 in a bank at 6% annual interest. What’s its value in 3 years? $$FV=50,000\times (1+0.06{)}^3=50,000\times 1.191=\text{Rs 59,550}$$

B. Present Value (PV)

Converts future money to today’s value (discounting). Formula: $$PV=\frac{FV}{(1+r{)}^n}$$ Example: You need Rs 100,000 in 5 years. How much should you invest today at 7% interest? $$PV=\frac{100,000}{(1+0.07{)}^5}=\frac{100,000}{1.403}=\text{Rs 71,270}$$

C. Annuities: Regular Payments

An annuity is a series of equal payments (e.g., loan EMIs, insurance premiums). Two types:

1. Ordinary Annuity: Payments at the end of the period (e.g., bank loan EMIs).
2. Annuity Due: Payments at the start of the period (e.g., rent paid in advance).

Formulas:

• Future Value of Annuity (FVA): $$FVA=PMT\times \frac{(1+r{)}^n-1}{r}$$
• Present Value of Annuity (PVA): $$PVA=PMT\times \frac{1-(1+r{)}^{-n}}{r}$$ Where $PMT$ = Payment per period.

Example: You want to save Rs 20,000 annually for 5 years at 8% interest. How much will you have? $$FVA=20,000\times \frac{(1.08{)}^5-1}{0.08}=20,000\times 5.867=\text{Rs 117,340}$$

3. Loan Amortization: How EMIs Work

Banks (e.g., Nabil Bank, Global IME) use amortization to structure loans. Each EMI payment covers:

• Interest (calculated on the remaining principal).
• Principal repayment (reduces the loan balance).

Amortization Schedule Example: Rs 500,000 Loan at 10% for 5 Years

Key Observations:

• Early EMIs mostly cover interest; later payments reduce the principal.
• Total interest paid = Total EMIs – Loan Amount = (11,560 × 60) – 500,000 = Rs 193,600.

4. Break-Even Analysis: When Does a Business Profit?

Used by retailers (Daraz, Hamrobazar), manufacturers (Butwal Manufacturing Company), and service providers (Pathao) to determine the minimum sales volume needed to cover costs.

Break-Even Point (BEP) Formula:

$$BEP(units)=\frac{FixedCosts}{SellingPriceperUnit-VariableCostperUnit}$$ $$BEP(Rs)=BEP(units)\times SellingPriceperUnit$$

Example: Kathmandu Manufacturing Company (KMC)

• Selling price per unit: Rs 200
• Variable cost per unit: Rs 120
• Fixed costs (rent, salaries): Rs 600,000

Step 1: Calculate Contribution Margin per Unit $$\text{Contribution Margin}=\text{Selling Price}-\text{Variable Cost}=200-120=\text{Rs 80}$$

Step 2: Calculate Break-Even in Units $$BEP=\frac{600,000}{80}=7,500\text{units}$$

Step 3: Calculate Break-Even in Rs $$BEP(Rs)=7,500\times 200=\text{Rs 1,500,000}$$

Interpretation:

• KMC must sell 7,500 units to cover costs.
• If they sell 8,000 units, profit = (8,000 – 7,500) × 80 = Rs 40,000.

5. Net Present Value (NPV) and Profitability Index

Used to evaluate investment projects (e.g., NTC’s road expansion, a new factory for BMC).

NPV Formula:

$$NPV=\sum \frac{C{F}_t}{(1+r{)}^t}-InitialInvestment$$ Where:

• $C{F}_t$ = Cash flow at time $t$
• $r$ = Discount rate (cost of capital, e.g., 12%)

Profitability Index (PI):

$$PI=\frac{PVofFutureCashFlows}{InitialInvestment}$$

• PI > 1: Accept the project (profitable).
• PI < 1: Reject the project.

Example: Project Alpha (from past exam)

• Initial Investment: Rs 20 million
• PV of Future Cash Flows: Rs 25 million $$PI=\frac{25}{20}=1.25(\text{Accept, as PI > 1})$$

6. Real-World Applications in Nepal

A. eSewa and Khalti: Discounting Future Payments

• How it works: When you pay utility bills (NTC, NEA) via eSewa/Khalti, you get a discount for early payment. This is an application of present value: paying Rs 950 today is better than Rs 1,000 later.
• Example: NTC offers a 2% discount if paid within 10 days. If the discount rate is 15% annually, the discount period is effectively: $$1-(1+0.15{)}^{-10/365}\approx 0.004(0.4\%)$$ (The actual 2% discount is a marketing incentive, not purely based on TVM.)

B. Daraz and Hamrobazar: Break-Even and Pricing

• Problem: Daraz sellers must determine the minimum order quantity to cover costs.
• Example: A seller in Kathmandu sells Rs 500 handmade masks with:
  • Variable cost: Rs 200
  • Fixed costs (website fees, storage): Rs 10,000/month $$BEP=\frac{10,000}{500-200}=33.33\text{masks}$$
  • Reality: Sellers aim for higher margins (e.g., Rs 800/mask) to account for risk.

C. Ncell and NTC: Loan Amortization for Customer Financing

• Ncell’s "Pay in Installments": When you buy a smartphone for Rs 20,000 with a 12-month EMI plan at 15% interest, the bank calculates: $$EMI=\frac{P\times r\times (1+r{)}^n}{(1+r{)}^n-1}=\frac{20,000\times 0.0125\times (1.0125{)}^{12}}{(1.0125{)}^{12}-1}\approx \text{Rs 1,820/month}$$
  • Total paid: 1,820 × 12 = Rs 21,840 (Rs 1,840 extra as interest).

D. NEPSE Stocks: Discounted Cash Flow (DCF) Valuation

• Investors use DCF to value stocks (e.g., NMB Bank, CG Group).
• Example: If a stock pays Rs 50 dividend annually and grows at 8%, its intrinsic value at a 12% discount rate is: $$PV=\frac{50}{0.12-0.08}=\frac{50}{0.04}=\text{Rs 1,250}$$
  • If the stock trades at Rs 1,500, it’s overvalued; if below Rs 1,250, it’s undervalued.

E. Banks (Nabil, Standard Chartered): Loan Approval

• Banks use Debt Service Coverage Ratio (DSCR) to check if a business can repay loans. $$DSCR=\frac{\text{Net Operating Income}}{\text{Total Debt Payments}}$$
  • Example: A restaurant has Rs 500,000 annual profit and a Rs 200,000 loan EMI. If DSCR > 1.25, the loan is approved.

7. Common Mistakes and Exam Pitfalls

8. Worked Example: Butwal Manufacturing Company (BMC)

Problem: BMC sells tea kettles at Rs 60 each. Variable cost = Rs 40, fixed costs = Rs 200,000.

1. What’s the break-even point in units and Rs?
2. If BMC sells 10,000 units, what’s the profit?
3. If BMC wants a Rs 50,000 profit, how many units must it sell?

Solution:

1. Break-Even Point:

   • Contribution margin = Rs 60 – Rs 40 = Rs 20
   • BEP (units) = Rs 200,000 / Rs 20 = 10,000 units
   • BEP (Rs) = 10,000 × Rs 60 = Rs 600,000

2. Profit at 10,000 Units:

   • Total Revenue = 10,000 × Rs 60 = Rs 600,000
   • Total Variable Cost = 10,000 × Rs 40 = Rs 400,000
   • Total Cost = Fixed + Variable = Rs 200,000 + Rs 400,000 = Rs 600,000
   • Profit = Revenue – Cost = Rs 600,000 – Rs 600,000 = Rs 0 (exactly at BEP).

3. Units for Rs 50,000 Profit: $$\text{Required Sales}=\frac{\text{Fixed Costs}+\text{Desired Profit}}{\text{Contribution Margin}}=\frac{200,000+50,000}{20}=12,500\text{units}$$

9. The Accounting Cycle and TVM

flowchart TD
    A["Start: Record Transactions"] --> B["Journal Entries\n(Cash, Sales, Expenses)"]
    B --> C["Post to Ledger\n(T-accounts)"]
    C --> D["Prepare Trial Balance"]
    D --> E["Adjusting Entries\n(Depreciation, Accruals)"]
    E --> F["Adjusted Trial Balance"]
    F --> G["Financial Statements\n(Income Statement, Balance Sheet)"]
    G --> H["Close Temporary Accounts\n(Revenue, Expenses to Retained Earnings)"]
    H --> I["Post-Closing Trial Balance"]
    I --> J["End: Prepare for Next Period"]

How TVM Fits In:

• Depreciation: Spreads the cost of an asset (e.g., a machine) over its useful life (e.g., Rs 500,000 over 5 years = Rs 100,000/year).
• Accrued Interest: Future interest on loans (e.g., a Rs 1,000,000 loan at 10% accrues Rs 100,000/year).
• Retained Earnings: Represents the PV of past profits reinvested in the business.

10. Exam Tip: How to Score Full Marks

1. Show All Steps: Examiners deduct marks for missing intermediate calculations (e.g., calculating contribution margin before BEP).
2. Label Clearly: Use Dr/Cr for journal entries and units/Rs for break-even answers.
3. Round Correctly: Use 2 decimal places for financial calculations (e.g., Rs 1,234.56, not Rs 1,235).
4. Interpret Results: Always explain what the answer means (e.g., "The project is profitable because NPV is positive").
5. Use Real-World Examples: If asked about applications, tie answers to Nepali businesses (e.g., "Like Ncell’s EMI plans...").
6. Watch for Tricks:
   • Annuity vs. Perpetuity: Perpetuity formula is $PV=\frac{PMT}{r}$ (no $n$).
   • Effective vs. Nominal Rates: Convert monthly rates to annual (e.g., 1% monthly = 12.68% annually).

11. Quick Revision Table

12. Practice Questions (From Past Exams)

1. Butwal Manufacturing Company (BMC):

   • Selling price = Rs 60, variable cost = Rs 40, fixed costs = Rs 200,000.
   • a. Calculate BEP in units and Rs.
   • b. If BMC sells 12,000 units, what’s the profit?

2. City Bank’s Intercom System:

   • Cost = Rs 550,000 + Rs 20,000 (transport) = Rs 570,000.
   • Salvage value after 5 years = Rs 50,000.
   • Annual savings = Rs 150,000.
   • a. Calculate NPV at 12% discount rate.
   • b. Should the bank buy it?

3. Stock Returns (Historical):

   Year	Stock A (%)	Stock B (%)
   2021	5	30
   2022	10	15
   2023	-5	5

   • a. Calculate average return for both stocks.
   • b. Which stock is less risky? (Hint: Use standard deviation.)

13. Final Checklist Before the Exam

• Can you calculate FV and PV for single sums and annuities?
• Do you know the difference between ordinary annuity and annuity due?
• Can you construct an amortization schedule for a loan?
• Do you understand break-even analysis and margins?
• Can you compute NPV and PI for investment decisions?
• Are you familiar with real-world applications (eSewa, Ncell, Daraz)?