Time Value of Money: Concepts, Calculations & Applications

TAKEAWAYS:

• Money’s value changes over time due to time preference and inflation, so $1 today ≠ $1 in the future.
• Simple interest grows linearly (I = P×r×t), while compound interest grows exponentially (FV = PV×(1+r)^n).
• Annuities (fixed payments over time) and perpetuities (infinite payments) have distinct formulas for valuation.
• NPV and IRR help compare investment projects by converting future cash flows to present value.
• Nepali apps like eSewa (loan repayments) and Ncell (installment plans) rely on these calculations for pricing.
• Exam focus: Memorize formulas, trace cash flows, and solve for missing variables (PV, FV, r, or n).

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Core Concepts: Why Time Matters

Money’s value changes over time due to three key factors:

1. Time preference: People prefer money now over later (opportunity cost).
2. Inflation: Purchasing power erodes (e.g., ₹100 in 2023 buys less in 2024).
3. Interest: Lenders charge a premium for delayed payments.

Visual 1: The Power of Compounding

Assumption: ₹10,000 invested at 10% and 15% annual compounding. Key takeaway: Higher rates and longer horizons amplify returns exponentially.

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1. Simple vs. Compound Interest

Worked Example: Kathmandu Retail Shop’s Loan Scenario: A shop owner borrows ₹50,000 for 2 years at 12% simple interest to buy inventory.

• Calculation: $I=50,000\times 0.12\times 2=₹12,000$ Total repayment = ₹50,000 + ₹12,000 = ₹62,000.
• Real-world tie: NTC charges simple interest on delayed utility bill payments.

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2. Future Value (FV) and Present Value (PV)

Future Value (FV)

Converts today’s money to its worth in the future. Formula: $FV=PV\times (1+r{)}^n$ Example: Invest ₹20,000 at 10% compounded annually for 4 years. $FV=20,000\times (1.10{)}^4=₹29,376.24$

Present Value (PV)

Converts future money to today’s worth. Formula: $PV=\frac{FV}{(1+r{)}^n}$ Example: What’s the PV of ₹50,000 received in 3 years at 8%? $PV=\frac{50,000}{(1.08{)}^3}=₹41,322.30$

Visual 2: PV vs. FV Flow

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3. Annuities and Perpetuities

Ordinary Annuity

Fixed payments at end of periods (e.g., loan EMIs). FV of Annuity: $FV=PMT\times \frac{(1+r{)}^n-1}{r}$ PV of Annuity: $PV=PMT\times \frac{1-(1+r{)}^{-n}}{r}$

Example: Pathao Driver’s Savings Plan

• Saves ₹5,000/month for 5 years at 9% annual interest (compounded monthly).
• Steps:
  1. Convert annual rate to monthly: $r=\frac{0.09}{12}=0.0075$.
  2. $n=5\times 12=60$ months.
  3. $PV=5,000\times \frac{1-(1.0075{)}^{-60}}{0.0075}=₹263,500$.

Perpetuity

Infinite payments (e.g., preferred stocks). PV Formula: $PV=\frac{PMT}{r}$ Example: A stock pays ₹1,000/year forever. At 6% discount rate, its PV = $\frac{1,000}{0.06}=₹16,666.67$.

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4. Loan Amortization (EMIs)

Breaks down payments into principal + interest. Example: Ncell Phone Loan

• Loan: ₹40,000, 5 years, 12% annual interest (compounded monthly).
• Monthly EMI: $EMI=\frac{P\times r\times (1+r{)}^n}{(1+r{)}^n-1}$ $r=\frac{0.12}{12}=0.01$, $n=60$. $EMI=\frac{40,000\times 0.01\times (1.01{)}^{60}}{(1.01{)}^{60}-1}=₹816.32$.

Visual 3: Amortization Schedule (First 3 Months)

| Month | Payment (₹) | Principal | Interest | Remaining Balance |
|-------|-------------|-----------|----------|--------------------|
| 1     | 816.32      | 383.68    | 432.64   | 39,616.32          |
| 2     | 816.32      | 392.84    | 423.48   | 39,223.48          |
| 3     | 816.32      | 402.06    | 414.26   | 38,821.42          |

Observation: Early payments cover more interest; later payments reduce principal faster.

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5. Net Present Value (NPV) and Internal Rate of Return (IRR)

NPV

Measures an investment’s profitability by discounting cash flows. Formula: $NPV=\sum \frac{C{F}_t}{(1+r{)}^t}-InitialInvestment$ Example: Daraz Delivery Hub Expansion

• Cost: ₹500,000.
• Cash Flows: ₹150,000/year for 4 years.
• Discount Rate: 10%.
• Calculation: $NPV=-500,000+\frac{150,000}{1.10}+\frac{150,000}{1.10^2}+\frac{150,000}{1.10^3}+\frac{150,000}{1.10^4}=₹5,600$.
• Decision: Accept (NPV > 0).

IRR

Rate that makes NPV = 0. Used to compare projects. Example: For the Daraz project, IRR ≈ 11.5% (found via trial or financial calculator).

• Rule: If IRR > required return, invest.

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## In the Real World

1. eSewa Loan Repayments

   • Uses annuity formulas to calculate EMIs for personal loans.
   • Example: A ₹200,000 loan at 14% for 3 years has EMIs of ₹6,840/month.

2. Ncell Postpaid Plans

   • Offers installment schemes (e.g., pay ₹5,000/month for 24 months for a ₹100,000 phone).
   • PV calculation: $PV=5,000\times \frac{1-(1.0075{)}^{-24}}{0.0075}=₹100,000$ (matches phone cost).

3. Nepal Investment Bank (NIBL) Fixed Deposits

   • Advertises compound interest (e.g., 8% annually).
   • ₹100,000 for 5 years grows to $100,000\times (1.08{)}^5=₹146,933$.

4. NEPSE Stock Valuation

   • Perpetuity model used for dividend stocks (e.g., NMB Bank).
   • If a stock pays ₹5/share annually and investors demand 12%, its price = $\frac{5}{0.12}=₹41.67$.

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## Exam Tip

1. Memorize Formulas:

   • PV/FV: $PV=\frac{FV}{(1+r{)}^n}$, $FV=PV\times (1+r{)}^n$.
   • Annuity PV: $PV=PMT\times \frac{1-(1+r{)}^{-n}}{r}$.
   • NPV: Sum of discounted cash flows minus initial cost.

2. Watch for Traps:

   • Compounding frequency: Annual vs. monthly (e.g., 12% annual = 1% monthly).
   • Ordinary vs. Annuity Due: Payments at end vs. start (use $(1+r{)}^n$ vs. $(1+r{)}^{n-1}$).
   • Signs in NPV: Positive NPV = good; negative = reject.

3. Practical Steps for Problems:

   • Draw a timeline (e.g., cash inflows/outflows).
   • Label all variables (PV, FV, r, n).
   • Use a calculator for complex IRR/NPV (but show steps).

4. Common Exam Questions:

   • Calculate missing PV/FV/r/n.
   • Compare two investments using NPV/IRR.
   • Amortization schedules (show first/last payments).
   • Case study: "A shopkeeper wants to buy equipment costing ₹300,000. He can borrow at 10% or save ₹25,000/year for 10 years. Which is better?" (Use PV to compare.)

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Final Note: Time value of money is the foundation of investing, lending, and financial planning. Master these concepts, and you’ll ace calculations in loans, savings, and project evaluations—just like the apps and banks in Nepal!