Time Value of Money: PV, FV, Annuities, Interest Rates & Applications

TAKEAWAYS:

• Money’s value changes over time due to interest, inflation, and opportunity cost—this is the time value of money (TVM).
• Present Value (PV) = Future money discounted back to today; Future Value (FV) = Today’s money grown with compounding.
• Annuities (fixed payments over time) are classified as ordinary (end-of-period) or annuity due (beginning-of-period).
• Interest rates (nominal vs. effective) and compounding periods (annual, monthly, daily) drastically alter calculations.
• Excel functions (PV, FV, PMT, RATE) and financial calculators automate TVM problems.
• Real-world applications include loan repayments (e.g., bank loans), investment decisions (e.g., NEPSE stocks), and cost-benefit analysis (e.g., hotel equipment purchases).

1. Core Concept: Time Value of Money (TVM)

Money today is worth more than the same amount in the future because:

• Opportunity cost: You could invest it and earn returns.
• Inflation: Future money buys less.
• Risk: Uncertainty reduces future value.

Formula: $$\text{FV}=\text{PV}\times (1+r{)}^n\text{or}\text{PV}=\frac{\text{FV}}{(1+r{)}^n}$$ Where:

• $r$ = interest rate per period
• $n$ = number of periods

Key Insight:

• Rule of 72: Years to double money ≈ $72/\text{interest rate}$. Example: At 15% interest, money doubles in $72/15=4.8$ years.

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

A. Future Value (FV) – Growing Money Over Time

Example: You invest Rs. 50,000 in a bank at 12% annual interest. What will it be worth in 3 years? $$\text{FV}=50,000\times (1+0.12{)}^3=50,000\times 1.4049=\text{Rs. 70,245}$$

B. Present Value (PV) – Discounting Future Money

Example: You need Rs. 100,000 in 5 years. How much must you invest today at 10%? $$\text{PV}=\frac{100,000}{(1+0.10{)}^5}=\frac{100,000}{1.6105}=\text{Rs. 62,092}$$

Real-World Tie-In:

• NEPSE Stocks: Investors use PV to decide if a stock’s future dividends justify today’s price.
• Bank Loans: Lenders discount future repayments to determine today’s loan amount.

3. Annuities: Fixed Payments Over Time

An annuity is a series of equal cash flows (e.g., rent, loan payments, pensions). Two types:

1. Ordinary Annuity: Payments at end of each period (e.g., monthly rent due on the 1st of next month).
2. Annuity Due: Payments at beginning of each period (e.g., lease payments due immediately).

Formulas:

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

Example (Ordinary Annuity): A hotel deposits Rs. 20,000 annually into a sinking fund for 10 years at 8%. How much will it accumulate? $$\text{FVA}=20,000\times \frac{(1.08{)}^{10}-1}{0.08}=20,000\times 14.4866=\text{Rs. 289,732}$$

Example (Annuity Due): If the hotel deposits Rs. 20,000 at the start of each year: $${\text{FVA}}_{\text{due}}={\text{FVA}}_{\text{ordinary}}\times (1+r)=289,732\times 1.08=\text{Rs. 312,310}$$

flowchart TD
    A["Ordinary Annuity"] -->|"Payments"| B["End of Period"]
    C["Annuity Due"] -->|"Payments"| D["Beginning of Period"]
    B -->|"Example"| E["Rent Paid on 1st of Next Month"]
    D -->|"Example"| F["Lease Payment Due Immediately"]

4. Interest Rates: Nominal vs. Effective

• Nominal Rate: Stated rate (e.g., 12% per year).
• Effective Rate: Actual rate earned after compounding. $$\text{Effective Rate}={\left(1+\frac{\text{Nominal Rate}}{m}\right)}^m-1$$ Where $m$ = compounding periods per year.

Example: A bank offers 12% nominal interest compounded monthly. $$\text{Effective Rate}={\left(1+\frac{0.12}{12}\right)}^{12}-1=1.01^{12}-1=0.1268\text{ or }12.68\%$$

Real-World Tie-In:

• Khalti/Esawa Loans: Banks advertise nominal rates but charge effective rates after daily compounding.
• NTC Bills: Utility companies use effective rates to calculate late fees.

5. Excel and Financial Calculators

Excel functions simplify TVM calculations:

Example (Loan Amortization): A hotel takes a Rs. 500,000 loan at 12% for 5 years. Monthly payment? $$\text{PMT}=\frac{500,000\times 0.01}{(1-(1+0.01{)}^{-60})}=\text{Rs. 11,254/month}$$

In the Real World

1. Pathao/Khalti Loans:

   • Concept: Present Value of Annuity
   • How: When you take a Rs. 50,000 loan to be repaid in 12 monthly installments at 2% per month, the app calculates your monthly payment using PVA. For example: $$\text{PMT}=\frac{50,000\times 0.02}{1-(1.02{)}^{-12}}=\text{Rs. 4,564/month}$$
   • Why It Matters: Ensures lenders earn a fair return while borrowers know exact repayment amounts.

2. NEPSE Stock Investments:

   • Concept: Future Value and Discounted Cash Flow (DCF)
   • How: Investors use FV to project future dividends and PV to compare stocks. For example, if a stock pays Rs. 5,000 annually for 5 years at 10% discount rate: $$\text{PV}=5,000\times \frac{1-(1.10{)}^{-5}}{0.10}=\text{Rs. 19,255}$$
   • Why It Matters: Helps decide if buying a stock at Rs. 20,000 is justified.

3. Hotel Equipment Purchases (e.g., Lumbini Hotel):

   • Concept: Net Present Value (NPV)
   • How: If a Rs. 17,000 truck saves Rs. 5,000/year in costs for 4 years at 15% discount rate: $$\text{NPV}=-17,000+5,000\times \frac{1-(1.15{)}^{-4}}{0.15}=-17,000+14,065=\text{Rs. -2,935 (Reject)}$$
   • Why It Matters: Ensures hotels only buy equipment that adds long-term value.

6. Common Mistakes and Pitfalls

Example of a Trap:

• Question: "What is the PV of Rs. 10,000 due in 3 years at 10%?"
• Wrong Answer: $10,000\times 0.90=9,000$ (simple interest).
• Correct Answer: $10,000/(1.10{)}^3=\text{Rs. 7,513}$ (compound discounting).

Exam Tip

1. Memorize Formulas:

   • Write down PV/FV/Annuity formulas before exams. Examiners often test recall.
   • Example: For NPV, always use: $$\text{NPV}=\sum \frac{{\text{Cash Flows}}_t}{(1+r{)}^t}-\text{Initial Investment}$$

2. Watch for Annuity Types:

   • 90% of errors come from mixing ordinary vs. annuity due. Always check if payments are at the start or end.

3. Excel Shortcuts:

   • Use =NPV(rate, cash_flow_series) for uneven cash flows.
   • For loans, =PMT(rate, nper, pv) is your best friend.

4. Real-World Scenarios:

   • 20% of questions tie TVM to business decisions (e.g., "Should Hotel X buy a new AC unit?").
   • Always calculate NPV for investment decisions.

5. Unit Consistency:

   • Ensure rates and periods match (e.g., annual rate for yearly periods, monthly rate for monthly payments).

6. Past Exam Patterns:

   • 2022 TU Question: "Calculate EOQ for a hotel’s supply order" → Not TVM, but working capital (Unit 5). Stick to PV/FV/Annuity for this unit.
   • 2023 PU Question: "Find the PV of Rs. 200,000 due in 10 years at 12%" → Direct PV formula application.

Final Worked Example (Nepali Business Context): Problem: Kathmandu Spice Mart needs Rs. 200,000 in 4 years to expand. The bank offers 9% annual interest. How much should they invest today?

Solution: $$\text{PV}=\frac{200,000}{(1.09{)}^4}=\frac{200,000}{1.4116}=\text{Rs. 141,684}$$ Interpretation: The mart must deposit Rs. 141,684 today to have Rs. 200,000 in 4 years at 9% interest.

Mermaid Diagram: Accounting Cycle (TVM in Business)

flowchart TD
    A["Cash Flow"] --> B["Invest Today\n(PV)"]
    B --> C["Earn Interest\n(Compounding)"]
    C --> D["Grow to Future Value\n(FV)"]
    D --> E["Business Use\n(Expansion, Loan Repayment)"]
    E --> F["New Cash Flows\n(Annuities, Dividends)"]
    F --> B