Time Value of Money: Present Value, Future Value, Annuities & Valuation

TAKEAWAYS:

• Time Value of Money (TVM) is the foundation of all financial decisions—whether calculating loan repayments, valuing stocks, or comparing investment options.
• Present Value (PV) discounts future cash flows to today’s dollars using the formula $PV=\frac{FV}{(1+r{)}^n}$, where $r$ is the discount rate (e.g., interest rate or required return).
• Future Value (FV) compounds cash flows forward using $FV=PV\times (1+r{)}^n$, critical for retirement planning (e.g., NTC’s fixed deposit schemes).
• Annuities (ordinary vs. annuity due) are fixed payments over time (e.g., monthly EMIs on a Daraz loan or a Pathao driver’s lease payments).
• Perpetuities (infinite cash flows) simplify stock valuation (e.g., NEPSE’s dividend-paying stocks like NMB Bank).
• Valuation techniques (NPV, IRR, Payback Period) help businesses like eSewa or Khalti decide whether a project (e.g., expanding a branch) is profitable.

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1. Why Money’s Value Changes Over Time

Money loses purchasing power due to:

• Inflation: Rs 1,000 today buys less in 5 years (e.g., a Kathmandu shop’s rent doubles).
• Opportunity Cost: Investing Rs 1,000 today could earn interest instead of leaving it idle.
• Risk: Future cash flows are uncertain (e.g., a Daraz seller’s future sales depend on market trends).

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2. Core TVM Formulas (With Visuals)

A. Single Sum: Future Value (FV)

If you invest Rs 5,000 today at 8% annual interest, how much will it grow to in 3 years? Formula: $$FV=PV\times (1+r{)}^n$$ Worked Example (Nepali Context): A Kathmandu retail shop owner deposits Rs 20,000 in a bank at 6% annual interest. What’s its value after 5 years?

Answer: Rs 26,809 (rounded).

B. Single Sum: Present Value (PV)

What’s the today’s value of Rs 30,000 received in 4 years if the discount rate is 10%? Formula: $$PV=\frac{FV}{(1+r{)}^n}$$ Worked Example (eSewa Loan): eSewa offers a 4-year loan of Rs 50,000 at 12% annual interest. What’s the equivalent lump-sum payment today? $$PV=\frac{50,000}{(1+0.12{)}^4}=\frac{50,000}{1.5735}=\text{Rs 31,770}$$ Interpretation: You’d need to pay Rs 31,770 now to avoid the loan’s future burden.

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3. Annuities: Fixed Payments Over Time

A. Ordinary Annuity (Payments at End of Period)

Example: Ncell’s 3-year savings plan deposits Rs 2,000 at the end of each year at 7% interest. Formula: $$F{V}_{\text{annuity}}=PMT\times \frac{(1+r{)}^n-1}{r}$$ Worked Example: Calculate the future value of Ncell’s plan after 3 years. $$FV=2,000\times \frac{(1.07{)}^3-1}{0.07}=2,000\times 3.217=\text{Rs 6,434}$$

B. Annuity Due (Payments at Start of Period)

Example: Pathao driver’s monthly lease payment of Rs 15,000 at the start of each month for 2 years at 1% monthly interest. Formula: $$F{V}_{\text{annuity due}}=PMT\times \frac{(1+r{)}^n-1}{r}\times (1+r)$$ Worked Example: $$FV=15,000\times \frac{(1.01{)}^{24}-1}{0.01}\times 1.01=\text{Rs 396,000}$$

Comparison Table:

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4. Perpetuities: Infinite Cash Flows

Used to value dividend stocks (e.g., NMB Bank) or government bonds (e.g., NTC’s perpetual securities). Formula: $$P{V}_{\text{perpetuity}}=\frac{D}{r}$$ Where:

• $D$ = Annual dividend (e.g., Rs 10/share)
• $r$ = Required return (e.g., 12%)

Worked Example (NEPSE Stock): A stock pays a dividend of Rs 8 and grows at 5%. If investors require a 10% return, what’s its price? $$PV=\frac{8}{0.10}=\text{Rs 80}$$ (Note: For growing perpetuities, use $PV=\frac{D_1}{r-g}$, where $g$ = growth rate.)

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5. Valuation Techniques for Business Decisions

A. Net Present Value (NPV)

Measures profitability of an investment by discounting all cash flows to today. Formula: $$NPV=\sum \frac{C{F}_t}{(1+r{)}^t}-\text{Initial Investment}$$ Worked Example (Khalti Expansion): Khalti wants to expand to Pokhara. Initial cost: Rs 500,000. Expected cash flows:

• Year 1: Rs 150,000
• Year 2: Rs 200,000
• Year 3: Rs 250,000 Discount rate: 10%.

Decision: Reject (NPV < 0).

B. Internal Rate of Return (IRR)

The discount rate where NPV = 0. Used to compare projects. Worked Example (Daraz Loan): A Daraz seller borrows Rs 100,000 for 3 years with repayments of Rs 40,000/year. What’s the IRR? Solve for $r$ in: $$0=-100,000+\frac{40,000}{(1+r{)}^1}+\frac{40,000}{(1+r{)}^2}+\frac{40,000}{(1+r{)}^3}$$ (Use financial calculator or Excel’s =IRR() function.) Answer: IRR ≈ 15.9%.

C. Payback Period

How long until cash inflows cover the initial investment? Worked Example (NTC Project): Initial cost: Rs 200,000. Annual cash flows: Rs 50,000/year. $$\text{Payback Period}=\frac{200,000}{50,000}=4\text{ years}$$

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## In the Real World

1. eSewa Loans:

   • Concept: Present Value of Loan Repayments
   • How it works: When you take a Rs 50,000 loan at 12% for 2 years, eSewa calculates the PV of your future EMIs to determine your borrowing limit. If your monthly income’s PV (discounted at 12%) covers the loan’s PV, you qualify.
   • Example: A monthly EMI of Rs 2,500 for 24 months has a PV of Rs 48,000 (using $PV=PMT\times \frac{1-(1+r{)}^{-n}}{r}$). Thus, eSewa approves a Rs 48,000 loan.

2. Ncell Savings Plans:

   • Concept: Future Value of Annuities
   • How it works: Ncell’s "Monthly Invest" plan lets you deposit Rs 2,000/month for 3 years at 7% interest. The FV of this annuity is calculated as Rs 6,434 (as shown earlier), which you receive at maturity.

3. NEPSE Stock Valuation (e.g., NMB Bank):

   • Concept: Gordon Growth Model (Perpetuity with Growth)
   • How it works: NMB Bank pays a dividend of Rs 12/share growing at 4%. If investors demand a 10% return: $$PV=\frac{12}{0.10-0.04}=\text{Rs 180/share}$$
   • Real Impact: This model helps traders decide whether to buy/sell NMB stock on NEPSE.

4. Pathao Driver’s Lease Decision:

   • Concept: Annuity Due vs. Ordinary Annuity
   • How it works: A driver leases a bike for Rs 15,000/month at the start of each month (annuity due). If interest is 1%/month, the PV of the lease is higher than if payments were at the end (ordinary annuity), making it a costlier option.

5. Khalti’s Expansion to Pokhara:

   • Concept: NPV and IRR for Capital Budgeting
   • How it works: Khalti uses NPV to decide whether expanding to Pokhara is worth Rs 500,000. If the NPV is negative (as in the worked example), they reject the project unless they can increase cash flows (e.g., by offering higher transaction fees).

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## Exam Tip

1. Memorize the 4 Core Formulas:

   • FV of a single sum: $FV=PV(1+r{)}^n$
   • PV of a single sum: $PV=\frac{FV}{(1+r{)}^n}$
   • FV of an annuity: $FV=PMT\times \frac{(1+r{)}^n-1}{r}$
   • PV of an annuity: $PV=PMT\times \frac{1-(1+r{)}^{-n}}{r}$

2. Watch for Annuity Due vs. Ordinary Annuity:

   • Annuity due = Ordinary annuity × (1 + r). Always check the question!

3. NPV vs. IRR:

   • NPV tells you profitability (accept if > 0).
   • IRR tells you return rate (compare to cost of capital).
   • Never choose a project solely on IRR if it has multiple sign changes in cash flows (use NPV instead).

4. Real-World Applications:

   • Loans (eSewa, banks): Use PV of annuity to calculate EMI.
   • Investments (NEPSE): Use Gordon Growth Model for stocks.
   • Savings (NTC, banks): Use FV of annuity for fixed deposits.

5. Common Pitfalls:

   • Ignoring compounding periods: If interest is monthly, adjust $r$ and $n$ (e.g., 12% annual = 1% monthly).
   • Miscounting cash flows: Ensure you’re discounting all inflows/outflows.
   • Forgetting taxes: In bond problems (like past exam Q1a), subtract tax savings from interest.

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## Practice Questions (Exam-Style)

1. Zero-Coupon Bond (Like Past Exam Q1a): A bond matures in 5 years with a face value of Rs 1,000 and sells for Rs 700. Tax rate: 30%. Calculate the after-tax cost of debt. Hint: First find the pre-tax yield, then adjust for taxes.

2. Perpetuity with Growth: A stock pays a dividend of Rs 6 today, growing at 3%. If the required return is 9%, what’s its price? Answer: Rs 100 (use $PV=\frac{D_0(1+g)}{r-g}$).

3. NPV Decision: A project costs Rs 100,000 and generates Rs 40,000/year for 4 years. Discount rate: 10%. Should you accept it? Answer: Calculate NPV = Rs 34,870 → Accept.

4. Annuity Due: You lease a car by paying Rs 20,000 at the start of each year for 3 years at 8% interest. What’s the PV? Answer: Rs 51,840 (ordinary annuity PV × 1.08).