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

TAKEAWAYS:

• Money’s value changes over time due to interest and inflation; time value of money (TVM) quantifies this using present value (PV) and future value (FV).
• Discounting (PV) and compounding (FV) are inverse processes; formulas like $PV=\frac{FV}{(1+r{)}^n}$ and $FV=PV(1+r{)}^n$ are essential for valuing future cash flows.
• Annuities (equal payments over time) and perpetuities (infinite payments) have distinct PV/FV formulas; semiannual compounding adjusts rates and periods.
• Bonds are valued using coupon payments and par value; stocks (preferred and common) rely on dividend discount models (DDM).
• Real-world applications: Loan approvals (banks), investment decisions (NEPSE), and retirement planning (eSewa savings) all use TVM.
• Exam focus: 60% numerical problems (PV/FV/annuity calculations), 30% conceptual (e.g., why PV < FV), 10% application (e.g., bond/stock valuation).

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1. Introduction to Time Value of Money

Money today is worth more than the same amount in the future because it can earn interest or be invested. This concept is called time value of money (TVM).

Key Principles

• Opportunity Cost: Investing money today means giving up current consumption for future returns.
• Risk: Future cash flows are uncertain; investors demand compensation (interest rate) for waiting.
• Inflation: Reduces purchasing power of future money.

Basic Terms

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

Single Cash Flow

• Future Value (FV): $$FV=PV\times (1+r{)}^n$$ Example: If you deposit Rs 1,000 at 8% annual interest, its value after 5 years: $$FV=1000\times (1+0.08{)}^5=1000\times 1.4693=\text{Rs 1,469.30}$$

• Present Value (PV): $$PV=\frac{FV}{(1+r{)}^n}$$ Example: What is the PV of Rs 2,000 due in 3 years at 6%? $$PV=\frac{2000}{(1+0.06{)}^3}=\frac{2000}{1.1910}=\text{Rs 1,679.56}$$

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FIGURE: Single cash flow PV/FV relationship

PV (Today)
       ↑
       |
       | (1+r)^n
       ↓
FV (Future)

Caption: Discounting (PV) and compounding (FV) are inverse processes.*

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Multiple Cash Flows

Use discounted cash flow (DCF) to value streams of payments: $$PV=\sum _{t=1}^n\frac{C{F}_t}{(1+r{)}^t}$$ Example: Calculate PV of Rs 500/year for 3 years at 10%: $$PV=\frac{500}{(1.10{)}^1}+\frac{500}{(1.10{)}^2}+\frac{500}{(1.10{)}^3}=454.55+413.22+375.66=\text{Rs 1,243.43}$$

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EXAMPLE: Daraz’s Inventory Financing Daraz offers sellers 30-day payment terms on orders. If a seller receives Rs 50,000 in 30 days and the bank offers 12% annual interest, what is the PV of this payment? Assumption: 30 days ≈ 0.083 years (360-day year). $$PV=\frac{50000}{(1+0.12{)}^{0.083}}=\frac{50000}{1.01}=\text{Rs 49,505}$$ Implication: Delaying payment costs the seller Rs 495 in lost interest.

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3. Annuities and Perpetuities

Annuities

Equal payments at fixed intervals (e.g., loan installments, rent).

• Ordinary Annuity: Payments at end of each period. $$P{V}_{annuity}=PMT\times \frac{1-(1+r{)}^{-n}}{r}$$
• Annuity Due: Payments at start of each period (multiply ordinary annuity PV by $(1+r)$).

Example: PV of Rs 200/month for 5 years at 6% annual interest (compounded monthly):

• Monthly rate $r=0.06/12=0.005$, $n=5\times 12=60$. $$PV=200\times \frac{1-(1.005{)}^{-60}}{0.005}=200\times 51.8415=\text{Rs 10,368.30}$$

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FIGURE: Ordinary Annuity vs. Annuity Due

Perpetuities

Infinite series of equal payments (e.g., preferred stock dividends). $$PV=\frac{PMT}{r}$$ Example: A preferred stock pays Rs 15/year with a 12% required return: $$PV=\frac{15}{0.12}=\text{Rs 125}$$

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COMPARISON TABLE: Annuity vs. Perpetuity

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4. Compounding Frequencies

Interest can be compounded annually, semiannually, quarterly, or monthly. Adjust the rate and periods:

• Effective Annual Rate (EAR): $$EAR={\left(1+\frac{r}{m}\right)}^m-1$$ where $m$ = compounding periods/year.

Example: 10% nominal rate, compounded semiannually:

• Semiannual rate = $0.10/2=0.05$.
• EAR = $(1.05{)}^2-1=0.1025$ or 10.25%.

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EXAMPLE: Ncell’s Loan Plan Ncell offers a phone loan with 18% annual interest, compounded monthly. What is the EAR? $$EAR={\left(1+\frac{0.18}{12}\right)}^{12}-1=1.1956-1=0.1956\text{ or }19.56\%$$ Implication: The true cost of borrowing is 19.56%, not 18%.

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5. Bond Valuation

Bonds pay coupon interest (fixed payments) and return par value at maturity. $$BondPV=\sum _{t=1}^n\frac{C}{(1+r{)}^t}+\frac{F}{(1+r{)}^n}$$ where:

• $C$ = annual coupon payment,
• $F$ = face/par value,
• $r$ = investor’s required yield.

Example: Rs 1,000 par, 9% coupon bond, 5 years to maturity, 10% required return:

• Coupon $C=0.09\times 1000=\text{Rs 90}$.
• PV of coupons: $$\frac{90}{1.10}+\frac{90}{1.10^2}+\frac{90}{1.10^3}+\frac{90}{1.10^4}+\frac{90}{1.10^5}=81.82+74.38+67.62+61.47+55.88=341.17$$
• PV of par value: $$\frac{1000}{1.10^5}=620.92$$
• Total Bond PV = Rs 341.17 + Rs 620.92 = Rs 962.09

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FIGURE: Bond Cash Flows

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WORKED EXAMPLE: Ragmati Textile’s Bond Ragmati Textile issues a Rs 1,000 par, 10% coupon bond with 7 years to maturity, paying semiannual interest. Investors require 12% yield. Calculate the bond’s value.

1. Adjust for semiannual periods:

   • Coupon per period = $0.10\times 1000/2=\text{Rs 50}$.
   • Periodic yield = $0.12/2=0.06$.
   • Total periods $n=7\times 2=14$.

2. PV of coupons: $$P{V}_{coupons}=50\times \frac{1-(1.06{)}^{-14}}{0.06}=50\times 9.7122=\text{Rs 485.61}$$

3. PV of par value: $$P{V}_{par}=\frac{1000}{(1.06{)}^{14}}=\text{Rs 417.27}$$

4. Total Bond Value: $$\text{Rs 485.61 + Rs 417.27 = Rs 902.88}$$

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6. Stock Valuation

Preferred Stock

Dividends are fixed and perpetual: $$P_0=\frac{D}{r}$$ Example: Rs 15 dividend, 12% required return: $$P_0=\frac{15}{0.12}=\text{Rs 125}$$

Common Stock (Dividend Discount Model - DDM)

Assumes dividends grow at a constant rate $g$: $$P_0=\frac{D_1}{r-g}$$ Example: Next dividend $D_1=\text{Rs 5}$, growth rate $g=5\%$, required return $r=12\%$: $$P_0=\frac{5}{0.12-0.05}=\frac{5}{0.07}=\text{Rs 71.43}$$

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COMPARISON TABLE: Bond vs. Stock Valuation

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7. Working Capital and TVM

TVM applies to working capital decisions:

• Inventory Financing: Delaying payment for inventory (e.g., Daraz) costs interest.
• Accounts Receivable: Faster collection = earlier cash flow = higher PV.
• Accounts Payable: Delaying payments (e.g., suppliers) reduces cash outflows.

Example: A Kathmandu retail shop buys inventory worth Rs 50,000 with 30-day credit. If the bank offers 12% annual interest, what is the cost of not paying immediately?

• PV of Rs 50,000 due in 30 days (as above) = Rs 49,505.
• Cost = Rs 50,000 - Rs 49,505 = Rs 495.

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FIGURE: Working Capital Cycle

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In the Real World

1. eSewa Savings Account

   • Idea: Future Value of Savings
   • How: eSewa’s savings plan compounds interest monthly. If you deposit Rs 1,000/month at 8% annual interest, your PV of future savings grows over time using annuity formulas. eSewa’s app shows this growth, helping users plan for emergencies.

2. NEPSE Stock Valuation

   • Idea: Dividend Discount Model (DDM)
   • How: Investors use DDM to value stocks like Nepal Bank Limited. If NBL pays Rs 10/year dividend, grows at 6%, and investors require 12% return, its value is: $$P_0=\frac{10}{0.12-0.06}=\text{Rs 166.67}$$ This guides buying/selling decisions on NEPSE.

3. Pathao’s Ride-Hailing Loans

   • Idea: Present Value of Loan Payments
   • How: Pathao offers driver loans with 20% annual interest, compounded monthly. A driver taking Rs 50,000 must repay: $$PV=\frac{50000}{(1+0.20/12{)}^{12}}=\text{Rs 43,450}$$ The PV of repayments shows the true cost of borrowing, helping drivers compare loan options.

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WORKED EXAMPLE: Kathmandu Traffic Signal Timing Scenario: A traffic engineer wants to optimize signal timing for Kathmandu’s Thapathali intersection. Vehicles wait 2 minutes on average before crossing. If the opportunity cost of time is Rs 50/minute (e.g., lost productivity), what is the PV of delay for 100 vehicles/day over 5 years at 6% annual interest?

1. Daily Cost: $$100\text{ vehicles}\times 2\text{ min}\times \text{Rs 50}=\text{Rs 10,000/day}$$

2. PV of 5-Year Cost: $$PV=10000\times \frac{1-(1.06{)}^{-5}}{0.06}=10000\times 4.2124=\text{Rs 42,124}$$

Implication: Reducing wait time by 30 seconds saves Rs 2,625/year, justifying signal upgrades.

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Exam Tip

1. Master the Formulas:

   • Memorize PV/FV of single cash flows, annuities, and perpetuities.
   • Know how to adjust for compounding frequencies (e.g., semiannual rates).

2. Show Work Clearly:

   • For bond/stock valuation, list all cash flows and discount them step-by-step.
   • Use tables for annuity calculations (e.g., PVIF or FVIF tables).

3. Watch for Units:

   • Interest rates must match compounding periods (e.g., 10% annual vs. 0.5% monthly).
   • Time $n$ must be in the same units as the rate (e.g., years for annual rates).

4. Real-World Linkage:

   • Connect problems to Nepali businesses (e.g., Daraz, Ncell loans) or NEPSE stocks.
   • Explain why a calculation matters (e.g., "This shows the true cost of a delay in payment").

5. Common Pitfalls:

   • Annuity Due vs. Ordinary Annuity: Multiply by $(1+r)$ for annuity due.
   • Growing Perpetuity: Use $P_0=\frac{D_1}{r-g}$, not $\frac{D_0}{r}$.
   • Zero-Coupon Bonds: Only discount the par value (no coupon payments).

6. Time Management:

   • 60% of marks go to numerical problems (PV/FV/annuity calculations). Spend 30 minutes on each.
   • 30% for concepts: Be ready to explain why PV < FV or how compounding works.
   • 10% for application: Relate to NEPSE, loans, or savings (e.g., "This is how eSewa calculates your savings growth").

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FINAL REMINDER:

• Practice past papers: Focus on Unit 3 questions from TU/PU exams (e.g., bond valuation, annuity FV).
• Use a calculator: Most exams allow financial calculators (e.g., BA-II+). Learn shortcuts like PV, FV, N, I/Y.
• Draw diagrams: Sketch cash flow timelines for annuities/bonds to visualize discounting.