Time Value of Money: Valuation, Annuities, Bonds & NPV

Core Concepts

1. Time Value of Money (TVM) Principle

Money has a time value because:

• Opportunity cost: Rs 100 today can earn interest if invested.
• Inflation risk: Rs 100 today buys more than Rs 100 in 5 years.
• Uncertainty: Future cash flows are riskier.

Key Formula: $$\text{Future Value (FV)}=\text{PV}\times (1+r{)}^n$$ $$\text{Present Value (PV)}=\frac{\text{FV}}{(1+r{)}^n}$$ where:

• $r$ = discount rate (required return),
• $n$ = number of periods.

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2. Annuities: Regular Cash Flows

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

• Ordinary annuity: Payments at end of period (e.g., bank loan EMIs).
• Annuity due: Payments at start of period (e.g., rent paid in advance).

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

Worked Example: Pathao Driver’s Savings Plan A Pathao driver saves Rs 5,000 every 3 months (quarterly) for 5 years at 8% annual interest (compounded quarterly).

• PMT (Payment) = Rs 5,000
• r (quarterly rate) = $\frac{8\%}{4}=2\%$
• n (quarters) = $5\times 4=20$

Calculate FV of annuity: $$FV=5000\times \frac{(1+0.02{)}^{20}-1}{0.02}=5000\times 24.297=\text{Rs 1,21,485}$$ Mermaid Diagram: Annuity Types

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3. Bond Valuation: How Bonds Are Priced

A bond is a loan to a company/government with fixed interest payments (coupons) and a par value (face value, e.g., Rs 1,000). Bond Price Formula: $$\text{Bond Price}=\sum \frac{C}{(1+r{)}^t}+\frac{F}{(1+r{)}^n}$$ where:

• $C$ = coupon payment,
• $F$ = face value,
• $r$ = required return (market interest rate),
• $n$ = years to maturity.

Worked Example: Ncell Bond Issue Ncell issues a 7-year bond with:

• Par value = Rs 1,000,
• 10% annual coupon (paid semiannually),
• Market required return = 12%.

Step 1: Calculate semiannual coupon and rate

• Coupon payment = $\frac{10\%\times 1000}{2}=\text{Rs 50}$,
• Semiannual rate = $\frac{12\%}{2}=6\%$,
• Total periods = $7\times 2=14$.

Step 2: Calculate PV of coupons (ordinary annuity) $$PVA=50\times \frac{1-(1+0.06{)}^{-14}}{0.06}=50\times 8.764=\text{Rs 438.20}$$

Step 3: Calculate PV of face value $$P{V}_{face}=\frac{1000}{(1+0.06{)}^{14}}=\text{Rs 456.39}$$

Step 4: Total Bond Price $$\text{Bond Price}=438.20+456.39=\text{Rs 894.59}$$ Since Rs 894.59 < Rs 1,000 (par value), the bond is trading at a **discount** (because market rate > coupon rate).

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4. Net Present Value (NPV) and Investment Decisions

NPV compares the present value of cash inflows to outflows to decide if a project is profitable. $$NPV=\sum \frac{C{F}_t}{(1+r{)}^t}-\text{Initial Investment}$$ Decision Rule:

• NPV > 0: Accept (project adds value).
• NPV < 0: Reject (project destroys value).

Worked Example: Daraz Warehouse Expansion Daraz considers expanding its warehouse in Kathmandu with:

• Initial cost = Rs 5,00,000,
• Annual cash inflows = Rs 1,50,000 for 5 years,
• Discount rate (r) = 10%.

Step 1: Calculate PV of cash inflows (ordinary annuity) $$PVA=150000\times \frac{1-(1+0.10{)}^{-5}}{0.10}=150000\times 3.791=\text{Rs 5,68,650}$$

Step 2: Calculate NPV $$NPV=568650-500000=\text{Rs 68,650}$$ Since NPV > 0, Daraz should expand the warehouse.

Mermaid Diagram: NPV Decision Flowchart

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5. Perpetuities and Preferred Stock Valuation

A perpetuity is an annuity with infinite payments (e.g., some preferred stocks, consols). Present Value of Perpetuity: $$PV=\frac{CF}{r}$$ Worked Example: NEPSE Preferred Stock A preferred stock pays a dividend of Rs 15 per share, and investors require a 12% return. $$PV=\frac{15}{0.12}=\text{Rs 125 per share}$$

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In the Real World

1. eSewa Loans

   • Uses TVM to calculate EMI installments for personal loans.
   • Example: A Rs 50,000 loan at 12% for 2 years has EMIs calculated using the PV of annuity formula.

2. Khalti Savings Account

   • Offers compound interest on savings (e.g., 6% annually).
   • A Rs 10,000 deposit grows to Rs 11,910 in 2 years using $FV=PV(1+r{)}^n$.

3. NTC’s Bond Issuances

   • NTC sells bonds to fund infrastructure. Investors use bond valuation to decide if the bond’s coupon rate (e.g., 9%) matches their required return (e.g., 10%).

4. Pathao Driver’s Retirement Plan

   • Drivers use FV of annuity to plan savings (e.g., Rs 3,000/month for 10 years at 7% → Rs 6,50,000).

5. Bank Loan Approvals

   • Banks use NPV to assess loan viability. Example: A Rs 2,00,000 loan for a shop with Rs 60,000/year profit for 5 years at 10% discount rate: $$NPV=60000\times 3.791-200000=\text{Rs 24,466 (Accept)}$$

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Exam Tip

1. Memorize Formulas:

   • Know PV/FV of single sum, annuity, and perpetuity formulas by heart.
   • Example: For a bond, always break it into PV of coupons + PV of face value.

2. Watch Units:

   • Coupons are semiannual unless stated otherwise (e.g., 10% annual coupon → Rs 50 semiannual).
   • Annuity due = Ordinary annuity × (1 + r).

3. Numerical Accuracy:

   • Use financial calculators or Excel (=PV(), =FV()) for complex problems.
   • Round to 2 decimal places for final answers (e.g., Rs 894.59, not 894.592).

4. Real-World Scenarios:

   • Exams often link TVM to loans (EMIs), investments (NPV), or bonds (discount/premium).
   • Example: If a bond’s coupon rate < market rate, it sells at a discount.

5. Common Pitfalls:

   • Miscounting periods: 5 years at semiannual = 10 periods, not 5.
   • Ignoring compounding: Simple interest ≠ compound interest in most cases.
   • Sign errors: Cash inflows are +, outflows are -.

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Summary Table: Key Formulas

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Final Note:

• Practice past exam questions on bond valuation and NPV—they appear frequently.
• Draw t-accounts for bond pricing to visualize coupon vs. face value contributions.
• Always label units (e.g., "Rs", "years") in answers to avoid deduction.