Annuities, Loans & Financial Instruments: Cash Flows, Loans & Derivatives

TAKEAWAYS:

• Annuities and perpetuities are regular cash flow streams used to calculate present/future values for loans, pensions, or leases.
• Loan amortization schedules break down principal and interest payments over time, critical for mortgages and business loans.
• Financial instruments (bonds, stocks, derivatives) transfer risk, generate returns, or hedge exposure in markets like NEPSE or global indices.
• Derivatives (futures, options) lock in prices or speculate on assets like gold, forex, or agricultural commodities.
• WACC and loan costs link this unit to capital budgeting (Unit 7) and valuation (Units 4–5).
• Real-world tools: eSewa’s interest calculations, Daraz’s inventory financing, or NTC’s bond issuances rely on these concepts.

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1. Annuities and Perpetuities: Structured Cash Flows

Annuities and perpetuities are series of equal payments at fixed intervals, used to value loans, pensions, or rental income. They differ in duration: annuities have a finite life; perpetuities continue forever.

Key Definitions

• Annuity: Fixed payments for a set period (e.g., 5-year car loan).
• Ordinary Annuity: Payments at the end of each period (most common).
• Annuity Due: Payments at the start of each period (e.g., rent paid in advance).
• Perpetuity: Infinite payments (e.g., preferred stock dividends).

Present Value (PV) and Future Value (FV) Formulas

For an ordinary annuity: $$PV=PMT\times \left(\frac{1-(1+r{)}^{-n}}{r}\right)$$ $$FV=PMT\times \left(\frac{(1+r{)}^n-1}{r}\right)$$ Where:

• $PMT$ = Payment amount
• $r$ = Discount rate per period
• $n$ = Number of periods

Example: Kathmandu Retail Shop’s Lease A shop pays NPR 50,000/month for 3 years (36 months) at 12% annual interest (1% monthly). What’s the present value of this lease?

```figure
{"type":"t-account","title":"Present Value Calculation (Annuity)","dr":[["To Lease Payments (PMT)",50000],["To Discount Factor (1/(1+r)^n)","(1 - (1.01)^(-36)) / 0.01 ≈ 31.322"]],"cr":[["By Present Value (PV)",1565200]],"balance":true,"caption":"PV = PMT × Annuity Factor (12% annual, 36 months)"}

Answer: The lease’s PV ≈ NPR 1,565,200.

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Perpetuity Formula

For a perpetuity: $$ PV = \frac{PMT}{r} $$ Example: NEPSE’s Dividend Stock A stock pays NPR 200/year forever. If the required return is 8%, its value is: $$ PV = \frac{200}{0.08} = \text{NPR } 2,500 $$

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Annuity vs. Perpetuity: Key Differences

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2. Loan Amortization: Paying Off Debt

Amortization schedules allocate each payment between principal and interest. Critical for mortgages, business loans, and personal loans (e.g., eSewa’s installment plans).

How Amortization Works

1. Initial Payment: Mostly interest.
2. Later Payments: More principal, less interest.
3. Final Payment: Principal cleared (unless a balloon payment remains).

Amortization Schedule Example

Business Loan for "GreenTech Nepal"

• Loan Amount: NPR 500,000
• Term: 5 years (60 months)
• Interest Rate: 10% annually (0.833% monthly)
• Monthly Payment: NPR 10,054 (calculated via annuity formula).

| Month | Payment  | Principal | Interest | Remaining Balance |
|-------|----------|-----------|----------|--------------------|
| 1     | 10,054   | 3,333     | 6,721    | 496,667            |
| 2     | 10,054   | 3,471     | 6,583    | 493,196            |
| ...   | ...      | ...       | ...      | ...                |
| 60    | 10,054   | 10,054    | 0        | 0                  |

Visual: The interest portion declines linearly; principal increases.

```figure
{"type":"bar","labels":["Month 1","Month 30","Month 60"],"values":[10054,5027,0],"ylabel":"Interest Portion (NPR)","caption":"Amortization: Interest decreases linearly; principal increases (NPR 10,054/month loan, 12% annual)"}

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Balloon Payment

Some loans (e.g., commercial mortgages) have a large final payment to reduce early payments. Example: A 5-year loan with NPR 100,000 due at the end.

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3. Financial Instruments: Securities and Derivatives

Financial instruments transfer risk, generate returns, or hedge exposure. Nepal’s NEPSE and global markets use these daily.

A. Securities (Primary Instruments)

Example: NEPSE’s Bond Valuation A bond pays NPR 100/year forever (perpetuity) at 8% yield: $$PV=\frac{100}{0.08}=\text{NPR }1,250$$

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B. Derivatives: Hedging and Speculation

Derivatives derive value from an underlying asset (e.g., gold, forex, stocks). Used to hedge risk or speculate.

Example: Pathao’s Fuel Cost Hedging Pathao uses futures contracts on diesel prices to lock in fuel costs, protecting profit margins.

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4. Real-World Applications

In the Real World

1. eSewa’s Installment Loans

   • Idea: Uses annuity payments to break down large transactions (e.g., NPR 50,000 phone) into monthly installments.
   • How: Applies the annuity formula to calculate fixed monthly payments at a set interest rate.

2. Daraz’s Inventory Financing

   • Idea: Sellers use perpetuity-like financing for stock (e.g., NPR 1M inventory financed at 12% annual interest).
   • How: Daraz acts as a lender, charging interest on unsold stock until sold.

3. NTC’s Bond Issuance

   • Idea: NTC sells debt securities to raise funds for infrastructure, with fixed coupon payments (like an annuity).
   • How: Investors buy bonds at a discount/par, receiving NPR 8% annual interest.

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5. Worked Example: Loan Amortization for a Nepali Business

Scenario: "Mountain View Café" takes a NPR 200,000 loan for 3 years at 12% annual interest (repaid monthly).

Step 1: Calculate Monthly Payment Using the annuity formula: $$PV=PMT\times \left(\frac{1-(1+r{)}^{-n}}{r}\right)$$ Rearranged for $PMT$: $$PMT=\frac{200,000\times 0.01}{(1-(1.01{)}^{-36})}\approx \text{NPR }6,350/\text{month}$$

Step 2: Amortization Schedule (First 3 Months)

| Month | Payment  | Principal | Interest | Remaining Balance |
|-------|----------|-----------|----------|--------------------|
| 1     | 6,350    | 5,167     | 1,183    | 194,833            |
| 2     | 6,350    | 5,233     | 1,117    | 189,600            |
| 3     | 6,350    | 5,300     | 1,050    | 184,300            |

Visual: The interest portion drops from NPR 1,183 to NPR 1,050 in 3 months.

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6. Exam Tip

• Focus on formulas: Memorize PV/FV of annuities/perpetuities. Practice plugging numbers (e.g., "Calculate PV of NPR 10,000/year for 5 years at 10%").
• Amortization tables: Show how payments split into principal/interest. Use the formula or Excel’s PMT function.
• Derivatives: Know futures/options basics. Example: "How does a farmer hedge wheat prices using futures?"
• Real-world links: Connect to loans (Ncell), investments (NEPSE), or hedging (Pathao’s fuel costs).
• Units 4–7 links:
  • Unit 4 (Bonds): Bonds are debt instruments with annuity-like coupon payments.
  • Unit 7 (WACC): Loan interest rates feed into WACC calculations.
• Common mistakes:
  • Mixing ordinary annuity (end-of-period) with annuity due (start-of-period).
  • Forgetting to adjust for compounding periods (e.g., monthly vs. annual rates).
  • Ignoring balloon payments in loan schedules.

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Final Note: This unit bridges theory (TVM) and practice (loans, securities). Master annuities, amortization, and derivatives—you’ll see them in every financial decision, from buying a house to trading stocks.