Financial Mathematics 💰

Perpetuity, bonds, NPV, IRR, stocks — smart financial decisions

1. Perpetuity

Perpetuity ek aise annuity hai jo forever payments karta hai — no end date!

PV of Perpetuity = C / r C = periodic cash flow r = interest rate per period (decimal) Growing Perpetuity: PV = C / (r − g) where g = growth rate

Udaharan: Perpetuity

Ek endowment fund mein ₹10,000/year milne chahiye. Interest rate 8%. Fund size?
PV = 10000/0.08 = ₹1,25,000

2. Bonds — Valuation

Bond ek debt instrument hai. Issuer periodic coupon payments karta hai aur maturity par face value return karta hai.

Bond Price = C×[1−(1+r)⁻ⁿ]/r + F/(1+r)ⁿ C = Annual coupon payment = Face Value × Coupon Rate F = Face value (par value) r = Required yield (discount rate) n = Years to maturity

💡 Rule: Coupon Rate > Required Yield → Bond trades at PREMIUM (Price > F)
Coupon Rate < Required Yield → Bond trades at DISCOUNT (Price < F)

Udaharan: Bond Pricing

Face Value = ₹1000, Coupon = 8%, Maturity = 5 years, Required yield = 10%
C = 1000×0.08 = ₹80 per year
PV of coupons = 80×[1−(1.10)⁻⁵]/0.10 = 80×3.7908 = ₹303.26
PV of face value = 1000/(1.10)⁵ = 1000/1.6105 = ₹620.92
Bond Price = 303.26 + 620.92 = ₹924.18 (discount par, kyunki yield > coupon)

3. Net Present Value (NPV) & IRR

NPV = Σ[CFₜ/(1+r)ᵗ] − Initial Investment CFₜ = Cash flow at time t, r = discount rate Decision Rule: NPV > 0 → Accept project (profitable) NPV < 0 → Reject project NPV = 0 → Indifferent IRR: Rate at which NPV = 0 If IRR > Required Rate → Accept If IRR < Required Rate → Reject

Udaharan: NPV Calculation

Initial investment = ₹1,00,000. Cash flows: Y1=₹30k, Y2=₹40k, Y3=₹50k. r=12%
NPV = 30000/1.12 + 40000/1.12² + 50000/1.12³ − 100000
= 26786 + 31888 + 35589 − 100000
= 94263 − 100000 = −₹5,737
NPV < 0 → Reject this project!

Udaharan: IRR Approximation

Try r=10%: NPV = 30000/1.1 + 40000/1.21 + 50000/1.331 − 100000
= 27273 + 33058 + 37566 − 100000 = −2103
Try r=9%: NPV = 27523 + 33663 + 38609 − 100000 = −205 ≈ 0
IRR ≈ 9% → If required rate is 12% → Reject

4. Stock Valuation

4.1 Dividend Discount Model (DDM)

Constant Dividend: P₀ = D/r (Gordon Growth — no growth) Gordon Growth: P₀ = D₁/(r − g) D₁ = next dividend, r = required return, g = growth rate

Udaharan: DDM

Stock expected dividend next year = ₹5. Growth rate = 4%. Required return = 12%.
P₀ = 5/(0.12 − 0.04) = 5/0.08 = ₹62.5

4.2 P/E Ratio aur EPS

EPS = (Net Profit − Preferred Dividends) / Shares Outstanding P/E Ratio = Market Price per Share / EPS Intrinsic Value = EPS × Industry P/E

5. Leasing

Financial Lease: Long-term, asset ownership effectively transfers Operating Lease: Short-term, lessor owns the asset Lease vs Buy Decision: NPV of Leasing = Σ[Lease Payment × (1−t)]/(1+r)ᵗ + Salvage Value benefit Compare with: Cost of buying − tax depreciation benefits

Udaharan: Lease vs Buy

Machine cost ₹5,00,000. Lease: ₹1,20,000/year for 5 years. r=10%, tax=30%
After-tax lease payment = 1,20,000 × (1−0.3) = ₹84,000
PV of lease = 84,000 × PVIFA(10%,5) = 84,000 × 3.7908 = ₹3,18,427
PV of buying net of tax = 5,00,000 − PV(tax shield on depreciation)
Compare totals → whichever is lower, choose that!

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