Financial Mathematics — Applied Maths 11

💸 1. Simple Interest vs Compound Interest

Jab aap bank mein paisa rakhte ho ya loan lete ho, toh interest milta/lagta hai. Do tarike hain — SI aur CI.

Simple Interest (SI)

Sirf original principal par interest lagta hai. Interest har saal same rehta hai. Linear growth!

Compound Interest (CI)

Interest on interest! Pichle period ka interest bhi principal mein jud jaata hai. Exponential growth!

Simple Interest Formula

SI = (P × R × T) / 100 Amount = P + SI = P(1 + RT/100) Jahan: P = Principal (mool rashi) R = Rate % per annum T = Time (years mein)

Example: SI Nikaalte Hain

Naveen ne ₹12,000 invest kiye 8% p.a. par 3 saal ke liye. SI aur total amount kya hoga?

SI = (12000 × 8 × 3)/100 = ₹2,880

Amount = 12000 + 2880 = ₹14,880

Compound Interest Formula

A = P × (1 + r/n)^(n×t) CI = A − P Jahan: r = rate per annum (decimal mein, e.g. 8% = 0.08) n = compounding periods per year (annually=1, semi-annually=2, quarterly=4, monthly=12) t = time in years Continuous compounding: A = P × e^(rt)

Example: CI Nikaalte Hain

Same ₹12,000, 8% p.a., 3 saal — lekin quarterly compounded.

A = 12000 × (1 + 0.08/4)^(4×3) = 12000 × (1.02)^12

(1.02)^12 = 1.2682

A = 12000 × 1.2682 = ₹15,218.40

CI = 15218.40 − 12000 = ₹3,218.40 (vs SI ₹2,880 — CI deta hai ₹338 zyada!)

Nominal aur Effective Rate of Interest

Nominal Rate (r): Jo rate quoted hoti hai (e.g. 12% p.a. quarterly compounded) Effective Rate (E): Ek saal mein actually kitna % bana E = (1 + r/n)^n − 1 Example: r = 12% quarterly → n = 4 E = (1 + 0.12/4)^4 − 1 = (1.03)^4 − 1 = 1.1255 − 1 = 0.1255 = 12.55%

💡 Yaad rakho: Jitna zyada compounding periods, utna zyada effective rate — aur utna zyada CI milta hai!

Continuous Compounding

Jab n → ∞ (infinitely frequent compounding): A = P × e^(rt) e ≈ 2.71828 Example: ₹10,000 at 10% continuously for 2 years: A = 10000 × e^(0.10 × 2) = 10000 × e^0.2 = 10000 × 1.2214 = ₹12,214

Compounding	Formula	Amount at 10% on ₹1000 for 1yr
Annually	(1+0.1)¹	₹1,100.00
Semi-annually	(1+0.05)²	₹1,102.50
Quarterly	(1+0.025)⁴	₹1,103.81
Monthly	(1+0.1/12)¹²	₹1,104.71
Continuous	e^0.1	₹1,105.17

📅 2. Annuity, EMI aur Present/Future Value

Annuity matlab: equal time intervals par equal payments. Ghar ka kiraya, insurance premium, EMI — sab annuity ke examples hain.

Annuity ke Types

Ordinary Annuity

Payment period ke end mein hoti hai.
e.g. Most home loans — EMI month end par

Annuity Due

Payment period ke beginning mein hoti hai.
e.g. Insurance premium — mahine ke shuru mein

Future Value of Annuity (FV)

FV = R × [(1+i)^n − 1] / i Jahan: R = regular payment amount (annuity amount) i = interest rate per period (decimal) n = total number of periods Annuity Due: FV_due = FV × (1+i)

Example: FV of Annuity

Riya har mahine ₹5,000 invest karti hai 12% p.a. (monthly) par, 5 saal ke liye. 5 saal baad kitna milega?

R = 5000, i = 12%/12 = 0.01, n = 5×12 = 60

FV = 5000 × [(1.01)^60 − 1] / 0.01

(1.01)^60 = 1.8167

FV = 5000 × [1.8167 − 1] / 0.01 = 5000 × 81.67 = ₹4,08,350

Total invested: 5000 × 60 = ₹3,00,000. Interest earned = ₹1,08,350!

Present Value of Annuity (PV)

PV = R × [1 − (1+i)^(-n)] / i Matlab: Aaj kitna lump sum chahiye taaki future mein R/period pay kar sakein? Annuity Due: PV_due = PV × (1+i)

Example: PV of Annuity

Retirement ke baad 20 saal tak ₹20,000/month chahiye. 6% p.a. par aaj kitna invest karein?

R = 20000, i = 0.06/12 = 0.005, n = 20×12 = 240

PV = 20000 × [1 − (1.005)^(-240)] / 0.005

(1.005)^240 = 3.3102 → (1.005)^(-240) = 0.3021

PV = 20000 × [1 − 0.3021] / 0.005 = 20000 × 139.58 = ₹27,91,600

EMI (Equated Monthly Instalment)

EMI = P × i × (1+i)^n / [(1+i)^n − 1] Jahan: P = Principal loan amount i = monthly interest rate = annual rate / 12 n = loan tenure in months Yahi formula banks use karte hain!

Example: Home Loan EMI

Rahul ne ₹25 lakh ka home loan liya, 8.5% p.a., 20 saal ke liye. Monthly EMI kya hoga?

P = 25,00,000, i = 0.085/12 = 0.007083, n = 240

(1.007083)^240 = 5.3516

EMI = 25,00,000 × 0.007083 × 5.3516 / (5.3516 − 1)

= 25,00,000 × 0.03791 / 4.3516 = 94,775 / 4.3516 = ₹21,779 approx

Total paid: 21,779 × 240 = ₹52,26,960 → Interest = ₹27,26,960!

Sinking Fund

Sinking Fund: Future mein ek badi payment ke liye har period equal amount save karna. R = FV × i / [(1+i)^n − 1] ← How much to save per period? Example: Company ko 5 saal baad ₹10 lakh machine replace karni hai. 8% p.a. quarterly: i = 0.02, n = 20 R = 10,00,000 × 0.02 / [(1.02)^20 − 1] = 20,000 / [1.4859 − 1] = 20,000 / 0.4859 = ₹41,157/quarter

📄 3. Bills of Exchange

Jab ek vyapari doosre ko credit par maal bechta hai — payment future date par hogi. Is agreement ko document karne ke liye Bill of Exchange banta hai.

Important Terms

Term	Kaun hai	Example
Drawer	Jo bill banata hai (seller/creditor)	Manufacturer jo maal bechta hai
Drawee/Acceptor	Jo bill accept karta hai (buyer/debtor)	Shopkeeper jo maal kharidta hai
Payee	Jisko payment milegi	Usually drawer hi hota hai
Face Value (FV)	Bill par likha hua amount	₹50,000
Due Date	Payment ki taarikh	3 mahine baad + 3 grace days

Banker's Discount (BD)

Jab drawer bank se pehle paisa lena chahta hai (bill discount karna): BD = Face Value × r × t [simple interest on FV] True Discount (TD) = Present Value × r × t [SI on PV] BD > TD (hamesha!) Banker's Gain (BG) = BD − TD = SI on TD BG = BD × TD / FV Present Value (PV) = FV − TD

Example: Bill Discounting

₹10,000 ka bill, 3 mahine baad due, bank 15% p.a. par discount karta hai. BD, TD, BG aur PV nikalo.

BD = 10000 × 0.15 × 3/12 = ₹375

PV = 10000 − TD. Pehle TD: TD = FV×r×t / (1 + r×t) = 10000×0.15×0.25/(1+0.0375) = 375/1.0375 = ₹361.45

PV = 10000 − 361.45 = ₹9,638.55 (drawer ko milta hai)

BG = BD − TD = 375 − 361.45 = ₹13.55 (bank ka profit)

🧾 4. GST — Goods & Services Tax

July 2017 se India mein GST lagu hua. Yeh ek destination-based, multi-stage tax hai jo goods aur services dono par lagta hai.

GST ke Types

CGST + SGST
Intra-state supply (same state ke andar). GST aadha aadha centre aur state mein jaata hai.
e.g. Delhi se Delhi

IGST
Inter-state supply (alag states ke beech). Poora tax centre leta hai phir state ko transfer karta hai.
e.g. Mumbai se Delhi

GST Rate Slabs

Rate	Items
0%	Essential items — fresh fruits, vegetables, milk, eggs, bread
5%	Basic food items — sugar, edible oil, coal, domestic LPG
12%	Processed food, medicines, mobile phones, computers
18%	Most services — restaurants, hotels, soaps, toothpaste, AC
28%	Luxury/sin goods — cars, tobacco, aerated drinks, cement

GST Calculation

GST Amount = Taxable Value × GST Rate / 100 Total Invoice = Taxable Value + GST Amount Intra-state: CGST = GST/2, SGST = GST/2 Inter-state: IGST = Full GST

Example 1: Intra-state Transaction

Delhi ki ek dukaan mein ₹8,000 ka sofa (GST 18%) becha. CGST, SGST aur invoice total nikalo.

GST = 8000 × 18/100 = ₹1,440

CGST = 1440/2 = ₹720, SGST = ₹720

Invoice Total = 8000 + 720 + 720 = ₹9,440

Example 2: Input Tax Credit (ITC)

Manufacturer ne ₹50,000 + 18% GST par raw material kharida. Phir ₹80,000 + 18% GST par maal becha.

Input GST paid = 50000 × 18% = ₹9,000

Output GST collected = 80000 × 18% = ₹14,400

Net GST payable = 14400 − 9000 = ₹5,400 (sirf value addition par tax!)

💡 ITC ka matlab hai ki har stage par sirf VALUE ADDITION par tax dena hota hai — isi liye GST efficient tax hai!

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