Chapter Overview
🧭Your visual roadmap through the chapter
Simple Interest
Fixed % of the original principal; I=Prn
Compounding
Interest earned on principal + accrued interest; annual, by parts, or continuous
Discounting
Finding the present value of a future cash flow — the reverse of compounding
Interest is the extra amount paid for the use of borrowed money — simple (fixed on the principal) or compound (accumulated over time). Discounting reverses compounding to find present value.
Compounding and discounting let a business compare and evaluate present and future cash flows for investment and borrowing decisions — the basis of later discounted cash flow topics.
Apply simple and compound interest formulas; adjust rates for sub-annual compounding; calculate effective annual rates; discount future cash flows to present value.
Key Definitions
📘Source-based terminology, exam-ready
Interest
The extra amount an investor receives (or a borrower pays) after investing/borrowing a sum, at a certain rate, for a certain time.
Key word: cost of money
Simple Interest
Annual interest that is a fixed percentage of the original amount (the principal) borrowed or invested.
Key word: fixed on principal
Compounding
The process of accumulating interest on an investment over time, so that interest is then earned on that interest too.
Key word: interest on interest
Future Value (FV / Sn)
The accumulated investment plus interest — the amount paid by a borrower or received by an investor at the end of the period.
Key word: value later
Present Value (PV / So)
The initial investment or principal amount; also the current-day equivalent of a future cash flow.
Key word: value today
Discounting
Estimating the present-day equivalent of a future cash flow at a given interest rate; the reverse of compounding.
Key word: reverse of compounding
Discount Factor
The inverse of a compounding factor for the same period and rate; multiplying by it is the same as dividing by the compounding factor.
Key word: 1/(1+r)ⁿ
Nominal vs Effective Rate
Nominal (stated) rate is the loan's "face value" rate; effective rate accounts for the compounding frequency and gives the true annual cost.
Key word: stated vs true rate
Core Concept
🎯Explained in exam-oriented language
CONCEPT → Simple vs Compound Interest
MEANING
Simple interest stays constant each period; compound interest grows each period since it's charged on principal + accrued interest.
PURPOSE
Model how a loan or investment accumulates value over time.
APPLICATION
Loans, deposits, and investments repaid/accumulated periodically or in a lump sum.
CONCEPT → Discounting
MEANING
Finding today's equivalent value of money to be received/paid in the future.
PURPOSE
Compare cash flows that occur at different points in time on a like-for-like basis.
APPLICATION
Evaluating investment offers, loan-repayment alternatives, and future purchase goals.
Key Rules & Principles
📏Every source rule, exam-ready
RULE 1 Rates are annual unless stated
For periods under a year, pro-rate the annual simple interest rate by the fraction of the year
RULE 4 Nominal vs effective rate
Nominal rate is not directly comparable across different compounding periods
Restate to effective annual rate before comparing loans/investments
Effective rate is generally higher than nominal — more frequent compounding raises it further
RULE 2 Compounding by parts
Divide the annual rate by the number of parts in the year (m)
Multiply the number of years by m to get total periods
RULE 5 Discounting is the reverse of compounding
Discount factor = inverse of the compounding factor for the same r and n
Multiplying by the discount factor = dividing by the compounding factor
RULE 3 Continuous compounding
Interest calculated at every smallest instance, using ern
Not realistic in practice, but useful theoretically
RULE 6 Comparing cash flows
Discount every cash flow to its present value before comparing
The option with the higher present value is preferable
Formula Zone
🧮Every formula supported by the source text
SIMPLE INTEREST
P=principal, r=period interest rate, n=number of periods, S=amount repayable, I=interest charge.
COMPOUND INTEREST
(1+r)ⁿ is the compounding factor. So=principal/present value, Sn=future value.
COMPOUNDING BY PARTS
m = number of compounding parts per year.
CONTINUOUS COMPOUNDING
Interest compounded at every smallest instant.
EFFECTIVE ANNUAL RATE
r=known rate for a known period, n=number of times that period fits into one year.
PRESENT VALUE (DISCOUNTING)
1⁄(1+r)ⁿ is the discount factor — the inverse of the compounding factor.
Calculation Roadmap
🗺️General path for any financial mathematics question
1· Read Question
Identify P, r, n and what's required
2· Identify Type
Simple, compound, or discounting?
3· Select Formula
Adjust r and n for the compounding frequency
4· Substitute
Put known values into the formula
5· Calculate & Check
Solve, recheck, state final answer
Step-by-step Methods
🪜Kept separate — do not mix methods
METHOD A · Simple Interest
- 1Identify P, r (annual) and n (in years, pro-rated if less than a year)
- 2For interest only: substitute into I = Prn
- 3For amount repayable: substitute into S = P(1+rn)
- 4If P, r or n is unknown, rearrange and solve algebraically
METHOD C · Compounding by Parts / Continuously
- 1Identify the annual rate r and parts per year m (or use e for continuous)
- 2By parts: divide r by m, multiply n by m
3 Substitute into Sn=So(1+r/m)n×m or Sn=So ern
- 4Calculate Sn
METHOD B · Compound Interest (Annual)
- 1Identify So, r and n
- 2Substitute into Sn = So(1+r)ⁿ
- 3Calculate the compounding factor (1+r)ⁿ first
- 4Multiply by So to find Sn; subtract So for interest earned
METHOD D · Effective Annual Rate
- 1Identify the known rate r and how many times that period fits in a year (n)
- 2Substitute into (1+r)ⁿ − 1
- 3Calculate and express as a percentage
- 4Use this to compare rates quoted for different periods
METHOD E · Present Value / Discounting
- 1Identify the future value (FV), rate r and number of periods n
- 2Calculate the discount factor 1/(1+r)ⁿ, or read it from a discount table
- 3Multiply FV by the discount factor
- 4State the present value; compare PVs if evaluating alternatives
Different Cases / Scenarios
🔀Recognise the situation, pick the right route
IF interest is a fixed % of the ORIGINAL principal only
→ USE simple interest: I=Prn, S=P(1+rn)
IF interest is earned on principal PLUS accrued interest
→ USE compound interest: Sn=So(1+r)ⁿ
IF the period given is less than a year
→ Pro-rate the annual rate by the fraction of the year
IF compounding happens m times per year (e.g. quarterly)
→ USE r/m for the rate and n×m for the periods
IF compounding is described as continuous
→ USE Sn=So ern
IF comparing loans/rates quoted for different periods
→ Convert each to its effective annual rate first
IF asked for the value TODAY of a future amount
→ USE discounting: PV=FV×1/(1+r)ⁿ
IF a discount-rate table is provided
→ Read the factor at the intersection of the rate column and period row
Comparison Tables
⚖️Don't confuse these!
| Simple Interest | Compound Interest | |
|---|---|---|
| Basis of interest | Original principal only | Principal + all accrued interest to date |
| Formula | S = P(1+rn) | Sn = So(1+r)ⁿ |
| Growth pattern | Same interest amount every period | Interest amount increases every period |
| Compounding | Discounting | |
|---|---|---|
| Direction | Present value → Future value | Future value → Present value |
| Formula | Sn = So(1+r)ⁿ | PV = FV × 1/(1+r)ⁿ |
| Factor used | Compounding factor (1+r)ⁿ | Discount factor = inverse of compounding factor |
| Nominal Rate | Effective Rate | |
|---|---|---|
| Meaning | The "face value" / stated rate of a loan | The true annual cost, accounting for compounding frequency |
| Comparability | Not directly comparable across different compounding periods | Always comparable on a like-for-like annual basis |
| Relative size | Generally lower | Generally higher (rises with compounding frequency) |
Reading a Discount-factor Table
📌How to interpret the source table
Each column of a discount table represents a discount rate (r); each row represents a number of periods (n). The value at the intersection of the correct rate column and period row is the discount factor for that r and n — multiply it directly by the future cash flow to get its present value, instead of calculating 1/(1+r)ⁿ by hand.
Worked Calculation Logic
✍️Method students must reproduce in the exam
GIVEN → FORMULA → SUBSTITUTION → CALCULATION → ANSWER
A person expects to receive Rs 13,310 in 3 years. At an interest rate of 10%, what is the present value of this amount?
GIVEN: FV = 13,310, r = 0.1, n = 3
FORMULA: PV = FV × 1/(1+r)ⁿ
SUBSTITUTION: PV = 13,310 × 1/(1.1)³
CALCULATION: PV = 13,310 × 1/1.331 = 13,310 × 0.7513
FINAL ANSWER: PV = Rs. 10,000 — the person would be indifferent between Rs. 10,000 today and Rs. 13,310 in 3 years
Exam Question Patterns
📝Direct calculation — find I, S or Sn from given P, r, n
Solve for unknown — rearrange to find P, r or n
Compounding frequency — annual, by parts, or continuous
Effective rate — calculate and compare loan/investment options
Present value — discount a single future cash flow
Comparing offers — evaluate which alternative has the higher PV
Exam Traps
⚠️⚠ WATCH OUT
- ✗ Simple vs compound mix-up — applying S=P(1+rn) when interest should compound, or vice versa.
- ✗ Forgetting to pro-rate the rate (or periods) for sub-annual timeframes.
- ✗ Using the annual rate directly in compounding by parts instead of dividing by m (and multiplying n by m).
- ✗ Nominal/effective confusion — comparing loans by their nominal rates instead of effective annual rates.
- ✗ Wrong table cell — swapping the rate column and period row when reading a discount table.
Quick Decision Tree
🌳Chapter 5 in 60 Seconds
⚡Simple: I=Prn, S=P(1+rn)
Compound: Sn=So(1+r)ⁿ
By parts: Sn=So(1+r/m)n×m
Continuous: Sn=So ern
Effective rate: (1+r)ⁿ−1
PV: FV×1/(1+r)ⁿ Discount factor = inverse of compounding factor
Trap: always adjust r & n for compounding frequency
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