Chapter Overview
🧭Your visual roadmap through the chapter
Annuities & Perpetuities
Regular equal payments — ordinary, due, and forever
Net Present Value
PV of inflows − PV of outflows; accept if positive
Internal Rate of Return
The discount rate at which NPV = 0
Payback Period
Time to recover the initial investment
Discounted cash flow (DCF) evaluates whether proposed investments are financially worthwhile, using annuities, perpetuities, NPV, IRR and payback period techniques.
DCF techniques are the standard tools for capital-investment decisions — comparing project returns against the cost of capital and each other.
Apply annuity/perpetuity formulas; compute NPV and accept/reject a project; estimate IRR by interpolation; calculate payback and discounted payback period.
Key Definitions
📘Source-based terminology, exam-ready
Annuity
A series of regular periodic payments of equal amount — a constant cash flow for a given number of time periods.
Key word: equal, regular, finite
Ordinary Annuity / Annuity Due
Ordinary: payments in arrears (end of each period). Due: payments in advance (beginning of each period).
Key word: end vs start
Perpetuity
A constant annual cash flow "forever" — an annuity in which the cash flows continue into the long-term future indefinitely.
Key word: forever
Sinking Fund
Setting aside a fixed sum at regular intervals to achieve a specific target sum at a future point in time.
Key word: save toward a target
Net Present Value (NPV)
The difference between the present value of cash inflows and the present value of cash outflows over a project's life.
Key word: PV in − PV out
Internal Rate of Return (IRR)
The discount rate that makes a project's NPV exactly equal to zero.
Key word: NPV = 0 rate
Payback Period
The length of time it takes for an investment's cash inflows to recover its initial cost.
Key word: time to break even
Discounted Payback Period
Like payback period, but the future cash flows are discounted to present value first, accounting for the time value of money.
Key word: payback, discounted
Core Concept
🎯Explained in exam-oriented language
CONCEPT → Annuities
MEANING
A stream of equal payments; can be valued as a future sum (FV) or a present value (PV).
PURPOSE
Avoid discounting/compounding each payment individually — use one annuity factor instead.
APPLICATION
Loan installments, savings plans, sinking funds, project cash flows.
CONCEPT → NPV / IRR
MEANING
NPV measures value created at a given cost of capital; IRR finds the rate at which value created is zero.
PURPOSE
Decide whether a project is worth undertaking.
APPLICATION
Capital budgeting / investment appraisal decisions.
Key Rules & Principles
📏Every source rule, exam-ready
RULE 1 Recognising annuity due
If payments start now/immediately, or the first payment is at the beginning of the first period, it is an annuity due
RULE 4 NPV decision rule
NPV > 0 → accept (value created)
NPV = 0 → break-even
NPV < 0 → reject (value destroyed)
RULE 2 Annuity due vs ordinary relationship
Sum (FV) of annuity due = (1+i) × sum of ordinary annuity
PV of annuity due = (1+i) × PV of ordinary annuity
Annuity due is always the larger figure, all else equal
RULE 5 IRR decision rule
IRR > required rate of return → accept
IRR = required rate → break-even (NPV = 0)
IRR < required rate → reject
RULE 3 Perpetuities
Sum (FV) of a perpetuity cannot be determined (it is infinite)
PV of an ordinary perpetuity = R ÷ i
PV of a perpetuity due = R + (R ÷ i)
RULE 6 Payback period
A shorter payback period is generally preferred
Ignores the time value of money, and any cash flows after payback — use discounted payback to correct for time value
Formula Zone — Annuities & Perpetuities
🧮Every formula supported by the source text
SUM (FV) OF ORDINARY ANNUITY
R=periodic payment, i=rate per period, n=number of payments; payments at end of each period.
SUM (FV) OF ANNUITY DUE
Payments at the start of each period; always > sum of ordinary annuity.
PV OF ORDINARY ANNUITY
Used for loan repayments and valuing a stream of receipts/payments in arrears.
PV OF ANNUITY DUE
Always > PV of ordinary annuity, all else equal.
PV OF PERPETUITY (ORDINARY)
Sum (FV) of a perpetuity cannot be determined.
PV OF PERPETUITY DUE
The first payment (R) is received immediately, plus the PV of the remaining ordinary perpetuity.
Formula Zone — NPV, IRR & Payback
🧮Every formula supported by the source text
NET PRESENT VALUE
Discount each period's cash flow at the cost of capital, then sum. Initial investment is a negative (outflow) at year 0.
IRR BY INTERPOLATION
NPVA at rate A% should ideally be positive, NPVB at rate B% negative, for best accuracy.
PAYBACK PERIOD (EVEN CASH FLOWS)
For uneven cash flows, accumulate cash flows year by year until the investment is recovered.
DISCOUNTED PAYBACK PERIOD
Accumulate discounted cash flows year by year until they cover the initial investment.
Calculation Roadmap
🗺️General path for any discounted cash flow question
1· Read Question
List every cash flow and its timing
2· Identify Type
Annuity, perpetuity, NPV, IRR, or payback?
3· Select Formula
Ordinary or due? Even or uneven cash flows?
4· Substitute
Put known values into the formula
5· Calculate & Interpret
Solve, apply decision rule, state answer
Step-by-step Methods
🪜Kept separate — do not mix methods
METHOD A · Sum (FV) or PV of an Annuity
- 1Identify R, i (per period) and n
- 2Decide ordinary (end of period) or due (start of period)
- 3Substitute into the matching Sn or P formula
- 4If "due", multiply the ordinary result by (1+i)
METHOD D · Net Present Value (NPV)
- 1List every cash flow by year, including the initial investment (negative)
- 2Discount each year's cash flow at the cost of capital
- 3Sum all discounted cash flows to get the NPV
- 4Apply the decision rule: accept if NPV > 0
METHOD B · Sinking Fund (Solve for R)
- 1Identify the target future sum (Sn), i and n
- 2Substitute into Sn = R[(1+i)ⁿ−1]÷i
- 3Rearrange to solve for R (the periodic saving)
- 4Alternative: find PV of the target, then divide by the annuity factor
METHOD E · Internal Rate of Return (IRR)
- 1Calculate NPV at a first discount rate, A%
- 2Calculate NPV at a second rate, B%, ideally giving the opposite sign
- 3Substitute both NPVs into the interpolation formula
- 4Compare the estimated IRR to the required rate of return
METHOD C · PV of a Perpetuity
- 1Identify R (periodic payment) and i (period rate)
- 2If payments are less frequent than annual (e.g. every 2 years), find the equivalent period's effective rate first
- 3Substitute into P=R÷i (ordinary) or P=R+(R÷i) (due)
- 4State the present value
METHOD F · Payback / Discounted Payback
- 1List cash flows by year (discount them first if using discounted payback)
- 2Build the cumulative cash flow, year by year
- 3Find the year the cumulative total turns positive
- 4Add the fraction of that year needed: remaining amount ÷ that year's cash flow
Different Cases / Scenarios
🔀Recognise the wording, pick the right route
IF payments occur at the END of each period
→ USE ordinary annuity formulas
IF payments start NOW / immediately / at the beginning
→ USE annuity due: multiply ordinary result by (1+i)
IF the cash flow continues forever
→ USE perpetuity: P=R÷i (only PV can be found)
IF a target future sum must be reached by saving
→ USE a sinking fund: solve for R
IF NPV > 0
→ Accept the project
IF NPV < 0
→ Reject the project
IF cash flows are equal every year
→ Payback = Investment ÷ Annual Cash Flow
IF cash flows are uneven, or time value of money matters
→ USE the cumulative (discounted) cash flow method
Comparison Tables
⚖️Don't confuse these!
| Ordinary Annuity | Annuity Due | |
|---|---|---|
| Payment timing | End of each period (in arrears) | Start of each period (in advance) |
| Sum (FV) | So = R[(1+i)ⁿ−1]÷i | SD = (1+i) × So |
| Present Value | Po = R[1−(1+i)⁻ⁿ]÷i | PD = (1+i) × Po |
| Net Present Value (NPV) | Internal Rate of Return (IRR) | |
|---|---|---|
| What it finds | The rupee value created at a given cost of capital | The discount rate at which NPV = 0 |
| Decision rule | Accept if NPV > 0 | Accept if IRR > required rate of return |
| Method | Discount all cash flows at one given rate | Interpolate between a positive and negative NPV |
| Payback Period | Discounted Payback Period | |
|---|---|---|
| Time value of money | Ignored | Accounted for (cash flows discounted first) |
| Cash flows used | Raw (undiscounted) cash flows | PV of cash flow = Cash Flow ÷ (1+i)ⁿ |
| Result | Shorter, since money isn't discounted | Longer, since future cash flows are worth less today |
Reading Cash-flow Timelines & Tables
📌How to interpret the source's diagrams
Cumulative cash flow timeline (payback)
Plot the running (cumulative) total below each year. The payback point is where the cumulative total crosses from negative to positive. Interpolate within that year: fraction = remaining negative balance ÷ that year's cash flow.
NPV discount-factor table
Each row is a year's cash flow, multiplied by that year's discount factor (from the rate column) to give its present value. Sum the present-value column (inflows and the negative initial outflow) to get the NPV.
Annuity factor tables
An annuity factor is the sum of the individual discount factors for a run of periods. Read it at the intersection of the correct rate column and period row, then multiply by the periodic payment (or divide the target/loan by it to find the payment).
IRR interpolation diagram
NPV is plotted against discount rate; the true relationship curves, but interpolation assumes a straight line between the two calculated NPVs to estimate where the line crosses zero (the IRR).
Worked Calculation Logic
✍️Method students must reproduce in the exam
GIVEN → FORMULA → SUBSTITUTION → CALCULATION → ANSWER
A company with a 10% cost of capital considers a project: initial payment Rs. 10,000; yields Rs. 6,000/year for 2 years. Should it be undertaken?
GIVEN: Year 0 = (10,000); Year 1 & 2 = 6,000 each; r = 10%
FORMULA: NPV = Σ (Cash Flow × Discount Factor)
SUBSTITUTION: Year 1: 6,000×1/(1.1)¹=5,456; Year 2: 6,000×1/(1.1)²=4,959
CALCULATION: NPV = −10,000 + 5,456 + 4,959 = 415
FINAL ANSWER: NPV = Rs. 415 (positive) — the project is viable and should be undertaken
Exam Question Patterns
📝Annuity/perpetuity calculation — FV, PV, ordinary vs due
Sinking fund — solve for the periodic saving amount
NPV calculation — compute and apply accept/reject rule
IRR estimation — interpolate between two discount rates
Payback / discounted payback — even and uneven cash flows
Reading annuity/discount factor tables
Exam Traps
⚠️⚠ WATCH OUT
- ✗ Ordinary vs due mix-up — forgetting to multiply by (1+i) for an annuity/perpetuity due.
- ✗ Perpetuity sum — trying to calculate a future sum for a perpetuity; only PV can be found.
- ✗ IRR sign errors — mixing up NPVA and NPVB or their signs in the interpolation formula.
- ✗ Payback ignoring time value of money when the question specifically asks for discounted payback.
- ✗ Wrong cash flow sign — treating the initial investment as positive instead of a negative outflow.
Quick Decision Tree
🌳Chapter 6 in 60 Seconds
⚡Ordinary Sn: R[(1+i)ⁿ−1]/i Due Sn/Pn = (1+i) × ordinary
Ordinary Pn: R[1−(1+i)⁻ⁿ]/i
Perpetuity PV: R÷i NPV >0 accept, <0 reject
IRR: A%+[NPVa/(NPVa−NPVb)]×(B−A)% Payback = Investment ÷ Annual CF
Trap: "starts now" = annuity due
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