NISM-Series-X-A: Investment Adviser (Level 1) · Time Value of Money

Rate of Return, NPV, IRR and Rule of 72 Explained

Updated 11 October 2026 · Fact-checked

Rate of return finds the yearly growth rate linking a present value to a future value. CAGR = (FV ÷ PV)^(1/n) − 1. NPV adds discounted cash flows minus the outlay. IRR is the discount rate that makes NPV zero. Rule of 72 estimates doubling time as 72 ÷ rate.

Understand Rate of Return, NPV, IRR and Rule of 72

Every question here uses one idea: money today and money later are linked by a rate. If you know any three of present value, future value, rate and time, you can find the fourth. This topic asks you to solve for the rate or the time.

CAGR (compound annual growth rate) is the single steady yearly rate that takes an investment from its starting value to its ending value. Actual returns jump around. CAGR smooths them into one number. It is not the simple average of yearly returns.

NPV (net present value) discounts every future cash flow back to today at a chosen rate, then subtracts the initial outlay. A positive NPV means the project earns more than the discount rate. A negative NPV means it earns less.

IRR (internal rate of return) is the discount rate at which NPV equals zero. It is the project's own break-even return. Compare it with your required rate. If IRR is higher, accept. For a standard project, NPV and IRR give the same accept or reject answer.

The Rule of 72 is a shortcut. Divide 72 by the annual rate in percent to get the approximate years to double money. It is an approximation and works best for rates around 6% to 10%.

Key formulas to remember

Solving for the rate: r = (FV ÷ PV)^(1/n) − 1; For a single lump sum with annual compounding. This is also the CAGR formula.
Solving for the number of periods: n = ln(FV ÷ PV) ÷ ln(1 + r); Exam questions usually give options, so test them instead of using logs.
CAGR: CAGR = (Ending value ÷ Beginning value)^(1/years) − 1; n is the number of years between the two values, not the number of data points.
Net present value: NPV = Σ [CFt ÷ (1 + r)^t] − Initial outlay; Use the same r for all flows. Accept if NPV > 0.
IRR condition: NPV at IRR = 0; Accept if IRR is greater than the required rate of return.
Rule of 72: Years to double ≈ 72 ÷ rate (in %); Approximate. Rearranged, rate ≈ 72 ÷ years.
Real rate of return (Fisher): (1 + real) = (1 + nominal) ÷ (1 + inflation); The simple approximation is real ≈ nominal − inflation. Use the exact form when asked.

How to solve Rate of Return, NPV, IRR and Rule of 72 questions

Use this order for any rate, period, NPV or IRR question.

1Identify what is unknown: rate, number of years, NPV or IRR.
2List the known values: PV, FV, cash flows, rate, years. Check that time and rate use the same unit, such as years and annual rate.
3For rate or CAGR, divide FV by PV, raise to the power 1/n, then subtract 1.
4For doubling or tripling time, use the Rule of 72 if the question says approximately. Otherwise test the options with compounding.
5For NPV, discount each cash flow by (1 + r)^t, add them, then subtract the outlay at time 0.
6For IRR, find the rate where NPV is zero. Test the answer options: a higher rate gives a lower NPV.
7Convert the answer to a percentage and check it is sensible against the options and the sign of NPV.

Quickest way: Test the options and use 72

When to use it: Use when the question is multiple choice and gives rate or time options.

For doubling time, divide 72 by the rate. Pick the nearest option.
For CAGR where FV is a round multiple of PV, such as 2x in 5 years, use 72 ÷ years to estimate the rate, then confirm with an option.
For IRR, put one option into the NPV formula. If NPV is positive, the true IRR is higher. If negative, it is lower.
For NPV, use a PV factor that the question may provide. Avoid recomputing powers by hand.
For real return, subtract inflation first to see which option is close, then apply the exact formula if two options are near.

Common mistakes in Rate of Return, NPV, IRR and Rule of 72

Using n as the number of data points instead of the number of years in CAGR
A table of 6 year-end values looks like 6 periods.
Fix: Count the gaps. Values at the end of years 0 to 5 mean n = 5.
Forgetting to subtract 1 after raising to the power 1/n
The bracket result such as 1.15 is read as the answer.
Fix: Always subtract 1 and then multiply by 100 to get the percentage.
Not subtracting the initial outlay in NPV
Students discount only the inflows and stop.
Fix: NPV = present value of inflows − outlay. Write the outlay as a negative at time 0.
Accepting a project because IRR is high without comparing it with the required rate
A large IRR looks good by itself.
Fix: Accept only if IRR is greater than the required rate of return.
Treating Rule of 72 as exact
It gives clean numbers, so it feels precise.
Fix: Use it as an estimate. If the question asks for exact value, use the compounding formula.
Subtracting inflation from nominal return when the exact real rate is asked
The approximation is easy to remember.
Fix: Use (1 + nominal) ÷ (1 + inflation) − 1 when options are close or the question says exact.

Worked examples

Example 1

An investment of ₹1,00,000 grows to ₹1,46,410 in 4 years. What is the CAGR?

Show the solution

FV ÷ PV = 1,46,410 ÷ 1,00,000 = 1.4641.
n = 4, so take the fourth root of 1.4641.
1.1^2 = 1.21 and 1.21^2 = 1.4641. So the fourth root is 1.1.
CAGR = 1.1 − 1 = 0.10 = 10%.

Answer: 10% per year

Example 2

A project needs an outlay of ₹1,00,000 today and returns ₹60,500 at the end of year 1 and ₹72,600 at the end of year 2. The required rate is 10%. Find the NPV and say whether to accept.

Show the solution

PV of year 1 flow = 60,500 ÷ 1.10 = ₹55,000.
PV of year 2 flow = 72,600 ÷ 1.21 = ₹60,000.
Total PV of inflows = 55,000 + 60,000 = ₹1,15,000.
NPV = 1,15,000 − 1,00,000 = ₹15,000.
NPV is positive, so the project earns more than 10%. Its IRR is above 10%.

Answer: NPV = ₹15,000 (positive), so accept the project

Exam tips

Read whether the question asks for exact or approximate. Approximate points to Rule of 72.
Check the sign of NPV first. It tells you whether IRR is above or below the discount rate.
In caselets, work out the CAGR or NPV once and reuse it for the later questions.
Watch for trap options: the simple average return, the unsubtracted 1.15 style value, and the NPV without the outlay.
With 25% negative marking, skip a calculation only if you cannot narrow the options. Test options with the NPV formula before guessing.

Practice questions from Time Value of Money

Rate of Return, NPV, IRR and Rule of 72 in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Rate of Return, NPV, IRR and Rule of 72: frequently asked questions

How do I calculate CAGR?

Divide the ending value by the beginning value, raise the result to 1 ÷ number of years, and subtract 1. Multiply by 100 for the percentage. For example, a value that doubles in 5 years has a CAGR of about 14.87%.

What is the difference between NPV and IRR?

NPV gives the rupee value created after discounting at a chosen rate. IRR gives the rate at which NPV is zero. For a standard project, both lead to the same accept or reject decision.

How does the Rule of 72 work?

Divide 72 by the annual rate in percent to estimate years to double. At 8%, money doubles in about 9 years. It is an approximation and is less accurate at very high or very low rates.

What is the real rate of return formula?

The exact form is (1 + nominal) ÷ (1 + inflation) − 1. The quick approximation is nominal minus inflation. Use the exact form when options are close.