NISM-Series-XV: Research Analyst · Fundamentals of Risk and Return

Holding Period, Arithmetic and Geometric Returns for NISM Research Analyst

Updated 11 October 2026 · Fact-checked

Return measures the gain on an investment as a percentage of its starting value. Holding period return = (income + ending price − beginning price) ÷ beginning price. Arithmetic mean is the simple average of period returns. Geometric mean compounds them, so it gives the true average growth rate (CAGR). Real return removes inflation from nominal return.

Understand Returns: Holding Period, Arithmetic and Geometric

A return tells you how much you gained or lost on an investment, as a percentage of what you put in. It has two parts: income (dividend or interest) and capital gain (change in price).

The holding period return (HPR) is the total return over the whole time you held the investment, whether that is a day or ten years. It is not annualised. A 40% HPR over four years is not 40% a year.

When you have returns for several periods, you need one average. The arithmetic mean adds the returns and divides by the number of periods. It is a good estimate of the expected return for a single future period. But it ignores compounding, so it overstates the growth you actually earned over many periods, especially when returns swing a lot.

The geometric mean multiplies the growth factors (1 + return), takes the n-th root and subtracts 1. It gives the constant yearly rate that would produce the same ending value. When it is measured between a start and an end value, it is called CAGR. The geometric mean is never higher than the arithmetic mean. They are equal only when all returns are identical.

A nominal return is the return in rupee terms, before inflation. A real return is what is left after inflation, so it shows the gain in purchasing power. The exact link is (1 + real) = (1 + nominal) ÷ (1 + inflation). The quick approximation is real ≈ nominal − inflation.

Key formulas to remember

Holding period return: HPR = (Ending value − Beginning value + Income) ÷ Beginning value; Equivalent form: (Income + Ending value) ÷ Beginning value − 1. Not annualised.
Arithmetic mean return: AM = (R1 + R2 + … + Rn) ÷ n; Simple average. Overstates compound growth when returns vary.
Geometric mean return: GM = [(1 + R1) × (1 + R2) × … × (1 + Rn)]^(1/n) − 1; Use the factors 1 + R, not R. A loss of 20% is a factor of 0.80.
CAGR: CAGR = (Ending value ÷ Beginning value)^(1/n) − 1; n is the number of years. It assumes no interim cash flows.
Annualising a holding period return: Annualised return = (1 + HPR)^(1/years) − 1; For periods under one year, use the exponent 1/years with years as a fraction, e.g. 6 months = 0.5.
Real return (exact): Real return = (1 + Nominal) ÷ (1 + Inflation) − 1; Approximation: Nominal − Inflation. Use the exact form if the question gives options that differ slightly.
AM and GM relationship: GM ≤ AM; Equal only if all period returns are the same.

How to solve Returns: Holding Period, Arithmetic and Geometric questions

Use this order for any returns question. Most mistakes come from picking the wrong measure or skipping the conversion to growth factors.

1Identify what is asked: total return over the period (HPR), simple average (arithmetic), compounded average (geometric or CAGR), or inflation-adjusted (real).
2Write down the inputs: beginning value, ending value, income received, number of years, and inflation if given.
3For HPR, add income to the price change and divide by the beginning value.
4For a multi-period average, convert each return to a factor (1 + R). Use the arithmetic mean only if the question says simple or average of returns. Use the geometric mean for compounded or annualised growth.
5For CAGR, divide ending by beginning value, raise to 1/n, subtract 1.
6For real return, divide (1 + nominal) by (1 + inflation) and subtract 1, or subtract if the question allows approximation.
7Check reasonableness: GM must not exceed AM, and a negative year must pull the factor below 1.
8Match your result to the options, watching for percentage versus decimal.

Quickest way: Factor shortcut with option elimination

When to use it: Use when you have little time and the options are numerically spread out.

Convert every return to a factor first. Work with 1.10, 0.80 and so on.
For two-period geometric mean, multiply the two factors and take the square root. Test option values by squaring them.
For CAGR, test the rule of 72 style checks: if value doubles in 6 years, CAGR is about 12%.
Eliminate any option where GM is above AM, or where real return is above nominal when inflation is positive.
If the question says average annual return and gives year-by-year data, check whether it wants arithmetic or compounded before calculating.

Common mistakes in Returns: Holding Period, Arithmetic and Geometric

Averaging returns arithmetically when the question asks for compounded annual growth.
The arithmetic average is quicker and looks natural.
Fix: If the words are CAGR, compounded, annualised or growth rate between two values, use the geometric mean.
Plugging returns directly into the geometric formula, e.g. multiplying 10% and −20%.
Students forget to add 1 to each return.
Fix: Always convert to factors: 1.10 and 0.80. Multiply the factors, take the root, then subtract 1.
Treating holding period return as an annual figure.
HPR is a percentage, so it looks like a yearly rate.
Fix: Check the time span. Divide or compound it to annualise: (1 + HPR)^(1/years) − 1.
Leaving out dividends or interest from HPR.
Focus stays on the price change.
Fix: Add all income received during the holding period to the numerator.
Using the approximation nominal − inflation when options are close, or reversing it to nominal + inflation.
Speed, and confusion about which direction inflation works.
Fix: Real is lower than nominal when inflation is positive. Use the exact formula when answers are close.
Using the wrong n in CAGR, such as counting data points instead of years.
Five year-end values cover only four years of growth.
Fix: n is the number of intervals between the first and last value.

Worked examples

Example 1

A share was bought at ₹200. Over 2 years the investor received dividends of ₹10 in total and then sold the share at ₹250. Find the holding period return and the annualised return (nearest 0.1%).

Show the solution

Price gain = 250 − 200 = ₹50.
Total gain = 50 + 10 = ₹60.
HPR = 60 ÷ 200 = 0.30 = 30%.
Annualised = (1.30)^(1/2) − 1.
√1.30 = 1.1402, so annualised ≈ 14.0%.

Answer: HPR = 30%; annualised return ≈ 14.0% per year.

Example 2

A fund returned +50% in year 1 and −20% in year 2. Find the arithmetic mean and geometric mean annual returns, and say which one shows the actual growth of ₹1,00,000.

Show the solution

Arithmetic mean = (50 + (−20)) ÷ 2 = 15%.
Factors are 1.50 and 0.80.
Product = 1.50 × 0.80 = 1.20.
Geometric mean = √1.20 − 1 = 1.0954 − 1 ≈ 9.5%.
Check: ₹1,00,000 × 1.50 × 0.80 = ₹1,20,000, a total gain of 20% over 2 years, which matches 9.5% a year compounded (1.0954² = 1.20).
At 15% a year, ₹1,00,000 would reach about ₹1,32,250, which overstates the actual outcome.

Answer: Arithmetic mean = 15%; geometric mean ≈ 9.5%. The geometric mean reflects actual growth, as the final value is ₹1,20,000.

Exam tips

Read the keyword: average return usually means arithmetic, while compounded, annualised or CAGR means geometric.
Remember GM is never above AM. Use this to remove wrong options fast.
Check whether income is included in the return before you pick an HPR option.
For real return questions, confirm the direction: real is below nominal when inflation is positive.
Since NISM-Series-XV has 25% negative marking, avoid guessing on calculation items where you can eliminate at least two options.

Practice questions from Fundamentals of Risk and Return

Returns: Holding Period, Arithmetic and Geometric in other exams

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

CFA Level IHolding Period Return and Return Measures

Returns: Holding Period, Arithmetic and Geometric: frequently asked questions

What is the holding period return formula for NISM Research Analyst?

HPR = (Ending value − Beginning value + Income) ÷ Beginning value. It measures total return over the entire time held and is not annualised unless you convert it.

What is the difference between arithmetic and geometric mean return?

The arithmetic mean is the simple average of period returns. The geometric mean compounds the returns and shows the constant growth rate that gives the same ending value. The geometric mean is lower unless all returns are equal.

What is the difference between nominal return and real return?

Nominal return is the return before inflation. Real return is after inflation and shows the gain in purchasing power. The exact formula is (1 + nominal) ÷ (1 + inflation) − 1.

How do I calculate CAGR for an investment?

Divide the ending value by the beginning value, raise the result to the power 1/n where n is the number of years, and subtract 1. For example, ₹1,00,000 growing to ₹1,44,000 in 2 years gives 1.44^(0.5) − 1 = 20%.