NISM-Series-X-A: Investment Adviser (Level 1) · Introduction to Modern Portfolio Theory

Risk and Return of Individual Assets for NISM Investment Adviser Level 1

Updated 11 October 2026 · Fact-checked

Expected return is the probability-weighted average of possible returns on a security. Variance is the probability-weighted average of squared deviations from that expected return. Standard deviation is the square root of variance and measures risk. Historical return is what actually happened; expected return is a forward-looking estimate.

Understand Risk and Return of Individual Assets

Every investment has a return and a risk. Return is the gain or loss as a percentage of the amount invested. Risk is the chance that the actual return differs from what you expected. In modern portfolio theory, risk is measured by how widely returns are spread around their average.

Start with expected return. Suppose a stock could give different returns in different economic conditions, each with a probability. Multiply each return by its probability and add them up. The result is the expected return. It is a long-run average, not a promise. The actual outcome will usually be different.

Now measure the spread. For each scenario, find how far its return is from the expected return, square that gap, and weight it by the scenario probability. The sum is the variance. Squaring stops positive and negative gaps from cancelling out. The side effect is that variance is in squared percentage units, which is hard to read. So you take the square root. That gives the standard deviation, in the same units as return. A higher standard deviation means a riskier security.

There are two ways to get these numbers. With expected (ex-ante) return, you use scenarios and probabilities about the future. With historical (ex-post) return, you use actual past returns, usually give each period equal weight, and take the simple average. For historical data that is a sample, the variance is usually calculated by dividing the sum of squared deviations by n − 1. For a full population of scenarios with probabilities, you use the probabilities as weights.

Historical figures are facts about the past. Expected figures are estimates about the future. Past averages are often used to estimate the future, but there is no guarantee that they will repeat.

Key formulas to remember

Expected return (with probabilities): E(R) = Σ pᵢ × Rᵢ; Probabilities must add up to 1 (100%). Multiply each return by its probability and sum.
Variance (with probabilities): σ² = Σ pᵢ × (Rᵢ − E(R))²; Use deviations from the expected return, not from zero.
Standard deviation: σ = √σ²; Same units as return. This is the usual measure of total risk of a single asset.
Historical average return: R̄ = Σ Rₜ ÷ n; Simple (arithmetic) average of n past period returns.
Sample variance of historical returns: s² = Σ (Rₜ − R̄)² ÷ (n − 1); Use n − 1 for a sample. If the question says to treat the data as the whole population, divide by n.
Single-period return: R = (P₁ − P₀ + D) ÷ P₀; P₀ is the starting price, P₁ the ending price, D the dividend or income received.

How to solve Risk and Return of Individual Assets questions

Use this order for any question on the risk and return of a single security. Check first whether the data are scenarios with probabilities or past returns.

1Read the data. Are there probabilities (expected return) or equal-weight past periods (historical return)?
2If probabilities are given, check they add up to 1. If not, something is wrong or a scenario is missing.
3Compute the average: Σ p × R for scenarios, or the simple average for history.
4Subtract the average from each return to get the deviations. Square each deviation.
5Weight the squared deviations by probability, or divide the sum by n − 1 for a sample (n if told it is a population). This is the variance.
6Take the square root for standard deviation.
7Check units and the question. Give the answer in % and check whether the question asked for variance or standard deviation.

Quickest way: Shortcut for scenario questions

When to use it: Use when there are 2 or 3 scenarios with round probabilities and you need standard deviation fast.

Compute E(R) = Σ p × R in your head or on the rough sheet.
Write each deviation from E(R) and square it. Use whole numbers (in % points) to keep arithmetic easy.
Multiply each squared deviation by its probability and add.
Take the square root. Use the answer options: perfect-square variances usually point to the correct option.
If two options differ only by variance versus standard deviation, re-read what was asked.

Common mistakes in Risk and Return of Individual Assets

Giving the variance when the question asks for standard deviation.
You stop one step early under time pressure.
Fix: Underline the word asked. Always finish with the square root if standard deviation is asked.
Measuring deviations from zero instead of from the expected return.
Squaring returns looks like the simplest route.
Fix: Always compute E(R) first, then subtract it from every return.
Forgetting to weight by probability and using a simple average of scenario returns.
Equal-weight averaging is familiar from history questions.
Fix: If probabilities are given, multiply each return by its probability. A simple average is right only for equally likely outcomes.
Treating expected return as a guaranteed or most likely return.
The word expected sounds like a forecast of what will occur.
Fix: Remember it is a probability-weighted average. The actual return may never equal it.
Using n instead of n − 1 for sample variance of historical returns (or the reverse).
Students mix the scenario method with the sample method.
Fix: For past returns given as a sample, divide by n − 1. For probability scenarios, use probabilities as weights.
Mixing up historical and expected returns.
Both are averages and both are quoted as percentages.
Fix: Historical uses past actual data (ex-post). Expected uses probabilities of future outcomes (ex-ante).

Worked examples

Example 1

A stock has three scenarios: boom (probability 0.3, return 20%), normal (probability 0.5, return 10%), recession (probability 0.2, return −5%). Find the expected return.

Show the solution

Check probabilities: 0.3 + 0.5 + 0.2 = 1.0.
Boom: 0.3 × 20 = 6.0
Normal: 0.5 × 10 = 5.0
Recession: 0.2 × (−5) = −1.0
Add: 6.0 + 5.0 − 1.0 = 10.0

Answer: Expected return = 10%.

Example 2

Using the same stock (expected return 10%), find the variance and standard deviation of returns, with returns in % points.

Show the solution

Boom deviation: 20 − 10 = 10. Squared = 100. Weighted: 0.3 × 100 = 30
Normal deviation: 10 − 10 = 0. Squared = 0. Weighted = 0
Recession deviation: −5 − 10 = −15. Squared = 225. Weighted: 0.2 × 225 = 45
Variance = 30 + 0 + 45 = 75 (in %²)
Standard deviation = √75 ≈ 8.66

Answer: Variance = 75 (%²); standard deviation ≈ 8.66%.

Exam tips

Read whether the question wants expected return, variance or standard deviation. Options often include all three.
Use the 'probabilities add up to 1' check to catch typo-style trap data.
Keep deviations in whole % points to avoid decimals, and convert at the end.
Know the conceptual difference: historical is ex-post and actual; expected is ex-ante and probability-based. Theory questions test this directly.
Remember that on 2-mark caselet questions a wrong answer costs twice as much, so finish the calculation before answering rather than guessing.

Practice questions from Introduction to Modern Portfolio Theory

Risk and Return of Individual Assets in other exams

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

Risk and Return of Individual Assets: frequently asked questions

What is the difference between historical return and expected return?

Historical return is calculated from actual past prices and income. Expected return is a forward-looking, probability-weighted average of possible future returns. Historical data is often used to help estimate expected return, but the two are not the same.

Why do we use standard deviation instead of variance to describe risk?

Variance is in squared units, which is hard to interpret. Standard deviation is its square root and is in the same units as return, such as percent. It is easier to compare with the expected return.

Does a higher standard deviation always mean a worse investment?

No. It means the returns are more spread out, so the investment is riskier. Whether it is worth it depends on the expected return and on the investor's risk tolerance.

Do I divide by n or n − 1 for historical returns?

For a sample of past returns, the usual method is to divide the sum of squared deviations by n − 1. Divide by n only if the question says to treat the data as the entire population.