NISM Series XIX-E · Chapter 3: Introduction to Modern Portfolio Theory and Capital Market Theory

L1 · Foundation

Modern Portfolio Theory - Measuring a Single Security

How Markowitz turned an old proverb into mathematics, the five assumptions of Modern Portfolio Theory, ex-ante against ex-post returns, and how to compute the expected return and the variance of an individual security from a probability distribution.

24 min read · Worth 1% of the paper · Written from pages 50–68 of the official NISM workbook · Updated 18 September 2026

Modern Portfolio Theory - Measuring a Single Security

3.1 The proverb that became mathematics

Conventional wisdom has always dictated NOT PUTTING ALL THE EGGS IN ONE BASKET. This adage is addressing the BENEFITS OF DIVERSIFICATION.

And the problem before 1952: prior to 1950, the investing community was FAMILIAR with the benefits of holding a diversified portfolio. However, they had NO WAY OF QUANTIFYING the benefits of diversification.

What changed: in 1952, the Journal of Finance published an article titled "PORTFOLIO SELECTION", authored by HARRY MARKOWITZ. Decades later, in 1990, Harry Markowitz was honoured with the NOBEL PRIZE IN ECONOMICS for his "Portfolio Theory".

The ideas in that article came to form the foundations of what is now popularly referred to as MODERN PORTFOLIO THEORY.

What MPT actually does

In simple words, MPT provides a FRAMEWORK for CONSTRUCTING AND SELECTING PORTFOLIOS based on the EXPECTED PERFORMANCE of the investments and the RISK APPETITE of the investor.

Its three specific contributions:

It QUANTIFIED the concept of diversification by introducing the statistical notion of COVARIANCE, or CORRELATION, between investment assets
Markowitz MATHEMATICALLY DEMONSTRATED that the VARIANCE OF THE RATE OF RETURN is a MEANINGFUL MEASURE OF PORTFOLIO RISK
He DERIVED THE FORMULA for computing the variance of a portfolio, showing how to EFFECTIVELY DIVERSIFY a portfolio

The proverb said diversify. Markowitz said by how much, with what, and how you would know.

3.2 The five assumptions of MPT

1. An investor wants to MAXIMISE THE RETURN FOR A GIVEN LEVEL OF RISK. That means given a choice between two assets with EQUAL rate of return, investors will select the asset with LOWER RISK.

2. Investors consider each investment alternative as being presented by a PROBABILITY DISTRIBUTION OF EXPECTED RETURNS over some holding period.

3. Investors MAXIMISE ONE-PERIOD EXPECTED UTILITY. Investors choose an action or event with the MAXIMUM EXPECTED UTILITY, and assign UTILITY SCORES to the various portfolio choices available to them.

4. Investors base decisions SOLELY ON EXPECTED RETURN AND RISK.

5. Investors ESTIMATE THE RISK OF THE PORTFOLIO on the basis of the VARIABILITY OF EXPECTED RETURNS OF CONSTITUENT ASSETS.

Two of these deserve a second look.

Assumption 1 is what makes the whole theory work. It says investors are risk averse - not that they refuse risk, but that between two equal returns they take the safer one. Everything that follows, including the shape of the efficient frontier, depends on it.

Assumption 4 is the theory's greatest simplification and its greatest vulnerability. Real investors care about liquidity, taxes, regret, and what their brother-in-law bought. MPT says they care about two numbers only.

3.3 Expected rate of return for an individual security

Expected rate of return of an individual investment opportunity is the SUM OF THE EXPECTED RETURNS MULTIPLIED WITH THEIR CORRESPONDING PROBABILITIES.

The two words that get confused in exams

Term	Meaning
Ex-Ante Return	The forward-looking expected return, computed from a probability distribution of possible outcomes
Ex-Post Returns	Returns which are calculated based on HISTORICAL DATA

"Ex-ante" is before the event; "ex-post" is after it. Everything in MPT is computed ex-ante, using estimates - which is precisely why the theory has an estimation problem, covered in the next lesson.

The workbook's Table 3.1

State		Probability	A	B
I	Boom	0.3	15%	25%
II	Normal	0.5	10%	10%
III	Recession	0.2	2%	1%

Note the probabilities sum to 1.0 - 0.3 + 0.5 + 0.2. They must, in any valid probability distribution.

The expected return of stock A:

R(A) = 0.3(15%) + 0.5(10%) + 0.2(2%) = 4.5% + 5.0% + 0.4% = 9.9 per cent

And the workbook's computation for stock B:

R(B) = 0.3(25%) + 0.5(20%) + 0.2(1%) = 7.5% + 10.0% + 0.2% = 17.7 per cent

Watch the workbook's own mismatch here. Table 3.1 prints stock B's Normal-state return as 10%, but the workbook's working above uses 20%. With the table's 10% the answer would be 0.3(25%) + 0.5(10%) + 0.2(1%) = 12.7 per cent; with the working's 20% it is 17.7 per cent. The workbook (Chapter 3, Table 3.1) states both and does not say which is the misprint. If an exam question gives you the state returns, compute from the figures in the question rather than recalling 17.7.

The method, in one line: multiply each possible return by its probability, and add. That is all an expected return is - a probability-weighted average of the outcomes you think possible.

What it is not. It is not a return anyone will actually receive. In this example stock A returns 15, 10 or 2 per cent - never 9.9. The expected return is the centre of the distribution, not a forecast of any single outcome.

3.4 Variance of return for an individual security

Risk of investing in an asset is defined as THE VARIABILITY IN ITS RETURN.

One of the best-known measures of risk is the STANDARD DEVIATION of expected returns - the SQUARE ROOT OF VARIANCE of expected returns. It is a STATISTICAL MEASURE OF THE DISPERSION of returns around the expected value.

And the reading is intuitive: LARGER VARIANCE (larger standard deviation) indicates GREATER DISPERSION, hence LARGER RISK.

The formula

Variance = the sum over all states of [R(i) - E(R(i))] squared, multiplied by P(i)

where P(i) is the probability of the possible rate of return R(i).

In words, four steps:

Compute the expected return E(R)
For each state, find the deviation from the expected return
Square each deviation - which removes the sign, so that a shortfall and an excess both count as dispersion
Weight each squared deviation by its probability and add them

The square root of this measure is known as the EX-ANTE RISK.

The workbook's worked example - Table 3.2

Four equally likely returns of 8, 10, 12 and 14 per cent, each with probability 0.25.

Step 1 - the expected return:

E(R) = (8% x 0.25) + (10% x 0.25) + (12% x 0.25) + (14% x 0.25) = 11 per cent

Step 2 to 4 - the variance table:

Return R(i)	E(R(i))	R(i) - E(R(i))	(R(i) - E(R(i)))²	P	(R(i) - E(R(i)))² x P
8%	11%	-3%	0.0009	0.25	0.000225
10%	11%	-1%	0.0001	0.25	0.000025
12%	11%	+1%	0.0001	0.25	0.000025
14%	11%	+3%	0.0009	0.25	0.000225
				Variance	0.0005

Ex-Ante or Expected Variance = 0.00050

Ex-Ante or Expected Standard Deviation = 0.0224, or 2.24 per cent - the square root of 0.0005.

Three things to notice about this computation

First, work in decimals, not percentages, when squaring. A deviation of -3 per cent is -0.03, whose square is 0.0009. Squaring "3" gives 9, which is out by a factor of ten thousand. This is the commonest arithmetical error in the whole topic.

Second, the deviations are symmetric here and still contribute equally. The -3 per cent and the +3 per cent both add 0.000225. Risk in this framework is two-sided - an outcome above expectation counts as dispersion just as much as one below it.

Third, the standard deviation is in the same units as the return. Variance of 0.0005 means nothing intuitively; 2.24 per cent can be compared directly against the 11 per cent expected return. That is why the square root is taken at all.

Think of it like this

Markowitz measured the proverb your grandmother already knew

Every Indian household has been told not to keep all the gold in one cupboard. What nobody could say was whether two cupboards halved the risk or reduced it by a third. That is exactly the gap the workbook describes: prior to 1950 the investing community was familiar with the benefits of diversification but had no way of quantifying them. Markowitz supplied the arithmetic, and got a Nobel Prize thirty-eight years later.

Ex-ante is the weather forecast, ex-post is yesterday's rainfall

The Met department says there is a 70 per cent chance of rain tomorrow - that is ex-ante, built from probabilities. The gauge in the garden says 42 millimetres fell yesterday - that is ex-post, built from history. Modern Portfolio Theory runs entirely on forecasts, which is why the workbook warns separately about estimation risk.

Expected return is the average marks of a student who never scores the average

A student scores 90 in maths, 60 in history and 75 in science. His average is 75 - a mark he did not get in any paper. An expected return of 9.9 per cent works the same way: the stock will return 15, 10 or 2 per cent, and never 9.9. The expected return is the centre of the distribution, not a prediction of any one outcome.

Probabilities must add to one, like shares of a plot

If three brothers divide a plot and their shares add to 90 per cent, someone has miscounted. In the workbook's table, boom, normal and recession carry 0.3, 0.5 and 0.2 - exactly 1.0, because those three states are all that can happen. Any probability distribution that fails this test is wrong before the arithmetic begins.

Squaring the deviation is why a fever of 39 and 35 are both alarming

A doctor does not subtract 37 from your temperature and take the answer at face value - two degrees above and two below are both departures from normal, and both matter. Squaring the deviation does the same job: it strips the sign, so that a return 3 per cent above expectation and one 3 per cent below both register as dispersion. In this framework risk is two-sided.

Standard deviation is the variance brought back into rupees

If you measure a room's size by squaring its length, you get a number in square metres that tells you nothing about whether the sofa fits. Take the square root and you are back in metres. Variance of 0.0005 is meaningless to compare against an 11 per cent return; its square root, 2.24 per cent, sits in the same units and can be read directly against it.

Risk aversion is choosing the same-fare taxi with the safer driver

Two taxis quote identical fares; one driver is known to be reckless. Everyone takes the other. That is the first assumption of MPT, and everything else rests on it: given a choice between two assets with equal rate of return, investors will select the asset with lower risk. Not a refusal of risk - a refusal of unpaid risk.

MPT's narrow view is the marriage biodata with two columns

A proposal listing only income and height leaves out temperament, family and whether the person can cook - all things people actually decide on. The fourth assumption of MPT is equally bare: investors base decisions solely on expected return and risk. It is the theory's greatest simplification and its greatest vulnerability, and it is worth remembering when a model's output looks more certain than the world.

Key terms in this chapter

Assumptions of MPT: Investors maximise return for a given level of risk and prefer lower risk at equal return; they view each alternative as a probability distribution of expected returns; they maximise one-period expected utility and assign utility scores to portfolio choices; they base decisions solely on expected return and risk; and they estimate portfolio risk from the variability of expected returns of constituent assets.
Assumption that cannot be relaxed: That capital markets are in equilibrium, meaning all investments begin properly priced in line with their risk levels. The workbook describes this as a very important assumption that cannot be relaxed as such.
Beta: The ratio of the covariance of returns on a security and the market portfolio to the variance of the market portfolio - a good proxy of the non-diversifiable portion of risk. A stock's return moves by beta times the change in the market portfolio's return.
Beta of the market portfolio: Always exactly 1, because the risk of the market portfolio is its covariance with itself, which is its own variance - making numerator and denominator identical in the beta formula.
Borrowing at the risk-free rate in practice: It is always possible to lend at the nominal risk-free rate by buying government T-bills, but not to borrow at that rate - an investor's borrowing rate is higher. However, assuming a higher borrowing rate does not change the general results of the theory.
Capital Asset Pricing Model: The first coherent framework for answering how the risk of an investment should affect its expected return, and the dominant model for valuation of risky assets and estimation of required rate of return. Developed concurrently in the early 1960s by Jack Treynor (1962), William Sharpe (1964), John Lintner (1965) and Jan Mossin (1966).
Capital Market Line: The line connecting the risk-free asset and the tangent on the efficient frontier at the market portfolio M. In the presence of a risk-free asset this line IS the efficient frontier, and all other risky portfolios lying below it are no longer desirable or efficient.
Capital Market Theory: Theory that builds on Modern Portfolio Theory and develops a model for pricing all risky assets, by introducing a risk-free asset while forming efficient portfolios of risky assets - which has significant implications for portfolio choices.
CAPM equation: E(Ri) = RFR + Beta(i) x (E(Rm) - RFR) - the required return equals the risk-free rate plus beta times the market risk premium. It defines risk in terms of the security's beta.
Correlation coefficient: Covariance standardised by dividing it by the product of the individual standard deviations. It varies from -1 to +1, where +1 is perfect positive correlation with returns moving together in a completely linear manner, and -1 is perfect negative correlation with returns moving in opposite directions.
Covariance: A measure of the degree to which two variables move together relative to their individual mean values.
Drivers of portfolio risk: Three variables affect portfolio risk - the weights of investments, the risk of the individual investments, and the co-movement between investments. Only the first two affect portfolio return.
Effect of leverage on the CML: Borrowing at the risk-free rate generates the market portfolio return on every unit invested including the borrowed portion, while paying only the risk-free rate on the borrowing - so the investor saves the difference between the market return and the risk-free return. The debt component is subtracted in the return computation, being a payment and not a receipt.
Efficient frontier: The umbrella-shaped curve representing the set of portfolios that provides the maximum rate of return for a given level of risk, or the minimum risk for a given level of return. A portfolio inside the curve is dominated on one or both tests.
Estimation risk: The potential source of error arising from estimating returns, risks and correlations. Since the output of the portfolio allocation depends on the accuracy of the statistical inputs, and correlations number (n² - n)/2 - 1,225 for a 50-security portfolio - the estimation burden is substantial.
Ex-ante return: The forward-looking expected rate of return, computed as the sum of the possible returns multiplied by their corresponding probabilities.
Expected return of a portfolio: The weighted average of the expected rates of return for the individual investments, where the weights are the proportions of the portfolio held in each asset.
Ex-post return: Returns which are calculated based on historical data, as distinct from the forward-looking ex-ante return.
Homogeneous expectations: The assumption that all investors estimate identical probability distributions for future rates of return. It can be relaxed - as long as differences in return expectations are not large, the effects are minor.
Identifying mispriced securities on the SML: Under market equilibrium all assets should plot on the SML. Any security with an estimated return plotting above the SML is undervalued, having a higher expected return than warranted by its market risk; any security plotting below is overvalued.
Infinite divisibility: The assumption that it is possible to buy or sell fractional shares of any asset or portfolio, which allows investment alternatives to be discussed as continuous curves. Changing it would have little impact on the theoretical models.
Lending and borrowing portfolios: Combinations to the left of point M on the CML represent lending at the risk-free rate and investing the balance in the market portfolio; combinations to the right represent borrowing at the risk-free rate and investing more than one's own wealth in the market portfolio.
Market portfolio: The most desirable risky portfolio, at the point of tangency between the efficient frontier and a line drawn from the risk-free rate. It consists of all risky assets - domestic and foreign stocks and bonds, real estate, coins, stamps, art, antiques and any other marketable risky asset worldwide.
Market proxy problem: The theoretical market portfolio cannot be constructed in practice, so proxies such as broad based equity market indices are used. There is no unanimity about which proxy to use, and an incorrect proxy affects both the beta risk measures and the position and slope of the SML.
Modern Portfolio Theory: The framework introduced by Harry Markowitz in "Portfolio Selection" (Journal of Finance, 1952) for constructing and selecting portfolios based on the expected performance of investments and the risk appetite of the investor. He received the Nobel Prize in Economics for it in 1990.
Optimum portfolio: A combination of investments having desirable individual risk-return characteristics for a given set of constraints, selected from the feasible combinations as the one meeting the investor's investment objectives.
Portfolio optimisation inputs: The expected return of every asset in the investment universe; the standard deviation of each asset's expected returns; and the correlation coefficients among the entire set, taken a pair at a time - plus any investor constraints.
Risk-free asset: An asset whose returns are certain, and therefore whose standard deviation is zero, which has zero correlation with all other risky assets, provides the risk-free rate of return, and lies on the vertical axis of a portfolio graph as it has no risk.
Risk of a portfolio containing a risk-free asset: Because the variance of a risk-free asset is zero and its covariance with any risky asset is also zero, portfolio risk reduces to the weight in the risky asset multiplied by the risk of the risky asset - a linear relationship, so such portfolios plot on a straight line.
Security Market Line: The graphical representation of CAPM, depicting the relationship between systematic risk and the expected rate of return on an asset traded on the capital market. Plotting a security's beta on the x-axis gives its expected return directly from the line.
Standard deviation as a risk measure: A statistical measure of the dispersion of returns around the expected value, expressed in the same units as the return itself - which is why the square root of variance is taken.
Variance-covariance matrix: The matrix required to compute portfolio risk where more than two securities are involved. Several methods exist for calculating it, and it can be easily computed on a spreadsheet or visual basic application.
Variance of return for a security: The sum over all states of the squared deviation of each possible return from the expected return, weighted by that state's probability. Its square root is the standard deviation, known as ex-ante risk; larger variance indicates greater dispersion and hence larger risk.
What MPT quantified: It quantified diversification by introducing the statistical notion of covariance or correlation between assets, demonstrated mathematically that the variance of the rate of return is a meaningful measure of portfolio risk, and derived the formula for computing portfolio variance.
Why only systematic risk is priced: When investors hold the market portfolio, which has only systematic risk, they need to be compensated only for bearing systematic risk - because unsystematic risk is diversified away, there is no need for any compensation for it.
Why the frontier curves: When the correlation between assets is less than perfect 1, the various weight combinations plot on a curve rather than a straight line - and that curvature is the diversification benefit made visible.

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Original explanatory writing based on the official NISM workbook for Series XIX-E, which is published and copyrighted by NISM. Check figures against the current edition at nism.ac.in.

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