NISM Series XIX-C · Chapter 4: Introduction to Modern Portfolio Theory

L1 · Foundation

Foundation — Markowitz, the assumptions, and utility

What Modern Portfolio Theory added to an old proverb, its five assumptions, the meaning of risk aversion, and the utility score that distinguishes one investor from another.

17 min read · Worth 1% of the paper · Written from pages 53–62 of the official NISM workbook · Updated 18 September 2026

4.1 What Markowitz actually contributed

Conventional wisdom has always dictated not putting all the eggs in one basket. This adage is addressing the benefits of diversification. Prior to 1950, investing community was familiar with the benefits of holding a diversified portfolio. However, they had no way of quantifying the benefits of diversification.

Read that last sentence again, because it is the whole point. Diversification was not discovered in 1952 — every grandmother knew it. What was missing was a way to measure it.

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 introduced in this article have come to form the foundations of what is now popularly referred as Modern Portfolio Theory (MPT).

The names and dates to memorise: Harry Markowitz, "Portfolio Selection", Journal of Finance, 1952; Nobel Prize in Economics, 1990.

What MPT 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.

How it did it:

MPT quantified the concept of diversification by introducing the statistical notion of covariance, or correlation between investment assets. Harry 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.

Three contributions in one sentence: he introduced covariance or correlation as the tool, showed that variance of the rate of return is a meaningful measure of portfolio risk, and derived the formula for computing the variance of a portfolio.

4.2 The five assumptions

Some of the assumptions of the Modern Portfolio Theory (MPT) are as under:

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.
Investors consider each investment alternative as being presented by a probability distribution of expected returns over some holding period.
Investors maximise one-period expected utility. Investors choose an action or event with the maximum expected utility. Investors assign utility scores to the various portfolio choices available to them.
Investors base decisions solely on expected return and risk.
Investors estimate the risk of the portfolio on the basis of the variability of expected returns of constituent assets.

Assumption 4 is the one most worth noticing — solely on expected return and risk. Nothing about taxes, liquidity needs, sentiment or the investor's opinion of the management. It is a deliberate simplification, and every criticism of MPT starts here.

Assumption 2 is the second-most important: an investment is not a number but a probability distribution of expected returns over some holding period. That is what makes variance meaningful at all.

4.3 Risk averse, risk seeking and risk neutral

The presence of risk means that more than one outcome is possible. Investors demand risk premium for bearing the risk. The question whether a given risk premium is adequate is a central theme in investment management. Measurement of risk and determination of risk premium is an important area of work in finance.

Risk averse investors will invest in risk-free investment opportunities or in investment opportunities with positive expected risk premium. The greater the risk, greater the demand for risk premium.

That defines the three types by how they respond to risk:

Investor type	Behaviour
Risk averse	Invests in risk-free opportunities or those with a positive expected risk premium; the greater the risk, greater the demand for risk premium
Risk neutral	Indifferent to risk — judges only by expected return, demanding no premium
Risk seeking	Willing to accept a lower expected return for the chance of a large gain

Sample question 5 asks who expects higher return for higher risk — that is the risk averse investor. This surprises candidates, who assume that demanding more return sounds aggressive. It is the opposite: only someone who dislikes risk needs to be paid extra to bear it. A risk seeker would take the gamble without additional compensation.

The utility score

Investors assign utility score to competing portfolios given their expected return and risk involved. Utility scores are assigned to those portfolios for their risk-return profile. Higher utility scores are assigned to portfolios with higher expected return and lower risk, and lower utility scores are assigned to portfolios with higher risk and lower returns.

The formula:

U = E(r) − 0.5 × A × σ²

Where: U is the utility value, A is an index of investor's risk aversion. σ² is the variance of the returns of the investment.

The worked example

Say for instance an investor has a risk aversion index of 3 and is evaluating an investment alternative that generates an expected return of 15%, with a variance of 10%. The ongoing risk free rate of return is 8%.

U = 0.15 - (0.5 * 3 * 0.10 * 0.10) = 0.135 or 13.5%

Now typically an investor would compare this Utility with the risk free rate of return, because that is the comparable risk free investment. In the above case we have assumed that the ongoing risk free rate of return is 8%, and therefore the excess Utility is 13.5% - 8% = 5.5%. Because Utility of the investment is greater than the risk free rate of return, the investor should choose this investment.

What the formula is doing

It may be noted that the above equation supports the thought process that utility is enhanced by expected return and reduced by risk. The return generated by a particular investment alternative is penalised for the risk involved according to the unique risk aversion of the investor. Therefore, Utility is indirectly measuring the risk free return of that investment. So, logically, if Utility is greater than the risk free rate, the investment is worthwhile investing.

Read the formula as a subtraction. You start with the expected return, and then you penalise it for the risk involved according to the unique risk aversion of the investor. What remains is the certain return that would make this investor equally happy — which is why comparing it with the risk free rate is the right test.

Note also that a higher A produces a bigger penalty. The same investment yields a different utility for a cautious investor and a bold one.

Why utility is not the Sharpe ratio

It is interesting to note that Utility is not like risk reward ratio or Sharpe Ratio, which also measures the excess return of a portfolio over its risk. In the risk to reward ratio or Sharpe Ratio, there is no involvement of investors' risk aversion. It goes by the assumption that every investor is similar and their risk perceptions about an investment are homogenous. Therefore, higher the Sharpe Ratio better the investment is.

But in the practical world, a good portfolio or investment is not equally attractive to all the investors. What explains this heterogeneity? The answer to this question is Utility that explains that it is the unique individual risk aversion, which makes investors differ in their choices.

This is an excellent distinction and a likely examination point:

	Sharpe Ratio	Utility
Investor's risk aversion	No involvement	A is an index of investor's risk aversion
Assumption	every investor is similar and their risk perceptions... are homogenous	Investors differ
Reading	higher the Sharpe Ratio better the investment	Compare against the risk free rate; higher utility is better for that investor

The Sharpe ratio ranks investments the same way for everybody. Utility explains why two sensible people, looking at the same fund, reach opposite conclusions.

Think of it like this

The grandmother knew; the accountant measured

Every grandmother in India has said not to keep all the eggs in one basket. None of them could tell you how many baskets, which baskets, or by exactly how much the risk falls when you use two instead of one.

That gap is what Markowitz closed. Prior to 1950, investing community was familiar with the benefits of holding a diversified portfolio. However, they had no way of quantifying the benefits of diversification.

His contribution was arithmetic, not wisdom: MPT quantified the concept of diversification by introducing the statistical notion of covariance, or correlation between investment assets. Harry 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.

The paper was "Portfolio Selection" in the Journal of Finance in 1952; the Nobel Prize came in 1990.

The umbrella shop and the ice-cream cart

A man who owns two umbrella shops has a bad summer and a bad summer again. A man who owns one umbrella shop and one ice-cream cart has an ordinary summer every summer — the rain that fills one shop empties the other.

That is negative correlation, and it is the engine of the whole theory. With low (less than +1 correlation) or negative correlation it is possible to derive portfolios that have lower risk than either of the assets. This is the essence of diversification. As long as the correlation is less than perfect 1, benefits of diversification occurs. Lower the correlation, higher the benefits of diversification.

At the extreme, the two businesses cancel out exactly: when the correlation between the two assets is -1, the complete benefits of diversification is realized. The negative correlation term is completely offsetting the combined variances terms. Hence, a unique combination of such perfectly negatively correlated securities could yield a zero risk portfolio.

Two shops selling the same goods on the same street

Buying a second shop on the same street, selling the same goods to the same customers, does not spread your risk at all. When trade is bad it is bad in both. You have doubled the size of the business and changed nothing about its safety.

That is perfect positive correlation. If two securities are perfectly correlated, all the possible combinations of these two assets are represented by a straight line... The graph in Exhibit 4.1 clearly depicts that there was no possibility to create a portfolio, that generates a higher return than that of security B, or a lower risk than that of security A. Indirectly conveying that there is no benefit due to combining them in any proportion.

And the workbook flags its importance: It also means that there was no possibility for risk diversification when two assets having perfect positive correlation between them are combined to form a portfolio. This is a very fundamental understanding in the Modern Portfolio Theory.

The same wedding hall, judged by a nervous father and a confident one

Two fathers are offered the same hall at the same price, with the same small chance that the caterer will let them down. One accepts happily. The other pays extra for a hall with a guaranteed caterer. Neither is being foolish — they simply price the possibility of disaster differently.

That is exactly what the utility formula captures. U = E(r) − 0.5 × A × σ², where A is an index of investor's risk aversion. The same investment, the same expected return, the same variance — and a different answer, because The return generated by a particular investment alternative is penalised for the risk involved according to the unique risk aversion of the investor.

And this is precisely where utility parts company with the Sharpe ratio: In the risk to reward ratio or Sharpe Ratio, there is no involvement of investors' risk aversion. It goes by the assumption that every investor is similar... But in the practical world, a good portfolio or investment is not equally attractive to all the investors.

The best price in the bazaar, once you fix the quality

Walk through a cloth bazaar and for every quality of cotton there is one shop selling it cheapest. Join those shops together and you have the best line available in the bazaar. Any shop not on that line is charging more for the same cloth, or selling worse cloth at the same price — and no sensible buyer goes there.

That line is the efficient frontier: The efficient frontier represents set of portfolios that provides maximum rate of return for a given level of risk or the minimum risk for a given level of return.

The workbook's own illustration shows both ways of failing the test: portfolio A stands on efficient frontier as against portfolio C, because portfolio C has the same rate of return of portfolio A but much higher risk. Similarly, portfolio B also stands on the efficient frontier, because it has a higher return than C, with the same risk. Portfolio C is the shop nobody should visit.

Fifty guests, and every pair who might quarrel

Seating twenty guests at a wedding, you must consider not just each guest but every pair who might quarrel — 190 pairs. Add thirty more guests and the pairs jump to 1,225. The work does not grow with the guests; it grows with the pairs.

That is the estimation problem in MPT. The manager must estimate the expected return of every asset class, the standard deviation of each asset's expected returns, and the correlation coefficients among the entire set... taking a pair of them at a time. The footnote gives the formula: (n²−n)/2, so for a portfolio of 50 securities, the number of correlation estimates is 1225.

And every one of those 1,225 numbers is a guess: the output of the portfolio allocation depend on the accuracy of the statistical inputs... The potential source of error that arises from these estimations is referred to as estimation risk.

Key terms in this chapter

Correlation Coefficient: Covariance standardised by dividing by the product of the two standard deviations, so that it always lies between −1 and +1. A value of +1 means returns move together in a completely linear manner; −1 means they move in opposite directions.
Covariance: A measure of the degree to which two variables move together relative to their individual mean values. It is unstandardised, so its magnitude cannot be compared meaningfully across different pairs of assets.
Efficient Frontier: The umbrella-shaped curve of portfolios providing the maximum rate of return for a given level of risk, or the minimum risk for a given level of return. Portfolios below it are dominated — offering the same return with more risk, or the same risk with less return.
Estimation Risk: The potential source of error arising from having to estimate expected returns, standard deviations and pairwise correlations. The correlation count is (n²−n)/2, so a 50-security portfolio needs 1,225 correlation estimates and the output is only as good as those inputs.
Ex-Ante and Ex-Post Returns: Ex-ante return is the forward-looking expected return computed as the sum of possible returns weighted by their probabilities. Ex-post returns are those calculated from historical data.
Expected Portfolio Return: The weighted average of the expected returns of the individual investments, each weight being that asset's proportion of portfolio value. Unlike portfolio risk, it involves no term for co-movement.
Modern Portfolio Theory (MPT): The framework introduced by Harry Markowitz in "Portfolio Selection" (Journal of Finance, 1952) for constructing and selecting portfolios based on expected performance and the investor's risk appetite. 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, chosen from the feasible combinations as the one meeting the investor's investment objectives.
Perfect Negative Correlation: At a correlation of −1 the complete benefits of diversification are realised, the negative correlation term completely offsetting the combined variance terms, so that one unique combination of weights yields a zero risk portfolio.
Perfect Positive Correlation: At a correlation of +1 all combinations of two securities lie on a straight line, no combination beats either security on both dimensions, and there is no possibility of risk diversification — described by the workbook as a very fundamental understanding in MPT.
Portfolio Risk Variables: Three things determine a portfolio's risk — the weights of investments, the risk of the individual investments, and the co-movement between them. The third has no counterpart in the return formula, which is why portfolio risk is not a simple weighted average.
Risk Averse Investor: One who invests only in risk-free opportunities or those offering a positive expected risk premium, and whose demand for risk premium rises with risk. An investor who expects higher return for higher risk is risk averse, not risk seeking.
Risk Aversion Index (A): The personal parameter in the utility formula by which an investment's return is penalised for its risk. A higher index means a larger penalty, so the same investment yields different utility for different investors.
Utility Score: A measure assigning higher scores to portfolios with higher expected return and lower risk, computed as U = E(r) − 0.5 × A × σ², where A is an index of the investor's risk aversion. If utility exceeds the risk-free rate, the investment is worthwhile.
Utility versus Sharpe Ratio: The Sharpe ratio measures excess return over risk without any involvement of the investor's risk aversion, assuming all investors are alike. Utility incorporates individual risk aversion, which is what explains why the same portfolio is not equally attractive to everyone.
Variance-Covariance Matrix: The table of variances and pairwise covariances required to compute the risk of a portfolio containing more than two securities, calculable on a spreadsheet or through a visual basic application.
Variance of a Security: The probability-weighted sum of squared deviations from the expected return, Σ [Ri − E(Ri)]² × Pi. Its square root is the standard deviation, or ex-ante risk, and a larger value indicates greater dispersion and hence larger risk.

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

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