Foundation — Markowitz, the assumptions of MPT, and how investors feel about risk

14.1 The old wisdom, and what was missing

Every Indian household already knows the rule. Do not put all your eggs in one basket. Your grandmother did not call it portfolio theory. She called it common sense. She kept some gold, some money in the post office, a little in a chit fund, and a plot of land in the village. If one failed, the others would hold.

That is diversification — spreading investments across different areas or asset classes to reduce risk, because every asset will not behave similarly at all times.

Here is the problem. Before 1950, the investing community knew the benefits of diversification but had no way of quantifying them. Everybody agreed that holding ten stocks was safer than holding one. Nobody could say how much safer. Nobody could say whether ten cement stocks were as good as ten stocks spread across cement, software, banking and pharma. Everybody felt it. Nobody could measure it.

It is like a cook who knows that adding tamarind balances sweetness, but has no measuring spoon. The dish comes out differently every day.

The 1952 paper that changed everything

In 1952 The Journal of Finance published an article titled "Portfolio Selection", authored by Harry Markowitz. It ran to a few pages. It changed the whole of investment management.

Decades later, in 1990, Harry Markowitz was honoured with the Nobel Prize in Economics for his portfolio theory. Thirty-eight years is a long wait, and it tells you how slowly the profession absorbed the idea.

The ideas in that article became the foundation of what we now call Modern Portfolio Theory (MPT).

Read that sentence twice, because it is the whole chapter in one line. MPT is a framework. It does two things — construct portfolios and select among them. It uses two inputs — expected performance of the investments and risk appetite of the investor.

What exactly did Markowitz add?

Three things, and the exam likes all three.

One — he quantified diversification. MPT quantified the concept of diversification by introducing the statistical notion of covariance, or correlation, between investment assets. Until then diversification was a proverb. Markowitz turned it into a number. Correlation tells you whether two investments tend to move together, move oppositely, or move independently. That single number is what separates real diversification from fake diversification.

Ten cement stocks are highly correlated with one another. When the construction cycle turns, all ten fall together. You held ten baskets, but all ten were tied to the same bullock cart. Correlation is the tool that exposes this.

Two — he showed that variance of the rate of return is a meaningful measure of portfolio risk. Harry Markowitz mathematically demonstrated that the variance of the rate of return is a meaningful measure of portfolio risk. Before this, "risk" was a word people used loosely — sometimes meaning the chance of losing money, sometimes meaning the company being weak, sometimes just a feeling in the stomach. Markowitz pinned it down. Risk equals variability of return. Variability can be measured. Variance is that measure.

Three — he derived the portfolio variance formula. He derived the formula for computing the variance of a portfolio, showing how to effectively diversify a portfolio. Note the word effectively. Anybody can diversify carelessly. The formula shows which combinations actually cut risk and which merely add names to the statement.

14.2 The assumptions of the theory

Every theory stands on assumptions. If the assumptions are shaky, the conclusions wobble. The workbook lists five, and the exam has asked about them directly.

Assumption 1 — An investor wants to maximise return for a given level of risk. Stated the other way round, given a choice between two assets with equal rate of return, investors will select the asset with lower risk. Investors are also assumed to consider each investment alternative as being represented by a probability distribution of expected returns over some holding period.

That last clause matters. The investor is not thinking "this share will give 14 per cent". The investor is thinking "there is a 30 per cent chance of 25 per cent, a 50 per cent chance of 12 per cent, and a 20 per cent chance of minus 5 per cent". A whole distribution, not a single number.

Assumption 2 — Investors maximise one-period expected utility. Investors choose an action or event with the maximum expected utility, and they assign utility scores to the various portfolio choices available to them.

Utility is simply a satisfaction score. Give the same lakh rupees to two people and the satisfaction differs. MPT assumes each investor can score every portfolio and will pick the highest score. Note the words one-period — the framework as taught here looks at a single holding period, not at a lifetime of rolling decisions.

Assumption 3 — Utility curves demonstrate diminishing marginal utility of wealth. Though investors have different rates of decrease in the marginal utility of wealth, the principle is that per rupee increment to utility decreases with wealth.

This is the most human assumption in the list. Your first ten lakh rupees changes your life. Your hundredth ten lakh rupees changes your weekend. The extra satisfaction from each additional rupee gets smaller as you get richer. Different people slow down at different rates — but everybody slows down.

Assumption 4 — Investors estimate the risk of the portfolio on the basis of the variability of expected returns of constituent assets. Risk is not credit rating, not brand name, not the promoter's reputation. Risk is variability of expected returns.

Assumption 5 — Investors base decisions solely on expected return and risk. Therefore their utility curves are a function of expected return and the expected variance (or standard deviation) of returns only.

That word solely is where the theory takes its biggest bite. In real life a Gujarati family may refuse to invest in a liquor company however good the numbers. A trustee may be barred from unrated paper. A promoter may hold his own company's shares out of pride. MPT sets all of that aside and says: only two numbers matter, expected return and variance.

14.3 Risk premium and the fair game

The presence of risk means more than one outcome is possible. If only one outcome is possible, you are looking at a certainty, not an investment decision.

Because more than one outcome is possible, investors demand a risk premium for bearing the risk. The question whether a given risk premium is adequate is a central theme in investment management. In fact, measurement of risk and determination of risk premium is an important area of work in finance.

Now the key definition:

A fair game is a bet with no edge. Toss a coin — heads you gain one thousand, tails you lose one thousand. On average you end where you started. Expected gain is zero. There is risk, but there is no premium for bearing it.

The three investor types

Risk averse investor. Risk averse investors reject a fair game. They will not take risk unless they are paid for it. They will invest in risk-free investment opportunities or in investment opportunities with positive expected risk premium. And — the greater the risk, the greater the demand for risk premium.

Risk neutral investor. A risk neutral investor evaluates investment opportunities solely on the basis of expected return with no regard to risk. The amount of risk is irrelevant to the risk neutral person. He provides no penalty for risk. Show him two portfolios both expected to return 12 per cent — one a government bond, one a penny stock — and he is indifferent.

Risk seeking investor. A risk seeking investor is the one who will engage in a fair game. Where the risk averse investor makes a downward adjustment for utility, the risk seeker makes an upward adjustment for utility. He enjoys the ride itself. The thrill is worth something to him, so he will accept a bet with zero premium and sometimes even a negative one. This is the man buying a lottery ticket, where the expected value is clearly below the price.

Utility scores — how the theory scores portfolios

Investors assign utility scores to competing portfolios given their expected return and the risk involved.

The rule is simple and symmetric:

• Higher utility scores are assigned to portfolios with higher expected return.
• Lower utility scores are assigned to portfolios with higher risk.
• Higher utility scores are assigned to those portfolios with a better risk-return profile.

So return pushes the score up and risk pulls it down. How hard risk pulls depends on how risk averse the investor is.

Certainty Equivalent Rate — the single most examinable idea in 14.3

Investors are assumed to be comparing the utility value of the risky portfolio with a risk free investment offering the same utility, to determine the Certainty Equivalent Rate (CER).

Put it as a question you ask yourself. A risky portfolio is expected to give 14 per cent. What guaranteed, no-risk rate would make me equally happy, so that I genuinely cannot choose between the two? If your answer is 8 per cent, your CER for that portfolio is 8 per cent.

Now the decision rule, and the exam loves this comparison:

• A highly risk averse investor may assign a risky portfolio a CER below the risk-free rate of return, and will reject the risky portfolio. If a bank fixed deposit gives 7 per cent guaranteed, and your CER for the risky portfolio is only 5 per cent, why take the risk? The deposit wins.
• A less risk averse investor may assign a higher CER and accept the same risky portfolio. Same portfolio, same numbers, different person, opposite decision.
• For a risk neutral investor, the portfolio CER is the expected rate of return on the risky portfolio. No haircut at all. Expected return is 14 per cent, so CER is 14 per cent.

That last line is worth memorising in those exact words. The risk neutral investor applies no discount, so his CER equals the expected return.

The picture so far

Markowitz gave the profession a measuring spoon. Risk became variance. Diversification became correlation. And an investor's temperament became a utility score, summarised in a single number called the certainty equivalent rate. From the next lesson, we put actual figures into the formulas.