Probability distribution of expected returns
Also written Probability distribution of returns · Expected return distribution · Scenario probability distribution
The list of returns an investment might deliver, each with the chance of it happening — the form in which Modern Portfolio Theory assumes an investor sees every investment choice.
In plain language
Nobody knows what a share will return next year. But a sensible investor does not therefore give up and guess.
Instead, the investor lists the outcomes that could happen. Then puts a chance against each one.
That list is a probability distribution of expected returns. It might say: a boom is 30% likely and pays 15%, a normal year is 50% likely and pays 10%, a recession is 20% likely and pays 2%.
This is how Modern Portfolio Theory assumes every investment is presented to an investor. Not as a single number, but as a spread of possible numbers with odds attached.
Two useful things fall straight out of the list. Weight each return by its chance and add them up, and you have the expected return. Look at how far the outcomes sit from that average, and you have the risk.
So one table gives both halves of the problem. That is why the theory starts here.
How it works
Where the idea sits in the theory. Section 14.2 lists the assumptions of Modern Portfolio Theory. The first is that an investor wants to maximise return for a given level of risk, which means that given two assets with equal rate of return, investors will select the asset with lower risk. The assumption continues: investors consider each investment alternative as being presented by a probability distribution of expected returns over some holding period.
The rest of the assumption set builds on the same distribution: investors maximise one-period expected utility, assign utility scores to portfolio choices, have utility curves showing diminishing marginal utility of wealth, estimate portfolio risk from the variability of expected returns of the constituent assets, and base decisions solely on expected return and expected variance or standard deviation.
Computing the expected return from the distribution (section 14.4). The expected rate of return of an individual investment opportunity is the sum of the potential returns multiplied with the corresponding probability of those returns. The workbook's own two stocks:
| State | Probability | Stock A | Stock B |
|---|---|---|---|
| Boom | 0.3 | 15% | 25% |
| Normal | 0.5 | 10% | 20% |
| Recession | 0.2 | 2% | 1% |
RA = 0.3(15%) + 0.5(10%) + 0.2(2%) = 9.9%
RB = 0.3(25%) + 0.5(20%) + 0.2(1%) = 17.7%
Computing the risk from the same distribution. Risk for an asset is defined as variability in return. The workbook names standard deviation of expected returns — the square root of the variance of expected returns — as one of the best known measures, and describes it as a statistical measure of the dispersion of returns around the expected value. A larger variance, and so a larger standard deviation, indicates greater dispersion and hence larger risk.
For a portfolio, the expected rate of return is the weighted average of the expected rates of return of the individual investments. The workbook's illustration weights four holdings at 0.2, 0.1, 0.3 and 0.4 with expected returns of 0.09, 0.12, 0.15 and 0.18, giving a portfolio expected return of 0.147.
A worked example
Take the workbook's Stock A and Stock B and put Rs 10,00,000 behind each.
| State | Probability | A returns | A value | B returns | B value |
|---|---|---|---|---|---|
| Boom | 0.3 | 15% | Rs 11,50,000 | 25% | Rs 12,50,000 |
| Normal | 0.5 | 10% | Rs 11,00,000 | 20% | Rs 12,00,000 |
| Recession | 0.2 | 2% | Rs 10,20,000 | 1% | Rs 10,10,000 |
Expected value of A = 0.3(11,50,000) + 0.5(11,00,000) + 0.2(10,20,000) = Rs 10,99,000, which is the 9.9% the workbook computes.
Expected value of B = 0.3(12,50,000) + 0.5(12,00,000) + 0.2(10,10,000) = Rs 11,77,000, the 17.7%.
B is expected to earn Rs 78,000 more. But look at the spread, not the average. A ranges from 2% to 15%, a band of 13 points. B ranges from 1% to 25%, a band of 24 points. In a recession B pays 1% while A pays 2%, so B is actually the worse of the two in the bad state.
Now the point of the whole exercise. Hold Rs 5,00,000 of each instead:
| State | Probability | Portfolio return | Portfolio value |
|---|---|---|---|
| Boom | 0.3 | (15 + 25) ÷ 2 = 20% | Rs 12,00,000 |
| Normal | 0.5 | (10 + 20) ÷ 2 = 15% | Rs 11,50,000 |
| Recession | 0.2 | (2 + 1) ÷ 2 = 1.5% | Rs 10,15,000 |
Expected return = 0.3(20) + 0.5(15) + 0.2(1.5) = 13.8%, which is exactly the weighted average of 9.9% and 17.7%. Returns add up like that. Risk does not — that is what correlation decides, and it is the next thing the chapter does.
Why NISM asks about it
Chapter 14 (Introduction to Modern Portfolio Theory), section 14.2 for the assumption and section 14.4 for the arithmetic. This is one of the most reliably computed items in the paper. Expect a three-state table of probabilities and returns and a demand for the expected return; expect the weighted-average version for a portfolio; and expect a conceptual question asking in what form MPT assumes an investor considers each investment alternative, where the answer is a probability distribution of expected returns over some holding period.
Common exam traps
- The probabilities must sum to 1. 0.3 + 0.5 + 0.2 = 1.0. If a question's states do not add up, something has been left out.
- Weight by probability, do not take a simple average. The simple average of 15, 10 and 2 is 9.0%; the workbook's answer is 9.9%, because the normal state carries half the weight.
- A higher expected return is not a better investment. Stock B expects 17.7% and still pays less than A in a recession. The distribution is there so you can see that.
- Expected returns are weighted averages; risk is not. Portfolio return is the weighted average of constituent returns, but portfolio variance depends on covariance too.
- Standard deviation is the square root of variance, and it is the dispersion around the expected value, not around zero.
- MPT assumes the investor sees the distribution, not the outcome. Questions sometimes offer realised past returns as the answer; the assumption is about forward-looking probabilities.
Where this is taught
Free preparation for NISM Series X-ARelated terms
- Standard deviationA measure of how far returns typically stray from their own average — the standard statistic for total risk, counting company-specific and market-wide causes alike.
- VarianceA measure of how widely a scheme's periodic returns fluctuate around its own average return — the square of standard deviation, and a risk measure relevant to both debt and equity schemes.
- CorrelationA measure of the strength and direction of the relationship between two variables, running from -1 to +1, and the single factor that determines how much risk diversification actually removes.
- Diminishing marginal utility of wealthThe idea, underlying Modern Portfolio Theory, that each additional rupee of wealth adds a little less satisfaction than the rupee before it — the reason most investors are risk averse.
- Expected utilityThe satisfaction score an investor is assumed to assign to each possible outcome of an investment, weighted by the probability of that outcome — the quantity Modern Portfolio Theory assumes investors actually maximise.
- Modern Portfolio TheoryMarkowitz's framework for building portfolios on expected return and risk together, in which the co-movement between holdings — not their individual riskiness — decides the risk of the whole.
- Utility scoreA single number that scores an investment for one particular investor: expected return minus a penalty for risk, scaled by that investor's own risk aversion.