No sensible investor puts all savings in one share. If that company fails, everything is lost; if it prospers, there is no safety net. But holding two, ten or fifty shares does not simply spread the risk like butter on bread.
The surprising result of portfolio theory is that the risk of a collection of investments can be much less than the average of the risks of its parts, provided the investments do not all move together. A farmer who grows both crops that thrive in wet years and crops that thrive in dry years has steadier income than one who grows either alone.
This topic explains how to measure the return and risk of a single investment (expected return, variance and standard deviation), how to calculate the return of a portfolio, what covariance and correlation measure, how the correlation between two investments determines the risk of a combined portfolio, and why diversification reduces risk, with worked calculations.
The next topic separates risk that diversification can remove from risk that it cannot, and introduces the efficient frontier.
Portfolio theory is the theoretical foundation of the capital asset pricing model, studied in the topics that follow. Read cost of equity: the dividend valuation model first.
In 30 seconds
Investors are risk averse: for the same expected return they prefer less risk. The expected return of an investment is the probability-weighted average of its possible returns. Its risk is measured by the variance (the probability-weighted squared deviations from the expected return) and the standard deviation (the square root of the variance).
A portfolio’s expected return is the weighted average of the returns of its assets, with weights by value. Portfolio risk is not a weighted average: for two assets, σp2 = wA2σA2 + wB2σB2 + 2wAwBρABσAσB. The correlation coefficient ρ (between −1 and +1) shows how the assets move together; covariance = ρσAσB. The lower the correlation, the greater the diversification benefit; at ρ = +1 there is none.
Return and risk of a single investment
Investors expect a return for providing funds and a reward for bearing risk: the higher the risk, the higher the return expected. The risk of an investment is the risk that the actual return differs from the expected return, and it is measured statistically.
The expected return is the average of the possible returns weighted by their probabilities. The variance is the probability-weighted average of the squared differences from the expected return, and the standard deviation is the square root of the variance. Example: an investment returns 22% in a boom (probability 0.3), 12% in normal conditions (0.5) and −6% in a slump (0.2).
| State | p | Return % | p × return | p(return − 11.4)2 |
|---|---|---|---|---|
| Boom | 0.3 | 22 | 6.6 | 33.708 |
| Normal | 0.5 | 12 | 6.0 | 0.180 |
| Slump | 0.2 | (6) | (1.2) | 60.552 |
| Total | 11.4 | 94.44 |
The expected return is 11.4%, the variance is 94.44 and the standard deviation is √94.44 = 9.72%. A larger standard deviation means more volatile returns and more risk. Variance and standard deviation are the usual measures. The assumptions of the theory are that investors are rational and risk averse, base decisions on the expected return and the standard deviation, have the same information, and prefer more return to less for a given risk.
Risk aversion means that an investor needs more expected return to accept more risk (as the purchase of insurance shows).
The return of a portfolio
The expected return of a portfolio is the weighted average of the expected returns of the assets, where the weights are the proportions of the total value invested in each asset (by value, not by number of shares): Rp = wARA + wBRB, with the weights adding to 1. An investor puts ₦6m in A (expected return 14%) and ₦4m in B (9%): the weights are 0.6 and 0.4, so Rp = 0.6 × 14 + 0.4 × 9 = 12.0%.
To find the weights needed for a target return, use wB = 1 − wA: for a target of 11%, 11 = 14wA + 9(1 − wA), so 5wA = 2 and wA = 0.4, wB = 0.6. (A weight above 1 or below 0 means borrowing or short selling, that is selling an asset one does not own.)
Covariance and correlation
Whether two assets reduce each other’s risk depends on how their returns move together. Covariance measures the joint movement: it is positive if the returns tend to move in the same direction, negative if in opposite directions. It is hard to interpret, so it is divided by the two standard deviations to give the correlation coefficient: ρ = covariance ÷ (σAσB), so that covariance = ρσAσB.
The coefficient lies between +1 (perfect positive correlation: the returns move exactly together), 0 (no relationship) and −1 (perfect negative correlation: the returns move exactly opposite). Asset A has σ = 20% and B has σ = 12%; if ρ = 0.5 the covariance is 0.5 × 20 × 12 = 120.
The risk of a two-asset portfolio
Unlike the return, the risk of a portfolio is not the weighted average of the risks, unless the assets are perfectly correlated:
σp2 = wA2σA2 + wB2σB2 + 2 wA wB ρAB σA σB
With wA = 0.6, σA = 20%, wB = 0.4 and σB = 12%, the first two terms are 0.36 × 400 = 144 and 0.16 × 144 = 23.04. With ρ = 0.5 the third term is 2 × 0.6 × 0.4 × 0.5 × 20 × 12 = 57.6, so the variance is 224.64 and the standard deviation 14.99%. The effect of different correlations:
| Correlation ρ | Portfolio standard deviation | Effect |
|---|---|---|
| +1.0 | 16.8% (0.6 × 20 + 0.4 × 12) | No diversification: the weighted average |
| +0.5 | 15.0% | Some reduction |
| 0 | 12.9% | Close to the less risky asset alone (12%) |
| −0.5 | 10.5% | Below both assets’ risks (20%, 12%) |
| −1.0 | 7.2% (20 × 0.6 − 12 × 0.4) | Greatest reduction; zero is possible with the right weights |
The expected return of the portfolio is 12.0% in every case; only the risk changes. So diversification reduces risk without reducing the expected return, provided the correlation is less than +1. With ρ = −1 the weights that give zero risk make wAσA = wBσB, which here is wA = 12 ÷ 32 = 0.375.
Why companies and investors diversify
Harry Markowitz showed that investors should look not at the risk of each security alone but at how it affects the risk of the whole portfolio, by choosing securities with low or negative covariance. A portfolio of many shares spreads the risk of each company’s own fortunes.
There is an argument that companies need not diversify their activities for their shareholders’ sake, since the shareholders can diversify themselves more cheaply by buying shares in several companies. A company may still diversify to protect employees, lenders and its own survival.
The ideas in more depth
Why risk is not additive
When one asset does badly, another may do well, and the bad and good results offset each other. The cross-product term in the formula is where this happens: a negative correlation makes it negative and cuts the variance; a positive correlation makes it positive and adds to it.
Only when all assets move perfectly together does the cross-product term make the portfolio’s risk equal to the average of the parts.
What diversification cannot do
Even a portfolio of many shares retains the risk that comes from movements in the whole market, because most shares are positively, if not perfectly, correlated with each other. That part of the risk cannot be diversified away and is the subject of the next topic. It is also the part for which investors are rewarded with extra return.
Sources and references
- No accounting standard governs portfolio theory, which derives from the work of Harry Markowitz (1952). The formula for the risk of a two-asset portfolio is not normally provided on the formula sheet, so learn it.
These points are stated as at 5 November 2026.
Worked examples
Example 1: portfolio return. ₦6m in A (14%) and ₦4m in B (9%). Required: the expected return.
- Weights. 0.6 and 0.4.
- Return.
0.6 × 14 + 0.4 × 9 = 12.0%.
Example 2: covariance. σA = 20%; σB = 12%; ρ = 0.5. Required: the covariance.
- Covariance.
0.5 × 20 × 12 = 120.
Example 3: portfolio risk. Weights 0.6 and 0.4; ρ = 0.5; σ = 20% and 12%. Required: the standard deviation.
- Variance.
144 + 23.04 + 57.6 = 224.64. - Standard deviation. √224.64 = 14.99%.
How to answer in the exam
- Work out the weights by value before calculating the return.
- Write out the variance formula and show the three terms.
- State the effect of the correlation on diversification, not only the number.
- Link to the next ideas: the risk left after diversification is the market risk.
Common mistakes in the exam
- Taking the portfolio’s standard deviation as the weighted average of the standard deviations.
- Using the number of shares as the weights instead of their values.
- Forgetting the factor 2 in the cross-product term.
- Forgetting to take the square root of the variance.
- Using percentages inconsistently (squared percentages against decimals).
- Saying that diversification lowers the expected return.
Quick revision
- Expected return = Σ probability × return; variance = Σ p(r − mean)2; standard deviation = √variance.
- Portfolio return = weighted average; weights by value.
- Covariance = ρσAσB; ρ between −1 and +1.
- Portfolio variance = wA2σA2 + wB2σB2 + 2wAwBρσAσB.
Practice questions
These are original questions written in the style of the Institute’s exams. Try each one before you open the answer. Section A is a recall check, because the paper itself is written.
Recall check: multiple choice
Question 1. The expected return of an investment is:
- The highest possible return
- The lowest possible return
- The probability-weighted average of the possible returns
- The most likely return only
Show the answer
C. The sum of probability times return.
Question 2. The standard deviation is:
- The square of the variance
- The square root of the variance
- The mean return
- The covariance
Show the answer
B. It is in the same units as the return.
Question 3. The weights of a portfolio are based on:
- The value invested in each asset
- The number of shares
- The share price
- The risk of each asset
Show the answer
A. Value, not quantity.
Question 4. A risk-averse investor:
- Prefers more risk
- Ignores risk
- Prefers high variance
- Prefers less risk for the same expected return
Show the answer
D. And needs more return to accept more risk.
Question 5. If the correlation between two assets is +1, diversification gives:
- The maximum risk reduction
- No reduction below the weighted average risk
- Zero risk
- A negative return
Show the answer
B. The assets move exactly together.
Question 6. With a correlation of −1, the portfolio risk:
- Can be reduced to zero with the right weights
- Is the weighted average
- Is always the higher of the two
- Cannot be reduced
Show the answer
A. The assets move exactly opposite.
Question 7. Covariance equals:
- σA + σB
- ρ ÷ (σAσB)
- ρ × σA × σB
- σA ÷ σB
Show the answer
C. So ρ = covariance ÷ (σAσB).
Question 8. Diversification reduces risk provided the correlation is:
- Exactly +1
- Zero only
- Positive only
- Less than +1
Show the answer
D. The lower, the greater the benefit.
Question 9. The expected return of a portfolio is:
- The lowest of the asset returns
- The highest of the asset returns
- The weighted average of the asset returns
- Reduced by diversification
Show the answer
C. Diversification changes risk, not this average.
Question 10. Markowitz showed that investors should choose securities with:
- Low or negative covariance with the rest of the portfolio
- The highest returns only
- The highest standard deviation
- The same industry
Show the answer
A. To reduce the risk of the whole portfolio.
Section B style: written questions
Question 11 (10 marks). (a) Define expected return, variance and standard deviation. (3 marks) (b) State the formula for the variance of a two-asset portfolio. (3 marks) (c) State the effect of the correlation coefficient on diversification. (4 marks)
Question 12 (15 marks). Asset X has an expected return of 16% and a standard deviation of 25%; Y has 8% and 10%. A portfolio is half in each and the correlation is 0.2. (a) Calculate the expected return. (3 marks) (b) Calculate the variance and standard deviation. (7 marks) (c) Calculate the weighted average of the standard deviations and explain the difference. (5 marks)
A further written question
Question 13 (20 marks). An investor holds ₦250,000 in P (expected return 9%), ₦400,000 in Q (12%) and ₦350,000 in R (15%).
(a) Calculate the weights and the expected return. (6 marks) (b) The investor wants 13% from Q and R only; calculate the weights. (5 marks) (c) A two-asset portfolio is 70% in an asset with standard deviation 18% and 30% in one with 9%; the correlation is −0.3. Calculate the variance and standard deviation. (6 marks) (d) State why this is below the weighted average of the standard deviations. (3 marks)
Model answers
Show the model answer to Question 11
(a) Expected return: the probability-weighted average return (1); variance: the probability-weighted squared deviations from the expected return (1); standard deviation: the square root of the variance (1).
(b) σp2 = wA2σA2 + wB2σB2 + 2wAwBρσAσB (3).
(c) At +1 there is no benefit; the lower the correlation the greater the reduction in risk; at −1 risk can be eliminated (4).
Show the model answer to Question 12
(a) 0.5 × 16 + 0.5 × 8 = 12% (3).
(b) 0.25 × 625 = 156.25; 0.25 × 100 = 25; 2 × 0.5 × 0.5 × 0.2 × 25 × 10 = 25 (4); variance 206.25 (1); standard deviation 14.36% (2).
(c)
- 0.5 × 25 + 0.5 × 10 = 17.5% (2).
- The portfolio risk is lower because the correlation is less than +1, so the assets’ fluctuations partly offset each other (3).
Show the model answer to Question 13
(a) Total ₦1,000,000; weights 0.25, 0.40, 0.35 (3); return 0.25 × 9 + 0.4 × 12 + 0.35 × 15 = 12.3% (3).
(b) 13 = 12wQ + 15(1 − wQ) gives 3wQ = 2, so wQ = 0.667 and wR = 0.333 (5).
(c) 0.49 × 324 = 158.76; 0.09 × 81 = 7.29; 2 × 0.7 × 0.3 × (−0.3) × 18 × 9 = −20.41; variance 145.64 (4); standard deviation 12.07% (2).
(d) The weighted average is 0.7 × 18 + 0.3 × 9 = 15.3%; the negative correlation makes the cross-product term negative, so the portfolio risk is lower (3).
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