Two shares each carry risk. Put them in one portfolio and the risk of the pair can be far lower than the risk of either, as long as they do not move together. An umbrella seller and an ice-cream seller do well in different weather. The company that owns both has steadier income than either alone.
This idea, diversification, is the foundation of modern finance, and it explains why shareholders care about only part of a share’s risk.
This topic covers risk and return, measuring risk as variance and standard deviation, the return on a share, the expected return and risk of a two-asset portfolio, covariance and correlation, systematic and unsystematic risk, the efficient frontier, indifference curves, the market portfolio and the capital market line. The previous topic covered the cost of capital. The next covers the capital asset pricing model. The May 2014 Pathfinder paper tested efficient portfolios and correlation.
In 30 seconds
Risk is the volatility of returns, measured by variance = Σp(r − mean)2 and its square root, the standard deviation. A portfolio’s return is the weighted average of its assets’ returns. Its risk is lower than the weighted average unless the correlation is +1: σp = √[σA2x2 + σB2(1 − x)2 + 2x(1 − x)ρσAσB]. Correlation runs from +1 to −1. Unsystematic risk can be diversified away; systematic (market) risk cannot.
Efficient portfolios give the most return for their risk and lie on the efficient frontier. The tangent from the risk-free rate touches it at the market portfolio, and the line is the capital market line.
Key terms in plain English
| Term | What it means |
|---|---|
| Variance | The probability-weighted average of squared deviations from the expected return |
| Standard deviation | The square root of the variance; a measure of volatility |
| Covariance | A measure of how two returns move together |
| Correlation coefficient | Covariance scaled to lie between −1 and +1 |
| Systematic risk | Market-wide risk that cannot be diversified away |
| Unsystematic risk | Risk specific to one security that diversification removes |
| Efficient frontier | The set of portfolios offering the best return for each level of risk |
| Capital market line | The line of combinations of the risk-free asset and the market portfolio |
Syllabus check
This topic is in syllabus area C (financing decisions). It covers:
- risk measured as a variance and the two-asset portfolio;
- systematic and unsystematic risk;
- portfolio theory, the efficient frontier and the capital market line.
CAMA 2020 and portfolio risk
The Companies and Allied Matters Act 2020 (CAMA 2020) is the company law of Nigeria.
| Section | What it says | Link to this topic |
|---|---|---|
| s. 305 | A director must act in good faith in the best interests of the company | Risk should be managed for the benefit of the company and its investors |
| s. 308 | A director must use the care, diligence and skill of a reasonably prudent director | Directors should understand the risk of what the company holds |
| s. 377(1) and s. 378 | Directors prepare statements that follow IFRS and give a true and fair view | Decisions later affect reported assets, profit and cash |
| s. 385 | The directors’ report covers performance and risk | Major plans and risks are reported to shareholders |
Risk and return
Investors want a return as a reward for tying up funds, plus a return for risk. The higher the risk, the higher the expected return they require. Risk for bonds is default and changes in interest rates (higher rates lower bond prices). For shares it is liquidation and, above all, fluctuating profits and dividends. In an exam you may be told to treat government debt in domestic currency as risk-free.
Measuring risk
Variance = Σp(r − r̅)2, where p is the probability of return r and r̅ the expected return. The standard deviation (σ) is its square root. In ICAN’s example, returns of 2%, 6% and 8% with probabilities 0.2, 0.5 and 0.3 give an expected return of 5.8% and a variance of 0.2(−3.8)2 + 0.5(0.2)2 + 0.3(2.2)2 = 2.888 + 0.02 + 1.452 = 4.36, so a standard deviation of about 2.1%.
Return on a share
R = [(end value − start value) + dividend] ÷ start value. A share bought at ₦50, now worth ₦55 after a ₦2 dividend, gives (5 + 2) ÷ 50 = 14%: a capital gain yield of 10% plus a dividend yield of 4%.
The two-asset portfolio
Expected return: Rp = xRA + (1 − x)RB. It is a weighted average. For ₦1m split 75% in A (expected return 20%) and 25% in B (30%), Rp = 22.5%. To reach a target return, solve for x. ICAN shows that a 28% target needs 20% in A and 80% in B, and a 36% target needs borrowing 60% of funds and investing 160% in B.
Risk: σp = √[σA2x2 + σB2(1 − x)2 + 2x(1 − x)ρABσAσB], where ρ is the correlation. This formula is on the exam formula sheet. Take a 70% : 30% portfolio of A (return 7%, σ 2.19%) and B (9%, 4.1%). The return is 7.6%. The risk by correlation: +1: 2.763% (the simple weighted average); 0: 1.965%; −0.5: 1.406%; −1: 0.303%. So only perfect positive correlation gives no diversification gain.
Even a correlation of zero can lift return and cut risk at once, which is why investors look for investments with low or negative correlation.
Covariance and correlation
Covariance = Σp(Rx − R̅x)(Ry − R̅y), and ρ = covariance ÷ (σxσy). ICAN’s shares X and Y (returns in three states with probabilities 0.3, 0.5, 0.2): X has mean 20.9% and σ 4.64%; Y has mean 20.4% and σ 5.10%; covariance 17.64; correlation 17.64 ÷ (4.64 × 5.10) = about 0.75.
Using portfolio risk and return
Compare the portfolio with and without a new investment. Accept if return rises and risk falls, or one improves while the other stays the same. Reject if return falls and risk rises. If return and risk rise together, or both fall, it depends on whether the change in one is adequate compensation for the other.
Systematic and unsystematic risk
Diversification cannot remove all risk. Unsystematic risk belongs to one security and can be removed by holding many, because good and bad results cancel out. Systematic risk is market risk that affects all investments, such as a recession, and cannot be diversified away. In a well-diversified portfolio unsystematic risk is zero, so investors should not demand extra return for it.
They should be paid only for systematic risk. This leads to CAPM in the next topic.
The efficient frontier and the market portfolio
With perfect positive correlation, the possible portfolios of two assets form a straight line. With perfect negative correlation, they form two straight lines meeting at a risk-free point, and only the upper line is efficient. With other correlations, they form a curve. With many assets, the portfolios form an area. The top edge is the efficient frontier.
A portfolio on it is mean-variance efficient: it gives the highest return for its risk or the lowest risk for its return.
Indifference curves show combinations of risk and return that are equally attractive to an investor. A curve further up and to the left is better. Where the best indifference curve touches the efficient frontier is the investor’s preferred portfolio. In theory the preferred portfolio is the same for everyone, the market portfolio (M): all traded investments held in proportion to their market values, excluding the risk-free asset.
The capital market line
Investors can mix the risk-free asset with M. The straight line from the risk-free return, tangent to the frontier at M, is the capital market line (CML). Points before M mean lending at the risk-free rate; points beyond M mean borrowing at the risk-free rate to invest more in M. ICAN’s example: one third in a 9% risk-free asset and two thirds in a market portfolio returning 15% with σ 12%.
Return = 9% × 1/3 + 15% × 2/3 = 13%; risk = 12% × 2/3 = 8%. To earn 18%, an investor must borrow half his capital: return = 15% × 1.5 − 9% × 0.5 = 18%, risk = 12% × 1.5 = 18%.
The rules in more depth
Why risk falls but return does not
The expected return is a simple average, so it cannot be diversified. Risk depends on how the returns interact, so it can be reduced.
Why the correlation matters more than the number of assets
Adding assets that move together adds little protection. Adding assets that move differently adds a lot.
Why only systematic risk is rewarded
Investors can remove the rest at no cost. A market that paid a premium for avoidable risk would attract diversifiers until the premium disappeared.
Why a company may not need to diversify
Shareholders can diversify themselves, so a company diversifying for risk reasons adds little for them.
Effect on analysis
State the portfolio’s return and risk, compare with the alternative, comment on the correlation and say whether it is efficient.
Worked examples
Example 1: variance. Returns of 4%, 10% and 16% have probabilities 0.3, 0.4 and 0.3.
- Mean = 1.2 + 4.0 + 4.8 = 10%.
- Variance = 0.3 × 36 + 0.4 × 0 + 0.3 × 36 = 21.6.
- Standard deviation = √21.6 = 4.65%.
Example 2: a 60 : 40 portfolio. A returns 8% with σ 10%. B returns 14% with σ 20%. The correlation is 0.2.
- Return = 0.6 × 8 + 0.4 × 14 = 10.4%.
- Variance = 0.36 × 100 + 0.16 × 400 + 2 × 0.6 × 0.4 × 0.2 × 10 × 20 = 36 + 64 + 19.2 = 119.2.
- σ = 10.9%, below the weighted average of 14%.
Example 3: the capital market line. The risk-free rate is 5%. The market portfolio returns 12% with σ 10%. An investor wants a risk of 15%.
- Weight in the market.
15 ÷ 10 = 1.5, so he borrows 0.5 of his capital. - Return = 1.5 × 12% − 0.5 × 5% = 15.5%.
- Check.
5% + (7% ÷ 10%) × 15% = 15.5%.
Example 4: ICAN’s choice table. Portfolios (return %, risk %): A (6.0, 4.0); B (7.0, 3.0); C (9.0, 5.5); D (8.0, 6.0).
- B beats A: higher return and lower risk.
- C beats D: higher return and lower risk.
- B against C is undecided: C offers more return for more risk.
How to score full marks
- Write the formula and then substitute.
- Compute return and risk for the stated correlation.
- Compare with the weighted average to show the diversification effect.
- Link the correlation to the size of the benefit.
- Name systematic and unsystematic risk.
- Explain the CML or the efficient frontier if asked.
Common mistakes in the exam
- Forgetting to square the weights in the variance formula.
- Confusing variance and standard deviation.
- Averaging the standard deviations.
- Forgetting that ρ = +1 gives no gain.
- Mixing percentages and decimals.
- Claiming diversification removes all risk.
Quick revision
- Variance = Σp(r − mean)2; σ = √variance.
- Portfolio return = weighted average; risk uses ρ.
- ρ = +1: no gain; −1: risk can reach zero.
- Only systematic risk is rewarded.
- Efficient frontier → market portfolio → capital market line.
- CAMA 2020: ss. 305, 308, 377(1), 378, 385.
Practice questions
These are original questions written in the style of the Institute of Chartered Accountants of Nigeria (ICAN) exam. Try each one before you open the answer. Full model answers follow the written questions.
Recall check: multiple choice
Recent ICAN papers are written, although older papers such as May 2014 included multiple-choice questions. These questions are for quick recall. The written questions that follow are closer to the exam.
Question 1. Risk in portfolio theory is measured by:
- Standard deviation of returns
- Dividend yield
- Tax rate
- Share price
Show the answer
A. The volatility of returns.
Question 2. The expected return of a portfolio is:
- The weighted average of asset returns
- Always lower than any asset
- The sum of the returns
- The lowest return
Show the answer
A. Weighted by proportions.
Question 3. If ρ = +1, portfolio risk equals:
- The weighted average of the standard deviations
- Zero
- The larger standard deviation squared
- The variance
Show the answer
A. No diversification gain.
Question 4. The measure of how two returns move together is:
- Covariance
- Variance
- Range
- Mean
Show the answer
A. Scaled to give the correlation.
Question 5. Correlation ranges from:
- −1 to +1
- 0 to 100
- −5 to +5
- 0 to 1
Show the answer
A. Perfect negative to perfect positive.
Question 6. Risk that diversification can remove is:
- Systematic
- Unsystematic
- Market
- Interest rate only
Show the answer
B. It is specific to one security.
Question 7. The efficient frontier shows portfolios with:
- The highest return for a given risk
- The lowest return
- Zero risk
- No correlation
Show the answer
A. Or the lowest risk for a given return.
Question 8. The market portfolio is where:
- The best indifference curve touches the efficient frontier
- Return is zero
- Risk is zero
- Debt is highest
Show the answer
A. It is the tangent point.
Question 9. The capital market line combines:
- The risk-free asset and the market portfolio
- Two shares
- Debt and equity
- Two markets
Show the answer
A. A straight line.
Question 10. Which section of CAMA 2020 sets the director’s duty of care and skill?
- Section 308
- Section 400
- Section 405
- Section 409
Show the answer
A. Directors should test the assumptions behind a decision.
Written questions
Question 11 (10 marks). (a) Distinguish systematic and unsystematic risk. (4 marks) (b) Explain why diversification reduces risk and what limits it. (6 marks)
Question 12 (20 marks). A portfolio is 50% A (expected return 10%, σ 6%) and 50% B (16%, 12%). (a) Calculate the expected return. (3 marks) (b) Calculate the portfolio standard deviation if the correlation is +1, 0.5 and −1. (9 marks) (c) Comment on the results. (3 marks) (d) The market portfolio returns 14% with σ 10% and the risk-free rate is 6%. Calculate the weighting and borrowing needed to earn 20%, and the risk. (5 marks)
Question 13 (10 marks). (a) Explain the efficient frontier and the market portfolio. (5 marks) (b) Explain the capital market line. (5 marks)
Model answers
Show the model answer to Question 11
(a) Types of risk (4 marks). Systematic risk affects the whole market and cannot be diversified away. (2) Unsystematic risk is specific to one security and can be removed by diversification. (2)
(b) Diversification (6 marks). Good and bad results on different investments offset each other, so the variability of the portfolio is lower. (2) The gain depends on correlation: the lower or more negative, the larger the reduction. (2) It cannot remove systematic risk, which affects all investments together. (2)
Show the model answer to Question 12
(a) Return (3 marks). 0.5 × 10 + 0.5 × 16 = 13%. (3)
(b) Risk (9 marks).
- ρ = +1: 0.5 × 6 + 0.5 × 12 = 9.0%. (3)
- ρ = 0.5: variance = 0.25 × 36 + 0.25 × 144 + 2 × 0.5 × 0.5 × 0.5 × 6 × 12 = 9 + 36 + 18 = 63, so σ = 7.94%. (3)
- ρ = −1: |0.5 × 6 − 0.5 × 12| = 3.0%. (3)
(c) Comment (3 marks). Return is the same in all cases, 13%, but risk falls as correlation falls. (1.5) Only perfect positive correlation gives no diversification gain, and negative correlation reduces risk the most. (1.5)
(d) Capital market line (5 marks).
- Return = 6% + w × (14% − 6%) = 20%, so w = 14 ÷ 8 = 1.75. (2)
- The investor borrows 0.75 of his capital at the risk-free rate and invests 1.75 in the market portfolio. (1.5)
- Risk = 1.75 × 10% = 17.5%. (1.5)
Show the model answer to Question 13
(a) Frontier and market portfolio (5 marks). The efficient frontier is the top edge of all possible portfolios, giving the highest return for each level of risk. (2) Where the best indifference curve touches the frontier is the preferred portfolio, the market portfolio, made up of all traded investments in proportion to their market value. (3)
(b) CML (5 marks). The capital market line is the straight line from the risk-free return, tangent to the frontier at the market portfolio. (2) It shows all combinations of the risk-free asset and the market portfolio, with lending before the market portfolio and borrowing beyond it. (3)
Pathfinder on Portfolio risk, correlation and the capital market line
These are the ICAN Pathfinder questions that test this topic. ICAN owns the questions and the suggested solutions, so we point you to them and do not copy them. Open your Pathfinder, attempt each question under exam conditions, then check your answer against the notes above.
The multiple-choice question asked for the measure of how well the returns of two risky assets move together (covariance). The short-answer questions asked for a combination of securities giving the highest expected return for a given standard deviation (an efficient portfolio) and for the correlation of two projects that are completely and positively linearly dependent (+1). Read the sections on covariance and on the efficient frontier.
Revise first: The two-asset portfolio
Questions ask which risk diversification removes. See the section on systematic and unsystematic risk.
Revise first: Systematic and unsystematic risk
Source: ICAN Pathfinder, Strategic Financial Management. Diets covered: May 2014 multiple-choice 18 and short-answer 4 and 17.
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