Business Finance · Key principles of finance

Risk and Return Trade-off for IAI CB1 Business Finance

Q: What is the difference between systematic and unsystematic risk?

Systematic risk affects the whole market, such as interest rate changes or a recession, and cannot be diversified away. Unsystematic risk is specific to one firm or sector and can be reduced by holding many assets. Models such as CAPM reward only systematic risk.

Q: How does diversification reduce risk?

When assets do not move perfectly together, a loss on one is partly offset by a gain on another. This shows in the covariance term of portfolio variance. The lower the correlation, the greater the reduction.

Q: How do I find the expected return and standard deviation of a portfolio?

Expected return is the weighted average of the asset returns. Standard deviation is the square root of portfolio variance, which uses the weights, each variance and the covariance between the assets. Do not average the standard deviations unless the correlation is 1.

Q: Does higher risk always mean higher return?

No. It means investors require a higher expected return as compensation. Actual outcomes can be lower, which is exactly what risk means.

Updated 11 October 2026 · Fact-checked

The risk and return trade-off says investors need higher expected returns to accept higher risk. You measure return by the expected value and risk by standard deviation. Diversification removes unsystematic risk but not systematic risk. To solve questions, compute expected return, variance, covariance and portfolio standard deviation, then compare the premium over the risk-free rate.

Understand Risk and Return Trade-off

Return is the gain from an investment, as income plus capital growth, divided by the amount invested. Risk is the chance that the actual return differs from what you expected. In finance, risk is usually uncertainty in both directions, not only the chance of loss.

The trade-off is simple. Investors dislike risk (they are risk-averse). So they will hold a risky asset only if it is expected to pay more than a safe one. The extra expected return is the risk premium: expected return on the risky asset minus the risk-free rate. Government securities are often used as the risk-free proxy. Remember this is about expected returns. A risky asset does not always give a higher actual return.

The common measure of risk is the standard deviation of returns, the square root of the variance. For two assets, the risk of a portfolio also depends on how the assets move together. This is measured by covariance or correlation. The lower the correlation, the more risk is cancelled out.

Diversification means spreading money across assets that do not move perfectly together. Total risk splits into two parts. Unsystematic (specific) risk belongs to one company or sector, such as a strike or a product failure. Diversification can remove it. Systematic (market) risk affects all assets, such as interest rate changes or recession. It cannot be diversified away, so only this risk earns a premium in models such as CAPM. Link this topic to CAPM and to efficient markets.

Key rules to remember

Expected return of an asset: E(R) = Σ pᵢ × Rᵢ; pᵢ are the probabilities of each outcome and they must add to 1.
Variance and standard deviation: σ² = Σ pᵢ × (Rᵢ − E(R))²; σ = √σ²; Equivalent form: σ² = E(R²) − [E(R)]². Standard deviation is in the same units as return.
Risk premium: Risk premium = E(R) − Rf; Rf is the risk-free rate.
Portfolio expected return: E(Rp) = w₁E(R₁) + w₂E(R₂); Weights add to 1. It is a simple weighted average, so no diversification effect appears here.
Two-asset portfolio variance: σp² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρ₁₂σ₁σ₂; Here ρ₁₂σ₁σ₂ = Cov(R₁, R₂). Portfolio standard deviation is the square root.
Correlation: ρ₁₂ = Cov(R₁, R₂) ÷ (σ₁σ₂); Lies between −1 and +1. Portfolio risk falls as ρ falls.
Total risk split: Total risk = systematic risk + unsystematic risk; In variance terms the two parts add. Diversification reduces only the unsystematic part.

How to solve Risk and Return Trade-off questions

Use this order for any numerical or descriptive question on risk and return.

1Write down what is given: returns, probabilities or weights, standard deviations, and correlation or covariance.
2Compute the expected return of each asset using E(R) = Σ pᵢ × Rᵢ.
3Compute each variance and standard deviation. Use the deviations method or E(R²) − [E(R)]².
4For a portfolio, find the expected return as the weighted average of asset returns.
5Find portfolio variance with the two-asset formula. Take the square root only at the end.
6Compare to the risk-free rate or other assets: risk premium, or return per unit of risk.
7State which risk is diversifiable and which is not, and link the result to the trade-off.
8Check units: percentages versus decimals, and that variance has not been quoted as standard deviation.

Quickest way: Decimals and the covariance shortcut

When to use it: Use it in the MCQ section or when time is short on portfolio calculations.

Convert all percentages to decimals first.
Use the key results: ρ = 1 gives σp = w₁σ₁ + w₂σ₂; ρ = 0 drops the covariance term; ρ = −1 can give zero risk with the right weights.
If the question gives covariance, put it straight in as 2w₁w₂Cov.
Check with bounds: σp must lie between the lowest and highest possible values for ρ = −1 and ρ = +1.
Eliminate MCQ options that break these bounds, then calculate only what remains.

Common mistakes in Risk and Return Trade-off

Taking the square root of portfolio variance before adding the terms, or averaging standard deviations.
Students treat risk like return, which is a simple weighted average.
Fix: Add the variance terms first. Only when ρ = 1 does the standard deviation equal the weighted average of the standard deviations.
Forgetting the factor 2 in the covariance term.
The formula is memorised loosely.
Fix: Write the formula in full every time: w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂.
Saying diversification removes all risk.
Students overlook the systematic part.
Fix: State that it removes only unsystematic risk. Systematic risk remains, and perfect negative correlation is a special case.
Mixing percentages and decimals, such as using 12 in one term and 0.12 in another.
Time pressure and inconsistent notation.
Fix: Convert everything to decimals at the start and convert back for the final answer.
Claiming a high-risk asset always earns more.
The trade-off is read as a guarantee.
Fix: Say that investors expect higher returns for higher risk. Actual returns can be lower.
Using total risk to explain the premium on a well-diversified holding.
Students ignore what the market rewards.
Fix: Explain that, under models like CAPM, the premium rewards systematic risk only.

Worked examples

Example 1

An asset has the following possible returns: 4% with probability 0.3, 10% with probability 0.5 and 16% with probability 0.2. The risk-free rate is 3%. Find the expected return, standard deviation and risk premium.

Show the solution

E(R) = 0.3 × 4 + 0.5 × 10 + 0.2 × 16 = 1.2 + 5.0 + 3.2 = 9.4%.
Deviations: 4 − 9.4 = −5.4; 10 − 9.4 = 0.6; 16 − 9.4 = 6.6.
Variance = 0.3 × 29.16 + 0.5 × 0.36 + 0.2 × 43.56 = 8.748 + 0.18 + 8.712 = 17.64.
Standard deviation = √17.64 = 4.2%.
Risk premium = 9.4 − 3 = 6.4%.

Answer: Expected return 9.4%, standard deviation 4.2%, risk premium 6.4%.

Example 2

Asset A has expected return 8% and standard deviation 10%. Asset B has expected return 14% and standard deviation 20%. The correlation is 0.5. You invest 60% in A and 40% in B. Find the portfolio expected return and standard deviation.

Show the solution

E(Rp) = 0.6 × 8 + 0.4 × 14 = 4.8 + 5.6 = 10.4%.
Use decimals: w₁ = 0.6, w₂ = 0.4, σ₁ = 0.10, σ₂ = 0.20, ρ = 0.5.
w₁²σ₁² = 0.36 × 0.01 = 0.0036.
w₂²σ₂² = 0.16 × 0.04 = 0.0064.
2w₁w₂ρσ₁σ₂ = 2 × 0.6 × 0.4 × 0.5 × 0.10 × 0.20 = 0.0048.
σp² = 0.0036 + 0.0064 + 0.0048 = 0.0148.
σp = √0.0148 = 0.12166, about 12.2%.
Compare: the weighted average of standard deviations is 0.6 × 10 + 0.4 × 20 = 14%. The portfolio risk is lower, which shows diversification benefit because ρ < 1.

Answer: Expected return 10.4%; standard deviation about 12.2%, below the 14% weighted average.

Exam tips

Show every term of the portfolio variance formula on separate lines. Method marks are given even if arithmetic slips.
In written answers, name the type of risk (systematic or unsystematic) and say which one diversification affects.
State assumptions, such as risk-averse investors and that standard deviation is the risk measure.
In MCQs, test options against the ρ = 1 and ρ = −1 bounds before calculating in full.
Always finish with a sentence interpreting the result: what it says about risk against return.

Practice questions from Key principles of finance

Risk and Return Trade-off in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Risk and Return Trade-off: frequently asked questions

What is the difference between systematic and unsystematic risk?

Systematic risk affects the whole market, such as interest rate changes or a recession, and cannot be diversified away. Unsystematic risk is specific to one firm or sector and can be reduced by holding many assets. Models such as CAPM reward only systematic risk.

How does diversification reduce risk?

When assets do not move perfectly together, a loss on one is partly offset by a gain on another. This shows in the covariance term of portfolio variance. The lower the correlation, the greater the reduction.

How do I find the expected return and standard deviation of a portfolio?

Expected return is the weighted average of the asset returns. Standard deviation is the square root of portfolio variance, which uses the weights, each variance and the covariance between the assets. Do not average the standard deviations unless the correlation is 1.

Does higher risk always mean higher return?

No. It means investors require a higher expected return as compensation. Actual outcomes can be lower, which is exactly what risk means.