Risk and Return

Risk and Return

Syllabus tag: KASNEB CPA | Intermediate Level | CA22 Financial Management | Topic 8 Risk and Return

Lesson objectives

By the end of this topic, you will be able to:

• Compute expected return, variance and standard deviation from a probability distribution
• Use the coefficient of variation to compare investments of different sizes
• Compute the expected return and risk of a two-asset portfolio
• Explain how correlation drives the benefit of diversification
• Distinguish systematic from unsystematic risk and apply beta

Why this matters

Cost of Capital used beta without saying where it comes from or why shareholders should be paid for some risks and not others. This topic answers both.

Words to know

• Expected return — the probability-weighted average of possible returns.
• Standard deviation — how widely outcomes spread around that average.
• Coefficient of variation — risk per unit of return.
• Correlation — how closely two investments move together, from +1 to −1.
• Systematic risk — market-wide risk that cannot be diversified away.
• Unsystematic risk — company-specific risk that can be.

Expected return and risk

A share's return depends on the economy:

Economy	Probability	Return
Recession	0.25	−10%
Normal	0.50	15%
Boom	0.25	30%

Expected return = (0.25 × −10) + (0.50 × 15) + (0.25 × 30) = 12.5%

Variance = 0.25(−10 − 12.5)² + 0.50(15 − 12.5)² + 0.25(30 − 12.5)² = 206.25

Standard deviation = square root of 206.25 = 14.36%

Note that the expected return of 12.5% is not one of the three possible outcomes. It is an average, not a forecast.

Coefficient of variation

CV = Standard deviation / Expected return = 14.36 / 12.5 = 1.15

This matters when comparing investments of different sizes. A project with a standard deviation of 30% is not necessarily riskier than one at 14% — if it also returns three times as much, it may carry less risk per shilling of return. Always compare the CV, not the raw standard deviation.

Two-asset portfolios

Asset A returns 18% with a standard deviation of 20%. Asset B returns 10% with a standard deviation of 12%. Hold 60% in A and 40% in B.

Portfolio expected return = (0.6 × 18) + (0.4 × 10) = 14.8%

Return is a simple weighted average. Risk is not.

Portfolio variance = (wA·σA)² + (wB·σB)² + 2·wA·wB·σA·σB·ρ

With correlation of 0.3:

= (0.6 × 20)² + (0.4 × 12)² + 2(0.6)(0.4)(20)(12)(0.3) = 144 + 23.04 + 34.56 = 201.6

Portfolio standard deviation = square root of 201.6 = 14.20%

The weighted average of the two standard deviations is 16.80%. The portfolio achieves 14.20%. That gap is diversification, and it is free.

Correlation is what does the work

The same two assets, same weights, at different correlations:

Correlation	Portfolio standard deviation
+1.0	16.80%
+0.3	14.20%
0.0	12.92%
−1.0	7.20%

At +1.0 there is no benefit at all — the portfolio risk is exactly the weighted average. The lower the correlation, the greater the gain. This is why a portfolio of ten bank shares is far less diversified than a portfolio of three shares drawn from unrelated industries.

:::checkpoint Two assets each have a standard deviation of 20% and are perfectly negatively correlated. Without computing, say what portfolio standard deviation is achievable and explain in one sentence why. :::

Systematic and unsystematic risk

Unsystematic risk is specific to the company — a factory fire, a lost contract, a poor product launch. Holding enough different shares makes these cancel out, so it disappears in a diversified portfolio.

Systematic risk affects every company at once — interest rates, exchange rates, the general state of the economy. No amount of diversification removes it.

The consequence is the central idea of this topic: the market rewards only systematic risk. An investor who fails to diversify still bears unsystematic risk, but receives no extra return for it, because they could have removed it for nothing.

Beta

Beta measures systematic risk — how far a share moves when the market moves.

• β = 1.0 — moves with the market
• β > 1.0 — amplifies market movements
• β < 1.0 — dampens them

Ke = Rf + β(Rm − Rf)

Risk-free 9%, market 16%, beta 0.8:

Ke = 9 + 0.8(16 − 9) = 14.6%

A portfolio's beta is the weighted average of its holdings' betas. Half in a 1.4 beta share and half in a 0.6 gives a portfolio beta of 1.0 — market risk exactly.

:::checkpoint An investor holds a single share with a standard deviation of 40%, of which most is company-specific. Explain why the market will not pay them a higher return for carrying that risk. :::