Lesson 3: Risk, Return and Diversification
1. What is return?
For a single period, the total return on an investment combines price change and any income received (dividends on shares, coupons on bonds):
Worked example — single-period return
Buy at $100, sell a year later at $110, and receive a $2 dividend during the year. Return = (110 − 100 + 2) ÷ 100 = 12 ÷ 100 = 12%. Past return is a fact about the past; it is not a promise about the future.
When returns are uncertain, we summarise them with an expected return — the probability-weighted average of possible returns (Lesson 5 develops this fully). Expected return is a planning concept, not a guaranteed outcome.
2. What is risk?
In finance, risk most often means variability of outcomes: how widely actual returns may spread around the expected return. The standard measures are variance and standard deviation (Lesson 5). A wider spread means a less predictable outcome, even if the expected return is the same. Risk also includes other hazards — default, illiquidity, inflation, and the risk of needing to sell at a bad moment — but variability is the quantity the mathematics of this course formalises.
The usual working relationship is that higher expected return is generally associated with higher risk. It is an association, not a law: an investment can carry high risk and deliver low returns, and a low-risk investment can occasionally do well. Anyone promising high returns with low risk should be treated with scepticism.
3. Two kinds of risk
- Idiosyncratic (asset-specific) risk — events affecting one company or asset: a failed product, a lawsuit, a factory fire. This risk is, in principle, diversifiable.
- Systematic (market) risk — events affecting broad markets: recessions, interest-rate shifts, wars. This risk cannot be diversified away by holding more assets in the same market, because it moves many assets together.
4. Why diversification reduces risk
Portfolio return is the weighted average of the assets' returns, but portfolio risk is not the weighted average of the assets' risks. It depends on how the assets move together — their correlation (Lesson 6 gives the formula). When assets are not perfectly correlated, their ups and downs partly offset, so the portfolio's variability is lower than the average variability of its parts. Equal weights are used in the example below so the arithmetic is transparent.
Worked example — two assets, each with 20% standard deviation
Assets A and B each have standard deviation 20% (variance 0.04). Equal weights (0.5, 0.5), expected returns 8% and 12%.
Portfolio expected return = 0.5 × 8% + 0.5 × 12% = 10% in every case — diversification does not change expected return here.
| Correlation (ρ) | Portfolio variance | Portfolio standard deviation |
|---|---|---|
| 0 (uncorrelated) | 0.25 × 0.04 + 0.25 × 0.04 + 0 = 0.0200 | √0.0200 = 14.14% |
| +1 (move together) | 0.0400 | 20.00% — no benefit |
| −1 (move oppositely) | 0.0000 | 0.00% — perfect offset (a textbook limiting case, not a realistic promise) |
The lesson: the lower the correlation, the larger the risk reduction. With ρ = 0, risk falls from 20% to about 14.14% while expected return stays at 10%.
5. Limits of diversification
- It cannot remove systematic risk. In a broad market fall, correlations often rise — assets that usually move independently fall together — so diversification disappoints exactly when it is most wanted. This is a qualitative warning, not a measured statistic.
- It is not a guarantee against loss. A diversified portfolio can still fall; diversification reduces variability, it does not set a floor under value.
- More assets help less and less. Once the main idiosyncratic risks are spread, each extra holding adds little further reduction — and adds complexity and cost.
- Diversification across what matters. Holding many shares in the same industry or country diversifies less than holding assets whose fortunes genuinely differ. What counts is low correlation, not a large count of holdings.
Key terms
- Total return
- Price change plus income, divided by the starting price.
- Expected return
- Probability-weighted average of possible returns.
- Idiosyncratic risk
- Risk specific to one asset; diversifiable in principle.
- Systematic risk
- Market-wide risk; not diversifiable within that market.
References & further study
- Standard introductory corporate-finance and investments textbooks (Brealey / Myers / Allen; Bodie / Kane / Marcus; or equivalents) for the conceptual framework used in this course.
- Standard introductory probability and statistics textbooks for expected value, variance, covariance and correlation.
- MIT OpenCourseWare (generic entry point for free course materials in mathematics, probability and finance-related topics): ocw.mit.edu — use the site search for the relevant topic; no specific course is prescribed here.
📝 Quiz
1. Buy at $100, sell at $110, receive a $2 dividend. The total return is:
Show answer
b) (110 − 100 + 2) ÷ 100 = 12 ÷ 100 = 12%.
2. Two assets each have 20% standard deviation, equal weights, correlation 0. The portfolio standard deviation is about:
Show answer
b) Variance = 0.25 × 0.04 + 0.25 × 0.04 = 0.02; √0.02 ≈ 14.14%.
3. Diversification primarily reduces:
Show answer
b) Diversification spreads asset-specific risk; systematic risk remains, and no floor under value is created.