Central tendency, dispersion, conditional probability, simulation, and portfolio-risk interpretation for Level I.
This part of Quantitative Methods teaches you how to describe a return distribution before you try to infer anything from it. Level I questions often look mathematical, but they are usually asking a classification question first: what kind of distribution are you looking at, what kind of dependence exists, and what kind of portfolio effect should you expect?
| Measure | What it helps you see | Common exam use |
|---|---|---|
| Mean | Average return level | Comparing expected payoff across choices |
| Median | Middle observation | Spotting skewed distributions |
| Variance / standard deviation | Dispersion around the mean | Interpreting total risk |
| Skewness | Asymmetry | Recognizing tail shape and downside concentration |
| Kurtosis | Tail thickness | Identifying outlier-prone distributions |
| Correlation | Co-movement direction and strength | Judging diversification benefit |
The exam often uses skewness and kurtosis conceptually. You may not need a long calculation. You do need to recognize what a negatively skewed or high-kurtosis distribution says about downside risk.
Expected portfolio return is a weighted average:
$$ E(R_p) = \sum_{i=1}^{n} w_i E(R_i) $$
Portfolio variance is where diversification enters. For a two-asset case:
$$ \sigma_p^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2w_1w_2\text{Cov}(R_1,R_2) $$
That covariance term is the real story. Level I is frequently testing whether you understand that diversification comes from less-than-perfect positive correlation, not from simply owning more line items.
Probability trees and conditional expectations force you to read the path, not just the endpoint. If event B depends on event A, you cannot treat the branches as independent.
Bayes-style reasoning appears whenever the question asks you to update a prior belief after new evidence arrives:
$$ P(A \mid B) = \frac{P(B \mid A)P(A)}{P(B)} $$
You do not need to become a theorem specialist. You do need to see that new information changes the probability of the underlying state.
Simulation methods at Level I are less about deep model building and more about understanding why analysts generate many possible paths instead of assuming one certainty. A strong answer usually recognizes:
Two assets have similar expected returns, but one has negative correlation with the rest of the portfolio. Level I often wants you to see that the negatively correlated asset can improve the portfolio opportunity set even if it does not look obviously superior on a standalone basis.
A candidate says a new asset cannot reduce portfolio risk because its own standard deviation is higher than the portfolio’s current standard deviation. What is the strongest reply?
Best answer: That conclusion is incomplete because portfolio risk depends on covariance, not just standalone volatility.
Why: Level I often tests diversification through the interaction term. A volatile asset can still reduce overall portfolio risk if it moves differently enough from the assets already held.