Module 3 · Quantitative Methods

Statistical Measures of Asset Returns

EN: Central tendency, dispersion, skewness, kurtosis, downside risk and the Sharpe ratio.
VN: Đo lường thống kê mô tả lợi suất tài sản và rủi ro.

In this module

Quantiles & Percentile Position
Range & Mean Absolute Deviation
Variance and Standard Deviation
Coefficient of Variation
Skewness
Kurtosis & Excess Kurtosis
Target Downside Deviation
Sharpe Ratio
Covariance & Correlation

1. Quantiles & Percentile Position Core

About: A quantile is a cut-point that splits sorted data into equal-frequency groups (quartiles=4, quintiles=5, deciles=10, percentiles=100). \(L_y\) gives the position of the y-th percentile.Tóm tắt: Phân vị chia dữ liệu đã sắp xếp thành các nhóm có tần suất bằng nhau. \(L_y\) là vị trí của percentile thứ y.

EN: Position of the \(y\)-th percentile in a sorted dataset of \(n\) observations.
VN: Vị trí của percentile thứ \(y\) trong dataset đã sắp xếp.

\[ L_y = (n + 1)\,\frac{y}{100} \]

Components / Thành phần

\(L_y\) Position (linear interpolate if non-integer).
\(n\) Sample size.
\(y\) Percentile (e.g. 25 for Q1, 50 for median).

Quartiles split into 4, quintiles 5, deciles 10, percentiles 100.

Practice problem

Find the position of the 30th percentile in a sample of n = 19 observations.

Show solution

\(L_{30} = (19 + 1) \times 0.30 = 6\)

P30 = 6th value of the sorted data

2. Range & Mean Absolute Deviation (MAD) Core

About: Range = simplest dispersion measure (max − min). MAD = average absolute deviation from the mean — less sensitive to outliers than variance because it doesn't square deviations.Tóm tắt: Range là khoảng cách max-min. MAD là trung bình độ lệch tuyệt đối — ít nhạy với outlier hơn variance.

\[ \text{Range} = X_{\max} - X_{\min} \] \[ MAD = \frac{1}{n}\sum_{i=1}^{n}\,|X_i - \bar{X}| \]

Components / Thành phần

\(\bar{X}\) Sample arithmetic mean.
\(|X_i - \bar{X}|\) Absolute deviation from the mean.

Practice problem

Daily returns: 2%, −1%, 3%, 0%, 4%. Compute range and MAD.

Show solution

Range = 4% − (−1%) = 5%

Mean = (2 − 1 + 3 + 0 + 4)/5 = 1.6%

Abs deviations: 0.4, 2.6, 1.4, 1.6, 2.4 → sum = 8.4

MAD = 8.4/5

Range = 5%; MAD = 1.68%

3. Variance and Standard Deviation Core

About: Variance = average squared deviation from the mean. Standard deviation = square root of variance, in same units as data. Sample uses n−1 (Bessel's correction) for an unbiased estimator.Tóm tắt: Phương sai = trung bình bình phương độ lệch. Độ lệch chuẩn = căn của variance. Dùng n−1 để ước lượng không chệch.

EN: Population uses \(N\); sample uses \(n - 1\) in the denominator (Bessel's correction) for an unbiased estimate.
VN: Population chia \(N\); sample chia \(n - 1\) để ước lượng không chệch.

\[ \sigma^{2} = \frac{1}{N}\sum_{i=1}^{N}(X_i - \mu)^{2} \qquad s^{2} = \frac{1}{n - 1}\sum_{i=1}^{n}(X_i - \bar{X})^{2} \] \[ \sigma = \sqrt{\sigma^{2}} \qquad s = \sqrt{s^{2}} \]

Components / Thành phần

\(\mu\) Population mean / \(\bar{X}\) Sample mean.
\(N,n\) Population / sample size.

Practice problem

Sample returns: 10%, 12%, 8%, 6%, 14%. Compute the sample variance and standard deviation.

Show solution

\(\bar{X} = 50/5 = 10\%\)

Squared deviations: 0, 4, 4, 16, 16 → sum = 40

\(s^{2} = 40/(5 - 1) = 10\) → \(s = \sqrt{10}\)

s² = 10, s ≈ 3.16%

4. Coefficient of Variation (CV) Core

About: CV expresses risk per unit of return — a unit-free dispersion measure. Useful for comparing risk across assets with very different mean returns. Lower CV = better risk-adjusted profile (in this simple sense).Tóm tắt: CV đo rủi ro trên một đơn vị lợi suất — không có đơn vị. Dùng so sánh rủi ro tương đối giữa tài sản có lợi suất khác nhau.

EN: Relative dispersion — risk per unit of expected return. Lower is "better" (less risk per unit of reward).
VN: Độ phân tán tương đối — đo rủi ro trên mỗi đơn vị lợi suất.

\[ CV = \frac{s}{\bar{X}} \]

Practice problem

Fund A: mean 12%, sd 8%. Fund B: mean 18%, sd 14%. Which is less risky per unit of return?

Show solution

CV(A) = 8/12 ≈ 0.667

CV(B) = 14/18 ≈ 0.778

Fund A has lower CV → less risk per unit of return

5. Skewness Core

About: Skewness measures asymmetry. Positive skew → long right tail (mean > median > mode). Negative skew → long left tail (mean < median < mode). Most equity returns are negatively skewed (rare large losses).Tóm tắt: Skewness đo độ lệch của phân phối. Skew dương: đuôi phải dài. Skew âm: đuôi trái dài. Cổ phiếu thường skew âm.

Negative skew: long left tail. Positive skew: long right tail.

EN: Measures asymmetry. Positive skew → long right tail (mean > median). Negative skew → long left tail (mean < median).
VN: Đo độ lệch của phân phối.

\[ \text{Skew} = \frac{1}{n}\sum_{i=1}^{n}\frac{(X_i - \bar{X})^{3}}{s^{3}} \]

Interpretation

> 0 Right-skewed: mean > median > mode.
< 0 Left-skewed: mean < median < mode.
= 0 Symmetric (e.g. normal).

Rule of thumb: |skew| > 0.5 is considered material.

Practice problem

A return distribution has mean 8%, median 6%, mode 4%. What is the likely sign of skewness?

Show solution

Mean > median > mode → positive (right) skew.

Positive skew (long right tail)

6. Kurtosis & Excess Kurtosis Core

About: Kurtosis measures tail thickness vs. the normal distribution. Excess kurtosis = K − 3. Leptokurtic (K > 3) means fat tails — more frequent extreme outcomes than the normal predicts. Critical for VaR/risk modeling.Tóm tắt: Kurtosis đo độ dày đuôi so với phân phối chuẩn. K > 3 (leptokurtic) = đuôi béo, nhiều outlier hơn dự kiến.

Higher kurtosis → more outliers (fat tails). Most asset returns are leptokurtic.

EN: Peakedness and tail thickness vs. the normal distribution.
VN: Độ nhọn và độ "fat tail" so với phân phối chuẩn.

\[ K = \frac{1}{n}\sum_{i=1}^{n}\frac{(X_i - \bar{X})^{4}}{s^{4}} \] \[ K_E = K - 3 \]

Interpretation

\(K = 3\) Mesokurtic — same as normal (\(K_E = 0\)).
\(K > 3\) Leptokurtic — fat tails, more outlier risk (\(K_E > 0\)).
\(K < 3\) Platykurtic — thin tails (\(K_E < 0\)).

Most asset returns are leptokurtic and negatively skewed.

Practice problem

Sample excess kurtosis = 2.1. What does this imply about tail risk?

Show solution

Excess kurtosis > 0 → leptokurtic.

Fatter tails than normal → more frequent extreme outcomes than the bell curve predicts.

Leptokurtic — elevated tail / outlier risk

7. Target Downside Deviation Core

About: Like standard deviation but only counts deviations BELOW a target threshold (e.g. 0% or risk-free rate). Captures downside risk only — better aligned with how investors actually feel risk.Tóm tắt: Giống độ lệch chuẩn nhưng chỉ tính quan sát DƯỚI mức target. Đo rủi ro xuống — phù hợp hơn với cảm nhận của nhà đầu tư.

EN: Like standard deviation but only counts deviations below a target return.
VN: Như độ lệch chuẩn nhưng chỉ tính các quan sát dưới mức target.

\[ s_{\text{target}} = \sqrt{\frac{\displaystyle\sum_{X_i < B}(X_i - B)^{2}}{n - 1}} \]

Components / Thành phần

\(B\) Target return (e.g. 0% or risk-free rate).
\(X_i < B\) Only observations below the target are included.
\(n - 1\) Full sample size minus 1 (NOT just the count below \(B\)).

Practice problem

Sample of 10 monthly returns, 4 are below the 0% target with squared deviations summing to 0.0036. Compute target downside deviation.

Show solution

\(s_{target}^{2} = 0.0036/(10-1) = 0.0004\)

\(s_{target} = \sqrt{0.0004}\)

≈ 2.00%

8. Sharpe Ratio Core

About: Sharpe = excess return per unit of total risk. The most-cited risk-adjusted performance metric. Higher Sharpe = better. Compare across portfolios at the same risk-free rate.Tóm tắt: Sharpe = lợi suất vượt trên một đơn vị tổng rủi ro. Chỉ số đánh giá hiệu quả phổ biến nhất. Cao hơn = tốt hơn.

EN: Excess return per unit of total risk — the most cited risk-adjusted performance metric.
VN: Lợi suất vượt phi rủi ro trên 1 đơn vị tổng rủi ro.

\[ \text{Sharpe} = \frac{\bar{R}_p - R_f}{s_p} \]

Components / Thành phần

\(\bar{R}_p\) Portfolio mean return.
\(R_f\) Risk-free rate.
\(s_p\) Portfolio standard deviation.

Practice problem

Portfolio mean return 14%, sd 18%. Risk-free 3%. Compute the Sharpe ratio.

Show solution

\(\text{Sharpe} = (14 - 3)/18\)

≈ 0.611

9. Covariance & Correlation Core

About: Covariance measures joint variability (how two variables move together) — units = product of variable units, hard to interpret directly. Correlation standardizes covariance to [-1, 1] and is unit-free.Tóm tắt: Covariance đo độ biến thiên đồng thời. Correlation chuẩn hóa về [-1, 1] và không có đơn vị, dễ hiểu hơn.

\[ \text{Cov}(X,Y) = \frac{1}{n - 1}\sum_{i=1}^{n}(X_i - \bar{X})(Y_i - \bar{Y}) \] \[ \rho_{X,Y} = \frac{\text{Cov}(X,Y)}{s_X \cdot s_Y}, \qquad -1 \le \rho \le 1 \]

Components / Thành phần

Cov Joint variability (units = product of variable units).
\(\rho\) Standardized correlation in [-1, 1].
\(s_X, s_Y\) Sample standard deviations.

Practice problem

Cov(X,Y) = 0.0024, σX = 6%, σY = 8%. Compute correlation.

Show solution

\(\rho = 0.0024 / (0.06 \times 0.08) = 0.0024 / 0.0048\)

ρ = 0.50