Module 1 · Quantitative Methods

Rates and Returns

EN: Nominal vs real rates, multiple measures of return (HPR, arithmetic, geometric, harmonic, money-weighted, time-weighted) and how to annualize.
VN: Lãi suất danh nghĩa vs lãi suất thực, các loại tỷ suất sinh lời, và cách quy đổi lợi suất theo năm.

In this module / Trong module này

Holding Period Return (HPR)
Nominal vs Real Risk-free Rate (Fisher)
Arithmetic Mean Return
Geometric Mean Return
Harmonic Mean
Trimmed & Winsorized Mean
Money-Weighted Return (IRR)
Time-Weighted Return (TWR)
Annualized Return
Continuously Compounded Return
Gross / Net / After-tax / Leveraged Return

1. Holding Period Return (HPR) Core

EN: The return an investor earns from holding an asset over a period, including both price change (capital gain) and any income (dividend / coupon).
VN: Tỷ suất sinh lời khi nắm giữ tài sản trong một kỳ — gồm thay đổi giá và thu nhập định kỳ.

\[ R = \frac{P_1 - P_0 + I_1}{P_0} = \frac{P_1 + I_1}{P_0} - 1 \]

Components / Thành phần

$R$ Holding period return / Tỷ suất sinh lời kỳ nắm giữ.
$P_0$ Beginning-of-period price / Giá tài sản đầu kỳ.
$P_1$ End-of-period price / Giá tài sản cuối kỳ.
$I_1$ Income received during the period (dividend / coupon) / Thu nhập trong kỳ.

Practice problem

An investor buys a stock at $50. One year later she sells it for $54 and has received $2 in dividends during the year. What is the holding period return?

Show solution

$P_0 = 50,\ P_1 = 54,\ I_1 = 2$

$R = \dfrac{54 - 50 + 2}{50} = \dfrac{6}{50} = 0.12$

HPR = 12%

2. Nominal vs Real Risk-free Rate (Fisher Equation) Core

EN: The relationship between nominal and real interest rates after adjusting for inflation, plus any required risk premium.
VN: Quan hệ giữa lãi suất danh nghĩa và lãi suất thực sau khi điều chỉnh lạm phát.

\[ (1 + r_{\text{nominal}}) = (1 + r_{\text{real}})\,(1 + \pi)\,(1 + RP) \] \[ \text{Approximation: } r_{\text{nominal}} \approx r_{\text{real}} + \pi + RP \]

Components / Thành phần

$r_{nom}$ Nominal interest rate / Lãi suất danh nghĩa quan sát trên thị trường.
$r_{real}$ Real risk-free rate — reward for deferring consumption / Lãi suất thực phi rủi ro.
$\pi$ Expected inflation rate / Tỷ lệ lạm phát kỳ vọng.
$RP$ Risk premium — default, liquidity, maturity (= 0 for risk-free) / Phần bù rủi ro.

Practice problem

The real risk-free rate is 2% and expected inflation is 3%. Compute the approximate and the exact nominal risk-free rate.

Show solution

Approximate: $r_{nom} \approx 2\% + 3\% = 5.00\%$

Exact: $(1.02)(1.03) - 1 = 1.0506 - 1 = 0.0506$

Nominal ≈ 5.00% (exact 5.06%)

3. Arithmetic Mean Return Core

EN: The simple average of single-period returns. It is the unbiased estimator of the expected return for a single future period.
VN: Trung bình cộng của các kỳ — ước lượng không chệch cho lợi suất kỳ vọng của 1 kỳ trong tương lai.

\[ \bar{R}_a = \frac{1}{N}\sum_{i=1}^{N} R_i = \frac{R_1 + R_2 + \dots + R_N}{N} \]

Components / Thành phần

$\bar{R}_a$ Arithmetic mean return.
$R_i$ Return in period $i$ / Tỷ suất kỳ thứ $i$.
$N$ Number of periods / Số kỳ quan sát.

Practice problem

A stock had returns of 10%, –5%, 20%, 15% over four years. What is the arithmetic mean annual return?

Show solution

$\bar{R}_a = \dfrac{10 - 5 + 20 + 15}{4} = \dfrac{40}{4}$

Arithmetic mean = 10%

4. Geometric Mean Return Core

EN: The compound return actually earned by an investor over multiple periods. Always ≤ arithmetic mean (equal only when all returns are identical).
VN: Lợi suất compound thực tế qua nhiều kỳ. Luôn nhỏ hơn hoặc bằng arithmetic mean.

\[ \bar{R}_g = \left[\prod_{i=1}^{N}(1 + R_i)\right]^{1/N} - 1 \]

Components / Thành phần

$\bar{R}_g$ Geometric mean return — compound annual growth rate (CAGR).
$R_i$ Return in period $i$ (decimal).
$N$ Number of periods.

Note / Lưu ý: The gap between arithmetic and geometric mean grows with volatility — known as variance drag.

Practice problem

Returns over three years are 20%, –10%, and 15%. Compute the geometric mean return.

Show solution

$(1.20)(0.90)(1.15) = 1.2420$

$\bar{R}_g = 1.2420^{1/3} - 1 = 1.0750 - 1$

Geometric mean ≈ 7.50%

5. Harmonic Mean Core

EN: Used when averaging reciprocal quantities — classic uses: dollar-cost averaging (fixed dollar amount per period) and aggregating P/E ratios in an index.
VN: Dùng khi trung bình các đại lượng nghịch đảo — kinh điển: cost averaging và P/E ratio bình quân.

\[ \bar{X}_H = \frac{N}{\displaystyle\sum_{i=1}^{N}\frac{1}{X_i}} \]

Components / Thành phần

$\bar{X}_H$ Harmonic mean.
$X_i$ Observation $i$ (must be positive).
$N$ Number of observations.

Inequality / Quan hệ: $\bar{X}_H \le \bar{X}_g \le \bar{X}_a$ (equality when all observations are identical).

Practice problem

An investor buys $1,000 worth of a stock each month for two months at prices of $20 and $25. What is the average price per share paid (harmonic mean)?

Show solution

$\bar{X}_H = \dfrac{2}{\frac{1}{20} + \frac{1}{25}} = \dfrac{2}{0.05 + 0.04} = \dfrac{2}{0.09}$

≈ $22.22 per share

6. Trimmed & Winsorized Mean Concept

EN: Techniques to reduce the impact of outliers when computing a mean.
VN: Kỹ thuật xử lý outlier khi tính trung bình.

Definitions / Định nghĩa

Trimmed Remove a fixed % of extreme values then compute the mean. e.g. "5% trimmed mean" drops the bottom 2.5% and top 2.5%. / Loại bỏ một tỷ lệ % giá trị cực trước khi tính mean.
Winsor. Replace extreme values with the nearest unaffected percentile, then take the mean. e.g. "90% winsorized mean" caps below P5 at P5 and above P95 at P95. / Thay giá trị cực bằng percentile gần nhất.

Practice problem

A dataset is 1, 2, 3, 4, 5, 6, 7, 8, 9, 100. Compute the 10% trimmed mean (drop one lowest and one highest value).

Show solution

Drop 1 and 100, leaving 8 values: 2, 3, 4, 5, 6, 7, 8, 9

Sum = 44, mean = 44/8

10% trimmed mean = 5.5 (vs. arithmetic mean = 14.5)

7. Money-Weighted Return (IRR) Core

EN: The internal rate of return of a portfolio's cash flows. It is sensitive to the timing and size of contributions and withdrawals — measures the investor's actual experience.
VN: IRR của dòng tiền danh mục — chịu ảnh hưởng bởi thời điểm/quy mô tiền nộp-rút. Đo kết quả thực của nhà đầu tư.

\[ \sum_{t=0}^{N} \frac{CF_t}{(1 + MWR)^t} = 0 \]

Components / Thành phần

$CF_t$ Cash flow at time $t$ (negative = contribution, positive = withdrawal or terminal value).
$MWR$ Money-weighted return (= IRR).
$N$ Number of periods.

Practice problem

At t = 0 an investor buys 1 share for $100. At t = 1 she buys another share at $110 and receives a $4 dividend on the first share. At t = 2 she sells both shares at $120 each and receives $4 dividend per share. Compute the money-weighted return.

Show solution

CF₀ = –100; CF₁ = –110 + 4 = –106; CF₂ = 2(120) + 2(4) = 248

$-100 - \dfrac{106}{1+r} + \dfrac{248}{(1+r)^2} = 0$

Solve with a financial calculator (CF, IRR keys): IRR ≈ 9.39%

MWR ≈ 9.39%

8. Time-Weighted Return (TWR) Core

EN: The compound growth rate of $1 invested across sub-periods, independent of the timing/size of cash flows. Standard for evaluating manager skill (GIPS).
VN: Compound growth của $1 qua các sub-period — không phụ thuộc dòng tiền nộp-rút. Tiêu chuẩn đánh giá kỹ năng quản lý quỹ.

\[ TWR = \left[(1+R_1)(1+R_2)\cdots(1+R_N)\right]^{1/N} - 1 \]

Components / Thành phần

$R_i$ HPR of sub-period $i$ (between two adjacent cash flows).
$N$ Number of sub-periods (annualize if each < 1 year).

MWR vs TWR: If no interim cash flows → MWR = TWR. If the investor adds money before a high-return period → MWR > TWR.

Practice problem

Using the same data as the MWR problem above, compute the annualized time-weighted return.

Show solution

Period 1 HPR: $R_1 = \dfrac{110 + 4 - 100}{100} = 14\%$

Period 2 HPR: $R_2 = \dfrac{2(120) + 2(4) - 2(110)}{2(110)} = \dfrac{28}{220} \approx 12.73\%$

$TWR = [(1.14)(1.1273)]^{1/2} - 1 = (1.2851)^{0.5} - 1$

TWR ≈ 13.36%

9. Annualized Return Core

EN: Convert a sub-period return (week, month, quarter) into an equivalent annual return for comparison.
VN: Quy đổi lợi suất 1 kỳ ngắn sang lợi suất tương đương 1 năm.

\[ R_{annual} = (1 + R_{period})^{c} - 1 \]

Components / Thành phần

$R_{period}$ Return in one sub-period.
$c$ Number of sub-periods per year (12 months, 52 weeks, 252 trading days, etc.).

Practice problem

A portfolio returned 3% over one quarter. What is the annualized return?

Show solution

$R_{annual} = (1.03)^{4} - 1 = 1.1255 - 1$

≈ 12.55%

10. Continuously Compounded Return Core

EN: Log-return — additive over time, widely used in option pricing (Black-Scholes) and financial statistics.
VN: Log-return — có tính chất cộng dồn qua thời gian, dùng trong định giá option và thống kê tài chính.

\[ r_{cc} = \ln(1 + R) = \ln\!\left(\frac{P_1}{P_0}\right) \]

Components / Thành phần

$r_{cc}$ Continuously compounded return.
$R$ Simple HPR over the period.
$P_0,\ P_1$ Beginning / ending price.

Additivity: $r_{cc}^{(0,T)} = r_{cc}^{(0,1)} + r_{cc}^{(1,2)} + \dots + r_{cc}^{(T-1,T)}$.

Practice problem

A one-year HPR is 12%. What is the continuously compounded return?

Show solution

$r_{cc} = \ln(1.12)$

≈ 0.1133 = 11.33%

11. Gross / Net / After-tax / Leveraged Return Concept

EN: Variants of return commonly seen in fund reporting.
VN: Các biến thể lợi suất thường gặp trong báo cáo quỹ.

\[ R_{net} = R_{gross} - \text{fees} - \text{expenses} \] \[ R_{after\text{-}tax} = R_{pre\text{-}tax}\,(1 - t) \] \[ R_{leveraged} = \frac{V_E\,R_{portfolio} - V_B\,r_D}{V_E} \]

Components / Thành phần

Gross Before management/trading fees / Trước phí.
Net After all fees — what investor actually earns / Sau phí.
$t$ Marginal tax rate (capital gains / income).
$V_E,\ V_B$ Equity value and borrowed amount; $r_D$ = borrowing cost.

Practice problem

A $200,000 portfolio is funded with $150,000 of equity and $50,000 borrowed at 5%. The portfolio earns 10% on total assets. What is the leveraged return on equity?

Show solution

Total profit: 200,000 × 10% = 20,000

Interest expense: 50,000 × 5% = 2,500

Profit on equity: 20,000 – 2,500 = 17,500

Leveraged return = 17,500 / 150,000 ≈ 11.67%