**📅 Date:** ➤ ⌈ [[2025-10-22-Wed〚 TWRR▪ Fixed-Income Securities〛]]⌋
**💭 What:** Book01-p21
➤ 加投的情况下使用time weighted Return， 在个人的投资计划里也是常用的，比如定期投资
➤ 结果比平均数小因为考虑到了时间，有“惩罚收益小的时间，实际上收益小的时间是离现在更近的
➤ 用每一期投资的收益去寻找投资在时间加权下的回报率
**👀 Snap:**
![[IMG_8888 1.jpeg]]
⇩ 🅻🅸🅽🅺🆂 ⇩
**🏷️ Tags**: #💰/Economy #💰/Formula
**🗂 Menu**:
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➤ ⌈[[💰L075-02-Fixed-Income Securities (固定收益证券)]]⌋
---
## Overview
The **Time-Weighted Rate of Return (TWRR)** measures how efficiently an investment grows over time, **independent of cash inflows or outflows**.
It captures the **true performance of the portfolio manager**, not the investor’s timing of contributions or withdrawals.
> 💡 TWRR answers: _“How well did the portfolio itself perform, regardless of when money went in or out?”_
---
## 🧩 Definition
TWRR represents the **compound rate of growth per unit of time**, assuming **no external cash flows**.
It removes the distortion caused by investors adding or withdrawing funds.
---
## 🧮 Formula
1️⃣ Divide the total period into sub-periods between each cash flow.
2️⃣ Compute each sub-period’s **Holding Period Return (HPR)**:
$
HPR_i = \frac{V_{1,i} - V_{0,i}}{V_{0,i}}
$
3️⃣ Compound all sub-period returns:
$
TWRR = \left( \prod_{i=1}^{n} (1 + HPR_i) \right)^{\frac{1}{n}} - 1
$
Where:
- V_{0,i} = value at the beginning of sub-period _i_
- V_{1,i} = value at the end of sub-period _i_
- n = number of sub-periods
---
## **📊 Example**
|**Sub-period**|**Start Value**|**End Value**|**External Flow**|**HPR**|
|---|---|---|---|---|
|1|$100,000|$110,000|0|+10%|
|2|$160,000|$152,000|+$50,000 (added)|–5%|
Calculation:
$
TWRR = \sqrt{(1 + 0.10)(1 - 0.05)} - 1 = 2.47%
$
→ Portfolio’s _true performance_ = **+2.47%**, regardless of cash inflows.
---
## **⚖️ Interpretation**
|**Attribute**|**Description**|
|---|---|
|**Measures**|Portfolio performance, not investor behavior.|
|**Cash Flow Sensitivity**|Neutral — removes the impact of timing.|
|**Use Case**|Evaluate portfolio manager skill.|
|**Best For**|Comparing funds with irregular cash flows.|
---
## **🧠 Insights**
- **TWRR ≠ Money-Weighted Return (MWR)** — MWR includes investor timing.
- Ideal when **external cash flows** are **not controlled by the manager**.
- Reflects **pure investment performance** — comparable across funds.
- **Geometric mean**, not arithmetic — captures compounding effects.
---
## **🧭 Quick Recap**
- **TWRR = Compounded Growth Rate (no cash flow effect)**
- **MWR = IRR (cash flow dependent)**
- **Use TWRR** for fund performance; **MWR** for investor experience.
- Formula core:
$
TWRR = \left( \prod_{i=1}^{n} (1 + HPR_i) \right)^{1/n} - 1
$