quantitative-valuation

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Quantitative Valuation

量化估值

Purpose

用途

Provides frameworks and formulas for estimating the intrinsic value of a stock or company using quantitative models. Covers discounted cash flow (DCF), dividend discount models (DDM), comparable multiples, residual income, and sum-of-the-parts valuation. Enables rigorous, assumption-driven valuation that can be stress-tested through sensitivity analysis.

提供使用量化模型估算股票或公司内在价值的框架和公式，覆盖贴现现金流（DCF）、股利贴现模型（DDM）、可比乘数、剩余收益和分类加总估值法，支持严谨的、基于假设的估值，可通过敏感性分析进行压力测试。

Layer

层级

3 — Valuation

3 — 估值

Direction

适用方向

prospective

前瞻性

When to Use

适用场景

Valuing a stock or company from first principles
Building a discounted cash flow (DCF) model
Comparing a company's multiples to peers or its own history
Estimating intrinsic value using dividend discount models
Performing sum-of-the-parts valuation for a conglomerate
Running sensitivity analysis on valuation assumptions
Screening for undervalued or overvalued securities using relative multiples

从基础原理出发对股票或公司进行估值
搭建贴现现金流（DCF）模型
对比公司与同行或自身历史的乘数水平
使用股利贴现模型估算内在价值
对 conglomerate（集团企业）进行分类加总估值
对估值假设开展敏感性分析
使用相对乘数筛选被高估或低估的证券

Core Concepts

核心概念

Discounted Cash Flow (DCF)

贴现现金流（DCF）

The DCF model values a company as the present value of its future free cash flows plus a terminal value:

V = Σ FCF_t / (1 + WACC)^t + TV / (1 + WACC)^n

where FCF_t is the free cash flow in year t, WACC is the weighted average cost of capital, and TV is the terminal value at the end of the explicit forecast period.

DCF模型将公司价值计算为未来自由现金流的现值加上终值：

V = Σ FCF_t / (1 + WACC)^t + TV / (1 + WACC)^n

其中FCF_t为第t年的自由现金流，WACC为加权平均资本成本，TV为明确预测期期末的终值。

Terminal Value — Gordon Growth Model

终值 — Gordon增长模型

Estimates the value of all cash flows beyond the explicit forecast period assuming perpetual growth:

TV = FCF_n × (1 + g) / (WACC - g)

where g is the long-term sustainable growth rate (typically near nominal GDP growth, 2-4%).

假设现金流永续增长，估算明确预测期之后所有现金流的价值：

TV = FCF_n × (1 + g) / (WACC - g)

其中g为长期可持续增长率（通常接近名义GDP增速，2%-4%）。

Terminal Value — Exit Multiple Method

终值 — 退出乘数法

Estimates terminal value by applying a market multiple to the final-year financial metric:

TV = EBITDA_n × EV/EBITDA multiple

The exit multiple is typically based on current peer trading multiples or long-run sector averages.

通过对最后一年的财务指标应用市场乘数估算终值：

TV = EBITDA_n × EV/EBITDA multiple

退出乘数通常基于当前同行交易乘数或长期行业平均值。

Weighted Average Cost of Capital (WACC)

加权平均资本成本（WACC）

Blends the cost of equity and after-tax cost of debt weighted by their market-value proportions:

WACC = w_e × r_e + w_d × r_d × (1 - τ)

where w_e and w_d are equity and debt weights, r_e and r_d are their respective costs, and τ is the marginal tax rate.

按市值比例加权融合股权成本和税后债务成本：

WACC = w_e × r_e + w_d × r_d × (1 - τ)

其中w_e和w_d为股权和债务的权重，r_e和r_d分别为对应的成本，τ为边际税率。

Cost of Equity — CAPM

股权成本 — CAPM

The Capital Asset Pricing Model estimates the required return on equity:

r_e = R_f + β × (R_m - R_f)

where R_f is the risk-free rate, β is the stock's sensitivity to market returns, and (R_m - R_f) is the equity risk premium.

资本资产定价模型用于估算股权的必要回报率：

r_e = R_f + β × (R_m - R_f)

其中R_f为无风险利率，β为股票对市场收益的敏感度，(R_m - R_f)为股权风险溢价。

Dividend Discount Model (DDM)

股利贴现模型（DDM）

Values a stock as the present value of its future dividends. The Gordon Growth (single-stage) form:

P = D_1 / (r - g)

where D_1 is the next-period dividend, r is the required return, and g is the constant dividend growth rate.

将股票价值计算为未来股利的现值，Gordon增长（单阶段）形式为：

P = D_1 / (r - g)

其中D_1为下一期股利，r为必要回报率，g为固定股利增长率。

Multi-Stage DDM

多阶段DDM

Accommodates companies transitioning through growth phases:

Stage 1 (High growth): Dividends grow at g_1 for n years
Stage 2 (Transition): Growth declines linearly from g_1 to g_3
Stage 3 (Stable): Dividends grow at g_3 in perpetuity (valued via Gordon Growth)

适配经历不同增长阶段的公司：

第一阶段（高增长）： 股利以g_1的增速增长n年
第二阶段（过渡期）： 增速从g_1线性下降到g_3
第三阶段（稳定期）： 股利以g_3永续增长（通过Gordon增长模型估值）

Residual Income Model

剩余收益模型

Values a company as its book value plus the present value of economic profits:

V = BV_0 + Σ (ROE - r) × BV_{t-1} / (1 + r)^t

This model is useful when free cash flows are negative but the company earns above its cost of equity.

将公司价值计算为账面价值加上经济利润的现值：

V = BV_0 + Σ (ROE - r) × BV_{t-1} / (1 + r)^t

该模型适用于自由现金流为负但股权回报率高于成本的公司。

Comparable Multiples

可比乘数

Relative valuation uses pricing ratios from a peer group to infer value:

P/E (Price-to-Earnings): most common for profitable companies
EV/EBITDA (Enterprise Value to EBITDA): capital-structure neutral
P/S (Price-to-Sales): useful for unprofitable or early-stage companies
P/B (Price-to-Book): useful for asset-heavy businesses (banks, REITs)

Use the median of the peer group to reduce outlier effects. Adjust for differences in growth, margins, and risk.

相对估值使用同行组的定价比率推断价值：

P/E（市盈率）：最常用于盈利公司
EV/EBITDA（企业价值/息税折旧摊销前利润）：不受资本结构影响
P/S（市销率）：适用于未盈利或早期公司
P/B（市净率）：适用于重资产业务（银行、REITs）

使用同行组的中位数降低异常值影响，需针对增长、利润率和风险差异进行调整。

Relative Valuation

相对估值

Compare a stock's current multiple to:

Its own historical average (time-series comparison)
Sector or industry median (cross-sectional comparison)

A stock trading at a discount to both may be undervalued, or there may be fundamental deterioration.

将股票当前乘数与以下基准对比：

自身历史平均值（时间序列对比）
板块或行业中位数（横截面对比）

同时低于两个基准的股票可能被低估，也可能存在基本面恶化的情况。

Sum-of-the-Parts (SOTP)

分类加总估值法（SOTP）

Value each business segment separately using the most appropriate method (DCF, multiples, or asset-based), then sum. Subtract net debt and add non-operating assets to arrive at equity value.

使用最合适的方法（DCF、乘数或资产基础法）分别对每个业务板块估值后求和，减去净债务、加回非运营资产得到股权价值。

Sensitivity Analysis

敏感性分析

Vary key assumptions (WACC and terminal growth rate are the most impactful) in a two-way data table to understand the range of possible valuations. This exposes which assumptions drive the result.

在双向数据表中调整核心假设（WACC和终值增长率影响最大），以了解可能的估值区间，明确哪些假设对结果影响最大。

Key Formulas

核心公式

Formula	Expression	Use Case
DCF Value	V = Σ FCF_t/(1+WACC)^t + TV/(1+WACC)^n	Enterprise valuation from cash flows
Gordon Growth TV	TV = FCF_n×(1+g)/(WACC-g)	Terminal value assuming perpetual growth
Exit Multiple TV	TV = EBITDA_n × multiple	Terminal value using market multiples
WACC	WACC = w_e×r_e + w_d×r_d×(1-τ)	Blended discount rate
CAPM	r_e = R_f + β×(R_m - R_f)	Cost of equity estimation
Gordon Growth DDM	P = D_1/(r-g)	Stock value from dividends
Residual Income	V = BV_0 + Σ (ROE-r)×BV_{t-1}/(1+r)^t	Value from economic profit
Implied Value (Comps)	V = Metric × Peer Median Multiple	Relative valuation

公式	表达式	适用场景
DCF估值	V = Σ FCF_t/(1+WACC)^t + TV/(1+WACC)^n	基于现金流的企业估值
Gordon增长终值	TV = FCF_n×(1+g)/(WACC-g)	假设永续增长的终值计算
退出乘数终值	TV = EBITDA_n × multiple	使用市场乘数的终值计算
WACC	WACC = w_e×r_e + w_d×r_d×(1-τ)	混合贴现率计算
CAPM	r_e = R_f + β×(R_m - R_f)	股权成本估算
Gordon增长DDM	P = D_1/(r-g)	基于股利的股票估值
剩余收益	V = BV_0 + Σ (ROE-r)×BV_{t-1}/(1+r)^t	基于经济利润的估值
可比法隐含价值	V = 指标 × 同行中位数乘数	相对估值

Worked Examples

示例

Example 1: Two-Stage DCF

示例1：两阶段DCF

Given:

Current FCF: $100M
Stage 1: 15% FCF growth for 5 years
Terminal growth rate: 3%
WACC: 10%

Calculate: Enterprise value

Solution:

Projected free cash flows:

Year 1: $100M × 1.15 = $115.0M
Year 2: $115M × 1.15 = $132.3M
Year 3: $132.3M × 1.15 = $152.1M
Year 4: $152.1M × 1.15 = $174.9M
Year 5: $174.9M × 1.15 = $201.1M

PV of Stage 1 cash flows:

PV = $115.0/1.10 + $132.3/1.10² + $152.1/1.10³ + $174.9/1.10⁴ + $201.1/1.10⁵
PV = $104.5 + $109.3 + $114.3 + $119.5 + $124.9 = $572.5M

Terminal value (Gordon Growth):

TV = $201.1M × 1.03 / (0.10 - 0.03) = $207.2M / 0.07 = $2,959.6M
PV of TV = $2,959.6M / 1.10⁵ = $1,837.3M

Enterprise Value = $572.5M + $1,837.3M = $2,409.8M

Note: Terminal value represents 76% of total value, which is typical but underscores the importance of terminal assumptions.

已知条件：

当前自由现金流：1亿美元
第一阶段：5年自由现金流增速15%
终值增长率：3%
WACC：10%

计算： 企业价值

解答：

预测自由现金流：

第1年：1亿美元 × 1.15 = 1.15亿美元
第2年：1.15亿美元 × 1.15 = 1.323亿美元
第3年：1.323亿美元 × 1.15 = 1.521亿美元
第4年：1.521亿美元 × 1.15 = 1.749亿美元
第5年：1.749亿美元 × 1.15 = 2.011亿美元

第一阶段现金流现值：

现值 = 1.15/1.10 + 1.323/1.10² + 1.521/1.10³ + 1.749/1.10⁴ + 2.011/1.10⁵
现值 = 1.045 + 1.093 + 1.143 + 1.195 + 1.249 = 5.725亿美元

终值（Gordon增长法）：

TV = 2.011亿美元 × 1.03 / (0.10 - 0.03) = 2.072亿美元 / 0.07 = 29.596亿美元
终值现值 = 29.596亿美元 / 1.10⁵ = 18.373亿美元

企业价值 = 5.725亿美元 + 18.373亿美元 = 24.098亿美元

备注：终值占总价值的76%，这是常见情况，但也凸显了终值假设的重要性。

Example 2: Comparable P/E Analysis

示例2：可比P/E分析

Given:

Target company EPS: $5.00
Peer group P/E ratios: 15x, 17x, 18x, 19x, 22x

Calculate: Implied share price using peer median

Solution:

Peer median P/E = 18x (middle value of the sorted set)

Implied share price = EPS × Peer Median P/E = $5.00 × 18 = $90.00

If the stock trades at $75, it appears undervalued relative to peers (16.7% discount). Before concluding, check whether lower growth, margins, or higher risk justify the discount.

已知条件：

目标公司每股收益（EPS）：5美元
同行组P/E比率：15x、17x、18x、19x、22x

计算： 使用同行中位数估算隐含股价

解答：

同行P/E中位数 = 18x（排序后的中间值）

隐含股价 = EPS × 同行P/E中位数 = 5美元 × 18 = 90美元

如果股票当前交易价为75美元，看起来相对同行被低估（折价16.7%）。在得出结论前，需检查更低的增速、利润率或更高的风险是否能解释该折价。

Common Pitfalls

常见误区

Terminal value dominates DCF output (often 60-80% of total value) — scrutinize terminal assumptions carefully
Garbage-in-garbage-out: a DCF is only as good as its assumptions; false precision gives false confidence
Using trailing multiples when forward multiples are more relevant for fast-growing or cyclical companies
Not adjusting comparable multiples for differences in growth rates, margins, and capital structure
Circular reference when WACC depends on market cap which depends on the WACC-derived valuation — iterate or use target capital structure
Projecting high growth rates indefinitely without considering competitive dynamics and mean reversion
Ignoring dilution from stock-based compensation in per-share value estimates

终值在DCF结果中占比很高（通常为总价值的60-80%）—— 需仔细审核终值假设
垃圾进垃圾出：DCF的质量取决于假设的质量，虚假的精确性会带来错误的信心
对高速增长或周期性公司使用历史乘数，而远期乘数更具参考性
未针对增长率、利润率和资本结构差异调整可比乘数
WACC依赖市值，而市值又基于WACC计算的估值，会出现循环引用—— 可迭代计算或使用目标资本结构
无限期预测高增长率，未考虑竞争格局和均值回归
估算每股价值时忽略股权激励带来的稀释效应

Cross-References

交叉参考

historical-risk (wealth-management plugin, Layer 1a): historical beta estimation for CAPM inputs
forward-risk (wealth-management plugin, Layer 1b): cost of equity estimation via CAPM and factor models
financial-statements (wealth-management plugin, Layer 2): FCF and EBITDA derivation from financials
qualitative-valuation (wealth-management plugin, Layer 3): complements quantitative models with moat and quality analysis
asset-allocation (wealth-management plugin, Layer 4): valuation outputs feed into portfolio construction decisions

historical-risk（财富管理插件，层级1a）：CAPM输入项的历史beta估算
forward-risk（财富管理插件，层级1b）：通过CAPM和因子模型估算股权成本
financial-statements（财富管理插件，层级2）：从财务报表推导FCF和EBITDA
qualitative-valuation（财富管理插件，层级3）：通过护城河和质量分析补充量化模型
asset-allocation（财富管理插件，层级4）：估值输出可用于投资组合构建决策

Reference Implementation

参考实现

See

scripts/quantitative_valuation.py

for computational helpers.

计算辅助工具见

scripts/quantitative_valuation.py

。