FINA 3103: Intermediate Investments

Lecture 7. APT

Don Noh

Hong Kong University of Science and Technology

Spring 2025
APT
Going Back to CAPM

• From the lecture on CAPM, recall:

E ( Ri − Rb ) = 𝛽 i,m E ( Rm − Rb ) = 𝛽 i,m E ( Fm ) .

• This is a one-factor model.

• One interpretation is:

𝛽 i,m : Amount (quantity) of systematic risk you take on

E ( Fm ) : Reward (price) you get per unit of that risk.

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Multi-Factor Models

• We can generalize to multiple sources of systematic risk:

J
∑︁
E ( Ri − Rb ) = 𝛽 i,1 E ( F1 ) + · · · + 𝛽 i,J E ( FJ ) = 𝛽 i,j E ( Fj ) .
j=1

• The factors could be:

1. Portfolio returns on various trading strategies (e.g., market excess return).
2. Macro factors (GDP growth, inflation, etc.).
• We will focus on (1) in this class, mainly because they work well empirically.

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Betas of Portfolios

• Suppose we’re in the CAPM world for now.

• One algebra to remember for portfolio p’s return:
N
∑︁
E ( Rp − Rb ) = wi E ( Ri − Rb )
i=1
N
∑︁
= wi 𝛽 i,m E ( Rm − Rb )
i=1
N
∑︁
= E ( Rm − Rb ) wi 𝛽 i,m
i=1

so that the portfolio betas are weighted-averaged betas.

• You can extend this to the case with multiple factors.

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Interpreting a Multi-Factor Model

• When you go shopping for fruit,

Price of basket = Q ( bananas) × P ( banana) + Q ( apples) × P ( apple)

• The “price” of asset i is:

J
∑︁
E ( Ri − Rb ) = 𝛽 i,j E ( Fj )
j=1

◦ 𝛽 i,j is the quantity of risk factor j that asset i has.

◦ E ( Fj ) is the price of risk factor j.
• Recall that only systematic risk is priced (i.e., E ( Fj ) = 0 for idiosyncratic risk), because of the “portfolio
view” from the CAPM lecture.

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Arbitrage Pricing Theory (APT)

• The APT is one way to arrive at multi-factor models.

• Three main assumptions of APT:
1. Security returns can be described by a factor model (e.g., the market model).
2. There are sufficient securities to diversify away idiosyncratic risk.
3. Well-functioning security markets do not allow for the persistence of arbitrage opportunities.

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Arbitrage Pricing Theory (APT)

• Assumption (1) means (in a simplified two-factor case) that

R̃i − Rb = 𝛼 i + 𝛽 i,1 F̃1 + 𝛽 i,2 F̃2 + 𝜀 i ,

where E (𝜀 i ) = 0 and cov (𝜀 i , 𝜀 j ) = 0 for i ≠ j.

• Assumption (2) means that you have enough assets to diversify away the idiosyncratic risk 𝜀 i as in the
lecture on Diversification: !
N
1 ∑︁
var 𝜀i → 0
N i=1
• Assumption (3) implies that 𝛼 i = 0 for all i due to arbitrage (similar to CAPM).

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Arbitrage Pricing Theory (APT)

• Combining these assumptions, we get:

E ( Ri − Rb ) = 𝛽 i,1 E ( F1 ) + 𝛽 i,2 E ( F2 ) .

• No formal proof is provided, but here is a rough thought process:

1. Suppose assets have 𝛼 i ≠ 0.
2. Then, you can form a well-diversified portfolio P of assets with zero idiosyncratic risk (because we will put
many assets in P) and 𝛽 P,1 = 0 and 𝛽 P,2 = 0 (because we can choose the weights).
3. Most likely, 𝛼 P ≠ 0.
4. Then, this portfolio has zero risk but a non-zero expected excess return.
5. This is an arbitrage opportunity, thus we reached a contradiction.

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3-Factor Model
Fama-French 3-Factor Model

• Divide all stocks into 6 buckets.

ME < 50th ME > 50th
B/M > 70th Small Value Big Value
Small Neutral Big Neutral
B/M < 30th Small Growth Big Growth

• Then, average returns of stocks within the buckets are computed.

• The differences of the bucket returns will be used as factors.

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Fama-French 3-Factor Model

1. Market factor: market return minus riskless rate

2. Size factor: small minus big (SMB) market equity
1
SMB = ( Small Value + Small Neutral + Small Growth)
3
1
− ( Big Value + Big Neutral + Big Growth)
3
3. Value factor: high minus low (HML) book-to-market equity
1 1
HML = ( Small Value + Big Value) − ( Small Growth + Big Growth)
2 2

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Interpreting the 3-Factor Model

• According to the Fama-French 3-factor model, expected return on asset i is

E ( Ri − Rb ) = 𝛽 i,m E ( Rm − Rb ) + 𝛽 i,SMB E ( SMB) + 𝛽 i,HML E ( HML)

1. Market beta
2. SMB beta
◦ 𝛽 i,SMB > 0: Portfolio tilts toward small-cap stocks.
◦ 𝛽 i,SMB < 0: Portfolio tilts toward large-cap stocks.
3. HML beta
◦ 𝛽 i,HML > 0: Portfolio tilts toward value stocks.
◦ 𝛽 i,HML < 0: Portfolio tilts toward growth stocks.

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Estimating the 3-Factor Model

• Using time-series data on returns, estimate a linear regression for each asset i.

Ri,t − Rb,t = 𝛼 i + 𝛽 i,m ( Rm,t − Rb,t ) + 𝛽 i,SMB SMBt + 𝛽 i,HML HMLt + 𝜀 i,t

• 𝛼 i is the alpha.
◦ Test whether 𝛼 i = 0 for all assets
◦ If 𝛼 i > 0, average return is too high relative to risk.
• The residual 𝜀 i,t represents idiosyncratic risk that is uncorrelated with the factors.
• Data on factors and portfolio returns available at Ken French’s website.

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Constructing Portfolios Sorted by Market Equity

1. Sort stocks into 10 portfolios based on market equity (i.e., cutoff at each 10th percentile).
2. Compute returns on these portfolios over the subsequent year.
3. Rebalance annually.

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Example 1: Portfolios Sorted by Market Equity

Portfolio
Statistics Low 2 3 4 5 6 7 8 9 High
Mean (%) 1.13 1.00 0.99 0.95 0.90 0.91 0.83 0.81 0.74 0.62
SD (%) 9.90 8.69 7.92 7.40 7.01 6.76 6.39 6.11 5.77 5.03
Sharpe ratio 0.11 0.11 0.12 0.13 0.13 0.13 0.13 0.13 0.13 0.12
Market beta 1.00 1.07 1.08 1.04 1.06 1.07 1.05 1.05 1.03 0.98
SMB beta 1.56 1.27 1.00 0.89 0.71 0.50 0.41 0.24 0.06 − 0.22
HML beta 0.78 0.48 0.37 0.31 0.20 0.22 0.13 0.12 0.12 − 0.03
Alpha (%) − 0.16 − 0.16 − 0.09 − 0.05 − 0.04 0.00 0.00 0.01 − 0.01 0.02
Alpha (t-stat) − 1.63 − 3.11 − 2.35 − 1.34 − 1.12 0.09 − 0.11 0.37 − 0.32 1.64

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Interpreting the 3-Factor Regression

• Portfolio 1:

R1,t − Rb,t = −0.16% + 1.00 × ( Rm,t − Rb,t ) + 1.56 × SMBt + 0.78 × HMLt + 𝜀 1,t

• Portfolio strategy:
◦ 1.00 × ( Rm,t − Rb,t ) : long $1 in market
◦ 1.56 × SMBt : long $1.56 in Small and short $1.56 in Big
◦ 0.78 × HMLt : long $0.78 in High and short $0.78 in Low

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Interpreting the 3-Factor Regression

• Portfolio 10:

R10,t − Rb,t = 0.02% + 0.98 × ( Rm,t − Rb,t ) − 0.22 × SMBt − 0.03 × HMLt + 𝜀 10,t

• Portfolio strategy:
◦ 0.98 × ( Rm,t − Rb,t ) : long $0.98 in market and long $0.02 in riskless
◦ −0.22 × SMBt : short $0.22 in Small and long $0.22 in Big
◦ −0.03 × HMLt : short $0.03 in High and long $0.03 in Low

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Cumulative Returns on Low vs High Market Equity

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Cumulative Returns on Low vs High Market Equity (1980–2018)

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Constructing Portfolios Sorted by Book-to-Market Equity

1. Sort stocks into 10 portfolios based on book-to-market equity (i.e., cutoff at each 10th percentile).
2. Compute returns on these portfolios over the subsequent year.
3. Rebalance annually.

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Example 2: Portfolios Sorted by Book-to-Market

Portfolio
Statistics Low 2 3 4 5 6 7 8 9 High
Mean (%) 0.61 0.71 0.70 0.66 0.73 0.80 0.72 0.92 1.06 1.06
SD (%) 5.68 5.28 5.37 5.89 5.67 5.97 6.40 6.68 7.64 9.17
Sharpe ratio 0.11 0.13 0.13 0.11 0.13 0.13 0.11 0.14 0.14 0.12
Market beta 1.08 0.98 0.99 1.03 0.98 0.98 1.01 1.02 1.11 1.19
SMB beta − 0.07 0.00 − 0.04 − 0.04 − 0.08 − 0.06 0.02 0.11 0.22 0.53
HML beta − 0.34 − 0.20 − 0.05 0.15 0.27 0.43 0.55 0.67 0.85 1.09
Alpha (%) 0.03 0.13 0.07 − 0.08 0.00 0.00 − 0.16 − 0.03 − 0.04 − 0.25
Alpha (t-stat) 0.62 2.94 1.66 − 1.39 0.00 0.01 − 3.00 − 0.59 − 0.75 − 3.30

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Interpreting the 3-Factor Regression

• Portfolio 1:

R1,t − Rb,t = −0.03% + 1.08 × ( Rm,t − Rb,t ) − 0.07 × SMBt − 0.34 × HMLt + 𝜀 1,t

• Portfolio strategy:
◦ 1.08 × ( Rm,t − Rb,t ) : long $1.08 in market and short $0.08 in riskless
◦ −0.07 × SMBt : short $0.07 in Small and long $0.07 in Big
◦ −0.34 × HMLt : short $0.34 in High and long $0.34 in Low

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Interpreting the 3-Factor Regression

• Portfolio 10:

R10,t − Rb,t = −0.25% + 1.19 × ( Rm,t − Rb,t ) + 0.53 × SMBt − 1.09 × HMLt + 𝜀 10,t

• Portfolio strategy:
◦ 1.19 × ( Rm,t − Rb,t ) : long $1.19 in market and short $0.19 in riskless
◦ 0.53 × SMBt : long $0.53 in Small and short $0.53 in Big
◦ 1.09 × HMLt : long $1.09 in High and short $1.09 in Low

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Cumulative Returns on Low vs High Book-to-Market

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Cumulative Returns on Low vs High Book-to-Market (2000–2018)

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Constructing Portfolios Sorted by Prior Returns

1. Sort stocks into 10 portfolios based on prior annual return (i.e., cutoff at each 10th percentile).
2. Compute returns on these portfolios over the subsequent month.
3. Rebalance monthly.

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Example 3: Portfolios Sorted by Prior Returns

Portfolio
Statistics Low 2 3 4 5 6 7 8 9 High
Mean (%) 0.06 0.42 0.49 0.61 0.62 0.68 0.74 0.86 0.93 1.24
SD (%) 9.73 8.00 6.94 6.32 5.91 5.77 5.47 5.31 5.59 6.45
Sharpe ratio 0.01 0.05 0.07 0.10 0.10 0.12 0.13 0.16 0.17 0.19
Market beta 1.40 1.24 1.13 1.06 1.01 1.01 0.98 0.95 0.98 1.02
SMB beta 0.49 0.13 − 0.02 − 0.04 − 0.06 − 0.04 − 0.10 − 0.08 − 0.03 0.29
HML beta 0.43 0.39 0.33 0.25 0.24 0.16 0.07 0.00 − 0.06 − 0.31
Alpha (%) − 1.14 − 0.57 − 0.38 − 0.18 − 0.13 − 0.04 0.08 0.25 0.31 0.62
Alpha (t-stat) − 8.12 − 5.47 − 4.54 − 2.60 − 2.22 − 0.77 1.63 4.46 4.61 6.43

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Cumulative Returns on Winner vs Loser Portfolios

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Summary of Anomalies

1. Low beta: Low-beta stocks outperform high-beta stocks (relative to CAPM).

2. Low volatility: Low-volatility stocks outperform high-volatility stocks.
3. Value: High book-to-market equity stocks outperform low book-to-market equity stocks.
4. Momentum: Winner stocks (i.e., high prior annual return) outperform loser stocks.
• These “anomalies” also exist in international equity markets.
• Tradable strategies as ETFs.
• Warning: Past performance does not necessarily predict future results.

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