FINA 3103: Intermediate Investments

Lecture 6. Stocks

Don Noh

Hong Kong University of Science and Technology

Spring 2025
Valuation
S&P Composite Index

• Founded in 1860, a value-weighted portfolio of large and liquid US stocks.

• Became the S&P 500 Index in 1957.
• Price-dividend ratio: current price index divided by the dividends over the prior year
◦ Its inverse (i.e., dividend-price ratio) is often referred to as the dividend yield.

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Price-Dividend Ratio for the S&P Composite Index

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Interpreting the Price-Dividend Ratio

• Mean reversion: tendency to rise when low and fall when high
• “Buy low and sell high”
◦ Go into the stock market when the price-dividend ratio is low.
◦ Sell out of the stock market when the price-dividend ratio is high.
• Two distinct periods.
◦ Mean of 23 through 1981
◦ Higher mean of 47 since then
• Is the price-dividend ratio permanently higher since late 1980s?

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A Probability Refresher

• A random variable Xt+1 at future date t + 1

• Its expectation at date t is Et ( Xt+1 ) .
• The law of iterated expectations (LIE) says that

E0 ( Et ( Xt+1 )) = E0 ( Xt+1 ) .

• Rather than a formal proof, let’s look at an example.

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An Example of LIE
• Flip a coin today
◦ If heads, flip a coin again tomorrow and receive $16 if heads (and $0 if tails)
◦ If tails, flip a coin again tomorrow and receive $4 if heads (and $0 if tails)
• Two ways to compute the expected value today
1. Use the formula E0 ( E1 ( X2 ))

EV = 0.5 × ( 0.5 × $16 + 0.5 × $0) + 0.5 × ( 0.5 × $4 + 0.5 × $0)

| {z } | {z }
heads today tails today
= 0.5 × $8 + 0.5 × $2 = $5

2. Use the formula E0 ( X2 )

EV = 0.25 × $16 + 0.25 × $0 + 0.25 × $4 + 0.25 × $0

| {z } | {z } | {z } | {z }
heads-heads heads-tails tails-heads tails-tails
= $4 + $1 = $5

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A Math Refresher

• Let 0 < x < 1 be a constant

• A geometric sum is
∞
∑︁
xi = 1 + x + x2 + x3 + · · ·
i=0
1
= .
1−x
• If we start the sum from i = 1,
∞
∑︁ 1 x
xi = −1= .
i=1
1−x 1−x

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Dividend Discount Model

• Definition of return in year t + 1:

Pt+1 + Dt+1
1 + Rt+1 = .
Pt
• Assume constant expected returns to get:

Et ( Pt+1 + Dt+1 )
Et ( 1 + Rt+1 ) = 1 + R̄ = .
Pt
• Rewrite the above as:
Et ( Dt+1 ) Et ( Pt+1 )
Pt = + .
1 + R̄ 1 + R̄

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Dividend Discount Model

• Start at t = 0 and substitute out future price recursively:

E0 ( D1 ) E0 ( P1 )
P0 = +
1 + R̄ 1 + R̄  
E0 ( D 1 ) 1 E1 ( D2 ) E1 ( D2 )
= + E0 +
1 + R̄ 1 + R̄ 1 + R̄ 1 + R̄
E0 ( D1 ) E0 ( D2 ) E0 ( P2 )
= + + = ··· .
1 + R̄ ( 1 + R̄) 2 ( 1 + R̄) 2

• We arrive at the dividend discount model:

∞
∑︁ E0 ( Dt )
P0 = .
t=1
( 1 + R̄) t

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Gordon Growth Model

• Assume that dividend growth is a constant G < R̄ per year.

 
Dt
E0 = ( 1 + G) t
D0

• Using the dividend discount model, we get:

∞
P0 ∑︁ E0 ( Dt /D0 )
=
D0 t=1 ( 1 + R̄) t
∞ ∞  t
∑︁ ( 1 + G) t ∑︁ 1 + G
= =
t=1
( 1 + R̄) t t =1
1 + R̄
  −1
( 1 + G) 1+G 1+G
= 1− =
1 + R̄ 1 + R̄ R̄ − G

where we used the geometric sum formula with x = ( 1 + G)/( 1 + R̄) to get to the last line.

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Interpreting the Gordon Growth Model

P0 1+G
=
D0 R̄ − G
• When the price-dividend ratio is high, either:
1. Expected returns R̄ are low (forecasts low future returns).
2. Expected dividend growth G is high (forecasts high future dividend growth).
• Empirically, (1) is true: That is, a high price-dividend ratio forecasts low future returns.
• A high price-dividend ratio is a warning that a correction may be ahead (e.g., the dot-com bubble).
• A low price-dividend ratio signals a buying opportunity (e.g., the 2008 financial crisis).

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Why is the Price-Dividend Ratio Higher since the 1980s?

• Firms switched from cash dividends to share repurchases (which are tax advantaged) after the SEC
Rule 10b-18 in 1982.
◦ Dividends may no longer be a good signal of fundamental value.
◦ Need to look at earnings or adjusted dividends that incorporate share repurchases.
• Investors are less risk-averse, so willing to hold the stock market at lower expected returns.
◦ Recall that stock returns are a compensation for bearing (systematic) risk.

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Price-Earnings Ratio for the S&P Composite Index

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Predictability
S&P Composite Index (1920–2017)

• Equity premium: Average excess returns of stocks relative to T-bills is 8.6% annually.

Statistic T-bills Stocks Stocks−T-bills

Mean 3.4% 12.1% 8.6%
Standard deviation 3.0% 19.7% 19.8%

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Testing Return Predictability

• Write stock returns as

Rs,t+1 = Rb,t+1 + Rs,t+1 − Rb,t+1
|{z} | {z }
riskless return excess return

• The riskless return is forecastable, but not obvious that the excess return is.
• Test whether the price-dividend ratio forecasts future excess returns through a time-series regression:

Rs,t+1 − Rb,t+1 = 𝛼 + 𝛽 Xt + 𝜀 t+1

• Null hypothesis (𝛽 = 0): Price-dividend ratio does not forecast future returns.
• Alternative hypothesis (𝛽 < 0): Price-dividend ratio forecasts future returns.

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Example 1: Estimated Regression

Rs,t+1 − Rb,t+1 = 𝛼 + 𝛽 Xt + 𝜀 t+1

Variable Sample Estimate t-stat

P/D 1920–2016 −0.002 −1.91
1920–1981 −0.010 −2.63
1982–2016 −0.003 −2.09
P/E10 1920–2016 −0.007 −2.68

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P/E10 vs Subsequent 5-Year Returns

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Average 5-Year Returns by P/E10

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Example 2: Implications for Portfolio Choice

• Recall the optimal portfolio weight on stocks from the lecture on Risk and Return:

E ( Rs − Rb )
w∗ =
𝛾𝜎 2

• Historically, E ( Rs − Rb ) = 8.6% and 𝜎 = 19.7%.

• For an investor with risk aversion 𝛾 = 5,

8.6%
w∗ = = 0.44.
5 ( 19.7%) 2

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Market-Timing Strategy

• However, we just learned that expected excess returns are time varying:

Et ( Rs,t+1 − Rb,t+1 ) = 𝛼 + 𝛽 Xt .

• So the optimal portfolio weight should also be time-varying:

Et ( Rs,t+1 − Rb,t+1 ) 𝛼 + 𝛽 Xt
w∗t = = .
𝛾𝜎 2 𝛾𝜎 2

• Since 𝛽 < 0, you must “buy low and sell high.”

◦ Increase share in stocks when the price-earnings ratio is low
◦ Decrease share in stocks when the price-earnings ratio is high

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Expected Excess Return

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Optimal Portfolio Weight on Stocks

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Shorting the Stock Market in the Late 1990s?

• Greenspan warned of “irrational exuberance” in a televised speech on 5/12/1996.

• “Crash, dammit” (The Economist, 16/10/1997)
• But most investors had optimistic views...

Newsweek, 5/7/1999 Time, 27/9/1999

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History of Economic Thought

Robert Shiller (1946–): Nobel Prize in 2013 for return predictability and
behavioral finance

• Published “Irrational Exuberance” (March 2000), predicting the dot-com crash

• Published “Is There a Bubble in the Housing Market?” (Brookings Papers, 2003)

“So a very plausible scenario is that home-price increases continue for a couple more years, and
then we might have a recession and they continue down into negative territory and languish
for a decade.”
Robert Shiller (2005, NYT interview)

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Interest Rates
3-Month T-Bill Rate

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Yield Spread (10-Year Minus 3-Month)

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Example 1 (continued): Estimated Regression

Variable Sample Estimate t-stat

T-bill rate 1920–2016 −1.04 −1.55
Yield spread 1920–2016 2.51 1.66

• T-bill rate: A high nominal interest rate is related to high inflation, which is bad news for stocks.
• Yield spread: Yield curve steepens in recessions when prices of risky assets fall, including long-term
bonds and stocks.

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Cross-Section
Applying the Gordon Growth Model to Individual Stocks

• The dividend-price ratio for stock i according to the Gordon growth model:

Di R̄i − Gi
=
Pi 1 + Gi
• Stocks that are trading at high dividend-price ratios are expected to have:
1. high future returns
2. low future dividend growth.
• Empirically, (1) is true.
• Value strategy: buy stocks with high dividend-price ratios (and short stocks with low dividend-price
ratios)

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Various Flavors of Value

1. Dividend-price ratio (a.k.a. dividend yield)

◦ Not all firms pay dividends.
2. Earnings-price ratio (a.k.a. earnings yield)
◦ Younger firms may not have earnings yet.
3. Book-to-market equity (preferred measure among academics)
◦ Book equity is a measure of shareholders’ equity, based on historical cost accounting.

• Ratios of “fundamentals” to market value

◦ Value stocks have high dividend-price, earnings-price, or book-to-market equity.
◦ Growth stocks have low dividend-price, earnings-price, or book-to-market equity.

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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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Cumulative Returns on Low vs High Book-to-Market

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Cumulative Returns on Low vs High Book-to-Market (1928–1938)

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Cumulative Returns on Low vs High Book-to-Market (2006–2016)

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Example 3: 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 1.01 0.95 0.97 1.05 1.00 1.03 1.10 1.14 1.28 1.46
Alpha (%) − 0.07 0.07 0.06 − 0.04 0.06 0.11 − 0.01 0.16 0.20 0.08
Alpha (t-stat) − 1.29 1.57 1.27 − 0.68 1.11 1.52 − 0.19 1.88 1.95 0.57

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Alphas on Portfolios Sorted by Book-to-Market

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Summary

• Market-timing strategy: increase allocation to stocks relative to the riskless asset when the
price-earnings ratio is low
• Value strategy: invest in value stocks that are trading at high book-to-market equity

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Next Steps

• Multi-factor models that generalize the CAPM and allow more sources of systematic risk to be priced.
• Use the multi-factor model for performance evaluation and event studies.

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