Capital Asset Pricing
Advanced Valuation
Arnab Dutta
November 20, 24, and 27, 2023
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Overview
1. Capital Asset Pricing Model or CAPM
2. Beta Risk
3. Applications
4. Reading material: Geltner & Miller Ch. 22
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Introduction
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What is Capital Asset Pricing?
▶ Capital Asset Pricing Theory or Capital Asset Pricing Model
or simply CAPM
▶ A scientific way of determining asset prices
▶ Based on MPT and investors’ risk valuation
▶ Asset price or return = Risk-free rate + Risk premium
▶ CAPM =⇒ ex ante asset risk premium required by investors
▶ Used by investors to calculate asset prices across the board
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Why do investors care about CAPM?
1. Assets may be mispriced in markets in the short-run
▶ Relevant for any investor looking for short-run gains
2. Provides an idea of market’s risk valuation
▶ Individual investors can tilt their portfolios
▶ Individual preferences depart from aggregate preferences
3. Benchmarking investment performance
▶ Are investors beating markets?
▶ Are they simply risk premiums?
▶ CAPM reveals underlying risks in assets
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Sources of Risks
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Only covariance with market portfolio returns matters
▶ Stand-alone risks in individual asset investments
▶ Variance in asset returns
▶ Portfolios diversify stand-alone risks
▶ Risks from co-movement cannot be diversified
▶ Fundamental or systematic risk underlying markets
▶ Higher covariance =⇒ lower diversifiable risk
▶ As part of a portfolio, an asset’s risk is
▶ Solely its covariance with the portfolio returns
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Diversification of stand-alone risk
Source: Based on lecture notes by Dr. Chongyu Wang.
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Systematic risk
Source: Based on lecture notes by Dr. Chongyu Wang.
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Market risks in real estate
1. Space market risk
▶ Volatility in cash flows (based on NOI) and CapEx
▶ Caused by changes in space market characteristics
▶ Example, employment growth =⇒ rent growth
2. Asset market risk
▶ Volatility in discount rate or opportunity cost of capital
▶ Assets may be revalued in markets
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Trends in CFs and PVs
Source: Geltner and Mei (1995).
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Deriving the CAPM
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With the CAPM
▶ We will develop a formula to calculate an asset’s price
▶ Based on Modern Portfolio Theory
▶ Assuming that Efficient Market Hypothesis holds
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Three ingredients
1. Common expectations
▶ Markets are efficient =⇒ all information is public
▶ Everyone has same return-risk expectations
2. Two-fund theorem
▶ Investors choose Sharpe Ratio-maximizing risky-asset portfolio
▶ Risk preferences do not matter
▶ Aggregating =⇒ Market Portfolio of risky assets
3. Co-movement of asset returns with Market Portfolio
returns
▶ Only covariance of asset returns with Market Portfolio matters
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Classical Capital Asset Pricing Model
▶ Main formula:
▶ Asset price or return = Risk-free rate + Risk premium
▶ Total risk premium = Amount of non-diversifiable risk ×
Price of risk per unit risk
▶ Total risk premium = Covariance of asset with Market
Portfolio × Modified Sharpe Ratio
▶ Asset Price = Risk-free Rate + Covariance of asset with
Market Portfolio × Modified Sharpe Ratio
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CAPM equation
▶ Suppose an asset i’s return is ri
▶ Risk-free rate is rf
▶ Covariance of asset returns with Market Portfolio returns is
CoviM
▶ Market portfolio return and volatility are rM and SM
▶ Classical CAPM boils down to:
E [ri ] = rf + CoviM
E [rM ] − rf
2
SM
!
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Sharpe-Lintner CAPM
▶ Re-arranging terms:
CoviM E
[r
]
−
r
M
f
2
SM
=⇒ E[ri ] = rf + βiM E[rM ] − rf
E [ri ] = rf +
▶ βiM = Risk on asset i as part of portfolio M
▶ E [rM ] − rf = Market price or premium of risk
▶ Depends on perceived risk and investors’ preferences
▶ Ranges between 4-8% per year
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Beta Risk
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Interpretation of β
▶ Quantifies non-diversifiable risk inherent to an asset
Si
▶ βiM = SM
ρiM
Si
▶ SM
= asset i’s share of market risk
▶ ρiM = co-movement of i and M
▶ βiM = asset i’s contribution to market risk
▶ Intersection between market risk and asset risk, that is,
stand-alone and portfolio risk
▶ It is not idiosyncratic risk of an asset
▶ Idiosyncratic risk is an asset’s unsystematic risk component
▶ Si2 = CoviM + Var [ϵi ]
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Assessing risk with β
2
1. If β = 1 =⇒ CoviM = SM
▶ Asset risk the same as market portfolio risk
2
2. If β > 1 =⇒ CoviM > SM
▶ Asset risk greater than market portfolio risk
2
3. If β < 1 =⇒ CoviM < SM
▶ Asset risk lower than market portfolio risk
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Calculating β
▶ Calculation relies on historical asset returns and market
portfolio returns data
▶ Suppose an asset i’s return in period t in the past is rit
▶ Market portfolio return in the same period is rMt
▶ βiM can be estimated using the following regression:
rit = rf + βiM rMt + ϵit
▶ rf is the intercept or risk-free rate
▶ ϵit is an idiosyncratic error term
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Security Market Line
Source: Geltner et al. (2007).
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Returns and Betas of asset classes
Asset Class
Expected Return
Beta
Stocks
10%
1.6
Real Estate
7%
0.8
Bonds
6%
0.6
Source: Based on lecture notes by Dr. Chongyu Wang.
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Negative Beta
▶ Is it possible?
▶ Best example: Gold (Baur and McDermott, 2010)
▶
▶
▶
▶
Negative Beta in some OECD countries: US, Germany
Positive Beta in Australia, Canada
Beta sign changes over time within stock markets
Generally treated as zero-beta (risk-free) by investors
▶ Other examples: ETFs (Rosenberg, 2016)
▶ Vanguard Extended Duration Treasury Index Fund
▶ Ishares Gold Trust
▶ Exceedingly rare!
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Betas by industrial sectors in the US
Industry
Beta
Banks (Regional)
Investments & Asset Management
Utility (General)
Retail (Grocery and Food)
Power
Real Estate (General/Diversified)
Insurance (Prop/Cas.)
Reinsurance
Rubber & Tires
Telecom Services
0.5
0.62
0.64
0.67
0.73
0.79
0.8
0.83
0.84
0.88
Source: NYU Stern.
Note: Table shows 10 industry-sectors with lowest Beta-risk between 2019-2023.
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What does negative Beta mean?
▶ Rate of return below risk-free rate
▶ Sounds counterintuitive
▶ Risk is so low (negative) that risk premium is negative
▶ Expected return below risk-free rate
▶ Hedge (protection) against negative macroeconomic events,
like inflation
▶ Gold is treated as an inflation hedge
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Applications
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Is the CAPM wrong?
▶ No such thing as a riskless asset =⇒ Two-fund theorem
does not hold
▶ Markets are not perfectly efficient =⇒ different risk-return
expectations across investors
▶ Covariances between asset returns are not fixed (stable) over
time
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Security Market Line in the US
Source: From Geltner et al. (2007) based on Li and Price (2005).
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Pros and cons of CAPM
▶ Strength: assumptions are close approximations of reality
1. Government bonds are pretty close to being risk-free
2. Asset markets are somewhat efficient
3. Covariances remain relatively stable
▶ Weakness: does not account for many factors
1. Size of a firm
2. Book/market value ratio
3. Example: small firms with high book/market value ratios are
vulnerable to macroeconomic shocks
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Why do we keep using CAPM?
1. CAPM is a model replication
▶ Not meant to exactly represent reality
▶ Reasonably approximates underlying asset prices
2. Beta explains a large portion of risk
▶ After accounting for firm size and book/market ratios
▶ Widely used by investors and academics
3. Risk quantified by CAPM and by firm size and book/market
ratios do not negate each other
▶ Additional factors only refine the measure of risk
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Three Factor Model
▶ Developed by economists Eugene Fama and Kenneth French
▶ Considers three risk factors
1. Beta coefficient of asset
2. Firm size
3. Book/market value ratio
▶ Criticism
▶ Firm size and book/market ratios important to investors for
behavioral reasons
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Book/Market Value Portfolio’s monthly returns vs. Beta
Source: From Geltner et al. (2007) based on Fama and French (2004).
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Book-to-market value ratio
▶ Book/market value ratio =
Book value
Market value
=
Actual firm equity
Market price of equity
▶ High book-to-market value ratio
▶ “Value stocks” =⇒ undervalued by markets
▶ Higher returns regardless of beta
▶ Low book-to-market ratio
▶ “Growth stocks” =⇒ overvalued
▶ Market premium despite low returns
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Beta in NCREIF sub-indices
Source: Geltner et al. (2007).
Note: Medium-sized properties between 40 million and 110 million USD. Y-axis plots quarterly returns over T-bills.
X-axis plots Beta coefficients by sub-indices of NCREIF Property Index (NPI). R-squared of regression of Risk
Premium on Beta is 0.27.
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Exercise
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Exercise
▶ Timothy invested 30% of his wealth in government bonds and
70% in a Sharpe Ratio-maximizing risky-asset portfolio
▶ Government bonds offer risk-free rate of 2.5%
▶ Timothy’s expected return on risky-asset portfolio is 10%
▶ Private RE is part of the risky-asset portfolio with a beta
coefficient 1.2.
▶ Assume that the Efficient Market Hypothesis holds and there
are no short positions
▶ Answer the following:
1. What is the rate of return on the market portfolio?
2. What is the equilibrium expected return on Private RE?
3. What is the risk premium on Private RE?
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Solution
1. 10%
▶ Two-fund theorem =⇒ All investors choose Sharpe
Ratio-maximizing portfolio
▶ Efficient markets =⇒ similar risk-return expectations
▶ Aggregating =⇒ market portfolio = Sharpe
Ratio-maximizing portfolio
2. 11.5%
▶ E [rRE ] = rf + βRE ,M E [rM ] − rf
▶ E [rRE ] = 2.5 + 1.2(10 − 2.5) = 11.5%
3. 9%
▶ Risk premium = E [rRE ] − rf = 11.5% − 2.5% = 9%
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References
Baur, D. G. and McDermott, T. K. (2010) Is gold a safe haven?
International evidence, Journal of Banking & Finance, 34, 1886–1898.
Fama, E. F. and French, K. R. (2004) The capital asset pricing model:
Theory and evidence, Journal of Economic Perspectives, 18, 25–46.
Geltner, D. and Mei, J. (1995) The present value model with
time-varying discount rates: Implications for commercial property
valuation and investment decisions, The Journal of Real Estate
Finance and Economics, 11, 119–135.
Geltner, D. M., Miller, N. G., Clayton, J. and Eichholtz, P. (2007)
Commercial real estate: Analysis & investments, Cengage learning.
Li, N. and Price, S. M. (2005) Multiple asset class investing: Equilibrium
asset pricing evaluation of real estate risk and return across four
quadrants, Ph.D. thesis, Massachusetts Institute of Technology.
Rosenberg, A. (2016) These 5 ETFs are perfect for the bears, Consumer
News and Business Channel (CNBC).
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