Introduction to

Asset Pricing Models

Capital Asset Pricing Model

Introduction to Asset Pricing Models2

Model to price all risky assets based on existing portfolio

theory (e.g. Risk Aversion, Return Maximization)

Gives the required rate of return for any given risky asset.

Assumptions

Introduction to Asset Pricing Models3

All investors are Markowitz Efficient Investors.

All investors can borrow or lend at the risk free rate.

All investors have homogenous expectations.

All investors have the same investment time horizon.

All investments are infinitely divisible.

No tax and transaction costs in buy/sell.

No inflation or change in interest rates.

Capital markets are in equilibrium.

Risk Free Asset

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Risky Asset = Asset with uncertain returns.

Risk-Free Asset = Asset with σ = 0

Thus, for any investment, minimum return should be at least

equal to the risk-free rate.

In modelling, this is usually the 365-day T-Bill rate.

Risk-Free Asset in a Risky Portfolio

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Expected Return

Standard Deviation

𝐸 𝑅𝑃𝑜𝑟𝑡 = 𝑊𝑅𝐹 𝑅𝐹𝑅 + 1 −𝑊𝑅𝐹 𝐸(𝑅𝑖)

𝜎 = (𝑤𝑎𝜎𝑎)2 + (𝑤𝑏𝜎𝑏)

2 + 2𝜌𝑤𝑎𝑤𝑏𝜎𝑎𝜎𝑏

𝜎 = (𝑤𝑅𝐹𝜎𝑅𝐹)2 + [(1 − 𝑤𝑅𝐹)𝜎𝑖]

2 + 2𝜌𝑤𝑅𝐹𝑤𝑖𝜎𝑅𝐹𝜎𝑖

𝜎 = [(1 − 𝑤𝑅𝐹)𝜎𝑖]2 = (1 − 𝑤𝑅𝐹)𝜎𝑖

𝑆𝑖𝑛𝑐𝑒 𝜎𝑅𝐹 = 0,

Risk-Free Asset in a Risky Portfolio

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Linear combinations of risk-free and risky asset portfolio.

Point M = point of tangency with portfolio M.

𝐸 𝑅𝑃𝑜𝑟𝑡 = 0 𝑅𝐹𝑅 + 1 𝐸 𝑅𝑀 = 𝐸(𝑅𝑀)

𝐸 𝑅𝑃𝑜𝑟𝑡 =1

2𝑅𝐹𝑅 +

1

2𝐸(𝑅𝑀)

Risk-Free Asset in a Risky Portfolio

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What if you want a return higher than M ?

Higher than D, but with the same level of risk.

𝐸 𝑅𝑃𝑜𝑟𝑡 = −0.5 𝑅𝐹𝑅 + [1 − −0.5 ]𝐸(𝑅𝑖)

𝐸 𝑅𝑃𝑜𝑟𝑡 = −0.5 𝑅𝐹𝑅 + 1.5𝐸(𝑅𝑖)

Risk-Free Asset in a Risky Portfolio

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New efficient frontier = Capital Market Line (CML)

Market Portfolio

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Includes all risky assets - Completely Diversified Portfolio Stocks – Local and International

Bonds

Options

Real Estate

Physical Assets – antiques, coins, gold, art, etc.

Complete Diversification takes away all unsystematic (diversifiable and unique) risk.

Systematic Risk = caused by macroeconomic variables.

All assets are in proportion to their market value.

Security Market Line (SML)

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Given that the Market Portfolio is the ideal and completely

diversified portfolio, an individual asset’s risk can be attributed

to its variability, or covariance, with the market portfolio.

If asset is riskier than market portfolio, then a higher return is expected.

If asset is less risky than market portfolio, lower return is expected.

Capital Asset Pricing Model (CAPM)

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Beta – standardized measure of systematic risk.

1 = perfectly correlated with the market portfolio.

> 1 = more volatile than market portfolio.

< 1 = less volatile than market portfolio.

𝐸 𝑅𝑖 = 𝑅𝐹𝑅 +𝑅𝑀 − 𝑅𝐹𝑅

𝜎𝑀2 𝐶𝑜𝑣𝑖,𝑚

𝐸 𝑅𝑖 = 𝑅𝐹𝑅 +𝐶𝑜𝑣𝑖,𝑚

𝜎𝑀2 (𝑅𝑀 − 𝑅𝐹𝑅)

𝐸 𝑅𝑖 = 𝑅𝐹𝑅 + 𝛽(𝑅𝑀 − 𝑅𝐹𝑅)

Required vs Estimated Returns

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Required – ideal return given level of risk as indicated by

CAPM Model.

If RFR = 6% and Market Rate of Return = 12%, compute for

the required returns of each stock and determine whether

the stock is properly, under, or over valued.

Stock Beta Estimated Return

A 0.70 10.0%

B 1.00 6.2%

C 1.15 21.2%

D 1.40 3.3%

E -0.30 8.0%