COMM 309 | Chapter 9: The Capital Asset Pricing Model (CAPM) | Page 1
COMM 309 — Business Finance
CHAPTER 9: The Capital Asset Pricing Model (CAPM)
Comprehensive Exam Preparation Notes
9.1 THE NEW EFFICIENT FRONTIER
Chapter 9 extends Chapter 8's discussion of the efficient frontier by introducing the ability to borrow and
invest at a risk-free rate. This transforms the curved efficient frontier into a straight line — the new (or
super) efficient frontier — and forms the theoretical foundation of the Capital Asset Pricing Model
(CAPM).
The Efficient Frontier with Risk-Free Borrowing and Lending
Recall from Chapter 8 that the efficient frontier is a curve above the minimum variance portfolio (MVP),
along which portfolios of risky securities offer the highest expected return (ER) for any given level of
risk (measured by standard deviation σ). All rational, risk-averse investors choose portfolios on this
frontier.
Risk Aversion — The Foundational Assumption
The entire CAPM framework rests on the assumption that investors are risk averse — they will not
willingly undertake a fair gamble (a bet where the expected payoff is zero). Key implications of risk
aversion:
• A risk-averse investor will turn down any gamble with an expected payoff of $0, even if the
potential gain equals the potential loss.
• To be induced into a risky situation, a risk-averse investor requires a risk premium — an expected
payoff above zero that compensates for the possibility of loss.
• More risk-averse investors demand higher risk premiums per unit of risk. This is represented
graphically: the steeper the slope of the line from the origin to a portfolio on the efficient frontier,
the more risk averse the investor.
• Insurance premiums are the mirror image of risk premiums — investors pay to get out of risky
situations. The fact that insurance companies earn ~50% gross margins indicates how strongly
risk averse individuals are.
📌 Risk Premium vs. Insurance Premium
Risk Premium: The EXTRA expected payoff an investor requires to voluntarily enter a risky situation.
Insurance Premium: The amount an investor will pay to EXIT a risky situation they are already in.
Both concepts are two sides of the same coin — they both reflect the degree of risk aversion.
Capital markets generate risk premiums to induce risk-averse investors to hold risky assets.
Insurance markets generate insurance premiums from investors wanting to transfer risk away.
Given risk aversion, investors will only hold portfolios on the efficient frontier above the MVP. Where
exactly on the frontier they sit depends on their personal degree of risk aversion:
• Very risk-averse investor → chooses portfolio A (left side of frontier, steep slope from origin = high
return per unit of risk required).
• Less risk-averse investor → chooses portfolio B (right side, flatter slope = willing to accept less
return per unit of risk).
, COMM 309 | Chapter 9: The Capital Asset Pricing Model (CAPM) | Page 2
Risk-Free Investing
A risk-free asset (such as a Government of Canada Treasury Bill, or T-bill) has a guaranteed return
with zero variance and zero covariance with any risky asset. It is the starting point of the new efficient
frontier.
When an investor places a fraction w of wealth in a risky portfolio A and the remainder (1 − w) in the
risk-free asset (RF), the resulting portfolio's expected return and risk are:
Eq. 9-1 | Expected Return (Risky Asset ERp = Expected return on the combined portfolio
+ Risk-Free Asset) RF = Risk-free rate (return on T-bill)
ERp = RF + w(ERA − RF) w = Fraction invested in the risky portfolio A
(ERA − RF) = Risk premium of risky portfolio A
As w → 0: ERp = RF (100% in T-bill, no risk)
As w → 1: ERp = ERA (100% in risky portfolio A)
As w > 1: Borrowing at RF to invest more in A
Eq. 9-2 | Portfolio Standard Deviation σp = Standard deviation of the combined portfolio
(Risky Asset + Risk-Free Asset) w = Fraction invested in the risky portfolio
σp = w × σA σA = Standard deviation of the risky portfolio
Key insight: Risk increases LINEARLY with the weight
invested in the risky asset.
The risk-free asset has σ = 0 and ρ(A,RF) = 0,
so all cross-product terms cancel out, leaving only w²σA²
under the square root, which simplifies to wσA.
Since both ERp and σp are linear in w, all combinations of the risk-free asset and a risky portfolio lie on
a straight line. Substituting w = σp/σA into Eq. 9-1:
Eq. 9-3 | Expected Portfolio Return in This is the equation of a straight line:
Terms of Standard Deviation Y-intercept: RF (when σp = 0, all money in T-bill)
ERp = RF + [(ERA − RF) / σA] × σp Slope: (ERA − RF) / σA = risk premium per unit of risk
The slope represents how much additional expected
return an investor earns per additional unit of risk
taken by moving from the T-bill into risky portfolio A.
Example 9-1 | Expected Return and Standard Deviation (Risk-Free Asset + Risky Portfolio)
Given: Portfolio T: ER = 8%, σ = 25%; RF = 3%; 30% in RF, 70% in T
w = 0.70; (1 − w) = 0.30
ERp = (0.30)(3%) + (0.70)(8%) = 0.9% + 5.6% = 6.5%
, COMM 309 | Chapter 9: The Capital Asset Pricing Model (CAPM) | Page 3
σp = (0.70)(25%) = 17.5%
Interpretation: Mixing 70% risky and 30% risk-free gives a moderate return of 6.5% with a risk of 17.5%
— lower than the full risky portfolio's 25%.
The Tangent Portfolio and the New Efficient Frontier
Among all possible risky portfolios on the original efficient frontier, one is special: the tangent portfolio
T. It is the risky portfolio whose line from RF (Eq. 9-3) has the steepest slope — meaning it offers the
highest expected return per unit of risk for any portfolio that mixes RF with a risky asset.
• All portfolios composed of RF and portfolio T dominate all other portfolios composed of RF and
any other risky portfolio (e.g., portfolio A).
• Therefore, T replaces all other risky portfolios as the optimal risky holding — regardless of the
investor's risk aversion.
• The line from RF through T and beyond is the new (or super) efficient frontier — a straight line
that dominates the curved efficient frontier of risky portfolios at every point except T itself.
📌 Why the New Frontier is Superior
Any investor holding risky portfolio A (a point on the old curved frontier) can improve their position by
instead holding a mix of RF and portfolio T.
This new portfolio A2 has the SAME risk as A but a HIGHER expected return — it lies on the straight line
above the original curved frontier.
The straight line dominates the curve because any point on the curve can be 'beaten' by a corresponding
point on the line at the same σ level.
This is true for EVERY point on the old frontier except for point T itself (where the line and curve are
tangent).
Risk-Free Borrowing
Investing in the risk-free asset corresponds to w < 1 (some wealth in T-bills). Borrowing at the risk-free
rate corresponds to w > 1 — the investor takes a short position in the risk-free asset (i.e., borrows
money) and invests more than 100% of wealth in the risky portfolio T. This is done practically by buying
stocks on margin.
• Margin buying: Borrowing part of the purchase price from the stockbroker. Example: 30% margin
requirement means buying $1,000 in stocks with only $300 invested and $700 borrowed → w =
1,000/300 = 333%, weight in RF = −233%.
• Borrowing assumption: The CAPM simplifies by assuming investors borrow at the same risk-free
rate they can lend at. In practice, borrowing rates are higher, but this assumption can be relaxed
without fundamentally altering the model.
• Risks of short selling: Theoretically unlimited losses (stock price can rise indefinitely), margin calls
(broker demands additional funds), and obligation to cover dividends on shorted stock.
Example 9-2 | Expected Return and Standard Deviation (Risk-Free Borrowing)
Given: Investor has $1,000; borrows additional $700 at RF = 3%; invests total $1,700 in Portfolio T (ER =
8%, σ = 25%)
COMM 309 — Business Finance
CHAPTER 9: The Capital Asset Pricing Model (CAPM)
Comprehensive Exam Preparation Notes
9.1 THE NEW EFFICIENT FRONTIER
Chapter 9 extends Chapter 8's discussion of the efficient frontier by introducing the ability to borrow and
invest at a risk-free rate. This transforms the curved efficient frontier into a straight line — the new (or
super) efficient frontier — and forms the theoretical foundation of the Capital Asset Pricing Model
(CAPM).
The Efficient Frontier with Risk-Free Borrowing and Lending
Recall from Chapter 8 that the efficient frontier is a curve above the minimum variance portfolio (MVP),
along which portfolios of risky securities offer the highest expected return (ER) for any given level of
risk (measured by standard deviation σ). All rational, risk-averse investors choose portfolios on this
frontier.
Risk Aversion — The Foundational Assumption
The entire CAPM framework rests on the assumption that investors are risk averse — they will not
willingly undertake a fair gamble (a bet where the expected payoff is zero). Key implications of risk
aversion:
• A risk-averse investor will turn down any gamble with an expected payoff of $0, even if the
potential gain equals the potential loss.
• To be induced into a risky situation, a risk-averse investor requires a risk premium — an expected
payoff above zero that compensates for the possibility of loss.
• More risk-averse investors demand higher risk premiums per unit of risk. This is represented
graphically: the steeper the slope of the line from the origin to a portfolio on the efficient frontier,
the more risk averse the investor.
• Insurance premiums are the mirror image of risk premiums — investors pay to get out of risky
situations. The fact that insurance companies earn ~50% gross margins indicates how strongly
risk averse individuals are.
📌 Risk Premium vs. Insurance Premium
Risk Premium: The EXTRA expected payoff an investor requires to voluntarily enter a risky situation.
Insurance Premium: The amount an investor will pay to EXIT a risky situation they are already in.
Both concepts are two sides of the same coin — they both reflect the degree of risk aversion.
Capital markets generate risk premiums to induce risk-averse investors to hold risky assets.
Insurance markets generate insurance premiums from investors wanting to transfer risk away.
Given risk aversion, investors will only hold portfolios on the efficient frontier above the MVP. Where
exactly on the frontier they sit depends on their personal degree of risk aversion:
• Very risk-averse investor → chooses portfolio A (left side of frontier, steep slope from origin = high
return per unit of risk required).
• Less risk-averse investor → chooses portfolio B (right side, flatter slope = willing to accept less
return per unit of risk).
, COMM 309 | Chapter 9: The Capital Asset Pricing Model (CAPM) | Page 2
Risk-Free Investing
A risk-free asset (such as a Government of Canada Treasury Bill, or T-bill) has a guaranteed return
with zero variance and zero covariance with any risky asset. It is the starting point of the new efficient
frontier.
When an investor places a fraction w of wealth in a risky portfolio A and the remainder (1 − w) in the
risk-free asset (RF), the resulting portfolio's expected return and risk are:
Eq. 9-1 | Expected Return (Risky Asset ERp = Expected return on the combined portfolio
+ Risk-Free Asset) RF = Risk-free rate (return on T-bill)
ERp = RF + w(ERA − RF) w = Fraction invested in the risky portfolio A
(ERA − RF) = Risk premium of risky portfolio A
As w → 0: ERp = RF (100% in T-bill, no risk)
As w → 1: ERp = ERA (100% in risky portfolio A)
As w > 1: Borrowing at RF to invest more in A
Eq. 9-2 | Portfolio Standard Deviation σp = Standard deviation of the combined portfolio
(Risky Asset + Risk-Free Asset) w = Fraction invested in the risky portfolio
σp = w × σA σA = Standard deviation of the risky portfolio
Key insight: Risk increases LINEARLY with the weight
invested in the risky asset.
The risk-free asset has σ = 0 and ρ(A,RF) = 0,
so all cross-product terms cancel out, leaving only w²σA²
under the square root, which simplifies to wσA.
Since both ERp and σp are linear in w, all combinations of the risk-free asset and a risky portfolio lie on
a straight line. Substituting w = σp/σA into Eq. 9-1:
Eq. 9-3 | Expected Portfolio Return in This is the equation of a straight line:
Terms of Standard Deviation Y-intercept: RF (when σp = 0, all money in T-bill)
ERp = RF + [(ERA − RF) / σA] × σp Slope: (ERA − RF) / σA = risk premium per unit of risk
The slope represents how much additional expected
return an investor earns per additional unit of risk
taken by moving from the T-bill into risky portfolio A.
Example 9-1 | Expected Return and Standard Deviation (Risk-Free Asset + Risky Portfolio)
Given: Portfolio T: ER = 8%, σ = 25%; RF = 3%; 30% in RF, 70% in T
w = 0.70; (1 − w) = 0.30
ERp = (0.30)(3%) + (0.70)(8%) = 0.9% + 5.6% = 6.5%
, COMM 309 | Chapter 9: The Capital Asset Pricing Model (CAPM) | Page 3
σp = (0.70)(25%) = 17.5%
Interpretation: Mixing 70% risky and 30% risk-free gives a moderate return of 6.5% with a risk of 17.5%
— lower than the full risky portfolio's 25%.
The Tangent Portfolio and the New Efficient Frontier
Among all possible risky portfolios on the original efficient frontier, one is special: the tangent portfolio
T. It is the risky portfolio whose line from RF (Eq. 9-3) has the steepest slope — meaning it offers the
highest expected return per unit of risk for any portfolio that mixes RF with a risky asset.
• All portfolios composed of RF and portfolio T dominate all other portfolios composed of RF and
any other risky portfolio (e.g., portfolio A).
• Therefore, T replaces all other risky portfolios as the optimal risky holding — regardless of the
investor's risk aversion.
• The line from RF through T and beyond is the new (or super) efficient frontier — a straight line
that dominates the curved efficient frontier of risky portfolios at every point except T itself.
📌 Why the New Frontier is Superior
Any investor holding risky portfolio A (a point on the old curved frontier) can improve their position by
instead holding a mix of RF and portfolio T.
This new portfolio A2 has the SAME risk as A but a HIGHER expected return — it lies on the straight line
above the original curved frontier.
The straight line dominates the curve because any point on the curve can be 'beaten' by a corresponding
point on the line at the same σ level.
This is true for EVERY point on the old frontier except for point T itself (where the line and curve are
tangent).
Risk-Free Borrowing
Investing in the risk-free asset corresponds to w < 1 (some wealth in T-bills). Borrowing at the risk-free
rate corresponds to w > 1 — the investor takes a short position in the risk-free asset (i.e., borrows
money) and invests more than 100% of wealth in the risky portfolio T. This is done practically by buying
stocks on margin.
• Margin buying: Borrowing part of the purchase price from the stockbroker. Example: 30% margin
requirement means buying $1,000 in stocks with only $300 invested and $700 borrowed → w =
1,000/300 = 333%, weight in RF = −233%.
• Borrowing assumption: The CAPM simplifies by assuming investors borrow at the same risk-free
rate they can lend at. In practice, borrowing rates are higher, but this assumption can be relaxed
without fundamentally altering the model.
• Risks of short selling: Theoretically unlimited losses (stock price can rise indefinitely), margin calls
(broker demands additional funds), and obligation to cover dividends on shorted stock.
Example 9-2 | Expected Return and Standard Deviation (Risk-Free Borrowing)
Given: Investor has $1,000; borrows additional $700 at RF = 3%; invests total $1,700 in Portfolio T (ER =
8%, σ = 25%)
