FinTech 522
Asset Pricing
Efficient Frontier 2/5/25
marvin.chang@duke.edu / 917 940 0055
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Let’s start
with pizza…
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Objective: Quest for a perfect pizza
• The Goal: Imagine you're on a
quest for the perfect pizza. You
care about two things:
• Flavor: You want the most
delicious, flavorful pizza
possible.
• Health: You also want it to be
somewhat healthy (maybe
not loaded with grease).
• The Trade-off:
• You realize there's a trade-off.
• The most flavorful pizza
might be loaded with greasy
components, making it less
healthy.
• The healthiest pizza might be
bland with just veggies.
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The pizza efficient frontier is the
list of the best pizza options.
For every level of (lack of) health
you're willing to accept…
it shows you the most flavorful
pizza you can get.
Flavor
The Pizza Efficient Frontier
(Lack of) Health
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Choices…
A
B
C
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The Pizza Efficient Frontier (continued)
Flavor
[Expected Return]
B
A
C
(Lack of) Health
[Risk]
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The efficient frontier is a curve representing a
set of optimal portfolios that offer the highest
expected return for a given level of risk, or the
lowest risk for a given expected return.
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Portfolio Theory
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Risk Aversion
First Game: Choose
between $50 for certain and
an equally-weighted gamble
of either $ 100 or $ 0
What happens if we
increase the stakes a little?
Option A
$ 50
Option B
If heads
$100
If tails
For certain
Option A
$ 500
$0
Option B
$0
If heads
$1,000
If tails
For certain
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Mean & Standard Deviation: Definitions
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Annualizing Volatility
* Why 252 1/2 rather than 365 1/2 ?
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Correlation and Covariance
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Statistical Relationships for a two-asset portfolio
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Portfolio Risk, Return
and Diversification
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The Core Dilemma: Balancing Risk and Return
• Investors seek to maximize returns while minimizing risk.
• This presents a fundamental challenge: higher returns often come
with higher risk.
• How do we find the “best” portfolio given this trade-off?
• The Efficient Frontier provides a framework for addressing this
challenge.
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Quantifying the Trade-off
• Expected Return: (y-axis) The anticipated return on an investment,
often calculated as the weighted average of possible outcomes.
Formula: E(R) = Σ (pi * ri) where pi is the probability of outcome i and ri
is the return of outcome i.
• Risk (Standard Deviation): (x-axis) A measure of the dispersion of
possible returns around the expected return. It quantifies the
volatility of an investment.
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Portfolio Opportunity Set - Two Assets
rp = (wA)rA + (1-wA)rB
p2 = (wA)2A2 + (1-wA )2B2 + 2(wA) (1-wA )ABAB
rp
Expected Return
rA
<1
=1
rB
B
A
p
Risk
Why Two Assets?
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Portfolio Opportunity Set - Two Assets
rp = (wA)rA + (1-wA)rB
p2 = (wA)2A2 + (1-wA )2B2 + 2(wA) (1-wA )ABAB
rp
Expected Return
rA
<1
=1
rB
B
A
p
Risk
Why Two Assets?
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The Power of Diversification
• A portfolio is a collection of different assets.
• Diversification involves investing in a variety of asset classes to reduce
overall portfolio risk.
• Diversification benefits arise from the imperfect correlation between
asset returns.
• Diversification can reduce unsystematic (specific) risk, but not
systematic (market) risk.
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Portfolio Opportunity Set—Multiple Assets
The “Efficient Frontier”
• It can be shown (mathematically) that if we
have hundreds or thousands of different
assets, and combine those assets into
portfolios, that all possible portfolios will lie
on or below a hyperbola shaped like this.
Efficient Frontier
rp
X
X
X
X
X
X
X
X
X
X
X
X
• The hyperbola is called the Efficient Frontier
X
X
p
• Why should investors prefer to pick
portfolios that lie on, rather than below, this
Frontier?
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Understanding the Efficient Frontier
• All portfolios on the Efficient Frontier are considered “efficient” – one
cannot achieve higher returns for the same or lower risk for the same
return.
• Portfolios below the frontier are “inefficient” – once can achieve
higher returns for the same risk or lower risk for the same return.
• Portfolios to the above the frontier may offer higher returns, but at a
disproportionately higher risk.
• The shape of the efficient frontier is influenced by the correlation
between assets.
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Asset Allocation Over Two Assets
rC = w rp + (1 – w)rf or, equivalently,
rC = rf + w (rp - rf )
• Consider a portfolio C with
weight w in a risky asset and
(1 – w) in the riskless asset
C = w p
rC
Risk Premium:
rp – rf
rp
Slope of line:
rp - rf
p
rf
p
• Risky asset return is rp and
standard deviation (p)
• Assume return on riskless
asset is rf
C
Mathematical formula for the line:
rC = rf + rp - rf C
p
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An Example of Portfolio with Leverage
Let P be the S&P 500
Let rf = 5%
Now suppose w = 1.5. You start with $100,
borrow $50 at 5% p.a., put $150 into S&P
Rtns
Cash Flows
1986
1987
2%
20%
Mean
Std Dev
11%
12.73%
This is the mean and std dev
on a portfolio 100% invested
in the S&P i.e., w=1
150(1.02) – 50(1.05) = 100.50
150(1.20) – 50(1.05) = 127.50
Returns*
0.5%
27.5%
Mean 14.00%
Std 19.09%
This is the mean and std dev on a portfolio with 150%
(50% borrowed) in the S&P
* Returns = (CF – 100) /100
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Capital Market Line (CML)
• What proportion of the complete
portfolio (C) is held in the risky
versus the riskless asset at points W,
X, Y, and Z?
rC
CML
Y
rp
• To get to Z, what investment do you
need to make in the riskless asset?
rf
W
Z
X
p
C
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Why does the Capital Market Line (CML) matter?
• Investment Decisions: The CML helps investors understand the relationship between risk
and return when investing in a diversified portfolio that includes a risk-free asset.
• Performance Evaluation: The CML can be used to evaluate the performance of
investment portfolios. Portfolios that lie above the CML are considered to be
outperforming the market, while those below the line are underperforming.
• Theoretical Foundation: The CML is a key component of the Capital Asset Pricing Model
(CAPM), which describes the relationship between risk and expected return for assets.
• The CML is a theoretical concept based on certain assumptions, such as efficient markets
and rational investors.
• In actual circumstances, it can be challenging to construct a true market portfolio.
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CML and Capital Allocation Line (CAL)
• The CML is a specific CAL where the risky portfolio is the market
portfolio.
• The portfolio that includes all risky assets in proportion to their
market values.
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Efficient Frontier and Capital Market Line
rc
CML
P
rp
Efficient Frontier
Capital Market Line: tangent
to the efficient frontier;
steepest possible slope
Where Efficiency Meets
Diversification
rf
p
c
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Tangency Point: Where the Efficient Frontier and
the CML meet is key…
This point represents the market portfolio, which is:
• The most diversified portfolio you can create using all available risky assets.
• The optimal risky portfolio to combine with the risk-free asset.
Why this matters:
• Superior Efficiency: All portfolios on the CML offer superior risk-return
trade-offs compared to any portfolio on the efficient frontier except for the
market portfolio itself. This is because the CML incorporates the risk-free
asset, allowing for even better diversification and risk management
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Aligning with Risk Tolerance
• The CML suggests that investors should aim to hold a combination of
the risk-free asset and the market portfolio.
• Their specific allocation will depend on their individual risk tolerance.
• Risk Aversion: More risk-averse investors will choose portfolios
closer to the risk-free asset.
• Risk Tolerance: Investors with higher risk tolerance will choose
portfolios further up the CML.
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The Optimal Asset Allocation Decision: Steps (1/2)
1. Identify the universe of possible assets (including the riskless asset),
and estimate returns, standard deviations, and all crosscorrelations.
2. Identify the Efficient Frontier, i.e., the set of "efficient" portfolios
that maximize return for each possible level of volatility (or
equivalently, the portfolios that minimize volatility for each given
level of returns).
3. Draw the steepest possible line (i.e., the tangent line) from rf on the
y-axis to the Efficient Frontier. This is the CML because it has the
maximum possible return, at each risk level, that can be achieved
using the universe of risky as well as the riskless asset.
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The Optimal Asset Allocation Decision: Steps (2/2)
4. Identify P, the optimal risky portfolio. P represents the optimal risky
portfolio for all investors, regardless of their risk preferences
5. Assess the optimal asset allocation between the risky portfolio and
the riskless asset for any given investor, based on their risk
preferences (i.e., their minimum desired rate of return, or their
maximum tolerated volatility). All investor portfolios will lie on CML.
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Real-World Challenges
• Estimating expected returns, standard deviations, and correlations is
challenging.
• The efficient frontier is only as good as the inputs used to create it
(Garbage In, Garbage Out).
• Transaction costs and taxes can impact portfolio optimization.
• Market conditions change, so the efficient frontier needs to be reestimated periodically.
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Diversification: Multiple Assets
• What is diversification?
• Why do we diversify our portfolio?
• What type of risks do we avoid through diversification? What risk remains?
• How do we assess (quantitatively) the benefit of a given security to the
diversification of a portfolio?
• Expected Return and Standard Deviation formulas for portfolios with many
assets:
rp = wi ri
p2 = i j wiwj i j ij
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Diversification (continued)
rp = wi ri
p2 = i j wiwj i j ij
From the formulas, we see that:
• Portfolio return is the weighted average return of the assets in the portfolio
(regardless of their correlation to each other)
• Portfolio risk (standard deviation) is reduced when low (or, better still, negative)
assets are added to the portfolio
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Key Takeaways
• The efficient frontier is a powerful tool for portfolio optimization.
• It helps investors understand the risk-return trade-off and construct efficient
portfolios.
• Diversification is crucial for managing risk.
• The optimal portfolio depends on investor risk tolerance.
• Practical considerations and limitations should be taken into account.
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