Efficient Diversification
Chapter 6
McGraw-Hill/Irwin
1
Copyright © 2013 by The McGraw-Hill Companies, Inc. All rights reserved.
Portfolio Risk

So far we’ve been focusing on individual
assets, but what happens when we combine
individual assets into a portfolio?
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Diversification
A
Time
B
Time
A&B
Time
3
Portfolio v Single Asset
Investing everything in a single asset exposes
you to that investment’s total risk
 A portfolio allows us to spread out our
investment reducing the impact of swings in a
single asset
→Assets are unlikely to all move in the same
direction
 Intuition: “Don’t put all your eggs in one
basket”

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Historical Non-Diversifiers

People on record for saying: “Put all your eggs
in one basket and watch it closely”
 Mark
Twain: Died penniless
 Andrew Carnegie: Died known as “The Richest
Man in the World”
→Definition of Risk
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Diversification and Portfolio Risk

Market/Systematic/Non-diversifiable Risk
 Risk
factors common to whole economy
 Affect all firms, economy wide risk

Unique/Firm-Specific/Nonsystematic/
Diversifiable Risk
 Risk
that can be eliminated by diversification
 Affect individual or small groups of firms (industries)

What does σ measure?
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The Wonders of Diversifying
Portfolio
standard
deviation
Unique risk
Market risk
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10
Number of
securities
Total risk = Unique risk + market risk
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Risk versus Diversification
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How Diversification Works
Reduces/eliminates unsystematic risk.
 Why can’t we diversify away systematic risk?

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Aside: Why Diversified Investors
Set Prices

Investor A is undiversified, while B has a
diversified portfolio. Both investors want to buy
a share of Facebook. Who gets the share?


Which investor uses a 10% discount rate? 20%?
Facebook will pay a constant $20 dividend

Which investor offers a higher price for the share?
(Remember: Price = Div / discount rate)
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Measuring Co-Movement

Covariance: measures how two variables move
in relation to one another
 Positive: The
two variables move up together or
down together

Ex: Height and Weight
 Negative:
When one moves up the other moves
down

Ex: Sleep and Coffee consumption
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Calculation

Cov (rS, rB) = σSB = Σ p(i)*(rS(i)– μS)*(rB(i) – μb)
 σXY
= p1*(X1 – μS)*(Y1 – μB)+p2*(X2 – μS)*(Y2 –
μB)+….+pN*(XN – μS)*(YN – μB)

Note that σss = Σ p(i)*(Xs(i) – μs)*(Xs(i) – μs)
 Which

implies?
How does a risky asset’s return move with the
risk free?
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Covariance Strength

If the covariance of asset 1 and 2 is -1,000, and
between asset 1 and 3 is 500. Which asset is
more closely related to 1?
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Correlation Coefficient

Measures the strength of the covariance relation
 It
is a standardization of covariance
ρ SB

Cov( rS , rB )

σS  σB
Bounded by -1 & 1
1
means the two are perfectly positively correlated
 -1 means the two are perfectly negatively correlated
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Correlation Coefficient Formula
ρSB = σSB / (σS * σB)
 ρSB – Correlation Coefficient
 σSB – Covariance between S and B
 σS – Std Dev of S
 σB – Std Dev of B

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Comparing Strength
When determining the strength of a correlation
all we care about is the absolute value of the
correlation coefficient
 If ρ13 is -0.8, and ρ23 is 0.5, which asset is
more correlated with 3?

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Correlation Coefficient Example
σ13 is -1,000; σ23 is 500
 σ1 is 10; σ2 is 1,000; σ3 is 250
 Is 1 or 2 more strongly correlated with 3?

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Covariance & Correlation Aside

The covariance between a risky asset and a
risk free is 0. Why?
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Portfolio Return & Risk

Portfolio Return: weighted average of component
returns


Expected Return: weighted average of component
expected returns


A stock’s weight is the percent of the portfolio it represents
A stock’s weight is the percent of the portfolio it represents
Variance: depends on covariance between the assets
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Portfolio Return Example

If a portfolio comprised of Google {E(r) =
25%} and Acme {E(r) = 10%} has an E(r) =
20%, how much is invested in Google and
Acme?
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Portfolio Variance Formulas


Depends primarily on covariances between the assets
Two Stock Portfolio
σP2 = (wBσB)2+(wSσS)2+2wBwSσSB
 σP2 = (wBσB)2+(wSσS)2+2wBwSσSσBρSB


Three Stock Portfolio
σP2 = (wBσB)2+ (wSσS)2+ (wCσC)2+
2wBwSσSB + 2wBwCσBC + 2wSwCσSC
 σP2 = (wBσB)2+ (wSσS)2+ (wCσC)2+
2wBwSσSσBρSB + 2wBwCσBσCρBC +
2wSwCσSσCρSC

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Portfolio Variance Formulas


Depends primarily on covariances between the assets
Two Stock Portfolio
σP2 = (wBσB)2+(wSσS)2+2wBwSσSB
 σP2 = (wBσB)2+(wSσS)2+2wBwSσSσBρSB


Three Stock Portfolio
σP2 = (wBσB)2+ (wSσS)2+ (wCσC)2+
2wBwSσSB + 2wBwCσBC + 2wSwCσSC
 σP2 = (wBσB)2+ (wSσS)2+ (wCσC)2+
2wBwSσSσBρSB + 2wBwCσBσCρBC +
2wSwCσSσCρSC

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Portfolio Variance Example




Two stocks A and B have expected returns of 10%
and 20%.
In the past, A and B have had std dev of 15% and
25%, respectively, with a correlation coefficient of
0.2.
You decide to invest 30% in A and the rest in B.
Calculate the portfolio return and portfolio risk. Has
diversification been of any use? Explain.
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Calculations
wA = 30% wB =
 A = 10%
B = 20%
 A = 15%
B = 25%
AB = 0.2
 Portfolio Return =
 Portfolio Variance =
 Portfolio Standard Deviation =
 Weighted Average Standard Deviation

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Remarks on Diversification

Diversification reduces the p from 22% to
18.9%

What happens if AB = 1?

What happens as AB approaches -1?
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Portfolio Example

An investor wants to maximize his return, but
doesn’t want the risk of his portfolio to exceed
15%. He has two options, a stock fund with a
standard deviation of 35% and T-Bills, how
much does he invest in the stock fund?
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Individual Stock Allocation

Your offer a portfolio comprised of 70% stock
A and 30% stock B. If an investor has half
their wealth invested in your portfolio, how
much of her wealth is in stock A?
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Which Stock do you Prefer?
Stock A :



= 10%;  = 2%
Stock B :

= 10%;  = 3%
Stock C :

Which stock do you prefer?
= 12%;  = 2%
0.25
Probability

0.20
Stock A
Stock B
Stock C
0.15
0.10
0.05
0.00
0
5
10
15
20
Returns
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Fundamental Premise of Portfolio
Theory
Rational investors prefer the highest expected
return at the lowest possible risk
 Investors want Lower Risk & Higher Returns
→ Mean-Variance Criterion
 If E(rA) ≥ E(rB) and σA ≤ σB

 Portfolio

A dominates portfolio B
How can investors lower risk without
sacrificing return?
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Investment Opportunity Set

The set of possible risk-return pairings offered
by the portfolios that can be formed from the
available securities
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Investment Opportunity Set Example
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return
Investment Opportunity Set
Continued
Stock
Min Var.
Bonds
P
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Changing the Correlation Coefficient
How does the
correlation
coefficient
affect the
diversification
benefit?
Artificial Risk Free
Asset (-1)
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return
What portfolios do we prefer?
Efficient Portfolios:
-Maximize risk premium for any
level of standard deviation
-Minimize standard deviation for
any level of return
-Maximize Sharpe ratio for any
standard deviation or risk premium
P
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return
Including More Assets
Efficient Portfolios
Possible
Portfolios
P
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Including a risk free asset?
A risk free asset changes
our investment
opportunity set and our
efficient portfolio
return

Capital Allocation Line,
Capital Market Line
Optimal Risky
Portfolio (O)
rf
P
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Which Efficient Portfolio?

O, the Optimal Efficient Portfolio
 It
offers the best risk return trade off
 Highest SHARPE RATIO (ri - rf) / i

All portfolios on the CAL will have the same
Sharpe Ratio as O
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Optimal Risky Portfolio

Given two risky asset
wB 
[ E (rB )  rf ] S2  [ E (rs )  rf ] B S  BS
[ E (rB )  rf ] S2  [ E (rs )  rf ] B2  [ E (rB )  rf  E (rs )  rf ] B S  BS
wS  1  wB
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Capital Allocation Line
The set of efficient portfolios available once
we include a risk free asset
 Why only two assets?

O

 Rf
dominates all other risky assets
All other are either too risky or offer to low a return
is used to adjust for risk tolerance
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Tobin’s Separation Property

Implies investing can be separated into two
tasks
Find O (Optimal risky portfolio)
2. Determining where we want to be on the CAL?
1.

Known as Asset Allocation
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Asset Allocation

How do we allocate our portfolio across
asset classes


Bogle: This the most fundamental investment
question
Basically: How much risk do we want?

We can adjust our risk level through our risk free
investments
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Finding Your Spot
Notionally there is a formula for finding an
investors preferred location on the CAL, which is
based on an investors level of risk aversion
 Risk Aversion measures how investors feel about
risk

 An
investor that is more risk averse will hold more of
the risk free asset
 As risk aversion decreases (investor becomes more risk
tolerant, risk loving) the investor will hold more O
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Measuring Risk Aversion

We measure risk aversion with A
 The
risk premium an investor demands for
investing everything in a portfolio given its risk
 The Sharpe Ratio an investor requires to put
everything into a single diversified portfolio
A
E(rS ) - rf
σ
2
S
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How Much O Should an Investor
Hold
Y is the amount of an investors portfolio
invested in the Optimal Risky Portfolio
 To find Y divide the Available Sharpe Ratio
(which is offered by O) by that demanded by
the investors (A)

E(rO ) - rf
Y
σ
A
2
O

E(rO ) - rf
Aσ
2
O
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What Does Our Portfolio Look Like?
Where are we on the CAL?
What risk premium do you demand to invest
all your money in a diversified portfolio with a
σ of 20%?
 If you have $250,000 to invest, how much of
your total investment portfolio is invested in
O? O’s risk premium is 11% and its σ is 18%
 Where is the rest of your portfolio invested?

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Moving on the CAL

As risk averse increases Y falls → Hold more
T-Bills
 Purchasing
T-Bills is lending the government
money

As investors become more risk tolerant
(loving) Y increase → Hold more of O
 We
can move past O by shorting T-Bills
 Borrow T-Bills → Sell them →Use proceeds to
buy O

Latter return the T-Bill plus interest
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Composition of our portfolio (C)
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Our Portfolio (C)
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Alternative: Passive Investing
Based on the idea that securities are fairly
priced
 Steps involved

Buy a portfolio of assets that tracks the broad
economy
2. Relax
1.

Advantages: Simple, Low Cost, Outperform
the average active strategy
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