Investment Course - 2005
Day One:
Global Asset Allocation and Portfolio Formation
1-0
Two Important Concepts Involving Expected Investment Returns
1. Investors perform two functions for capital markets:
- Commit Financial Capital
- Assume Risk
so,
E(R) = (Risk-Free Rate) + (Risk Premium)
2. The expected return (i.e., E(R)) of an investment has a number of
alternative names: e.g., discount rate, cost of capital, cost of equity, yield
to maturity. It can also be expressed as:
k = (Nominal RF) + (Risk Premium)
= [(Real RF) + E(Inflation)] + (Risk Premium)
where:
Risk Premium = f(business risk, liquidity risk, political risk, financial risk)
1-1
Historical Real Returns, 1954-2003: The Global Experience
Historical Real Returns, 1954-2003
9%
8%
Annually Compounded Real Return, %
7%
6%
5%
Equities
Bonds
4%
3%
2%
1%
Chile: Returns 1/54 – 6/03
Chile*: Returns 1/54 – 12/71; 1/76 – 6/03
Source: Global Financial Data
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Annually Compounded Real Return, %
Global Historical Volatility Measures, 1954-2003
Historical Risk, 1954-2003
40%
35%
30%
25%
20%
Equities
Bonds
15%
10%
5%
0%
1-3
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Annually Compounded Real Return, %
Global Historical Risk Premia, 1954-2003
Risk Premia of Stocks and Bonds to Cash, 1954-2003
18%
16%
14%
12%
10%
8%
Equities-Cash
Bonds-Cash
6%
4%
2%
0%
-2%
1-4
Historical Returns and Risk for Various U.S. Asset Classes
1-5
Historical Global Stock Market Volatility
1-6
More on Historical Asset Class Returns: U.S. Experience
Stocks:
Bonds:
T-Bills:
Inflation:
1926-2004:
Avg. Return
Std. Deviation
12.39%
20.31%
6.19%
8.56%
3.76%
3.14%
3.13%
4.32%
1980-2004:
Avg. Return
Std. Deviation
14.73
16.33
11.05
11.51
6.09
3.26
3.75
2.43
1995-2004:
Avg. Return
Std. Deviation
14.00
21.09
9.45
9.32
3.92
1.90
2.49
0.81
2000-2004:
Avg. Return
Std. Deviation
-0.70
20.32
9.92
5.04
2.72
2.10
2.60
0.97
Source: Ibbotson Associates
1-7
Historical Risk Premia vs. T-bills: U.S. Experience
Stocks:
Bonds:
Stock - Bond
Difference:
1926-2004:
8.63%
2.43%
6.20%
1980-2004:
8.64
4.96
3.68
1995-2004:
10.08
5.53
4.55
2000-2004:
-3.42
7.20
-10.62
1-8
Performance of U.S.-Oriented Investment Strategies: 1975-2004
Growth
of $1
Avg.
Ann. Ret.
Std. Dev.
Sharpe
Ratio
100% Stock
$47.52
14.90%
16.13%
0.540
100% Bond
$16.06
10.23
11.26
0.359
100% Cash
$6.19
6.19
3.09
nm
“60-30-10”
Mix
$30.70
12.63
11.12
0.579
1-9
Portfolio Management Strategy: Broad View

Passive Management
Attempt to generate “normal” returns over time commensurate
with investor risk tolerance
 Typically achieved through diversified asset class selection and
asset-specific portfolio formation


Active Management
Attempt to generate above-normal returns over time relative to
acceptable risk level
 Typically achieved either through periodic asset class or securityspecific portfolio adjustments

1 - 10
Two Ways to Increase Returns
(i.e., “Add Alpha”):


Tactical Allocation Decisions
- Global Market Timing
- Asset Class Timing
- Style/Sector Timing
Security Selection Decisions
- Stock or Bond Picking
1 - 11
Allure of Tactical Market Timing

Suppose that on January 1st each year from 1975-2004,
you put 100% of your money in what turned out to be the
best asset class (stocks, bonds, or cash) at the end of
the year.

This is equivalent to owning a perfect lookback option
that entitles you to receive the return for the best
performing asset class each year.

What difference would that type of tactical portfolio
rebalancing make to your investment performance?
1 - 12
Allure of Tactical Market Timing (cont.)
Growth
of $1
Avg.
Ann. Ret.
Std. Dev.
Sharpe
Ratio
100% Stock
$47.52
14.90%
16.13%
0.540
100% Bond
$16.06
10.23
11.26
0.359
100% Cash
$6.19
6.19
3.09
nm
“60-30-10”
Mix
$30.70
12.63
11.12
0.579
“Perfect Foresight”
$237.68
20.48
10.92
1.309
1 - 13
Danger of “Missing the Boat” (i.e., Not Being Invested):
S&P 500 Annualized
Return: 1980-1989
S&P 500 Annualized
Return: 1990-1999
12.6%
15.3%
Less: 10 Best Days
7.7
11.1
Less: 20 Best Days
4.7
8.1
Less: 30 Best Days
2.1
5.6
Less: 40 Best Days
-0.3
3.4
Less: 10 Worst Days
21.0
20.1
Less: 20 Worst Days
24.7
23.4
Less: 30 Worst Days
27.5
26.4
Less: 40 Worst Days
30.5
28.9
Investment Period
Entire Decade
(2,528 Days)
1 - 14
The Asset Allocation Decision

A basic decision that every investor must make is how to distribute his or
her investable funds amongst the various asset classes available in the
marketplace:







Stocks (e.g., Domestic, Global, Large Cap, Small Cap, Value, Growth)
Fixed-Income (e.g., Government, Investment Grade, High Yield)
Cash Equivalents (e.g., T-bills, CDs, Commercial Paper)
Alternative Assets (e.g., Private Equity, Hedge Funds)
Real Estate (e.g., Residential, Commercial)
Collectibles (e.g., Art, Antiques)
The Strategic (or Benchmark) allocation is the proportion of wealth the
investor decides to place in each of these asset classes. It is sometimes
also referred to as the investor’s long-term normal allocation because it is
presumed to be the “baseline” allocation that will remain in place until the
investor’s life circumstances change appreciably (e.g., retirement)
1 - 15
The Importance of the Asset Allocation Decision

In an influential article published in Financial Analysts
Journal in July/August 1986, Gary Brinson, Randolph
Hood, and Gilbert Beebower examined the issue of how
important the initial strategic allocation decision was to
an investor

They looked at quarterly return data for 91 pension funds
over a ten-year period and decomposed the average
returns as follows:




Actual Overall Return (IV)
Return due to Strategic Allocation (I)
Return due to Strategic Allocation and Market Timing (II)
Return due to Strategic Allocation and Security Selection (III)
1 - 16
The Importance of the Asset Allocation Decision (cont.)

Graphically:

In terms of return performance,
they found that:
1 - 17
The Importance of the Asset Allocation Decision (cont.)

In terms of return variation:

Ibbotson and Kaplan support this conclusion, but argue that the
importance of the strategic allocation decision does depend on
how you look at return variation (i.e., 40%, 90%, or 100%).
1 - 18
Examples of Strategic Asset Allocations

Public Endowments:
1 - 19
Examples of Strategic Asset Allocations (cont.)

Public Retirement Fund:
1 - 20
Examples of Strategic Asset Allocations (cont.)
1 - 21
Asset Allocation and Building an Investment Portfolio
I. Global Market Analysis
- Asset Class Allocation
- Country Allocation Within Asset Classes
II. Industry/Sector Analysis
- Sector Analysis Within Asset Classes
III. Security Analysis
- Security Analysis Within Asset Classes
and Sectors
1 - 22
Asset Allocation Strategies

Strategic Asset Allocation: The investor’s “baseline” asset
allocation, taking into account his or her return requirements, risk
tolerance, and investment constraints.

Tactical Asset Allocation: Adjustments to the investor’s strategic
allocation caused by perceived relative mis-valuations amongst the
available asset classes. Ordinarily, tactical strategies overweight the
undervalued asset class. Also known as market timing strategies.

Insured Asset Allocation: Adjustments to the investor’s strategic
allocation caused by perceived changes in the investor’s risk
tolerance. Usually, the asset class that experiences the largest
relative decline is underweighted. Portfolio insurance is a wellknown application of this approach.
1 - 23
Sharpe’s Integrated Asset Allocation Model
C1
Capital Market
Conditions
I1
Investor Assets, Liabilities
and Net Worth
C2
Prediction
Procedure
I2
Investor's Risk Tolerance
Function
C3
Expected Returns, Risk
and Correlations
I3
Investor's Risk Tolerance
M1
Optimizer
M2
Investor's Asset Mix
M3
Returns
1 - 24
Sharpe’s Integrated Asset Allocation Model (cont.)

Notice that the feedback loops after the performance assessment
box (M3) make the portfolio management process dynamic in
nature.

The strategic asset allocation process can be viewed as going
through the model once and then removing boxes (C2) and (I2),
thus removing any temporary adjustments to the baseline allocation.

Tactical asset allocation effectively removes box (I2), but allows for
allocation adjustments due to perceived changes in capital market
conditions (C2).

Insured asset allocation effectively removes box (C2), but allows for
allocation adjustments due to perceived changes in investor risk
tolerance conditions (I2).
1 - 25
Measuring Gains from Tactical Asset Allocation

Example: Consider the following return and allocation characteristics for a
portfolio consisting of stocks and bonds only.
Allocation:
Strategic
Actual
Stock
60%
50
Returns:
Benchmark
Actual
10%
9

Bond
40%
50
7%
8
The returns to active management (i.e., tactical and security selection) are:
Policy Performance:
Actual Performance:
(.6)(.10) + (.4)(.07)
(.5)(.09) + (.5)(.08)
Active Return =
= 8.80%
= 8.50%
- 30 bp
1 - 26
Measuring Gains from Tactical Asset Allocation (cont.)
Also:
(Policy & Timing):
(.5)(.10) + (.5)(.07)
= 8.50%
(Policy & Selection): (.6)(.09) + (.4)(.08)
= 8.60%
so:
8.50 – 8.80
= -0.30%
Selection Effect: 8.60 – 8.80
= -0.20%
Other: 8.50 – 8.60 – 8.50 + 8.80
= +0.20%
Timing Effect:
Total Active
= -0.30%
1 - 27
Example of Tactical Asset Allocation: Fidelity Investments
1 - 28
Example of Tactical Asset Allocation: Texas TRS
1 - 29
Example of Tactical Asset Allocation: Texas TRS
1 - 30
Overview of Equity Style Investing

The top-down approach to portfolio formation involves prudent
decision-making at three different levels:

Asset class allocation decisions
 Sector allocation decisions within asset classes
 Security selection decisions within asset class sectors

The equity style decision (e.g., large cap vs. small cap, value vs.
growth) is essentially a sector allocation decision

There is tremendous variation in the returns produced by the
myriad style class-specific portfolios, so investors must pay attention
to this aspect of the portfolio management process
1 - 31
Defining Equity Investment Style
The investment style of an equity portfolio is typically defined by two
dimensions or characteristics:
-
Market Capitalization (i.e., Shares Outstanding x Price)
-
Relative Market Valuation (i.e., “Value” versus “Growth”)
1 - 32
Equity Style Classification: Specific Terminology

Market Capitalization
- Large (> $10 billion)
- Mid ($1 - $10 billion)
- Small (< $1 billion)

Relative Valuation
- Value (Low P/E, Low P/B, High Dividend Yield, Low
EPS Growth)
- Blend
- Growth (High P/E, High P/B, Low Dividend Yield, High
EPS Growth)
1 - 33
Equity Style Grid
Value
Large
Growth
Large-Cap
Value (LV)
Large-Cap
Blend (LB)
Large-Cap
Growth (LG)
Mid-Cap
Value (MV)
Mid-Cap
Blend (MB)
Mid-Cap
Growth (MG)
Small-Cap
Value (SV)
Small-Cap
Blend (SB)
Small-Cap
Growth (SG)
Small
1 - 34
Style Indexes & Representative Stock Positions: January 2005
Value
Growth
- Russell 1000 Value
- Russell 1000
- Russell 1000 Growth
- ExxonMobil
Citigroup
- General Electric
Pfizer
- Microsoft
Wal-Mart
- Russell Mid Value
- Russell Mid
- Russell Mid Growth
- Archer Daniels Midlan
Norfolk Southern
- Monsanto
Kroger
- Apple Computer
Adobe Systems
- Russell 2000 Value
- Russell 2000
- Russell 2000 Growth
- Goodyear Tire & Rubber
Energen
- First Bancorp
Crown Holdings
- Allegheny Technologies
Aeropostale
Large
Small
1 - 35
Comparative Classification Ratios: January 2005
(Source: Morningstar)
Value
Large
Growth
Fwd P/E: 15.3
P/B: 2.1
Div Yld: 2.2%
Fwd P/E: 17.8
P/B: 2.8
Div Yld: 1.6%
Fwd P/E: 21.6
P/B: 3.7
Div Yld: 0.9%
Fwd P/E: 16.1
P/B: 2.1
Div Yld: 1.8%
Fwd P/E: 18.4
P/B: 2.5
Div Yld: 1.2%
Fwd P/E: 22.9
P/B: 3.6
Div Yld: 0.4%
Fwd P/E: 13.7
P/B: 1.7
Div Yld: 1.7%
Fwd P/E: 13.1
P/B: 2.1
Div Yld: 1.1%
Fwd P/E: 12.3
P/B: 3.0
Div Yld: 0.3%
Small
1 - 36
Historical Equity Style Performance: 1991-2004
(Source: Frank Russell)
Style Class
Avg
Ann Ret
Std Deviation
Sharpe
Ratio
LV
13.75%
13.36%
0.731
LB
12.67
14.44
0.602
LG
11.38
17.78
0.416
MV
16.01
13.42
0.896
MB
15.25
15.02
0.751
MG
14.03
21.76
0.462
SV
16.98
14.51
0.896
SB
14.58
18.40
0.576
SG
12.34
23.78
0.352
1 - 37
Equity Style Rotation: 1991-2004
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
SG
SV
SV
LG
LV
LG
LV
LG
MG
SV
SV
MV
SG
MV
48.62%
26.52%
21.84%
3.12%
33.04%
21.72%
31.35%
35.84%
44.26%
21.54%
14.54%
-8.56%
41.54%
21.93%
MG
MV
SB
LB
LB
LB
MV
LB
SG
MV
SB
SV
SB
SV
41.10%
20.07%
17.86%
0.85%
32.60%
20.95%
30.55%
26.53%
39.03%
19.20%
5.15%
-9.81%
40.49%
21.04%
SB
SB
LV
SV
LG
LV
LB
MG
LG
MB
MV
LV
SV
MB
39.94%
18.02%
16.97%
-1.11%
32.18%
20.19%
29.85%
20.53%
30.33%
9.78%
3.36%
-14.84%
39.54%
19.07%
SV
MB
MV
SB
MV
SV
SV
LV
SB
LV
MB
MB
MG
SB
36.59%
15.57%
14.84%
-1.31%
30.54%
19.95%
28.50%
16.80%
21.01%
8.32%
-3.62%
-15.82%
36.84%
17.93%
MB
LV
MB
LV
MB
MV
LG
MB
LB
SB
SG
SB
MB
LV
36.44%
13.28%
13.70%
-1.53%
30.20%
19.00%
28.37%
12.35%
19.92%
0.52%
-4.14%
-20.21%
34.89%
15.60%
LG
SG
SG
MG
MG
MB
MB
MV
MB
MG
LV
LB
MV
MG
36.44%
9.04%
13.34%
-1.59%
29.96%
18.13%
26.60%
7.04%
17.83%
-4.73%
-4.95%
-22.26%
33.45%
15.14%
MV
LB
MG
MB
SG
MG
MG
SG
LV
LB
LB
MG
LV
SG
33.62%
8.94%
11.19%
-1.60%
28.01%
17.21%
21.76%
6.79%
7.92%
-6.66%
-11.32%
-29.08%
27.30%
14.74%
LB
MG
LB
MV
SB
SB
SB
SB
MV
SG
MG
LG
LB
LB
29.99%
8.89%
9.91%
-1.69%
25.73%
16.41%
21.51%
1.14%
0.70%
-17.29%
-15.09%
-30.16%
27.00%
11.12%
LV
LG
LG
SG
SV
SG
SG
SV
SV
LG
LG
SG
LG
LG
23.06%
5.19%
3.17%
-1.72%
23.43%
12.78%
14.44%
-4.35%
-0.54%
-22.07%
-18.09%
-32.39%
26.76%
8.96%
1 - 38
200412
200406
200312
200306
200212
200206
200112
-30.00
200106
40.00
200012
200006
199912
199906
199812
199806
199712
199706
199612
199606
199512
199506
199412
199406
199312
199306
199212
199206
199112
Annualized Return Difference (%)
Relative Return Performance:
Value vs. Growth
50.00
LV Outperforms
30.00
20.00
10.00
0.00
-10.00
-20.00
LG Outperforms
-40.00
-50.00
1 - 39
1 - 40
200412
200406
200312
200306
200212
200206
200112
200106
200012
200006
199912
199906
-20.00
199812
20.00
199806
199712
199706
199612
199606
199512
199506
199412
199406
199312
199306
199212
199206
199112
Annualized Risk Difference (%)
Relative Risk Performance:
Value vs. Growth
30.00
LV Riskier
10.00
0.00
-10.00
LG Riskier
-30.00
200406
200412
200312
200212
200306
200112
200206
200106
50.00
40.00
30.00
20.00
10.00
0.00
-10.00
-20.00
-30.00
-40.00
-50.00
200006
200012
199912
199812
199906
199806
199706
199712
199612
199512
199606
199506
199406
199412
199306
199312
199212
199112
199206
Annualized Return Difference (%)
Relative Return Performance:
Large Cap vs. Small Cap
LB Outperforms
SB Outperforms
1 - 41
200406
200412
200306
200312
200212
200112
200206
200012
200106
200006
-20.00
199906
199912
199806
199812
199712
199612
199706
199606
199506
199512
199406
199412
199312
199212
199306
199112
199206
Annualized Risk Difference (%)
Relative Risk Performance:
Large Cap vs. Small Cap
30.00
LB Riskier
20.00
10.00
0.00
-10.00
SB Riskier
-30.00
1 - 42
Value vs. Growth: Global Evidence (Source: Chan and Lakonishok,
Financial Analysts Journal, 2004)
1 - 43
Equity Style Investing: Instruments and Strategies

Passive Style Alternatives
- Index Mutual Funds
- Exchange-Traded Funds (ETFs)

Active Style Alternatives
- Investor Portfolio Formation
- Open-Ended Mutual Funds
1 - 44
Methods of Indexed Investing

Open-End Index Mutual Funds: There is a long-standing and active
market for mutual funds that hold broad collections of securities that
mimic various sectors of the stock market. Examples include the
Vanguard 500 Index Fund, which recreates the holdings and
weightings of the Standard & Poor’s 500, and the various Fidelity
Select Funds, which reproduce the profiles of different industry
sectors.

Exchange-Traded Funds (ETF): A more recent development in the
world of indexed investment products has been the development of
exchange-tradable index funds. Essentially, ETFs are depository
receipts that give investors a pro-rata claim on the capital gains and
cash flows of the securities held in deposit.
1 - 45
Index Fund Example: VFINX
1 - 46
Index Fund Example (cont.)
1 - 47
Top ETFs in the Large Blend Style Category
Name
Category
Style
Box
YTD
1 mo
Return % Return %
Consumer Staples Select Sector SPDR (XLP)
Large Blend
0.91
0.91
Industrial Select Sector SPDR (XLI)
Large Value
-4.06
iShares Dow Jones US Cons Goods (IYK)
Large Blend
iShares Dow Jones US Industrial (IYJ)
Large Blend
iShares Dow Jones US Total Market Ind (IYY) Large Blend
iShares Morningstar Large Core Index (JKD)
Large Blend
iShares NYSE Composite Index (NYC)
Large Blend
iShares Russell 1000 Index (IWB)
3 mo
Return %
1 yr
Return %
3 yr
Return %
Trading
Volume
7.18
9.53
-0.79
1,183,000
-4.06
4.98
11.68
5.81
452,700
-0.36
-0.36
10.52
10.83
8.48
61,500
-3.98
-3.98
5.06
10.06
4.85
101,300
-3.40
-3.40
5.11
6.01
3.70
22,100
-3.01
-3.01
5.54
---
---
6,400
-3.29
-3.29
5.81
---
---
600
Large Blend
-2.99
-2.99
4.77
5.84
3.42
136,900
iShares Russell 3000 Index (IWV)
Large Blend
-3.56
-3.56
4.58
5.45
3.78
166,000
iShares S&P 1500 Index (ISI)
Large Blend
-3.57
-3.57
4.55
6.31
---
33,400
iShares S&P 500 Index (IVV)
Large Blend
-3.11
-3.11
4.21
5.25
2.77
731,700
Rydex S&P Equal Weight (RSP)
Large Blend
-4.16
-4.16
5.71
9.22
---
43,400
SPDRs (SPY)
Large Blend
-2.85
-2.85
4.52
5.57
2.76 60,817,300
streetTRACKS DJ Global Titans (DGT)
Large Blend
-2.67
-2.67
4.27
3.01
0.87
1,600
streetTRACKS Fortune 500 Index (FFF)
Large Blend
-3.12
-3.12
4.98
5.63
2.55
8,700
Vanguard Consumer Staples VIPERs (VDC)
Large Blend
0.97
0.97
9.29
11.77
---
58,600
Vanguard Large Cap VIPERs (VV)
Large Blend
-3.84
-3.84
4.66
5.66
---
2,400
Vanguard Total Stock Market VIPERs (VTI)
Large Blend
-3.48
-3.48
4.87
6.12
4.46
85,700
1 - 48
ETF Example: SPY
1 - 49
ETF Example (cont.)
1 - 50
Growth of U.S. Equity Mutual Funds
Morningstar Mutual Fund Style Category:
Year
LV
LB
LG
MV
MB
MG
SV
SB
SG
Total
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
133
138
154
166
211
269
346
406
500
615
750
813
946
158
166
178
197
238
303
371
436
571
778
1036
1198
1407
117
119
124
137
174
228
293
351
421
651
808
1061
1245
60
60
65
67
69
87
102
126
167
212
259
269
301
46
48
53
54
62
69
95
101
121
140
209
268
349
79
78
78
82
106
150
183
221
288
415
559
670
784
25
28
31
38
47
62
79
96
120
193
230
259
270
29
30
30
37
52
71
97
123
147
201
228
299
384
42
44
49
59
77
112
152
206
262
383
472
570
672
689
711
762
837
1036
1351
1718
2066
2597
3588
4551
5407
6358
Source: Morningstar, Frank Russell
1 - 51
Mutual Fund Performance Characteristics:
1991-2003
Style Group
Avg. Annual
Fund Return (%)
Avg. Fund
Std. Dev. (%)
Sharpe
Ratio
Large Value
11.81
13.03
0.587
Large Blend
11.43
13.98
0.520
Large Growth
12.44
17.57
0.471
Mid Value
13.85
13.09
0.740
Mid Blend
13.88
14.91
0.652
Mid Growth
14.57
21.35
0.488
Small Value
16.18
14.33
0.839
Small Blend
14.89
16.00
0.671
Small Growth
15.86
22.81
0.513
1 - 52
Mutual Fund Performance Characteristics:
1991-2003 (cont.)
Style Group
Avg. Fund Firm
Size ($MM)
Avg. Fund
Expense Ratio
(%)
Avg. Fund
Turnover (%)
Median
Tracking Error
(%)
Large Value
24,966
1.41
69.24
4.72
Large Blend
38,137
1.32
73.32
4.24
Large Growth
37,596
1.54
100.41
6.28
Mid Value
6,672
1.56
87.06
6.60
Mid Blend
7,848
1.48
90.58
6.78
Mid Growth
5,924
1.64
135.76
7.88
Small Value
1,710
1.56
67.50
7.12
Small Blend
2,596
1.66
88.21
8.32
Small Growth
1,119
1.70
120.02
8.07
1 - 53
Notion of Tracking Error
When managing an active investment portfolio against a well-defined benchmark (such as the
Standard & Poor’s 500 or the IPSA index), the goal of the manager should be to generate a return
that exceeds that of the benchmark while minimizing the portfolio’s return volatility relative to the
benchmark. Said differently, the manager should try to maximize alpha while minimizing
tracking error.
Tracking error can be defined as the extent to which return fluctuations in the managed portfolio
are not correlated with return fluctuations in the benchmark. The concept is analogous to the
statistic (1 – R2) in a regression context.
A flexible and straightforward way of measuring tracking error can be developed as follows:
Let:
wi = investment weight of asset i in the managed portfolio
Rit = return to asset i in period t
Rbt = return to the benchmark portfolio in period t.
With these definitions, we can define the period t return to managed portfolio as:
R pt 
N
w
i 1
i
Rit
where:
N = number of assets in the managed portfolio
and:
N
w
i 1
i
 1 (i.e., the managed portfolio is fully invested).
1 - 54
Notion of Tracking Error (cont.)
We can then specify the period t return differential between the managed portfolio and the benchmark as:
t 
N
w R
i
i 1
it
- R bt  R pt - R bt .
Notice two things about the return differential . First, given the returns to the N assets in the managed portfolio and the
benchmark, it is a function of the investment weights that the manager selects (i.e.,  = f({wi}/{Ri}, Rb)). Second,  can be
interpreted as the return to a hedge portfolio where wb = -1.
With these definitions and a sample of T return observations, calculate the variance of  as follows:
T
 ( t -  ) 2
 2  t 1
(T - 1)
.
Then, the standard deviation of the return differential is:
 
 2 = periodic tracking error,
so that annualized tracking error (TE) can be calculated as:
TE =   P
where P is the number of return periods in a year (e.g., P = 12 for monthly returns, P = 252 for daily returns).
1 - 55
Notion of Tracking Error (cont.)

Generally speaking, portfolios can be separated into the following
categories by the level of their annualized tracking errors:

Passive (i.e., Indexed): TE < 1.0% (Note: TE < 0.5% is normal)

Structured: 1.0% < TE < 3%

Active: TE > 3% (Note: TE > 5% is normal for active managers)
1 - 56
“Large Blend” Active Manager: DGAGX
1 - 57
Tracking Errors for VFINX, SPY, DGAGX
1 - 58
Risk and Expected Return Within a Portfolio

Portfolio Theory begins with the recognition that the total risk and
expected return of a portfolio are simple extensions of a few basic
statistical concepts.

The important insight that emerges is that the risk characteristics of a
portfolio become distinct from those of the portfolio’s underlying assets
because of diversification. Consequently, investors can only expect
compensation for risk that they cannot diversify away by holding a
broad-based portfolio of securities (i.e., the systematic risk)

Expected Return of a Portfolio:
n
E(R p ) =
 w i * E(R i )
i = 1
where wi is the percentage investment in the i-th asset

Risk of a Portfolio:
 p2  [w1212  ...  w 2n n2 ]  [2w1w 21 2 1,2  ...  2w n-1w n n1 n n 1,n ]
Total Risk = (Unsystematic Risk) + (Systematic Risk)
1 - 59
Example of Portfolio Diversification:
Two-Asset Portfolio


Consider the risk and return characteristics of two stock positions:
E(R1) = 5%
1 = 8%
E(R2) = 6%
2 = 10%
1,2 = 0.4
Risk and Return of a 50%-50% Portfolio:
E(Rp) = (0.5)(5) + (0.5)(6) = 5.50%
and:
p = [(.25)(64) + (.25)(100) + 2(.5)(.5)(8)(10)(.4)]1/2 = 7.55%
Note that the risk of the portfolio is lower than that of either of the
individual securities
1 - 60
Another Two-Asset Class Example:
Suppose that a portfolio is divided into two different subportfolios consisting of stocks and bonds,
respectively. Further assume that the subportfolios have the following risk and expected return
characteristics:
E(Rstock) = 12.0%
E(Rbond) = 5.1%
stock = 21.2%
bond = 8.3%
 = 0.18
Then, an overall portfolio consisting of a 60%-40% mix of stocks and bonds would have the
following characteristics:
E(Rp) = (0.6)(0.120) + (0.4)(0.051) = 0.0924 or 9.24%
and
p = [(0.6)2(0.212)2 + (0.4)2(0.083)2] + [2(0.6)(0.4)(0.212)(0.083)(0.18)] = 0.0188
or
p = (0.0188)1/2 = 0.1371 or 13.71%
For different asset mixes and different levels of correlation between stocks and bonds, the portfolio
variance is given as:
( = 0.18)
( = 1)
( = -1)
Portfolio
wstock
wbond
E(Rp)
p
p
p
1
2
3
4
5
6
7
0.00
0.25
0.28
0.40
0.50
0.75
1.00
1.00
0.75
0.72
0.60
0.50
0.25
0.00
5.10%
6.83
7.04
7.86
8.55
10.28
12.00
8.30%
8.87
9.16
10.58
12.06
16.40
21.20
8.30%
11.53
11.93
13.46
14.75
17.98
21.20
8.30%
0.93
0.00
3.50
6.45
13.83
21.20
1 - 61
Example of a Three-Asset Portfolio:
Suppose that a portfolio is divided into three different subportfolios consisting of stocks, bonds, and
cash equivalents, respectively. Further assume that the subportfolios have the following risk and
expected return characteristics:
E(Rstock) = 12.0%
E(Rbond) = 5.1%
E(Rcash) = 3.6%
stock = 21.2%
bond = 8.3%
cash = 3.3%
stock,bond = 0.18
cash,stock = -0.07
cash,bond = 0.22
Then, an overall portfolio consisting of a 60%-30%-10% mix of stocks, bonds, and cash equivalents
would have the following characteristics:
E(Rp) = (0.6)(0.120) + (0.3)(0.051) + (0.1)(0.036) = 0.0909 or 9.09%
and:

p = [(0.6)2(0.212)2 + (0.3)2(0.083)2 + (0.1)2(0.033)2] +
{[2(0.6)(0.3)(0.212)(0.083)(0.18)] + [2(0.6)(0.1)(0.212)(0.033)(-0.07)]
+ [2(0.3)(0.1)(0.083)(0.033)(0.22)]} = 0.01793
or
p = (0.01793)1/2 = 0.1339 or 13.39%
1 - 62
Diversification and Portfolio Size: Graphical Interpretation
Total Risk
0.40
0.20
Systematic Risk
Portfolio Size
1
20
40
1 - 63
Advanced Portfolio Risk Calculations
Total Portfolio Risk
Suppose you have formed a portfolio consisting of N asset classes. Suppose also the portfolio
weight in the j-th asset class is denoted as wj while jk represents the covariance between assets j
and k (where jk equals the variance, j2, when j = k). With this notation, the return variance, p2,
of the portfolio is given by:
σ 2p 
N
N
 w w
j
j 1 k 1
k
σ jk
(1)
or, equivalently:
σ
2
p

N
w
j1
2
j
σ
2
j

N
N
 w
j1 k 1
j k
j
w k σ jk
(2)
The standard deviation of the portfolio is:
σp
N
  w 2j σ 2j 
 j1


w j w k σ jk 


j1 k 1

j k
N
1/ 2
N
(3)
1 - 64
Advanced Portfolio Risk Calculations (cont.)
Marginal Asset Risk
In order to compute the contribution of asset k’s risk to the overall risk of the portfolio, we can take
the derivative of equation (3) with respect to asset k’s weight in the portfolio:
σ p
w k

1N

w 2j σ 2j 


2 j1


w j w k σ jk 


j1 k 1
j k

1 / 2


N
2 w σ 2  2 w σ 

k k
j jk


j1
j k


N
 w 2σ 2 
j
j

j1


w j w k σ jk 


j1 k 1
j k

1 / 2

w σ 2 
 k k

=
 
= σ
2 -1/2
p
N
N

w σ 2 
 k k

N
N

w j σ jk 


j1
j k

N

w j σ jk 


j1
j k

N
which can be simplified to:
σ p
w k
 1  N
 1  N


    w j σ jk      w j σ j σ k ρ jk 
 σ p   j1
 σ p   j1


(4)
where j and k are the standard deviations of asset classes j and k, respectively, and jk is the
correlation coefficient between them.
1 - 65
Advanced Portfolio Risk Calculations (cont.)
Equation (4) shows that the marginal volatility of asset k in a portfolio is the weighted sum of the kth row (or, equivalently, the k-th column) of the return covariance matrix divided by the standard
deviation of the portfolio. Notice that the magnitude of this marginal risk contribution is determined
by three factors: (i) the volatility of the asset itself, (ii) the asset’s weight in the portfolio, and (iii) the
asset’s covariance with all of the other portfolio holdings and their investment weights.
A convenient property of marginal volatilities is that the weighted sum over all assets is, in fact, the
overall volatility of the portfolio. By contrast, recall that the standard deviation of a portfolio is not
simply a weighted average of the standard deviations of the underlying assets whenever the
correlations between the asset classes are less that +1.0. However, by redefining the risk of asset k
within the portfolio taking those correlations into account—which is what equation (4) does—it is
possible to view overall portfolio risk as an additive statistic. To see this notice that:
σ p
N
w
k 1
k
w k
 1  N
 
  w k   w j σ jk  
 σ p  j1
k 1
 
N
1
=
σp
N
N
 w w
j1 k 1
j
k
σ jk 
σ 2p
σp
 σp
(5)
1 - 66
Advanced Portfolio Risk Calculations (cont.)
This feature provides a mechanism for the portfolio manager to “roll-up” marginal volatilities to a
higher level (e.g., the sector or country level) without having to recompute the derivatives. In other
words, assume that the marginal volatility of each of the N assets has been calculated. Assume also
that we are interested in knowing the aggregate marginal volatility of a collection of M assets where k
= 1 to M < N (i.e., the M assets comprise a subset of the total portfolio). The marginal volatility of
this sub-portfolio is given by:
σ p
M
σ p
w M

w
k 1
k
w k
(6)
M
w
k 1
k
This additive property also allows the portfolio manager to interpret the weighted marginal volatilities
directly as the asset’s contribution to overall portfolio risk or as the contribution to tracking error if
asset class returns are defined in excess of the returns to a benchmark. That is:
σ p
Asset k’s Marginal Risk:
Asset k’s Total Contribution to Risk:
w k
wk
σ p
w k
(7)
(8)
Once again, equations (7) and (8) highlight two facts: (i) Asset k’s marginal volatility within the
portfolio depends not only on its own inherent riskiness (i.e. k) but also how it interacts with every
other asset held in the portfolio (i.e., jk), and (ii) Asset k’s total contribution to the risk of the overall
portfolio also depends on how much the manager invests in that asset class (wk).
1 - 67
Example of Marginal Risk Contribution Calculations
1 - 68
Fidelity Investment’s PRISM Risk-Tracking System:
Chilean Pension System – March 2004
PRISM (CR) / Return, Volatility and Tracking Error for 200401
Obs
1
2
3
4
5
6
7
8
AFP
LLLL
MMMM
NNNN
PPPP
QQQQ
RRRR
SSSS
SISTEMA
AFP
Id
Assets
Volatility
1
2
3
4
5
6
7
8
733
738
635
257
469
54
31
2918
0.081731
0.081423
0.080780
0.077193
0.079808
0.073797
0.061490
0.080525
Obs
1
2
3
4
5
6
7
8
AFP
LLLL
MMMM
NNNN
PPPP
QQQQ
RRRR
SSSS
SISTEMA
Assets
733
738
635
257
469
54
31
2918
0.3646
0.3514
0.4187
0.4713
0.3791
0.8308
2.0927
0.0000
Mean
Portfolio
Return
Tracking
Error
0.003646
0.003514
0.004187
0.004713
0.003791
0.008308
0.020927
0.000000
PRISM (CY) / AFP Value at Risk for 200401
Tracking
Error
(%)
March 17, 2004
0.23543
0.23476
0.22947
0.21804
0.22660
0.22981
0.20014
0.23089
Mean
Excess
Return
0.004531
0.003862
-0.001425
-0.012857
-0.004296
-0.001082
-0.030757
0.000000
March 17, 2004
50bp
Shortfall
Probability
(%)
8.5130
7.7410
11.6213
14.4383
9.3576
27.3640
40.5582
.
200bp
Shortfall
Probability
(%)
0.0000
0.0000
0.0001
0.0011
0.0000
0.8034
16.9613
.
1 - 69
Chilean Sistema Risk Tracking Example (cont.)
PRISM (CX) / Risk Diagnostics
Sistema-Relative Tracking Error for 200401
Obs
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
AFP
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
LLLL
Asset Class
PRD_BCD
PDBC_PRBC
BCP
RECOGN_BOND
CERO
PRC_BCU
BCE
PCD_PTF
ZERO
DEPOSITS_OFFNO
DEPOSITS_OFFUF
LHF
BEF
BSF
CC2
CFI
CORPORATE_BOND
CTC_A
ENDESA
COPEC
ENTEL
CMPC
SQM-B
CERVEZAS
COLBUN
D&S
ENERSIS
Chile_SMALL_CAP
US
EUROPE
UK
JAPAN
GLOBAL
ASIA_EX_JAPAN
LATIN_AMERICA
EASTERN_EUROPE
EMERGING_MARKETS
G7_BOND
HY
EMB
MM
March 17, 2004
Active
Weight
(%)
Worst Case
Contribution
(%)
Contribution
to
Tracking Error
(%)
-0.2144
-1.1800
0.1830
-1.2731
-0.2823
-1.1605
0.0385
-0.0002
-0.0184
0.4018
3.7074
-0.9636
-0.0080
-0.0395
-0.2398
-0.3106
-0.1932
-0.3847
-0.0505
0.1789
-0.0672
0.2043
-0.1966
0.0069
0.0184
-0.4237
0.3731
0.6830
0.8959
0.0364
-0.0954
0.0710
0.8999
-2.1238
-0.5469
-0.1620
1.8244
0.1123
0.0364
0.2535
0.0094
==========
0.0000
-0.0194
-0.0331
0.0111
-0.1965
-0.0026
-0.0822
0.0011
-0.0000
-0.0017
0.0041
0.1039
-0.0363
-0.0011
-0.0029
0.0000
-0.0212
-0.0084
-0.0826
-0.0095
0.0424
-0.0179
0.0436
-0.0356
0.0018
0.0032
-0.1334
0.1107
0.0677
0.1332
0.0063
-0.0158
0.0149
0.1219
-0.3475
-0.0898
-0.0299
0.2729
0.0062
0.0041
0.0304
0.0008
============
-0.1870
0.0023
-0.0097
-0.0003
0.1040
0.0002
0.0193
0.0003
-0.0000
0.0002
0.0001
0.0304
0.0074
0.0002
0.0005
0.0000
0.0004
0.0006
0.0148
0.0001
0.0031
0.0015
0.0104
-0.0010
-0.0001
-0.0001
0.0127
0.0286
0.0039
0.0607
0.0022
-0.0064
0.0029
0.0518
-0.0165
-0.0272
-0.0061
0.0638
0.0008
0.0011
0.0069
0.0002
==============
0.3640
1
Implied
View
(%)
-1.0527
0.8206
-0.1709
-8.1706
-0.0617
-1.6626
0.8206
0.8206
-1.0363
0.0347
0.8206
-0.7657
-2.4299
-1.3547
0.0000
-0.1428
-0.3317
-3.8358
-0.2352
1.7448
-2.1698
5.1077
0.5129
-1.2372
-0.4791
-2.9882
7.6550
0.5649
6.7785
6.0088
6.7521
4.0322
5.7571
0.7754
4.9767
3.7790
3.4975
0.7247
3.0627
2.7101
2.5109
1 - 70
Chilean Sistema Risk Tracking Example (cont.)
PRISM (CX) / Risk Diagnostics
Sistema-Relative Tracking Error for 200401
Obs
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
AFP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
PPPP
Asset Class
PRD_BCD
PDBC_PRBC
BCP
RECOGN_BOND
CERO
PRC_BCU
BCE
PCD_PTF
ZERO
DEPOSITS_OFFNO
DEPOSITS_OFFUF
LHF
BEF
BSF
CC2
CFI
CORPORATE_BOND
CTC_A
ENDESA
COPEC
ENTEL
CMPC
SQM-B
CERVEZAS
COLBUN
D&S
ENERSIS
Chile_SMALL_CAP
US
EUROPE
UK
JAPAN
GLOBAL
ASIA_EX_JAPAN
LATIN_AMERICA
EASTERN_EUROPE
EMERGING_MARKETS
G7_BOND
HY
EMB
MM
March 17, 2004
Active
Weight
(%)
0.5674
-1.1800
-0.5650
-0.2379
-0.2119
0.8570
-0.0128
-0.0002
-0.0184
6.9872
-3.3930
1.2495
-0.0080
-0.0790
-0.2401
-1.2552
-0.6841
-0.4095
-0.1138
-0.0081
0.0744
-0.0372
0.0650
0.0145
0.1316
-0.4237
0.2388
0.8843
-0.1981
-0.2826
-0.0419
0.0953
0.1448
0.0479
-0.5673
-0.3052
-1.7698
-0.1264
0.3431
-0.0563
0.5247
==========
-0.0000
Worst Case
Contribution
(%)
Contribution
to
Tracking Error
(%)
0.0515
-0.0331
-0.0343
-0.0367
-0.0020
0.0607
-0.0004
-0.0000
-0.0017
0.0717
-0.0951
0.0471
-0.0011
-0.0057
0.0000
-0.0858
-0.0298
-0.0880
-0.0213
-0.0019
0.0198
-0.0079
0.0118
0.0038
0.0232
-0.1334
0.0708
0.0876
-0.0295
-0.0492
-0.0069
0.0200
0.0196
0.0078
-0.0931
-0.0563
-0.2647
-0.0070
0.0386
-0.0068
0.0460
============
-0.5113
0.0224
0.0101
0.0010
0.0009
0.0001
0.0138
0.0001
0.0000
-0.0006
0.0131
0.0290
0.0084
-0.0002
-0.0010
0.0000
0.0207
-0.0005
0.0246
0.0045
0.0001
-0.0017
0.0007
-0.0013
0.0001
-0.0009
0.0434
-0.0028
-0.0165
0.0147
0.0297
0.0038
-0.0090
-0.0122
-0.0051
0.0635
0.0318
0.2148
0.0005
-0.0167
0.0023
-0.0150
==============
0.4708
4
Implied
View
(%)
3.9475
-0.8560
-0.1737
-0.3819
-0.0523
1.6098
-0.8560
-0.8560
3.3578
0.1876
-0.856
0.6692
1.9925
1.2991
0.0000
-1.6488
0.0663
-6.0021
-3.9277
-0.8357
-2.2350
-2.0070
-2.0004
0.4622
-0.6567
-10.2519
-1.1930
-1.8696
-7.4319
-10.5277
-8.9775
-9.4234
-8.3883
-10.5855
-11.1915
-10.4358
-12.135
-0.3814
-4.8702
-4.1709
-2.863
1 - 71
Notion of Downside Risk Measures:

As we have seen, the variance statistic is a symmetric measure of
risk in that it treats a given deviation from the expected outcome the
same regardless of whether that deviation is positive of negative.

We know, however, that risk-averse investors have asymmetric
profiles; they consider only the possibility of achieving outcomes that
deliver less than was originally expected as being truly risky. Thus,
using variance (or, equivalently, standard deviation) to portray
investor risk attitudes may lead to incorrect portfolio analysis
whenever the underlying return distribution is not symmetric.

Asymmetric return distributions commonly occur when portfolios
contain either explicit or implicit derivative positions (e.g., using a put
option to provide portfolio insurance).

Consequently, a more appropriate way of capturing statistically the
subtleties of this dimension must look beyond the variance measure.
1 - 72
Notion of Downside Risk Measures (cont.):

We will consider two alternative risk measures: (i) Semi-Variance, and (ii) Lower Partial Moments

Semi-Variance: The semi-variance is calculated in the same manner as the variance statistic,
but only the potential returns falling below the expected return are used:
E(R)
Semi - Variance
=

R p = -

p p (R p - E(R)) 2
Lower Partial Moment: The lower partial moment is the sum of the weighted deviations of each
potential outcome from a pre-specified threshold level (t), where each deviation is then raised to
some exponential power (n). Like the semi-variance, lower partial moments are asymmetric risk
measures in that they consider information for only a portion of the return distribution. The
formula for this calculation is given by:
t
LPM n =

R p = -
p p (t - R p ) n
1 - 73
Example of Downside Risk Measures:
To see how these alternative risk statistics compare to the variance consider the following
probability distributions for two investment portfolios:
Potential
Return
-15%
-10
-5
0
5
10
15
20
25
30
35
Prob. of Return for
Portfolio #1
Prob. of Return for
Portfolio #2
5%
8
12
16
18
16
12
8
5
0
0
0%
0
25
35
10
7
9
5
3
3
3
Notice that the expected return for both of these portfolios is 5%:
E(R)1 = (.05)(-0.15) + (.08)(-0.10) + ...+ (.05)(0.25) = 0.05
and
E(R)2 = (.25)(-0.05) + (.35)(0.00) + ...+ (.03)(0.35) = 0.05
1 - 74
Example of Downside Risk Measures (cont.):
Clearly, however, these portfolios would be viewed differently by different investors.
nuances are best captured by measures of return dispersion (i.e., risk).
These
1. Variance
As seen earlier, this is the traditional measure of risk, calculated the sum of the weighted squared
differences of the potential returns from the expected outcome of a probability distribution. For
these two portfolios the calculations are:
(Var)1 = (.05)[-0.15 - 0.05]2 + (.08)[-0.10 - 0.05]2 + ... + (.05)[0.25 - 0.05]2 = 0.0108
and
(Var)2 = (.25)[-0.05 - 0.05]2 + (.35)[0.00 - 0.05]2 + ... + (.03)[0.35 - 0.05]2 = 0.0114
Taking the square roots of these values leaves:
SD1 = 10.39%
and
SD2 = 10.65%
1 - 75
Example of Downside Risk Measures (cont.):
2. Semi-Variance
The semi-variance adjusts the variance by considering only those potential outcomes that fall
below the expected returns. For our two portfolios we have:
(SemiVar)1 = (.05)[-0.15 - 0.05]2 + (.08)[-0.10 - 0.05]2 + (.12)[-0.05 - 0.05]2 +
(.16)[0.00 - 0.05]2 = 0.0054
and
(SemiVar)2 = (.25)[-0.05 - 0.05]2 + (.35)[0.00 - 0.05]2 = 0.0034
Also, the semi-standard deviations can be derived as the square roots of these values:
(SemiSD)1 = 7.35%
and
(SemiSD)2 = 5.81%
Notice here that although Portfolio #2 has a higher standard deviation than Portfolio #1, it's semistandard deviation is smaller.
1 - 76
Example of Downside Risk Measures (cont.):
3. Lower Partial Moments
For these two portfolios, we will consider two cases (n = 1 and n = 2), both having a threshold level of 0% (i.e., t = 0):
(i) LPM1
(LPM1)1 = (.05)[0.00 - (-0.15)] + (.08)[0.00 - (-0.10)] + (.12)[0.00 - (-0.05)] = 0.0215
and
(LPM1)2 = (.25)[0.00 - (-0.05)] = 0.0125
(ii) LPM2
(LPM2)1 = (.05)[0.00 - (-0.15)]2 + (.08)[0.00 - (-0.10)]2 + (.12)[0.00 - (-0.05)]2
= 0.0022
and
(LPM2)2 = (.25)[0.00 - (-0.05)]2 = 0.0006
For comparative purposes, it is also useful to take the square root of the LPM2 values. These are:
(SqRt LPM2)1 = 4.72%
and
(SqRt LPM2)1 = 2.50%
Notice again that Portfolio #2 is seen as being less risky when the lower partial moment risk measures are used.
1 - 77
Overview of the Portfolio Optimization Process

The preceding analysis demonstrates that it is possible for investors
to reduce their risk exposure simply by holding in their portfolios a
sufficiently large number of assets (or asset classes). This is the
notion of naïve diversification, but as we have seen there is a limit to
how much risk this process can remove.

Efficient diversification is the process of selecting portfolio holdings
so as to: (i) minimize portfolio risk while (ii) achieving expected
return objectives and, possibly, satisfying other constraints (e.g., no
short sales allowed). Thus, efficient diversification is ultimately a
constrained optimization problem. We will return to this topic in the
next session.

Notice that simply minimizing portfolio risk without a specific return
objective in mind (i.e., an unconstrained optimization problem) is
seldom interesting to an investor. After all, in an efficient market,
any riskless portfolio should just earn the risk-free rate, which the
investor could obtain more cost-effectively with a T-bill purchase.
1 - 78
The Portfolio Optimization Process

As established by Nobel laureate Harry Markowitz in the 1950s, the
efficient diversification approach to establishing an optimal set of portfolio
investment weights (i.e., {wi}) can be seen as the solution to the following
non-linear, constrained optimization problem:
Select {wi} so as to minimize:
 p2  [w 12 12  ...  w 2n n2 ]  [2w 1 w 2 1 2 1,2  ...  2w n-1w n n 1 n  n 1,n ]
subject to:

(i) E(Rp) = R*
(ii) S wi = 1
The first constraint is the investor’s return goal (i.e., R*). The second
constraint simply states that the total investment across all 'n' asset
classes must equal 100%. (Notice that this constraint allows any of
the wi to be negative; that is, short selling is permissible.)

Other constraints that are often added to this problem include: (i) All wi > 0
(i.e., no short selling), or (ii) All wi < P, where P is a fixed percentage
1 - 79
Example of Mean-Variance Optimization:
(Three Asset Classes, Short Sales Allowed)
1 - 80
Example of Mean-Variance Optimization:
(Three Asset Classes, No Short Sales)
1 - 81
Mean-Variance Efficient Frontier With and Without Short-Selling
1 - 82
Efficient Frontier Example: Five Asset Classes
1 - 83
Example of Mean-Variance Optimization:
(Five Asset Classes, No Short Sales)
1 - 84
Efficient Frontier Example: 2003 Texas Teachers’ Retirement System
1 - 85
Efficient Frontier Example: Texas Teachers’ Retirement System (cont.)
1 - 86
Efficient Frontier Example: Texas Teachers’ Retirement System (cont.)
1 - 87
Efficient Frontier Example: Chilean Pension System
(Source: Fidelity Investments)
Risk Premium
Chile
Stock
7.19%
Chile
Bond
2.65%
Chile
Cash
0.00%
Developed
Stock
4.89%
Developed
Bond
1.66%
US Stock
5.83%
US Bond
1.41%
Real Cash Return
0.60%
0.60%
0.60%
0.60%
0.60%
0.60%
0.60%
Expected Real Return
7.79%
3.25%
0.60%
5.49%
1.66%
6.43%
2.01%
Volatility
25.02%
6.75%
1.50%
12.57%
3.33%
14.75%
5.05%
Base Case Assumptions:
-Expected real returns based on 1954 – 2003 risk premiums
-Real returns for developed market stocks and bonds are
GDP-weighted excluding US (equally-weighted returns for stocks and bonds are 5.73% and
1.39%, respectively)
- Chilean risk-premium volatility estimates exclude the period 1/72 – 12/75
1 - 88
Efficient Frontier Example: Chilean Pension System (cont.)
Domestic Stocks
Domestic Bonds
Domestic Cash
DM Stocks
DM Bonds
US Stocks
US Bonds
Chile
Stock
100.00%
Chile
Bond
21.85%
100.00%
Chile
Developed Developed
Cash
Stock
Bond
US Stock US Bond
13.51%
35.28%
-20.91%
38.78%
-23.13%
31.04%
2.71%
-0.98%
-1.39%
2.37%
100.00%
1.79%
10.31%
6.27%
3.77%
100.00%
26.01%
71.23%
11.06%
100.00%
7.54%
73.19%
100.00%
16.56%
100.00%
- Correlation matrix is based on real returns from the period
1/93 – 6/03 using Chilean inflation and based in Chilean pesos
- Real returns for developed market stocks and bonds are
GDP-weighted excluding US
1 - 89
Efficient Frontier Example: Chilean Pension System (cont.)
Unconstrained Frontier:
1 - 90
Efficient Frontier Example: Chilean Pension System (cont.)
Constraint Set:
Chile Stock
Chile Bond
Chile Cash
All Foreign Investments
Min Total Equity
Max Total Equity
Fund A
60%
40%
40%
Fund B
50%
40%
40%
Fund C
30%
50%
50%
Fund D
15%
70%
70%
Fund E
0%
80%
80%
30%
40%
80%
30%
25%
60%
30%
15%
40%
30%
5%
20%
30%
0%
0%
1 - 91
Efficient Frontier Example: Chilean Pension System (cont.)
Constrained Frontier for Fund A:
Point on
EF
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Chile
Stock
10.0%
10.6%
11.3%
12.1%
13.2%
14.4%
15.7%
17.2%
18.7%
20.3%
22.1%
23.9%
25.8%
27.8%
29.9%
32.3%
34.8%
37.6%
40.9%
45.6%
Chile
Bond
20.0%
21.0%
22.4%
23.8%
24.9%
25.7%
26.3%
26.9%
27.2%
27.5%
27.7%
27.8%
27.8%
27.6%
27.2%
26.7%
25.9%
25.1%
24.2%
23.0%
Chile
Cash
40.0%
38.4%
36.3%
34.1%
32.1%
30.2%
28.3%
26.4%
24.5%
22.7%
20.9%
19.3%
17.8%
16.4%
15.3%
14.1%
12.9%
11.6%
10.3%
8.0%
Developed Developed
Expected
Stock
Bond
US Stock US Bond Return Volatility
24.2%
0.0%
5.8%
0.0%
3.37
5.65
22.3%
0.2%
7.2%
0.3%
3.43
5.73
20.7%
0.5%
8.2%
0.6%
3.51
5.85
19.3%
0.7%
9.0%
0.9%
3.59
6.00
18.0%
0.9%
9.6%
1.3%
3.68
6.18
16.9%
1.1%
10.0%
1.7%
3.76
6.38
15.9%
1.4%
10.3%
2.0%
3.85
6.62
15.0%
1.6%
10.6%
2.3%
3.95
6.89
14.2%
1.9%
10.9%
2.6%
4.05
7.19
13.5%
2.1%
11.0%
2.8%
4.16
7.51
13.0%
2.3%
11.1%
3.0%
4.27
7.86
12.4%
2.4%
11.1%
3.2%
4.38
8.23
11.9%
2.3%
11.2%
3.3%
4.49
8.63
11.4%
2.3%
11.3%
3.2%
4.61
9.06
10.9%
2.1%
11.4%
3.2%
4.74
9.53
10.5%
2.0%
11.5%
3.1%
4.87
10.04
10.1%
1.8%
11.5%
2.9%
5.01
10.61
9.6%
1.7%
11.6%
2.8%
5.17
11.23
8.8%
1.4%
11.6%
2.7%
5.34
11.97
7.9%
1.2%
12.2%
2.2%
5.63
13.08
1 - 92
Efficient Frontier Example: Chilean Pension System (cont.)
Asset Allocations of Various Funds Using Point 20 on Unconstrained Frontier:
Unconstrained
Fund A
Fund B
Fund C
Fund D
Fund E
Chile
Stock
42.8%
45.6%
35.5%
18.6%
6.5%
0.0%
Chile
Bond
11.2%
23.0%
32.2%
40.3%
54.9%
63.8%
Chile Developed Developed US
US Expected
Cash Stock
Bond Stock Bond Return Volatility
0.0% 15.2%
1.6% 26.0% 3.2% 6.3%
13.9%
8.0%
7.9%
1.2% 12.2% 2.2% 5.6%
13.1%
10.1%
6.1%
1.7% 11.1% 3.3% 5.0%
10.6%
15.7%
6.0%
2.5% 10.7% 6.2% 4.0%
6.9%
14.5%
4.9%
5.4%
6.7% 7.1% 3.3%
4.9%
15.3%
0.0%
7.4%
0.0% 13.5% 2.6%
4.5%
1 - 93
Example of Mean-Lower Partial Moment Portfolio Optimization:
(Five Asset Classes, No Short Sales)
1 - 94
Estimating the Expected Returns and
Measuring Superior Investment Performance

We can use the concept of “alpha” to measure superior investment
performance:
a = (Actual Return) – (Expected Return) = “Alpha”

In an efficient market, alpha should be zero for all investments. That is,
securities should, on average, be priced so that the actual returns they
produce equal what you expect them to given their risk levels.

Superior managers are defined as those investors who can deliver
consistently positive alphas after accounting for investment costs

The challenge in measuring alpha is that we have to have a model
describing the expected return to an investment.

Researchers typically use one of two models for estimating expected
returns:


Capital Asset Pricing Model
Multi-Factor Models (e.g., Fama-French Three-Factor Model)
1 - 95
Developing the Capital Asset Pricing Model
Recall that one of the most fundamental notions in all of finance is that an
investor’s expected return can be expressed in terms of these activities:
E(R) = (Risk-Free Rate) + (Risk Premium)
Clearly, the practical challenge in measuring expected returns comes from
assessing the risk premium component properly. The Capital Market Line
(CML) offers one tractable definition of the risk premium by writing this
relationship as follows:
 E(R m )  RFR 
E(R p )  RFR   p 

m


While this is a reasonable first step, the CML expresses the risk-expected
return tradeoff that investors should expect in an efficient capital market if
they are purchasing entire portfolios of securities.
To be fully useful, financial theory must address the following question:
What is the appropriate risk-expected return relationship for individual
securities?
The problem posed by individual securities (compared to fully diversified
portfolio holdings) is that the risk of those securities contains both
systematic and unsystematic elements.
Simply put, investors cannot
expect to be compensated for risk that they could have diversified away
themselves (i.e., unsystematic risk).
1 - 96
Developing the Capital Asset Pricing Model (cont.)
One way to handle this problem in the context of the CML is to
adjust the number of “total risk” units that the investor assumes for
security i (i.e., i) to account for just the systematic portion of that
risk.
This can be done by multiplying i by the security’s
correlation with the market portfolio (i.e., rim):
 E(R m )  RFR 
E(R i )  RFR  (i rim ) 

m


Rearranging this expression leaves:
 i rim
E(R i )  RFR  
 

m


 [E(R m )  RFR]

or:
E(R i )  RFR  i [E(R m )  RFR]
This is the celebrated Capital Asset Pricing Model (CAPM).
Notice that the CAPM redefines risk in terms of a security’s “beta”
(i.e., i), which captures that stock’s riskiness relative to the
market as a whole.
The graphical representation of the CAPM is called the Security
Market Line (SML).
1 - 97
Using the SML in Performance Measurement: An Example

Two investment advisors are comparing performance. Over the last
year, one averaged a 19 percent rate of return and the other a 16
percent rate of return. However, the beta of the first investor was
1.5, whereas that of the second was 1.0.
a. Can you tell which investor was a better predictor of individual
stocks (aside from the issue of general movements in the market)?
b. If the T-bill rate were 6 percent and the market return during the
period were 14 percent, which investor should be viewed as the
superior stock selector?
c. If the T-bill rate had been 3 percent and the market return were 15
percent, would this change your conclusion about the investors?
1 - 98
Using the SML in Performance Measurement (cont.)
a.
To tell which investor was a better predictor of individual stocks we look at their alphas.
Alpha is the difference between their actual return an that predicted by the SML, given
the risk of their individual portfolios. Without information about the parameters of this
equation (risk-free rate and the market rate of return) we cannot tell which one is more
accurate.
b.
If RF = 0.06 and Rm = 0.14, then
Alpha1 = .19 – [.06+1.5(.14-.06)] = .19 - .18 = 0.01
Alpha2 = .16 – [.06+1(.14-.06)] = .16 - .14 = 0.02
Here, the second investor has the larger alpha and thus appears to be a more accurate
predictor. By making better predictions the second investor appears to have tilted his portfolio
toward undervalued stocks.
c.
If RF = 0.03 and Rm = 0.15, then
Alpha1 = .19 – [.03+1.5(.15 -.03)] = .19 - .21 = -0.02
Alpha2 = .16 – [.03+1(.15 -.03)] = .16 - .15 = 0.01
1 - 99
Using CAPM to Estimate Expected Return:
Empresa Nacional de Telecom
1. Expected Return/Cost of Equity (Assumes RF = 4.73%)
(i) RPm = 4.2%: E(R) = k = 4.73% + 0.79(4.2%) = 8.05%
(ii) RPm = 7.2%:
E(R) = k = 4.73% + 0.79(7.2%)
= 10.42%
2. Expected Price Change (Recall that E(R) = E(Capital Gain) + E(Cash Yield)):
(i) RPm = 4.2%: E(P1) = (4590)[1 + (.0805 - .0196)] = CLP 4869.53
(ii) RPm = 7.2%: E(P1) = (4590)[1 + (.1042 - .0196)] = CLP 4978.31
1 - 100
Estimating Mutual Fund Betas: FMAGX vs. GABAX
1 - 101
Estimating Mutual Fund Betas: FMAGX vs. GABAX (cont.)
1 - 102
Estimating Mutual Fund Betas: FMAGX vs. GABAX (cont.)
1 - 103
The Fama-French Three-Factor Model

The most popular multi-factor model currently used in practice was
suggested by economists Eugene Fama and Ken French. Their model
starts with the single market portfolio-based risk factor of the CAPM and
supplements it with two additional risk influences known to affect security
prices:

A firm size factor
 A book-to-market factor

Specifically, the Fama-French three-factor model for estimating expected
excess returns takes the following form:
(Rit – RFRt) = ai + bi1(Rmt – RFRt) + bi2SMBt + bi3HMLt + eit
where, in addition to the excess return on a stock market portfolio, two other risk factors are
defined:
SMB (i.e., “Small Minus Big”) is the return to a portfolio of small capitalization stocks less
the return to a portfolio of large capitalization stocks
HML (i.e., “High Minus Low”) is the return to a portfolio of stocks with high ratios of
book-to-market values (i.e., “value” stocks) less the return to a portfolio of low bookto-market value (i.e., “growth”) stocks
1 - 104
Estimating the Fama-French Three-Factor Return Model:
FMAGX vs. GABAX
1 - 105
Fama-French Three-Factor Return Model: FMAGX vs. GABAX (cont.)
1 - 106
Fama-French Three-Factor Return Model: FMAGX vs. GABAX (cont.)
1 - 107
Style Classification Implied by the Factor Model
Growth
Value
FMAGX *
Large
* GABAX
Small
1 - 108
Fund Style Classification by Morningstar

FMAGX

GABAX
1 - 109
Does Investment Style Consistency Matter?
Consider the style classification of two funds (A &B) over time:
1 - 110
Does Investment Style Consistency Matter? (cont.)

Study conducted using several thousand mutual funds
from all nine style classes over the period 1991-2003
(see K. Brown and V. Harlow, “Staying the Course: Performance Persistence and the
Role of Investment Style Consistency in Professional Asset Management”)

Calculates style consistency measure for each fund
using two different methods (i.e., R-squared from threefactor model, tracking error from style benchmark) and
correlated these statistics with several portfolio
characteristics, including returns

Estimated regressions of future fund returns on past
performance, style consistency, and other controls (e.g.,
fund expenses, turnover, assets under management)
1 - 111
Correlation of Style Consistency (i.e., R-Squared) With
Other Fund Characteristics
1 - 112
Regression of Future Predicted Returns on:
(i) Past Performance (i.e., Alpha), (ii) Style Consistency (i.e., RSQ), and
(iii) Portfolio Control Variables in both Up and Down Markets
1 - 113
Investment Style Consistency: Conclusions

In general, the findings strongly suggest that fund style consistency
does matter in evaluating future fund performance

Overall, there is a positive relationship between fund style
consistency and subsequent investment performance

However, the nature of how style consistency matters is somewhat
complicated:
In “up” markets, style-consistent funds outperform style- inconsistent
funds, everything else held equal
 The reverse is true in “down” markets: style-inconsistent funds
outperform style-consistent funds, everything else held equal
 “Up” and “down” markets are predictable in advance


Being able to maintain a style-consistent portfolio is a valuable skill
for a manager to have
1 - 114
Using Derivatives in Portfolio Management

Most “long only” portfolio managers (i.e., non-hedge fund managers)
do not use derivative securities as direct investments.

Instead, derivative positions are typically used in conjunction with
the underlying stock or bond holdings to accomplish two main tasks:
“Repackage” the cash flows of the original portfolio to create a more
desirable risk-return tradeoff given the manager’s view of future market
activity.
 Transfer some or all of the unwanted risk in the underlying portfolio,
either permanently or temporarily.


In this context, it is appropriate to think of the derivatives market as
an insurance market in which portfolio managers can transfer certain
risks (e.g., yield curve exposure, downside equity exposure) to a
counterparty in a cost-effective way.
1 - 115
The Cost of “Synthetic” Restructuring With Derivatives
Consider the relative costs of rebalancing a stock portfolio in two ways:
Cost Factor
(i)
physical rebalancing by trading the stocks themselves; or
(ii)
synthetic rebalancing using future contracts
United States
(S&P 500)
Japan
(Nikkei 225)
United Kingdom
(FT-SE 100)
France
(CAC 40)
Germany
(DAX)
Hong Kong
(Hang Seng)
A. Stocks
Commissions
Market Impact
Taxes
Total
0.12%
0.30
0.00
0.42%
0.20%
0.70
0.21
1.11%
0.20%
0.70
0.50
1.40%
0.25%
0.50
0.00
0.75%
0.25%
0.50
0.00
0.75%
0.50%
0.50
0.34
1.34%
0.01%
0.05
0.00
0.06%
0.05%
0.10
0.00
0.15%
0.02%
0.10
0.00
0.12%
0.03%
0.10
0.00
0.13%
0.02%
0.10
0.00
0.12%
0.05%
0.10
0.00
0.15%
B. Futures
Commissions
Market Impact
Taxes
Total
Source: Joanne M. Hill, “Derivatives in Equity Portfolios,” in Derivatives in Portfolio Management, edited by T. Burns, Charlottesville, VA: Association for
Investment Management and Research, 1998.
1 - 116
The Hedging Principle
Suppose a portfolio manager holds a $100 million position in U.S. equity securities and she is
concerned with the possibility that the stock market will decline over the next three months. How
can she hedge the risk that her portfolio will experience significant declines in value?
1) Hedging With Stock Index Futures:
Economic Event
Actual
Stock Exposure
Desired
Futures Exposure
Stock Prices Fall
Loss
Gain
Stock Prices Rise
Gain
Loss
2) Hedging With Stock Index Options:
Economic Event
Actual
Stock Exposure
Desired
Hedge Exposure
Stock Prices Fall
Loss
Gain
Stock Prices Rise
Gain
No Loss
1 - 117
The Hedging Principle (cont.)

Consider three alternative methods for hedging the downside risk of
holding a long position in a $100 million stock portfolio over the next
three months:
1) Short a stock index futures contract expiring in three months.
Assume the current contract delivery price (i.e., F0,T) is $101 and that
there is no front-expense to enter into the futures agreement. This
combination creates a synthetic T-bill position.
2) Buy a stock index put option contract expiring in three months with an
exercise price (i.e., X) of $100. Assume the current market price of the
put option is $1.324. This is known as a protective put position.
3) (i) Buy a stock index put option with an exercise price of $97 and (ii)
sell a stock index call option with an exercise price of $108. Assume
that both options expire in three months and have a current price of
$0.560. This is known as an equity collar position.
1 - 118
1. Hedging Downside Risk With Futures
Expiration Date Value of a Futures-Hedged Stock Position:
Potential
Portfolio Value
Value of Short
Futures Position
60
(101-60) = 41
0
(60+41) = 101
70
(101-70) = 31
0
(70+31) = 101
80
(101-80) = 21
0
(80+21) = 101
90
(101-90) = 11
0
(90+11) = 101
0
(100+0) = 101
100
0
Cost of
Futures Contract
Net Futures
Hedge Position
110
(101-110) = -9
0
(110-9) = 101
120
(101-120) = -19
0
(120-19) = 101
130
(101-130) = -29
0
(130-29) = 101
140
(101-140) = -39
0
(140-39) = 101
Notice that this net position can be viewed as a synthetic Treasury Bill (i.e., risk-free) holding with
a face value of $101.
1 - 119
1. Hedging Downside Risk With Futures (cont.)
Graphically, restructuring the long stock position using a short position in the futures contract
creates the following synthetic restructuring:
Now
Three Months
Long Stock
Short Futures
Net
Position:
Long T-Bill
(p = 0)
Long Stock
(p = 1)
1 - 120
2. Hedging Downside Risk With Put Options
Expiration Date Value of a Protective Put Position:
Potential
Portfolio Value
Value of
Put Option
Cost of
Put Option
Net Protective
Put Position
60
(100-60) = 40
-1.324
(60+40)-1.324 = 98.676
70
(100-70) = 30
-1.324
(70+30)-1.324 = 98.676
80
(100-80) = 20
-1.324
(80+20)-1.324 = 98.676
90
(100-90) = 10
-1.324
(90+10)-1.324 = 98.676
100
0
-1.324
(100+0)-1.324 = 98.676
110
0
-1.324
(110+0)-1.324 = 108.676
120
0
-1.324
(120+0)-1.324 = 118.676
130
0
-1.324
(130+0)-1.324 = 128.676
140
0
-1.324
(140+0)-1.324 = 138.676
1 - 121
2. Hedging Downside Risk With Put Options (cont.)
Long Stock Plus Long Put:
Terminal
Position
Value
Equals:
Terminal
Position
Value
Long Stock
Put-Protected
Stock Portfolio
98.676
98.676
100
-1.324
Expiration Date
Stock Value
Long Put
100
Expiration Date
Stock Value
-1.324
1 - 122
3. Hedging Downside Risk With An Equity Collar
Expiration Date Value of an Equity Collar-Protected Position:
Potential
Portfolio Value
Net Option
Expense
Value of
Put Option
Value of
Call Option
Net CollarProtected Position
60
(0.56-0.56)=0
(97-60)=37
0
60 + 37 = 97
70
(0.56-0.56)=0
(97-70)=27
0
70 + 27 = 97
80
(0.56-0.56)=0
(97-80)=17
0
80 + 17 = 97
90
(0.56-0.56)=0
(97-90)= 7
0
90 + 7 = 97
97
(0.56-0.56)=0
0
0
97 + 0 = 97
100
(0.56-0.56)=0
0
0
100 + 0 = 100
108
(0.56-0.56)=0
0
0
108 - 0 = 108
110
(0.56-0.56)=0
0
(108-110)= -2
110 - 2 = 108
120
(0.56-0.56)=0
0
(108-120)= -12
120 - 12 = 108
130
(0.56-0.56)=0
0
(108-130)= -22
130 - 22 = 108
140
(0.56-0.56)=0
0
(108-140)= -32
140 - 32 = 108
1 - 123
3. Hedging Downside Risk With An Equity Collar (cont.)
Terminal Position
Value
Collar-Protected
Stock Portfolio
108
97
97
108
Terminal
Stock Price
1 - 124
Zero-Cost Collar Example: IPSA Index Options
1 - 125
Zero-Cost Collar Example: IPSA Index Options (cont.)
1 - 126
Another Portfolio Restructuring

Suppose now that upon further consideration, the portfolio manager
holding $100 million in U.S. stocks is no longer concerned about her
equity holdings declining appreciably over the next three months.
However, her revised view is that they also will not increase in value
much, if at all.

As a means of increasing her return given this view, suppose she
does the following:


Sell a stock index call option contract expiring in three months with an
exercise price (i.e., X) of $100. Assume the current market price of the
at-the-money call option is $2.813.
The combination of a long stock holding and a short call option
position is known as a covered call position. It is also often referred
to as a yield enhancement strategy because the premium received
on the sale of the call option can be interpreted as an enhancement
to the cash dividends paid by the stocks in the portfolio.
1 - 127
Restructuring With A Covered Call Position
Expiration Date Value of a Covered Call Position:
Potential
Portfolio Value
Value of
Call Option
Proceeds from
Call Option
Net Covered
Call Position
60
0
2.813
(60+0)+2.813 = 62.813
70
0
2.813
(70+0)+2.813 = 72.813
80
0
2.813
(80+0)+2.813 = 82.813
90
0
2.813
(90+0)+2.813 = 92.813
100
0
2.813
(100+0)+2.813 = 102.813
110
-(110-100) = -10
2.813
(110-10)+2.813 = 102.813
120
-(120-100) = -20
2.813
(120-20)+2.813 = 102.813
130
-(130-100) = -30
2.813
(130-30)+2.813 = 102.813
140
-(140-100) = -40
2.813
(140-40)+2.813 = 102.813
1 - 128
Restructuring With A Covered Call Position (cont.)
Long Stock Plus Short Call:
Equals:
Terminal
Position
Value
Terminal
Position
Value
102.813
Long Stock
Covered Call
Portfolio
Expiration Date
Stock Value
2.813
100
Short
Call
2.813
100
Expiration Date
Stock Value
1 - 129
Some Thoughts on Currency Hedging and Portfolio Management
Question: How much FX exposure should a portfolio manager hedge?
Exchange R ate
Chi lean P eso p er U.S. Do llar
Monthly: Feb 2 9, 20 00 - Feb 28 , 200 5
Hi gh: 7 49
Lo w: 50 2
La st: 573
75 0
Weakening CLP
Strengthening CLP
70 0
65 0
60 0
55 0
50 0
00
01
02
03
04
1 - 130
Conceptual Thinking on Currency Hedging in Portfolio Management
There are at least three diverse schools of thought on the optimal amount of
currency exposure that a portfolio manager should hedge (see A. Golowenko,
“How Much to Hedge in a Volatile World,” State Street Global Advisors, 2003):
Completely Unhedged: Froot (1993) argues that over the long term, real
exchange rates will revert to their means according to the Purchasing Power
Parity Theorem, suggesting currency exposure is a zero-sum game. Further,
over shorter time frames—when exchange rates can deviate from long-term
equilibrium levels—transaction costs make involved with hedging greatly
outweigh the potential benefits. Thus, the manager should maintain an
unhedged foreign currency position.

Fully Hedged: Perold and Schulman (1988) believe that currency exposure
does not produce a commensurate level of return for the size of the risk; in fact,
they argue that it has a long-term expected return of zero. Thus, since the
investor cannot, on average, expect to be adequately rewarded for bearing
currency risk, it should be fully hedged out of the portfolio.

Partially Hedged: An “optimal” hedge ratio exists, subject to the usual caveats
regarding parameter estimation. Black (1989) demonstrates that this ratio can
vary between 30% and 77% depending on various factors. Gardner and
Wuilloud (1995) use the concept of investor regret to argue that a position
which is 50% currency hedged is an appropriate benchmark.
1 - 131

Hedging the FX Risk in a Global Portfolio: Some Evidence

Consider a managed portfolio consisting of five different asset
classes:



Monthly returns over two different time periods:



Chilean Stocks (IPSA), Bonds (LVAC Govt), Cash (LVAC MMkt)
US Stocks (SPX), Bonds (SBBIG)
February 2000 – February 2005
February 2002 – February 2005
Five different FX hedging strategies (assuming zero hedging
transaction costs):
#1: Hedge US positions with selected hedge ratio, monthly rebalancing
#2: Leave US positions completely unhedged
#3: Fully hedge US positions, monthly rebalancing
#4: Make monthly hedging decision (i.e., either fully hedged or completely
unhedged) on a monthly basis assuming perfect foresight about future
FX movements
#5: Make monthly hedging decision (i.e., either fully hedged or completely
unhedged) on a monthly basis assuming always wrong about future FX
movements
1 - 132
Investment Performance for Various Portfolio Strategies:
February 2000-February 2005
1 - 133
Investment Performance for Various Portfolio Strategies:
February 2002-February 2005
1 - 134
Sharpe Ratio Sensitivities for Various Managed Portfolio Hedge Ratios
1 - 135
Currency Hedging and Global Portfolio Management: Final Thoughts

Foreign currency fluctuations are a major source of risk that the global
portfolio manager must consider.

The decision of how much of the portfolio’s FX exposure to hedge is not
clear-cut and much has been written on all sides of the issue. It can depend
of many factors, including the period over which the investment is held.

It is also clear that tactical FX hedging decisions have potential to be a
major source of alpha generation for the portfolio manager.

Recent evidence (Jorion, 1994) suggests that the FX hedging decision
should be optimized jointly with the manager’s basic asset allocation
decision. However, this is not always possible or practical.

Currency overlay (i.e., the decision of how much to hedge made outside of
the portfolio allocation process) is rapidly developing specialty area in global
portfolio management.
1 - 136