PORTFOLIO CONCEPTS
MEAN–VARIANCE ANALYSIS
Mean–variance analysis is the fundamental implementation of modern portfolio
theory, and describes the optimal allocation of assets between risky and riskfree assets when the investor knows the expected return and standard
deviation of those assets.
Assumptions necessary for mean–variance efficiency analysis:
1.
All investors are risk averse; they prefer less risk to more for the same level
of expected return.
2.
Expected returns for all assets are known.
3.
The variances and covariances of all asset returns are known.
4.
Investors need know only the expected returns, variances, and covariances
of returns to determine optimal portfolios. They can ignore skewness,
kurtosis, and other attributes of a distribution.
5.
There are no transaction costs or taxes.
2
EFFICIENT PORTFOLIOS
Efficient portfolios (assets) offer the highest level of return for a given level
of risk as measured by standard deviation in modern portfolio theory.
• Because investors are risk-averse, by assumption, they will choose to allocate
their assets to portfolios that have the highest possible level of expected return
for a given level of risk.
• These portfolios are known as efficient portfolios.
- We can use optimization techniques to determine the necessary weights to
minimize the portfolio standard deviation for a specified set of expected
returns, standard deviations, and correlations for the assets comprising the
portfolio.
3
PORTFOLIO EXPECTED RETURN AND RISK
• We can calculate the expected return and variance of a two asset portfolio as:
𝐸 𝑅𝑝 = 𝑤1 𝑅1 + 𝑤2 𝑅2
σ2𝑝 = 𝑤12 σ12 + 𝑤22 σ22 + 2𝑤1 𝑤2 σ1 σ2 ρ1,2
• We can calculate the expected return and variance of a three asset portfolio
as:
𝐸 𝑅𝑝 = 𝑤1 𝑅1 + 𝑤2 𝑅2 + 𝑤3 𝑅3
σ2𝑝 = 𝑤12 σ12 + 𝑤22 σ22 + 𝑤32 σ23 + 2𝑤1 𝑤2 σ1 σ2 ρ1,2 + 2𝑤1 𝑤3 σ1 σ3 ρ1,3 + +2𝑤2 𝑤3 σ2 σ3 ρ2,3
• Standard deviation is, of course, the positive square root of variance in both
cases.
4
PORTFOLIO EXPECTED RETURN AND RISK
Focus On: Calculations
• You are examining three international indices. What is the expected return and
standard deviation of a portfolio composed of 50% French equities, 25%
English equities, and 25% German equities?
Correlation with:
Portfolio Exp Ret Std Dev French
English
French
7.028
17.447
1
English
0.724
5.870
0.2390
1
German
–0.077
7.431
0.2894
0.5830
• The E(r) is 3.6758%,
and the standard deviation
is 10.0191.
German
1
σ2𝑝
= 0.52 17.447 2 + 0.252 5.8702 + 0.252 7.4312
+ 2 0.5 0.25 17.447 5.870 0.2390
+ 2 0.5 0.25 17.447 7.431 0.2894
+ 2 0.25 0.25 (5.877.431)(5.870)(0.5830)
5
THE EFFICIENT FRONTIER
The efficient frontier is a plot of the set of expected returns and standard
deviations for all efficient portfolios (assets) above the global minimumvariance portfolio.
• The minimum-variance frontier
(solid green line) is the set of
all portfolios that represent the
lowest level of risk that can be
achieved for each possible level
of return.
- The portfolio with the lowest
variance of all the portfolios,
with the lowest level of risk
that can be achieved, is
known as the global
minimum-variance portfolio.
Efficient Frontier
E(r)
Standard Deviation
6
THE EFFICIENT FRONTIER
Portfolios on the efficient frontier provide the highest possible level of
return for a given level of risk.
• Because portfolios on the
efficient frontier use risk
efficiently to generate returns,
investors can restrict their
selection process to portfolios
lying on the frontier.
- This approach simplifies the
risky-asset selection process
and reduces selection cost.
- The light green portfolios in
the figure are inefficient
portfolios.
Efficient Frontier
E(r)
Standard Deviation
7
DIVERSIFICATION AND CORRELATION
The trade-off between portfolio risk as measured by standard deviation and
portfolio expected return is affected by asset returns, variances, and
correlations.
𝐸 𝑅𝑝 = 𝑤1 𝑅1 + 𝑤2 𝑅2
• Recall the expected return and variance
σ2𝑝 = 𝑤12 σ12 + 𝑤22 σ22 + 2𝑤1 𝑤2 σ1 σ2 ρ1,2
of a two-asset portfolio.
• All the terms in the variance calculation
are strictly positive, except the last
term, which includes the correlation,
which ranges from perfect negative (–1:
blue) to perfect positive (+1: purple)
with zero correlation in between (0:
green).
Range of Correlation Effects
30
25
20
E(r) 15
10
- As a correlation moves from perfect
positive toward perfect negative,
diversification benefits increase.
5
0
0
10
20
30
Standard Deviation
8
FINDING THE MINIMUM-VARIANCE FRONTIER
We can use an optimizer, such as the Solver in Excel, to solve for the
weights in the minimum-variance portfolios and thus the minimumvariance frontier.
n
• Recall that the set of weights in any portfolio
must sum to 1 and, if there are no short sales,
must all be positive.
• The expected return and variance for a given
set of weights are
• For every return, z, between zmin and zmax,
we solve for the set of weights that
minimizes the portfolio variance subject to
E(rp) = z.
w
i 1
i
1
E rp    wi E ri 
n
i 1
Var  rp    wi w j σi σ j ρi , j
n
n
i 1 j 1
- If we do so iteratively, we begin at zmin and
iterate by a fixed amount of E(rp) until we
reach zmax.
9
EQUAL-WEIGHTED PORTFOLIOS
• The expected return to an equally weighted portfolio is just the sum of the
expected returns to the assets divided by the number of assets.
• It can be shown that the variance of an equally weighted portfolio is:
1 2 𝑛−1
2
σ𝑝 = σ +
Cov
𝑛
𝑛
where n is the number of assets in the portfolio, σ2 is the average variance of
those assets, and Cov is the average covariance of the assets.
• Consider a 10-asset portfolio with average variance of 0.0225 and average
covariance of 0.04. The variance of such a portfolio will be
1
9
σ2𝑝 =
0.0225 +
(0.04) = 0.03825
10
10
10
THE CAPITAL ALLOCATION LINE
The capital allocation line (CAL) describes the optimal expected return and
standard deviation combinations available from combining risky assets
with a risk-free asset.
• This is a line originating at the expected return–standard deviation coordinates
of the risk-free asset and lying tangent to the efficient frontier.
- The slope of this line is known as the Sharpe ratio, and it represents the best
possible risk–return trade-off by construction.
- As can be seen from the equation for the CAL:
𝐸 𝑅𝑇 − 𝑅𝐹
𝐸 𝑅𝑝 = 𝑅𝐹 +
σ𝑃
σ𝑅𝑇
- The intercept is the risk–return coordinate for the risk-free asset or [RF,0].
- The slope is the excess return E(RT) – RF per unit of risk sRT.
11
THE CAPITAL ALLOCATION LINE
CAL
E(r)
Efficient Frontier
Standard Deviation
12
THE CAPITAL ALLOCATION LINE
Focus On: Calculations
Consider an investor facing a 3% risk-free rate with access to a tangency
portfolio with a 12% return and an 18% standard deviation.
- If the investor requires a 10% return, how much risk will she have to bear?
14%
𝐸 𝑅𝑝 = 𝑅𝐹 +
0.1 = 0.03 +
𝐸 𝑅𝑇 − 𝑅𝐹
σ𝑃
σ𝑅𝑇
0.12 − 0.03
σ𝑃
0.18
σ𝑃 = 0.14
13
THE CAPITAL MARKET LINE
When all investors share identical expectations about the expected
returns, variances, and covariances of assets, the CAL becomes the CML.
The capital market line (CML) represents the case in which all investors have the
same expectations and, therefore, hold the same risky portfolio as the tangency
portfolio.
- In equilibrium, this will be all risky assets in their market value weights;
hence, all investors will hold the market portfolio as part of their portfolio.
𝐸 𝑅𝑝 = 𝑅𝐹 +
𝐸 𝑅𝑀 − 𝑅𝐹
σ𝑃
σ𝑅𝑀
- The slope of the CML is known as the market price of risk and is the Sharpe
ratio for the market portfolio.
14
CAPITAL ASSET PRICING MODEL
The capital asset pricing model, or CAPM, describes the expected return to
any asset as a linear function of its “beta.”
•
The CAPM proposes that all security expected returns can be broken down into
two components:
𝐸 𝑅𝑖 = 𝑅𝐹 + β𝑖 [𝐸 𝑅𝑀 − 𝑅𝐹 ]
-
A risk-free component (in red).
-
A component received for bearing market risk (in blue).
-
This component is the amount of risk, bi, times the price of risk, E(RM) – RF.
-
bi is a measure of the asset’s sensitivity to market movements (market risk).
-
-
bi = 1 is the beta for the market, or bM.
-
bi > 1 is greater than the beta for the market and we would expect returns
in excess of market returns.
-
bi < 1 is less than the beta for the market and we would expect returns
lower than market returns.
-
bi = 0 is zero market risk (risk free) and we would expect the risk-free
return.
E(RM) – RF is known as the market risk premium.
15
CAPM ASSUMPTIONS
1. Investors need only know the expected returns, the variances, and the
covariances of returns to determine which portfolios are optimal for them.
-
This assumption appears throughout all of mean–variance theory.
2. Investors have identical views about risky assets’ mean returns, variances of
returns, and correlations.
3. Investors can buy and sell assets in any quantity without affecting price, and
all assets are marketable (can be traded).
4. Investors can borrow and lend at the risk-free rate without limit, and they can
sell short any asset in any quantity.
5. Investors pay no taxes on returns and pay no transaction costs on trades.
16
THE SECURITY MARKET LINE
The graphical depiction of the CAPM is often known as the security market
line, or SML.
SML
E(r)
E(rm)
rf
bm=1
b
17
MARKOWITZ DECISION RULE
The Markowitz decision rule provides several principles by which investors
can determine how to allocate their assets.
• When choosing to allocate all of your money to Asset A or Asset B, choose A
when
- The mean return on A is greater than or equal to that of B, but A has a
smaller standard deviation than B, or
- The mean return of A is strictly larger than that of B, and A and B have the
same standard deviation.
- When either of these is the case, we say that A “mean–variance dominates”
B.
• If we can borrow and lend at the risk-free rate, then
- The portfolio with the higher Sharpe ratio mean–variance dominates the
asset with the lower Sharpe ratio and should be chosen.
18
ADDING AN ASSET CLASS
We will add a new asset class to our existing portfolio when making that
addition provides a higher Sharpe ratio for the resulting portfolio.
• In order to determine whether we will have a higher Sharpe ratio, we need
- The Sharpe ratio of the new asset class;
- The Sharpe ratio of the existing portfolio, p; and
- The correlation between the new investment’s returns and those of our
existing portfolio.
𝐸 𝑅𝑛𝑒𝑤 − 𝑅𝐹 𝐸 𝑅𝑝 − 𝑅𝐹
>
ρ𝑅𝑛𝑒𝑤 ,𝑅𝑝
σ𝑛𝑒𝑤
σ𝑝
- If this condition is true, our risk–return relationship is improved by adding the
new asset class.
19
LIMITATIONS OF MEAN–VARIANCE ANALYSIS
Historical estimates of model parameters involve two potential problems: (1)
the large number of estimates needed, and (2) the quality of such estimates.
• The number of parameters needed for mean–variance efficiency analysis is
n2/2 + 3n/2.
- For even a small set of assets, this number is very large.
- For example, if we have 28 assets, that is 434 parameters that must be used
in the optimization process.
• The quality of the estimates themselves is generally low.
- The estimate of mean return has a large variance, and small changes can
dramatically effect mean–variance estimation outcomes.
- The estimate of the variance has a smaller relative variance, but is also
measured with error.
- The estimates of the covariance also have a large amount of measurement
error.
20
THE MARKET MODEL
The market model can be estimated via linear regression and is often used
to estimate unadjusted firm betas.
• The estimated regression equation for the market model is:
𝐸 𝑅𝑖 = α𝑖 + β𝑖 𝑅𝑀
• From this, we can calculate the expected return, variance, and standard
deviation of any stock as:
𝐸 𝑅𝑖 = α𝑖 + β𝑖 𝐸(𝑅𝑀 )
σ2𝑅𝑖 = β2𝑖 σ2𝑀 + σ2ϵ𝑖
σ𝑅𝑖,𝑅𝑗 = β𝑖 β𝑗 σ2𝑀
• Using the market model to determine the necessary mean–variance
parameters reduces the set of parameter estimates to 3n + 2.
21
THE MARKET MODEL
Focus On: Calculations
• We are examining two industry indices, one of which has a beta of 1.87 and a
residual standard deviation of 14.56. The other has a beta of 1.24 and a
residual standard deviation of 9.46.
• If the market portfolio has a variance of 30.2, what is the correlation between
the two assets?
β𝑖 β𝑗 σ2𝑀
σ1,2
ρ1,2 =
=
σ1 σ2
β2𝑖 σ2𝑀 + σ2𝜖1 β22 σ2𝑀 + σ2ϵ2
ρ1,2 =
(1.87)(1.24)(30.2)
1.872 (30.2) + 14.562 1.242 30.2 + 9.462
= 0.3370
22
BETA: TO ADJUST OR NOT ADJUST
Historical betas may not be as useful for predicting future behavior
because we know that betas change over time.
• We can model beta itself from past values of beta.
- Beta can be modeled as an AR(1) process as in Chapter 10, where
β𝑖,𝑡+1 = α0 + α1 β𝑖,𝑡 + ϵ𝑖,𝑡+1
- We know that adjusted betas predict better than unadjusted betas, and that
they are typically mean reverting (see Chapter 10).
- One common adjustment is
β𝑖,𝑡+1 = 0.333 + 0.667β𝑖,𝑡
which can easily be shown to mean revert to a beta of 1.
23
ADJUSTED BETA
Focus On: Calculations
• Use the beta adjustment model: β𝑖,𝑡+1 = 0.333 + 0.667β𝑖,𝑡
• What is the adjusted beta for a firm whose unadjusted beta is 1.87?
β𝑖,𝑡+1 = 0.333 + 0.667 1.87 = 1.5803
24
INSTABILITY IN THE EFFICIENT FRONTIER
When small changes in the input values lead to large changes in the
efficient frontier, it is called “instability in the efficient frontier.”
• Instability arises because we use parameter estimates as inputs rather than
the true underlying parameter values.
- If the differences in parameters are small (statistically or economically
insignificant), the optimization process will likely overfit the model.
- Large negative weights in the absence of short-selling restrictions may be
indicative of this problem.
- The model may indicate frequent rebalancing in response to only small
variable changes.
• Instability may also arise across time because of true changes in the
underlying parameters or because of the same estimation problem as already
noted.
25
MULTIFACTOR MODELS
Models of asset returns that use more than one underlying source of risk,
known as a factor, are known as multifactor models.
• Features of multifactor models:
- The underlying sources of risk are known as systematic factors and referred
to as priced risks.
- Multifactor models explain asset returns better than the market model.
- Multifactor models provide a more detailed analysis of risk than single-factor
models.
• Categories of multifactor models:
1. Macroeconomic  The factors are surprises in macroeconomic
variables.
2. Fundamental  The factors are attributes of stocks or companies.
3. Statistical  The factors are determined statistically and are often the
return on differing portfolios.
26
MACROECONOMIC FACTOR MODELS
A macroeconomic factor surprise is the component of the factor’s return
that was unexpected.
• The surprise is generally measured as the difference between the realized
value and the predicted value prior to realization.
• A k-factor macroeconomic model is expressed as:
𝑅𝑖 = 𝑎𝑖 + 𝑏𝑖,1 𝐹1 + 𝑏𝑖,2 𝐹2 + 𝑏𝑖,3 𝐹3 … 𝑏𝑖,𝑘 𝐹𝑘 + ϵ𝑖
where “a” is the expected return to the asset, the “b” terms are factor
sensitivities, and the “F” terms are the surprises in the macroeconomic
factors.
27
MACROECONOMIC FACTOR MODELS
Focus On: Calculations
• Suppose I believe a multifactor asset pricing model is a correct description of
the risk–return relationship for equity returns. The model takes the following
form:
Ri = ai + bi,f Forex + bi,dDefault + bi,sSize + ei
• I plan on buying two stocks with the following factor sensitivities:
Stock
ai
bi,m
bi,s
bi,v
X
0.05
1.1
1.4
.6
Y
0.02
0.8
1.2
1.0
• What is the expected return to a portfolio of 25% Stock X and 75% Stock Y?
ri = 0.0290 + 0.908Forex + 1.292Default + 0.9180Size
28
ARBITRAGE PRICING THEORY
The APT, as it is known, describes the expected return to an asset as a
linear function of the risk of the asset with respect to a set of factors.
• The APT is an equilibrium model
𝐸 𝑅𝑝 = 𝑅𝑓 + β𝑝,1 l1 + β𝑝,2 l2 + β𝑝,3 l3 … β𝑝,𝐾 l𝐾
where the bs represent factor sensitivities and the ls represent risk premiums.
• The APT relies on three assumptions:
1. A factor model describes asset returns.
2. There are many assets, so investors can form well-diversified portfolios that
eliminate asset-specific risk.
3. No arbitrage opportunities exist among well-diversified portfolios.
•
In contrast to multifactor models, the APT models the expected return in
equilibrium (the first term of the equation), in essence restricting the first term
in the general multifactor expression to the APT value for that term.
29
ARBITRAGE PRICING THEORY
Focus On: Calculations
• You are considering purchasing shares in Cleveland Corp., and you believe the
APT with three priced risk factors is an accurate description of the expected
return to Cleveland Corp. The first risk factor, Macro, has a risk premium of 3%
and Cleveland Corp. has a b for this risk factor of 1.1. The second risk factor,
Term, has a risk premium of 2% and Cleveland has a b of 0.74. Finally, the last
risk factor, Inflation, has a risk premium of 1.3% and Cleveland has a b of 0.27.
• If the current risk-free rate is 3.5%, what is the APT three-factor expected
return to Cleveland Corp. shares?
𝐸 𝑅𝐶 = 𝑅𝑓 + β𝑀,1 Macro + β 𝑇,2 Term + β𝐼,3 Inflation
30
THE APT AND ARBITRAGE
Focus On: Calculations
• Consider the following stock returns and factor sensitivities for a single factor
APT.
Stock
Expected Return
Sensitivity
X
0.10
1.625
Y
0.14
2.625
Z
0.11
2.375
• Can we combine X and Y to achieve an arbitrage possibility with Z?
- What weights create a portfolio with equal sensitivities so that the sensitivity
of the portfolio = the sensitivity of Z?
- Is the expected return to this portfolio the same as the expected return to Z?
31
THE APT AND ARBITRAGE
Focus On: Calculations
- What weights create a portfolio with equal sensitivities so that the sensitivity
of the portfolio = the sensitivity of Z?
β𝑝 = 𝑤𝑥 β𝑥 + (1 − 𝑤𝑥 )β𝑌
β𝑝 = 𝑤𝑥 1.625 + 1 − 𝑤𝑥 2.625 = 2.375
𝑤𝑥 = 0.25, 1 − 𝑤𝑥 = 0.75
- Is the expected return to this portfolio the same as the expected return to Z?
𝐸 𝑅𝑝 = 0.25 0.1 + 0.75 0.14 = 0.13
- No. Therefore, if we go short Z, we can use the proceeds to go long X
and Y in weights 25% and 75%, respectively. We will generate a risk-free
profit of 2% = 13% – 11%.
32
FUNDAMENTAL FACTOR MODELS
In contrast to macroeconomic models, fundamental models use expected
returns (instead of surprises) as factors.
• Because the expected returns no longer have an expected value of zero, as do the
surprises in macroeconomic factor models, the intercept, ai, is no longer an
expected return but the intercept term from a regression.
𝑅𝑖 = 𝑎𝑖 + 𝑏𝑖,1 𝐹1 + 𝑏𝑖,2 𝐹2 + 𝑏𝑖,3 𝐹3 … 𝑏𝑖,𝑘 𝐹𝑘 + ϵ𝑖
• The bi terms are typically factor sensitivities that have been standardized by
the sensitivity across all stocks.
- This is done by subtracting the average sensitivity across all stocks and then
dividing the result by the standard deviation of the attribute across all stocks.
- Doing this enables us to interpret all factor sensitivities as unitless and by
comparison with the “typical” stock.
- A factor sensitivity of 0.75 would then be interpreted as a sensitivity that is ¾
standard deviations above average.
33
THE INFORMATION RATIO
Often denoted IR, the information ratio takes a form similar to the Sharpe
ratio.
• The information ratio can be used to capture the mean active return per
unit of active risk.
- The historical IR is
𝑅𝑝 − 𝑅𝐵
IR =
σ𝑅𝑝−𝑅𝐵
where the subscripts p and B indicate the portfolio being evaluated
and the benchmark, respectively.
- The information ratio is the difference in mean return for the portfolio
and the benchmark divided by the standard deviation of the
difference in return for the portfolio and the benchmark.
• This can be used to set guidelines for the amount by which the portfolio
performance can deviate from its benchmark (tracking risk).
34
ASSESSING ACTIVE RETURN
Focus On: Calculations
• Returning to our previous example, consider a firm that uses the following
asset pricing model to determine expected return:
𝐸 𝑅𝐶 = 𝑅𝑓 + β𝑀,1 Macro + β 𝑇,2 Term + β𝐼,3 Inflation
• This is an empirical model such that the factor sensitivities used are
determined via regression and are not standardized.
• If the portfolio factor sensitivities, benchmark sensitivities, and factor returns
are as follows, how would you decompose the sources of active return for the
portfolio?
Factor Sensitivity
Factor
Return
Factor
Portfolio
Macro
1.1
1
0.1
3%
Term
0.74
1.1
–0.36
2%
Inflation
0.27
0.9
–0.63
1%
Benchmark Difference
35
ASSESSING ACTIVE RETURN
Focus On: Calculations
• If the manager achieved 3.4% active return from asset selection, the active
return sources are then:
Return Components
Absolute
Contribution
% of Total Active
Macro
0.30%
13.9%
Term
–0.72%
–33.3%
Inflation
–0.82%
–37.9%
Return from Factor Tilts
–1.24%
–57.3%
Asset Selection
3.40%
157.3%
Active Return
2.16%
100.0%
• This is an active asset selection manager, as seen by the large proportion of
return attributable to asset selection. The manager also had a positive
contribution from a macro tilt, but did poorly with the inflation and term tilts.
36
ASSESSING ACTIVE RISK
Focus On: Calculations
• Recall that Active risk squared = Active factor risk + Active specific risk
• Consider the following portfolios and their risk calculations:
Active Factor
Risk
Total
Portfolio Industry Index
Factor
Active
Specific
Active Risk
Squared
A
12.25
17.15
29.4
19.6
49
B
1.25
13.75
15
10
25
C
1.25
17.5
18.75
6.25
25
D
0.03
0.47
0.5
0.5
1
37
ASSESSING ACTIVE RISK
Focus On: Calculations
Active Factor (% of total active)
Risk
% Active
Portfolio Industry Index Total Factor Specific
A
B
C
25%
5%
5%
3%
35%
55%
70%
47%
60%
60%
75%
50%
40%
40%
25%
50%
Active Risk
7%
5%
5%
1%
D
• Portfolio D is effectively a passive portfolio with little or no tracking risk (last
column).
• Portfolio A gets most of its active risk from an industry component followed by a
stock-specific component, then a risk-index component.
• Portfolios B and C have similar levels of tracking error, but C has more from
risk factor selection and B from a stock-specific component.
38
ACTIVE RISK: MULTIFACTOR MODELS
Focus On: Calculations
• Factor marginal contribution to active risk squared is
b𝑗𝑎 3𝑖=1 b𝑗𝑎 Cov (𝐹𝑗 𝐹𝑖 )
FMCAR𝑗 =
Active risk squared
• Recall our three-factor model with active factor exposures of 0.1, –0.36, and
–0.63.
• You have calculated the variance–covariance matrix as:
Macro
Term
Inflation
Macro
128
—
—
Term
16
89
—
Inflation
18
24
67
39
ACTIVE RISK: MULTIFACTOR MODELS
Focus On: Calculations
• What are the factor marginal contributions to active risk squared if total active
risk squared is 158.87?
Factor
Num FMCAR
FMCAR
Macro
–0.43
–0.27%
Term
16.4016
10.32%
Inflation
30.9015
19.45%
Total
46.8731
29.50%
• Ex. calculation: NumFMCAR(Term) = –0.36[0.1(16) + –0.36(89) + –0.63(24)]
• Ex. calculation: FMCAR (Inflation) = 30.9015/158.87
• If 29.50% of the active risk is attributable to factor tilts, then (1 – 0.2950) =
0.705, or 70.5% of the active risk is attributable to security selection.
40
TRACKING RISK
Focus On: Calculations
Date
Jan 2001
Index
Return
Fund
Return
Tracking
Error
0.0960
0.0334
–0.0626
–0.0461
• Consider that a mutual fund
and its relevant benchmark
have the returns and tracking
error shown in the table.
Jan 2002
–0.0913 –0.1374
Jan 2003
0.0472
Jan 2004
–0.1495 –0.1951
–0.0456
• The client is a foundation that
wants to earn an active return
above the cost of managing its
account and keep tracking risk
below 5%. We currently
receive 1.5% for managing the
account.
Jan 2005
–0.0684
0.0169
0.0853
Jan 2006
0.1154
0.1896
0.0742
Jan 2007
–0.0221
0.0000
0.0221
Jan 2008
0.0208
0.0954
0.0746
Jan 2009
0.0242
–0.0494
–0.0736
Jan 2010
–0.0868 –0.1302
–0.0434
• Evaluate the performance of
the fund, calculate the IR, and
interpret it.
0.1812
0.134
Avg –0.0115
0.0004
0.0119
StdDev 0.0868
0.1315
0.0751
41
TRACKING RISK
Focus On: Calculations
• Evaluate the performance of the fund.
- The fund is currently earning slightly in excess of its benchmark, but it is
currently not meeting its active return objective because its average tracking
error is below current management fees (1.19 < 1.5).
- It is also not meeting its tracking risk objective because the tracking risk
calculated as the active risk of 7.5% in the prior example is greater than 5%.
• The information ratio for this portfolio is
0.0004 − 0.0015
IR =
= 0.1583
0.0751
The client is earning 15.83 bps per unit of active risk accepted.
42
FORMING A TRACKING PORTFOLIO
Focus On: Calculations
• Tracking portfolios are portfolios with factor sensitivities that match those of the
benchmark portfolio.
• We can formulate the weights for a tracking portfolio of n factors as long as we
have n + 1 well-diversified portfolios.
• Consider the following three well-diversified portfolios and target benchmark
weights:
Factors
Portfolio
Business
Cycle
Term
1
0.75
1.0
2
1.0
0.6
3
1.3
0.8
Benchmark
1.1
0.9
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FORMING A TRACKING PORTFOLIO
Focus On: Calculations
• What are the weights for a tracking portfolio that has the benchmark’s
sensitivities? Solve this set of equations:
which gives weights of w1 = 0.4117, w2 = –0.0882, and w3 = 0.6765.
• Confirming this matches the desired factor sensitivities:
44
MARKET RISK AND NONMARKET RISK PREMIUMS
Investors can earn substantial premiums from exposure to risks
unrelated to market risk when his or her factor risk exposures to other
sources of income and his or her risk aversion differs from the average
investor.
- In such cases, tilts away from indexed investments may be optimal.
- For example, human capital risk increases the factor sensitivity of an
investor who relies on earned employment income to recession risk.
Such an investor will bid up the price of countercyclical stocks and
sell down the price of countercyclical stocks, causing a recession risk
premium to exist for procyclical stocks.
45
SUMMARY
• Portfolio management has a host of quantitative techniques that are used to
- Select assets
- Assess expected returns and risks
- Track performance
• Mean–variance efficient analysis forms the foundation of modern portfolio
theory and describes how investors will choose between risky assets and how
they will weight a portfolio of risky and risk-free assets.
• Asset pricing models generally describe the expected return to assets
(portfolios) as a function of the types and levels of risk they bear and the
rewards due for bearing each type of risk.
46