Portfolio Management
Grenoble Ecole de Management
MSc Finance
Fall 2009
2
Risk is not a vague concept
The convention is to define risk as the variance of
returns.
with
3
Risk is not a vague concept
The standard deviation which is the square root of
the variance is easier to interpret because it has
the same dimension as returns
Standard-error is also referred as volatility
4
Covariance as a measure of
diversification
5
Beta
• one of the most important idea of this course: the risk of
a well-diversified portfolio depends on the sensitivity to
market risk (beta) of the securities included in the
portfolio not of the unique risk of each security in the
portfolio
• If we want to know the contribution of an individual
security to the risk of a well diversified portfolio, it is no
good thinking about how risky that security is if held in
isolation.
• We need to measure its sensitivity to market risk, market
movements. That is because unique risk can be
eliminated by diversification.
• This sensitivity is called Beta.
6
Betas and covariances
A statistician would define the beta of stock i as:
7
Efficient Portfolios – 2 assets
Expected
Returns
8.90%
8.80%
0% Energy - 100% Materials
30% Energy - 70% Materials
8.70%
8.60%
8.50%
8.40%
8.30%
Global minimum variance portfolio 70% Energy 30% Materials
8.20%
8.10%
8.00%
Inefficient portfolios
7.90%
7.80%
20%
21%
22%
23%
24%
25%
Variance
Portfolios which dominates are said to be efficient. The location of these portfolios
is the efficient frontier or minimum variance frontier. It runs from the global
minimum variance portfolio (70% Energy – 30% Materials) to the farthest on the
right.
8
Lending and Borrowing
Expected returns
lending
S
rf
borrowing
Sd dev
Since borrowing is negative lending we can extend the range of
possibilities to the right of the market portfolio
9
Capital allocation line
The capital allocation line describes the combinations of expected return
and standard deviation of return available to an investor from combining
her optimal portfolio of risky assets with the risk free asset.
Expected
returns
S
rf
Sd
dev
The CAL is the line starting at the risk free rate of return that is
tangent to the efficient frontier of risky assets.
10
Capital Asset Pricing Model
The use of the CAPM simplifies the estimation of the variancecovariance matrix. It has further implications: it is a single factor
equation describing the expected return on any asset as a linear
function of its beta.
Expected
returns
M
ri
i
rf
βi
1
β
Security market line: the relation between risk and returns for any asset
11
CAPM: in practice
The β representation of the investment universe:
• greatly reduce the computational task of providing the
inputs to a mean-variance optimization.
• enables one to rank assets according to their sensitivity
to market (systemic) risk.
• enables one to estimate expected return for any asset
knowing the risk-free rate, Rf, and the market risk
premium
• enables one to measure the marginal risk contribution
of an asset to an index or a portfolio
12
Sector weighting-Stock selection.
The preceding equation can be rearranged to form the
following relationship:
1 is pure sector allocation. It assumes that within each sector the manager
held the same securities as the benchmark and in the same proportions.
2 is allocation/selection interaction. Joint effect of the portfolio managers’
and security analysts’ decisions to assign weights to both sectors and individual
securities. It equals the difference between the weight of the portfolio in a given
sector and the portfolio’s benchmark for that sector, times the difference
between the portfolio’s and the benchmarks returns in that sector, summed
across all sectors.
13
Sector weighting-Stock selection.
3 is within sector selection. return implicitly assumes that the
manager weights each sector in the portfolio in the same proportion as in the
overall benchmark, although within the sector the manager may hold securities
in different from benchmark weights. Thus, the impact on relative performance
is now attributed only to the security selection decisions of the manager.
14
Ex post alpha
returns
α<0
S
α>0
rf
β
The level of skills depends on the sign and value of alpha which
measures the vertical distance to the SML. We seek managers with positive alpha:
managers that have excess returns compared to the risk of their portfolios.
15
Balanced portfolio
Weights
Beta
Beta P
Sector A
3%
0,7
0,021
Sector B
2%
0,9
0,018
Country 1
4%
1,3
0,052
Country 2
2%
1,7
0,034
Country 3
6%
0,8
0,048
Small Cap
11%
1,6
0,176
Value stocks
12%
1,1
0,132
Index ETF
10%
1
0,1
Future Index
-15%
1
-0,15
Option Index
-4,3%
1
-0,043
Weights
Sensitivity
Sensit P
Euro 2Y
25%
1,7
0,425
EURO 10Y
10%
7
0,7
Euro 30Y
2,5%
16
0,4
Cash
10,5%
0
0
2%
0
0
EUR IR
-3%
0
0
US IR
3%
0
Total
50%
Margin
0,39
3-month Fw 1,4
Total
50%
0
1,53
Overlay/alternative
L/S
Small Cap
20%
1,6
0,32
Large Cap
-20%
0,9
-0,18
Fund HugeReturns
10%
1,3
0,13
Future Index
-10%
1
-0,1
Portable alpha
Etc…
Total Equity
50%
0,56
Weights are useless, only risk
measure must be used: betas,
duration, credit risk…