SUPTECH WORKSHOP III
Tomáš Výrost
Background Session I – Introduction to modern portfolio theory
Microeconomics
A primer on microeconomics
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Microeconomics
Utility and choice
Preference relation
a b a ∼ b a b
Rationality assumptions:
Every investor possesses a complete preference relation.
The preference relation satisfies the property of
transitivity.
The preference relation is continuous.
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Microeconomics
Utility
Previous are sufficient to guarantee the existence of a
continuous function u : RN
→ R such that, for any
consumption bundles a and b,
a b ⇔ u(a) ≥ u(b)
This real-valued function u is called a utility function.
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Microeconomics
Risk aversion
Consider an investor with wealth Y and a fair-game lottery
L = (h, −h, 0.5) with h > 0.
We say an investor is risk averse iff
Y Y + L
This implies the utility function to be strictly concave:
E[U(Y )] > E[U(Y + L)]
U(Y ) >
1
2
U(Y + h) +
1
2
U(Y − h)
Thus, U (Y ) < 0 and we have decreasing marginal utility of
wealth.
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Microeconomics
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Linear algebra
Linear algebra basics
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Linear algebra
Linear algebra basics
A vector x ∈ Rn
and a matrix A ∈ Rn
× Rm
for n, m ∈ N.
1n = (1, 1, ..., 1) In =





1 0 · · · 0
0 1 · · · 0
0 0
... 0
0 0 · · · 1





A =


1 2 3
0 1 0
4 5 6

 , AT
=


1 0 4
2 1 5
3 0 6


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2 1 0
0 1 0


1 2
1 0
2 3

 =?
Linear algebra
Some examples
r = (r1, r2, ..., rn)
E(r) = (E(r1), E(r2), ..., E(rn))
wT r = (w1, w2, ..., wn)





r1
r2
...
rn





= w1r1 + w2r2 + ... + wnrn =
n
i=1
wiri
1T
n w = (1, 1, ..., 1)





w1
w2
...
wn





=
n
i=1
wi
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Linear algebra
More examples
Let w ∈ Rn
and Σ ∈ RN
× Rn
.
wT
Σw = (w1, w2, ..., wn)
cov(r1, r1) cov(r1, r2) · · · cov(r1, rn)
cov(r2, r1) cov(r2, r2) · · · cov(r2, rn)
.
.
.
.
.
.
.
.
.
.
.
.
cov(rn, r1) cov(rn, r2) · · · cov(rn, rn)
w1
w2
.
.
.
wn
wT
Σw =
n
i=1
n
j=1
wiwjcov(ri, rj)
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Linear algebra
Expected value
E(X) = pixi
Properties:
E1 E(a) = a
E2 E(aX) = aE(X)
E3 E(X + Y ) = E(X) + E(Y )
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Linear algebra
Covariance
cov(X, Y ) = E[(X − E(X))(Y − E(Y ))]
Properties:
C1 cov(a, X) = 0
C2 cov(X, Y ) = cov(Y, X)
C3 cov(a + bX, Y ) = bcov(X, Y )
C4 cov(X + Y, Z) = cov(X, Z) + cov(Y, Z)
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Linear algebra
Variance
var(X) = E[(X − E(X))2
] = cov(X, X)
Properties:
D1 var(a) = 0
D2 var(a + bX) = b2
var(X)
D3 var(X + Y ) = var(X) + var(Y ) + 2cov(X, Y )
Correlation
ρ(X, Y ) =
cov(X, Y )
var(X)var(Y )
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Mean-variance portfolio theory
Mean-variance portfolio
theory
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Mean-variance portfolio theory
Mean-variance preferences
The general n-variate problem is difficult, often simplified to
M-V. Taylor expansion of U(˜Y1) around E(˜Y1):
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Mean-variance portfolio theory
Mean-variance preferences
The general n-variate problem is difficult, often simplified to
M-V. Taylor expansion of U(˜Y1) around E(˜Y1):
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Mean-variance portfolio theory
Mean-variance preferences
The general n-variate problem is difficult, often simplified to
M-V. Taylor expansion of U(˜Y1) around E(˜Y1):
U(˜Y1) = U(E[˜Y1]) +
U (E[˜Y1]) · (˜Y1 − E[˜Y1]) +
1
2
U (E[˜Y1]) · (˜Y1 − E[˜Y1])2
+ ε
thus for the expected utility
E[U(˜Y1)] = U(E[˜Y1]) +
1
2
U (E[˜Y1]) · V ar[˜Y1] + E[ε]
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Mean-variance portfolio theory
The case with two assets
r1, r2 σ2
1, σ2
2 ρ1,2 = cov(r1, r2) w1, w2
rp = w1r1 + w2r2
E(rp) =??
σp =??
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Mean-variance portfolio theory
The case with two assets
r1, r2 σ2
1, σ2
2 ρ1,2 = cov(r1, r2) w1, w2
rp = w1r1 + w2r2
E(rp) = w1E(r1) + w2E(r2)
σ2
p = var(rp) = w2
1σ2
1 + w2
2σ2
2 + 2w1w2cov(r1, r2)
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Mean-variance portfolio theory
Portfolio – initial setup, n assets
Let w ∈ Rn
and Σ ∈ RN
× Rn
.
wT
Σw = (w1, w2, ..., wn)
cov(r1, r1) cov(r1, r2) · · · cov(r1, rn)
cov(r2, r1) cov(r2, r2) · · · cov(r2, rn)
.
.
.
.
.
.
.
.
.
.
.
.
cov(rn, r1) cov(rn, r2) · · · cov(rn, rn)
w1
w2
.
.
.
wn
wT
Σw =
n
i=1
n
j=1
wiwjcov(ri, rj)
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Mean-variance portfolio theory
The diversification/insurance principle
σ2
p =
n
i=1
n
j=1
wiwjcov(ri, rj)
=
n
i=1
w2
i var(ri) + 2
n
i=1 j>i
wiwjcov(ri, rj)
For mutually uncorrelated ri with equal variance σi = σ, and an
equal weights strategy wi = 1/n we have
σ2
p =
n
i=1
1
n2
σ2
=
n
n2
σ2
=
σ2
n
And for the limiting case lim
n→∞
σ2
p = lim
n→∞
σ2
/n = 0
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Mean-variance portfolio theory
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Mean-variance portfolio theory
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Mean-variance portfolio theory
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Mean-variance portfolio theory
Risk-free asset: transformation line
Assume a risk-free asset with E(r) = r and σr = 0.
Now we mix a risky asset (rm, E(rm), σm) with the risk free
asset. For the portfolio with w1 invested into the risky asset, we
get
E(rp) = wmE(rm) + (1 − wm)r
σp = wmσm
E(rp) =
σp
σm
E(rm) + 1 −
σp
σm
r
= r +
E(rm) − r
σm
σp
= δ0 + δmσp
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Mean-variance portfolio theory
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Mean-variance portfolio theory
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Mean-variance portfolio theory
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Mean-variance portfolio theory
Separation principle
The investor makes two separate decisions:
without any recourse to the individual’s preferences, the
investor determines the point of tangency, the market
portfolio.
the investor then determines how he will combine the
market portfolio of risky assets with the riskless asset
The slope of the CML is called market price of risk.
dE(rp)
dσp
=
E(rm) − r
σm
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Mean-variance portfolio theory
Capital Asset Pricing Model – CAPM
E(ri) = r + β[E(rm) − r]
β =
cov(ri, rm)
σ2
m
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Mean-variance portfolio theory
CAPM assumptions
Assumption 1 :
Investors agree in their forecasts of expected returns,
standard deviation and correlations
Therefore all investors optimally hold risky assets in the
same relative proportions
Assumption 2 :
Investors generally behave optimally. In equilibrium prices
of securities adjust so that when investors are holding their
optimal portfolio, aggregate demand equals its supply.
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Mean-variance portfolio theory
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Mean-variance portfolio theory
Systematic risk
σp =
n
i=1
n
j=1
wiwjcov(ri, rj)
=
n
i=1
w2
i var(ri) + 2
n
i=1 j>i
wiwjcov(ri, rj)
For constant covariance cov(ri, rj) = cov, i = j with equal
variance σ2
i = σ2
, and an equal weights strategy wi = 1/n we
have
σ2
p =
n
n2
σ2
+
n(n − 1)
n2
cov =
1
n
σ2
+ 1 −
1
n
cov
And for the limiting case lim
n→∞
σ2
p = cov
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Mean-variance portfolio theory
Properties of Betas
Betas represent an asset’s systematic (market or
non-diversifiable) risk.
Beta of the market portfolio : βm = 1
Beta of the risk-free asset: βr = 0
Portfolio beta: βp = wiβi
Applications of betas: market timing (bull/bear markets),
portfolio construction, performance measures, risk
management.
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Mean-variance portfolio theory
The Efficient Frontier: Markowitz Model
min
1
2
σ2
p =
1
2
wT
Ωw
wT
E(r) = E(rp)
wT
1 = 1
The Two-Fund Theorem: if w1 and w2 represent efficient
portfolios, then αw1 + (1 − α)w2 is also an efficient portfolio
for any α ∈ R.
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Mean-variance portfolio theory
Borrowing and Lending: Market Portfolio
max
E(rp) − r
σp
wT
E(r) = E(rp)
wT
1 = 1
σp = wT
Ωw
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Mean-variance portfolio theory
International Diversification
International investments:
Can you enhance your risk return profile ?
US investors seem to overweight US stocks
Other investors prefer their home country (Home country
bias)
International diversification is easy (and ‘cheap’)
Improvements in technology (the internet)
‘Customer friendly’ products : Mutual funds, investment
trusts, index funds
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Mean-variance portfolio theory
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Mean-variance portfolio theory
Benefits and Costs of Intl. Investments
Benefits:
Interdependence of domestic and international stock
markets
Interdependence between the foreign stock returns and
exchange rate
Costs:
Equity risk: could be more (or less than domestic market)
Exchange rate risk
Political risk
Information risk
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Mean-variance portfolio theory
Performance Measures / Risk Adjusted Rate
of Return
Sharpe ratio:
SRi = (E(ri) − r)/σi
Risk is measured by the standard deviation (total risk of
security). Aim: maximize. (CML)
Treynor ratio:
TRi = (E(ri) − r)/βi
Risk is measured by beta (market risk only). Aim:
maximize. (SML)
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Mean-variance portfolio theory
Estimating the Betas
Time series regression:
ri,t − rt = αi + βi(rm,t − rt) + εi,t
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Mean-variance portfolio theory
Arbitrage pricing theory
Ri,t = ai +
k
j=1
bi,jFj,t + εi,t
Note that the factors are common to all stocks, ie, they are
pervasive risk factors. The parameters
βi,j, called factor loadings, measure the sensitivity of security i
to factor j. The random variable εi,t is the residual, the part of
return not explained by the common factors (var[εi,j] is
idiosyncratic risk).
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Mean-variance portfolio theory
Exact factor pricing, single factor
Assume a single factor is responsible for return dynamics:
rj = aj + βjF
Construct a portfolio consisting of a risk-free asset and the factor
with weights (wf , wF )T = (1 − βj, βj)T . The portfolio return is then
rp = wf rf + wF F = (1 − βj)rf + βjF
The slopes are the same, and by no arbitrage condition, so must be the
intercepts, aj = (1 − βj)rf . Thus, rj = rf + βj(F − rf ) and for the
expectation
E(rj) = rf + βj(E(F) − rf )
When the risk factor is the market, we replace F with Rm and get
CAPM.
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Mean-variance portfolio theory
Fama and French (1993) 3-factor model
E(rj) − rf = βj[E(rm) − rf ] + βjsE[SMB] + βjhE[HML]
Small Value Big Value
Small Neutral Big Neutral
Small Growth Big Growth
Thresholds: size (median), book to market equity (30th & 70th
percentile)
Size premium: SMB = (SmallV alue + SmallNeutral +
SmallGrowth)/3 − (BigV alue + BigNeutral + BigGrowth)/3,
Value premium:
(SmallV alue + BigV alue)/2 − (SmallGrowth + BigGrowth)/2.
<Ken French data library>
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Mean-variance portfolio theory
Empirical testing of CAPM
Time-series tests:
E(Ri,t) − rt = αi + βi[E(Rm,t) − rt]
Cross-section: Fama-MacBeth (1973) rolling regression For any
single t, run
Rt = αT
t e + γtβ + θtZt + ε
If CAPM holds, then for any t, αt = θt = 0 and γt > 0. Thus, it is
easy to test for E(αt) = 0, E(θt) = 0 and E(γt) > 0 from a
sample from t = 1, 2, ..., T.
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