Goodness of Fit Tests

Theorem (admitted) and the Berra and Jarque test

Let X 1 , X 2 , • • • X n be i.i.d N(m, σ 2 ) and X = (X 1 , X 2 , • • • , X n ) then asymptotically: √ n Skew (X ) ∼ N(0, 6) √ n[ Kur (X ) -3] ∼ N(0, 24)
Let BJ(X ) = n 6 [ Skew (X )] 2 + n 24 [ Kur (X ) -3] 2 then asymptotically: BJ(X ) ≈ χ 2 (2) Let χ 2 1-α (2) be such that P(χ 2 (2) > χ 2 1-α (2)) = 1 -α Then for n large enough the Berra and Jarque test rejects at confidence level α the normality hypothesis iff: BJ(x) > χ 2 1-α (2)

Tests Based on Density Function Estimates Theorem of Parzen Rosenblatt (admitted)

Let X be a random variable of density function f (x) Let X i be i.i.d variables of the same law as X Let K be positive of integral 1 and (h n ) n∈N such that h n → 0 and

nh n → ∞ Let f n (x) be defined by f n (x) = 1 n n i=1 1 hn K ( X i -x hn ) then f n (x) is a density and √ nh n [f n (x) -f (x)] Law --→ N(0, f (x) +∞ -∞ K 2 (x)dx)
Exemples of Kernels

Rectangular Kernel K (u) = 1 2 1 |u|<1 Gaussian Kernel K (u) = 1 √ 2π exp(-u 2
2 ) Remark: "visual test" where the estimated density is usually compared to the density of a normal distribution with the same mean and variance as the empirical mean and variance of the sample. fitted normal with same mean and variance -2.09% -1.67% -1.25% -0.83% -0.41% 0.01% 0.43% 0.85% 1.27% 1.69% 2.11% -2.09% -1.67% -1.25% -0.83% -0.41% 0.01% 0.43% 0.85% 1.27% 1.69% 2.11% -2.09% -1.67% -1.25% -0.83% -0.41% 0.01% 0.43% 0.85% 1.27% 1.69% 2.11% -2.09% -1.67% -1.25% -0.83% -0.41% 0.01% 0.43% 0.85%

(x) = 1 n i=n i=1 1 X i ≤x and F n (x) -F (x) ∞ = sup x | F n (x) -F (x) |
Law of Large Numbers and Central Limit Theorem ∀x, F n (x) → F (x) p.s and

√ n(F n (x) -F (x)) Law --→ N(0, F (x)[1 -F (x)])
Glivenko Cantelli Theorem (admitted)

F n (x) -F (x) ∞ → 0 p.s (which is stronger than the Law of Large Numbers)

Kolmogorov Smirnov Theorem (admitted)

√ n F n (x) -F (x) ∞

Law

--→ K when n is large K is the Kolmogorov's law and is independant from F

Tests Based on Cumulative Distribution Function Estimates

Several Goodness of fit tests are based on Kolmogorov Smirnov's theorem with F being a normal cdf with the same mean and variance as the sample observed. Amongst these tests (available in SAS and with excel extended libraries): Kolmogorov Smirnov's test Cramer von Mises's test Anderson Darling's test Remark: The CLT result is not well adapted to test the normality hypothesis as it is not optimal to test an hypothesis based on the value of the empirical repartition function on one single point. The Kolmogorov Smirnov result is much more interesting for this matter.

Tests Based on Order Statistics

Theorem and Definition: if Z is a random variable and if we note q Z (α) = inf x {x, P(Z ≤ x) ≥ α} then P(Z ≤ q Z (α)) ≥ α and q Z (α) is called the α-quantile of Z Remark 1: if Z ∼ N(0, 1): q Z (0.5) = 0 and q Z (97.5%) = 1.96

Remark 2: (definition derived for a sample via the empirical distribution) if z = (z 1 , z 2 , • • • , z n ) is a sample with frequency (f 1 , f 2 , • • • f n ) the α quantile of z, qz (α) is the quantile of the r.v Z defined by the empirical probability P(Z = z i ) = f i Remark 3: if we have 10 observations:

{z i = i} i∈[1,10] then qz (1) = 10 and ∀i ∈ {1, • • • , 9} ∀α ∈] i 10 , i+1 10 ] qz (α) = i + 1
Tests Based on Order Statistics

Exercice 1: Show that if F Z is invertible then: q Z (α) = F -1 Z (α) Exercice 2: Show that if F Z is invertible then: F Z (Z ) ∼ U([0, 1]) Exercice 3: Show that if Z 1 ∼ N(m 1 , σ 2 1 ) and Z 2 ∼ N(m 2 , σ 2 2 ) then: {(q Z 1 (α), q Z 2 (α)), α ∈ [0, 1]} is a line.
Theorem: order statistics (admitted) and QQ-Plot test

Let N ∼ N(0, 1) and z(n) = (z 1 , z 2 , • • • , z n ) be
a sample for the r.v Z we want to test. Then when n is large:

qz(n) ( i n ) ≈ q Z ( i n ) and if Z is a normal Law:
The points (q N ( i n ), qz(n) ( i n )) should be "concentrated" around a line (called the Henry's line)

Tests Based on Order Statistics

Remark 1: The test can be used "visually" but can also be quantified through the Shapiro-Wilk test Exemple: QQ plot DAX daily returns for 2015

0.00% 0.50% 1.00% 1.50% 2.00% Dax 2015 QQ Plot -2.00% -1.50% -1.00% -0.50% 0.00% -4 -3 -2 -1 0 1 2 3
Parameter Estimations and Confidence Intervals (Reminder) Definition

if X ∼ N(0, 1) then X 2 ∼ χ 2 (1) if X i ∼ N(0, 1) i.i.d then i=n i=1 X 2 i ∼ χ 2 (n)
(Reminder) Theorem and Definition

If X 1 ∼ N(M 1 , Id R d ) and X 2 ∼ N(M 2 , Id R d ) then: M 1 = M 2 =⇒ X 1 2 and X 1 2 have the same law and this law is called χ 2 (d, M 1 2 )
Parameter Estimations and Confidence Intervals demonstration:

M 1 = M 2 =⇒ ∃ A othonormal such that AM 1 = M 2 Let's consider AX 1 then: X 1 has a Normal law =⇒ AX 1 has a Normal law E [AX 1 ] = AE [X 1 ] = AM 1 = M 2 Var [AX 1 ] = Cov (AX 1 , AX 1 ) = ACov (X 1 , X 1 )A = AId R d A = Id R d so,
AX 1 and X 2 are both normal with the same mean and variance so, Theorem: Student Law for Confidence Intervals

AX 1 ∼ X 2 and consequently AX 1 2 ∼ X 2 2 . As AX 1 2 = X 1 2 (because A is orthonormal) by transitivity X 1 2 ∼ X 2 2 . Q.E.D Exercise: Show that Cov (AX , BY ) = ACov (X , Y )B (
Let X = (X 1 , X 2 , • • • , X n ) with X i ∼ N(m, σ 2 ) i.i.d Let X = 1 n i=n i=1 X i and σ(X ) = 1 n i=n i=1 (X i -X ) 2 then: X and σ(X ) are independant √ n( X -m σ ) ∼ N(0, 1) n( σ(X ) σ ) 2 ∼ χ 2 (n -1) that we can write as X -X 2 σ 2 ∼ χ 2 (n -1) X -m σ(X ) √ n-1 ∼ t(n -1) that we can write as m = X -σ(X ) √ n-1 t(n -1)

Parameter Estimations and Confidence Intervals

Sketch of the proof: we just need to show the result for X i ∼ N(0, 1) i.i.d

Let h : R n -→ R n × R n be defined by h :

   X 1 . . . X n    -→ X X -X 1 n then: X Gaussian and h linear ⇒ h(X ) is Gaussian ∀i Cov (X , X i -X ) = 0 ⇒ X and X -X 1 n are independent because for Gaussian vectors zero covariance means independence < X -X 1 n , 1 n >= 0 ⇒ X -X 1 n ∈ (R1 n ) ⊥ Note that ∀u ∈ (R1 n ) ⊥ Var (< u, X -X 1 n >) = Var (< u, X > -X < u, 1 n , >) = Var (< u, X >) = u Var (X )u = u 2 so if we call 0 Z the components of X -X 1 n in an orthonormal basis of R n whose first vector is in R1 n we have Z ∼ N(0, Id R n-1 ) and so X -X 1 n 2 n = Z 2 n-1 ∼ χ 2 (n -1) Q.E.D.

Parameter Estimations and Confidence Intervals

Exemple: GDAX returns for 2015

X = 0.02% σ(X ) √
n-1 = 0.04% So at confidence level 95% the expected daily rate of return is in the interval [-0.06%, 0.09%] Remarks:

t(n -1) is a symmetric distribution from √ n-1(X -m) σ(X )
∼ t(n -1) and the law of large numbers we can deduct that t(n -1)

Law

--→ N(0, 1) (which means that when n is large the law/shape of a student distribution is very similar to the a normal distribution)

Preferred investments Definition: Utility functions

A utility function is any function u : R -→ R such that u is continuous, strictly increasing and two times differentiable.

Remark: based on the definition u is invertible

Definition: Preferred Investment and Risk Premium

If X and Y are two random payoffs:

X Y (X is preferred to Y for u) ⇔ E [u(X )] > E [u(Y )] Let C u (X ) be the constant defined by u[C u (X )] = E [u(X )] C u (X ) is called the certain equivalent to X Π u (X ) = E [X ] -C u (X )

Preferred investments

Properties (exercise)

u convex ⇔ ∀X , u(E [X ]) ≤ E [u(X )] ⇔ ∀X , Π u (X ) ≤ 0 ⇔ risk taker u concave ⇔ ∀X , u(E [X ]) ≥ E [u(X )] ⇔ ∀X , Π u (X ) ≥ 0 ⇔ risk adverse u affine ⇔ ∀X , u(E [X ]) = E [u(X )] ⇔ ∀X , Π u (X ) = 0 ⇔ risk neutral
Theorem: Risk Aversion Measure

Let u and v be two utility functions strictly concave or convex then:

(∀X discrete r.v, Π u (X ) ≥ Π v (X )) ⇔ (∀a, -u (a) u (a) ≥ -v (a) v (a) )
Remarks:

u u defines the risk aversion/concavity of u ∀λ = 0, λu and u have the same risk aversion the concavity measure we arrive at here differs from the geometric definition of curvature which is given by u (1+u 2 )

Preferred investments

Demonstration theorem: Let u be a utility function two times differentiable and strictly convex or concave (i.e u = 0). Demonstration ⇒: Let X a h be defined by: P(X a h = a) = 1 2 and P(X

a h = a + h) = 1 2 Let C a u (h) = C u (X a h ) i.e u(C a u (h)) = E [u(X a h )] = 1 2 u(a) + 1 2 u(a + h) by derivation on h of u(C a u (h)) = 1 2 u(a) + 1 2 u(a + h) we obtain: C a u (0) = a from u(C a u (0)) = 1 2 u(a) + 1 2 u(a) C a u (0) = 1 2 from u (C a u (0))C a u (0) = 1 2 u (a) C a u (0) = 1 4 u (a) u (a) from deriving u (C a u (h))C a u (h) = 1 2 u (a + h) Preferred investments Now, Π u ≥ Π v ⇒ ∀a, ∀h, Π u (X a h ) ≥ Π v (X a h ) ⇒ ∀h, E [X ] -C a u (h) ≥ E [X ] -C a v (h) ⇒ -C a u (0) ≥ -C a v (
0) (as in 0 the value and first derivatives are equal)

⇒ -u (a) u (a) ≥ -v (a) v (a) (and this is true for all a). Q.E.D Demonstration ⇐ Let X be a discrete variable with P(X = a i ) = p i Π u (X ) ≥ Π v (X ) ⇔ C v (X ) -C u (X ) ≥ 0 ⇔ v -1 [E (v (X ))] -u -1 [E (u(X ))] ≥ 0 ⇔ v -1 i=n i=1 p i v (a i ) -u -1 i=n i=1 p i u(a i ) ≥ 0 ⇔ i=n i=1 p i v (a i ) ≥ v • u -1 i=n i=1 p i u(a i ) (as v is increasing) which should be true if v • u -1 is convex.
Preferred investments

Let's calculate (v • u -1 ) to prove that v • u -1 is convex (v • u -1 ) = (v • u -1 )(u -1 ) (v • u -1 ) = (v • u -1 )[(u -1 ) ] 2 + (v • u -1 )(u -1 ) Now, (u -1 ) = 1 u •u -1 and 1 u •u -1 = -(u •u -1 )(u -1 ) (u •u -1 ) 2 = -(u •u -1 ) (u •u -1 ) 3 so, (v • u -1 ) ≥ 0 ⇔ (v •u -1 ) (u •u -1 ) 2 -(v • u -1 ) (u •u -1 ) (u •u -1 ) 3 ≥ 0 ⇔ (v • u -1 ) ≥ (v • u -1 ) (u •u -1 ) (u •u -1 ) ⇔ -(v •u -1 ) (v •u -1 ) ≤ -(u •u -1 ) (u •u -1 ) Q.E.D as we assumed here -v v ≤ -u u

Remarks:

If there is aversion to risk we use u concave to modelize For u(a) = 1 -exp(-λa) (which is often used) -u u = λ

Gaussian Laws and Mean Variance Implications

When investing W 0 the utility function criteria picks strategies π which maximizes:

E [u(W π T )]
where

W π T W 0 = 1 + R π T If we assume R π T ∼ N(m π , σ 2 π ) then: E [u(W π T )] = E [u(W 0 + W 0 R π T )] = E [u(W 0 + m π W 0 + σ π W 0 Z )] with Z ∼ N(0, 1) If we define U(m, σ) = E [u(W 0 + mW 0 + σW 0 Z )] then: max π E [u(W π T )] = max π U(m π , σ π )
So in practice the model consists in maximizing max π U(m π , σ π ) where U is derived from a utility function.

Remarks: For the DAX, if we assume that we are in an equilibrium where investors are therefore indifferent (when applying their utility functions to the expected returns) between investing into a 1 year zero coupon bond which yields approx 0% or into the DAX, which is expected to return approx 0.02%x260, then the Risk Premium is 5.20%...

Gaussian Laws and Mean Variance Implications

From now on we will modelize directly with a function U and we will assume that U is increasing in m and decreasing in σ which means that:

if the investor has the choice between two strategies with the same expected returns he prefers the one with the less variance on the returns if the investor has the choice between two strategies with the same variance on the returns he prefers the one with the higher expected returns

In practice an investor will define the level of risks he accepts and based on this will find the efficient portfolio (which maximizes the expected return). Based on this we make the following definition

Efficient Investment Strategies Definition: Efficient investment strategy

An investment strategy π e is efficient iff for all π:

E [R π ] > E [R πe ] ⇒ σ(R π ) > σ(R π e ) and E [R π ] = E [R πe ] ⇒ σ(R π ) ≥ σ(R π e )
Proposition:

π e is a solution of inf π σ(R π ) E [R π ] = E [R πe ]
Demonstration: If the solution of the constraint minimization was less than σ(R πe ) then we could find π such that σ(R π ) < σ(R πe ) and

E [R π ] = E [R πe ]
which would be in contradiction with the definition of π e . QED

Efficient Investment Strategies

Remarks: Definitions and Geometric interpretation If we note:

E = {π e efficient} and E(σ, m) = σ(R πe ) E (R πe ) , π e ∈ E Then any point σ(R π ) E (R π )

Notations and Definitions

We note: S i t : the value of asset i at time t. Here we consider only two periods (0 and T ) q i : the number of shares i held (q i > 0) or shorted (q i < 0) at time 0

R i T the return of asset i between 0 and T , i.e R i T = S i T S i 0 -1 R T =    R 1 T . . . R n T  
 the vector of returns of the n shares between 0 and T .

Definition:

At inception, An investment portfolio is a portfolio for which

i=n i=1 q i S i 0 > 0 A self financing portfolio is a portfolio for which i=n i=1 q i S i 0 = 0

Notations and Definitions

Theorem and Definition:

If for an investment portfolio we define x 0 = i=n i=1 q i S i 0 and ∀i,

π i = q i S i 0 x 0 then : (x 0 , π 1 , • • • , π n ) defines

Notations and Definitions

Theorem and Definition:

All self financing portfolios can be represented by a (n+1)-uplet

(x 0 , π 1 , • • • , π n ) such that : ∀i, q i = x 0 π i S i 0 i=n i=1 π i = 0 x 0 > 0 The representation is not unique as ∀λ > 0 ( 1 λ x 0 , λπ 1 , • • • , λπ n ) represents the same self financing portfolio as (x 0 , π 1 , • • • , π n ). For any chosen representation (x 0 , π 1 , • • • , π n ) of the self financing portfolio, x 0 is called the Notional and (π 1 , • • • , π n ) the allocation.

Notations and Definitions

Fom now on we will describe investment portfolios and self financing portfolios by pairs (x 0 , π) where π =

   π 1 . . . π n   
Theorem and Definition:

We note W t (x 0 , π) the value at time t of the portfolio (x 0 , π) and we have:

W 0 (x 0 , π) = x 0 W T (x 0 , π) = i=n i=1 x 0 π i S i T S i 0
Remark: π i < 0 means that the portfolio has a short position in asset i S i is called a risky asset iff R i T is not determinist i.e Var (R i T ) = 0

Notations and Definitions

Example: when we consider various allocations (x 0 , π) between two shares S 1 0 = 100 S 2 0 = 50 we get the following table:

x 0 π 1 π 2 q 1 q 2 W 0 100 1 -1 1 -2 0 1000 1
-1 10 -20 0 100 0.5 0.5 0.5 1 100 1000 0.5 0.5 5 10 1000

Pierre Brugière (copyrights Pierre Brugière ) Portfolio Management -version 2 May 4, 2020 41 / 127

Notations and Definitions

Theorem and Definition:

We define the return of a portfolio (x 0 , π) as: R T = W T (x 0 ,π)-W 0 (x 0 ,π)

x 0

For an investment portfolio this definition corresponds to the definition of the return of an asset For a self financing portfolio this quantity equals W T (x 0 ,π)

x 0

Proposition

For any investment or self financing portfolio (x 0 , π) the return R T verifies:

R T = i=n i=1 π i R i T .
As this expression depends only on π and not on x 0 we note it R π T .

Notations and Definitions

Proposition:

For any investment or self-financing portfolio (x 0 , π) we have:

R π T = π R T E (R π T ) = E (π R T ) = π E [R T ] Cov (R π 1 T , R π 2 T ) = Cov (π 1 R T , π 2 R T ) = π 1 Cov [R T , R T ]π 2
Exercise: demonstrate the proposition Notations: from now on we note:

σ π = σ(R π T ) m π = E (R π T )
Markowitz's Framework

We are going two consider two cases, each based on the existence or not of a risk-free investment strategy:

First case:

There are n risky assets (S i ) i∈ [1,n] and no risk-free asset.

Second case:

There are n risky assets (S i ) i∈[1,n] and one risk-free asset S 0 .

Remarks:

In both cases it is natural to assume that the R i T are such as : • (H1) there is no way to build a risk-free investment portfolio based on the (S i ) i∈[1,n] • (H2) there is no way to build a risk-free self-financing portfolio (other than the null portfolio) based on the (S i ) i∈ [1,n] Comments:

The reason why we assume (H1) in the first case is because otherwise we could build a risk-free asset. The reason why we assume (H1) in the second case is because otherwise we could build a second risk-free asset and then: either this second risk-free asset has the same return as the risk-free asset and in this case one risky asset could be replicated (and then should be disregarded). or this second risk-free asset has a different return from the risk-free asset, in which case there would be some arbitrage opportunities. The reason why we assume (H2) is because for a risk-free self-financing strategy:

if the return is different from zero there are some arbitrage opportunities if the return is zero one risky asset can be replicated (and then should be be disregarded)

Opimization without a risk-free asset

From the utility framework and in the context of normal distributions assumptions the efficient investment portfolios (x 0 , π) are the solutions of (P)

     min π π Σπ π M = m π 1 n = 1
where 1 n is the vector of R n with components equal to 1

We define a new scalar product in R n by < x, y > Σ -1 = x Σ -1 y We can write (P) as

(P)      min π < Σπ, Σπ > Σ -1 < Σπ, M > Σ -1 = m < Σπ, 1 n > Σ -1 = 1 (P)
can be solved either by writing its Lagrangien or geometrically by noticing that Σπ must be in Vect(M, 1 n ) in order to mimimize its <, > Σ -1 norm while satisfying the constraints. We note F all the investment portfolios solutions of (P) for any possible value of m.

We note:

a =< 1 n , 1 n > Σ -1 and b =< M, 1 n > Σ -1
Excluding for now the case Mb a 1 n = 0 and noticing that < Mb a 1 n , 1 n > Σ -1 = 0 we get that Mb a 1 n and 1 n form an orthogonal basis of Vect(M, 1 n ). So we can write:

Σπ e = λ1 n + ν(M -b a 1 n ) or π e = λΣ -1 1 n + νΣ -1 (M -b a 1 n )
We can now renormalize this decomposition. We note:

π a = 1 a Σ -1 1 n which satisfies π a 1 n = 1 and ω a,b = Σ -1 (M-b a 1n) M-b a 1n Σ -1 which satisfies ω a,b 1 n = 0 and var [R ω a,b T ] = 1 π a 1 n = 1 (so π a is an investment portfolio). m πa = b a σ πa = 1 √ a ω a,b 1 n = 0 (so ω a,b is a self-financing portfolio). m ω a,b = M -b a 1 n Σ -1 σ ω a,b = 1 cov (R πa , R ω a,b ) = 0 Pierre Brugière (copyrights Pierre Brugière )
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Opimization without a risk-free asset Demonstration:

π a 1 n = 1 n π a = 1 a 1 n Σ -1 1 n = 1 m πa = M π a = 1 a M Σ -1 1 n = b a σ πa = [π a Σπ a ] 1 2 = [ 1 a 2 1 n Σ -1 ΣΣ -1 1 n ] 1 2 = [ a a 2 ] 1 2 = 1 √ a ω a,b 1 n = 1 M-b a 1n Σ -1 1 n Σ -1 (M -b a 1 n ) = 0 m ω a,b = M ω a,b = M Σ -1 (M-b a 1n) M-b a 1n Σ -1 using 1 n Σ -1 (M -b a 1 n ) = 0 we get: M Σ -1 (M -b a 1 n ) = (M -b a 1 n ) Σ -1 (M -b a 1 n ) and so m ω a,b = M-b a 1n 2 Σ -1 M-b a 1n Σ -1 = M -b a 1 n Σ -1 (σ ω a,b ) 2 = 1 M-b a 1n 2 Σ -1 (M -b a 1 n ) Σ -1 ΣΣ -1 (M -b a 1 n ) = 1 cov (R πa , R ω a,b ) = 1 a 1 n Σ -1 ΣΣ -1 (M -b a 1 n ) 1 M-b a 1n 2 Σ -1 = 0 Q.E.D a) F = {π a + λω a,b , λ ∈ R} b) R πa+λω a,b = R πa + λR ω a,b c) m πa+λω a,b = m πa + λm ω a,b d) (σ πa+λω a,b ) 2 = (σ πa ) 2 + λ 2 e) min π∈F σ π = σ πa Demonstration: a)
We know that a solution of (P) is of the form απ a + βω a,b as it must be in Vect(π a , ω a,b ). The condition π 1 n = 1 implies απ a 1 n = 1 which implies α = 1. So the solutions of (P) are all of the forms π a + λω a,b .

Reciprocally the investment portfolio π a + λω a,b is the solution of (P) for m = m πa + λm ω a,b .

Optimization without a risk-free asset Optimization without a risk-free asset

b) R πa+λω a,b = (π a + λω a,b ) R T = π a R T + λω a,b R T = R πa + λR ω a,b . Q.E.D c) by definition m πa+λω a,b = E [R πa+λω a,b ] so c) is
Corollary If π e ∈ F π e = π a + m πe -m πa m ω a,b ω a,b (σ πe ) 2 = (σ πa ) 2 + ( m πe -m πa m ω a,b
) 2

m πe = m πa + m ω a,b (σ πe ) 2 -(σ πa ) 2 if m πe > m πa m πe = m πa -m ω a,b (σ πe ) 2 -(σ πa ) 2 if m πe < m πa

Demonstration:

Trivial when writing π e as π a + λω a,b and eliminating λ between the equations.

Optimization without a risk-free asset

Remarks:

The frontier F(σ, m) is an hyperbole For all assets and investment portfolios: (σ, m) is on or inside the hyperbole Geometrically: the line y = a + bx is above the hyperbole y = a + b √ x 2 -1 and is "asymptotically tangent" as:

a + bx -(a + b √ x 2 -1) x→∞ ---→ 0 (points of the curves converging)and b/[b x √ x 2 -1 ] x→∞ ---→ 1 (slopes of the curves converging)

Definitions:

We note: Optimization without a risk-free asset Two Funds Theorem:

F + = {π a + λω a,b , λ ≥ 0} F -= {π a + λω a,b , λ ≤ 0} F + (σ, m) = {(σ π , m π ), π ∈ F + } and call it the Efficient Frontier F -(σ, m) = {(σ π , m π ), π ∈ F -}
All the efficient portfolios can be built from two fixed distinct efficient investment portfolios π 1 and π 2 . For this reason to analyse the efficient frontier we just need to analyse a model based on two risky assets!

Demonstration:

Let

π 1 = π a + λ 1 ω a,b and π 2 = π a + λ 2 ω a,b be distinct on F Let π = π a + λω a,b be on F. For any α ∈ R, απ 1 + (1 -α)π 2 is an investment portfolio (as απ 1 1 n + (1 -α)π 2 1 n = 1) and if we choose α = λ-λ 2 λ 1 -λ 2 we get απ 1 + (1 -α)π 2 = π a + λω a,b = π. Q.E.D Pierre Brugière (copyrights Pierre Brugière )
Portfolio Management -version 2 May 4, 2020 56 / 127

Optimization without a risk-free asset

Exercise 1:

We consider 2 risky assets S 1 , S 2 :

m 1 = 5%, σ 1 = 15% m 2 = 20%, σ 2 = 30%
We note ρ the correlation between R 1 and R 2 . a) Plot F for: ρ = -1, ρ = 0, ρ = 0.5, ρ = 1. b) What is the portfolio of minimum standard deviation in each case? If we consider the portfolio π α = απ 1 + (1 -α)π 2 c) In which region of F(σ, m) is (σ πα , m πα ) in the following cases:

α < 0 0 ≤ α ≤ 1 α > 1 Pierre Brugière (copyrights Pierre Brugière ) Portfolio Management -version 2 May 4, 2020 57 / 127
Optimization without a risk-free asset Optimization without a risk-free asset Proposition:

F = 1 b-ma Σ -1 (M -m1 n ), m = b a 1 a Σ -1 1 n Demonstration: We know that F = 1 a Σ -1 1 n + λΣ -1 (M -b a 1 n ), λ ∈ R for λ = 0 we obtain the portfolio 1 a Σ -1
1 n for λ = 0 we can write λ in the form λ = 1 b-ma and by doing so we obtain

1 a Σ -1 1 n + 1 b-ma Σ -1 (M -b a 1 n ) = 1 b-ma Σ -1 (M -m1 n ) + 1 b-ma Σ -1 (m -b a )1 n + 1 a Σ -1 1 n = 1 b-ma Σ -1 (M -m1 n ) + 1 b-ma Σ -1 (am -b) 1 a 1 n + 1 a Σ -1 1 n = 1 b-ma Σ -1 (M -m1 n ) Q.E.D Remarks:
We will demonstrate later that the parameter m can be interpreted geometrically, by showing that the tangent to F at point (σ, m) intersects the axe {σ = 0} at point (0, m).

Optimization without a risk-free asset

Remarks: We have assumed so far that Mb a 1 n = 0. We analyse here what would happen if this was not the case. If Mb a 1 n = 0 then all portfolios would have the same returns equal to b a . In this case (P) would be a problem of minimizing, for an investment portfolio,the standard deviation of the return, i.e to solve:

(P) min π < Σπ, Σπ > Σ -1 < Σπ, 1 n > Σ -1 = 1
As previously, geometrically we see that the solution should verify Σπ e ∈ Vect(1 n ), so π e = λΣ -1 1 n . The only π e of this form satisfying

(π e ) 1 n = 1 is π e = 1 a Σ -1 1 n
Optimization with a risk-free asset

We note Π = π 0 π the allocation between the risk-free asset and the n risky assets.

We note Π 0 = 1 0 the risk-free asset of return r 0 .

For any investment portfolios Π we must have π 0 + (π) 1 n = 1. Replacing π 0 by 1 -π 1 n the problem we have to solve, to find the efficient portfolios, can now be written as:

(Q) min π π Σπ π M + (1 -π 1 n )r 0 = m Which we can also write as: (Q) min π < Σπ, Σπ > Σ -1 < Σπ, M -r 0 1 n > Σ -1 = m -r 0
Optimization with a risk-free asset

Geometrically, the solution has to be of the form Σπ = λ(M -r 0

1 n ) or equivalently π = λΣ -1 (M -r 0 1 n ). Note that 1 n Σ -1 (M -r 0 1 n ) = (b -r 0 a).
Until the end of this section we will consider that br 0 a = 0 To renormalize the problem we define:

π M = 1 b-r 0 a Σ -1 (M -r 0 1 n ) Note that Π M = 0 π M is an investment portfolio as π M 1 n = 1
Remark : for any portfolio

Π P = 1 -π P 1 d π P =⇒ m P = r o M 1 -π P 1 d π P =⇒ m P -r o = π P (M -r 0 1 d )

Theorem: Capital Market Line

The portfolios solutions of (Q) are the portfolios:

λΠ M + (1 -λ)Π 0 with λ ∈ R. We note C = {λΠ M + (1 -λ)Π 0 , λ ∈ R}
Optimization with a risk-free asset Corollaries and "Market Portfolio"

1 The portfolios Π of C (i.e the solutions of (Q)) verify:

m Π = λm Π M + (1 -λ)m Π0 σ Π = |λ|σ Π M
so their risk parameters (σ, m) are on a cone (i.e C(σ, m) is a cone).

2 All the efficient portfolios are built by allocating money only between Π 0 and Π M . For this reason to study optimal investments we just need to use a model with one single risky asset!

3 CAPM/MEDAF: if all the market participants are allocating efficiently and with the same parameters, then:

all the risky investments are in Π M and in Π M the weight in risky asset i is the % of the total Market Capitalization of risky assets that asset i represents.

For this reason Π M should be/is called the (risky) "Market Portfolio".

Optimization with a risk-free asset

Corollaries:

1 Usually in the model b a > r 0 otherwise all the efficient portfolios would short Π M which practically would not make sense.

2 Assuming b a > r 0 we define:

C + = {λm Π M + (1 -λ)m Π0 , λ ≥ 0} and call it the Cone Efficient Frontier (or Capital Market Line) C -= {λm Π M + (1 -λ)m Π0 , λ ≤ 0}

Tangent Portfolio

Lemma: Tangent Portfolio

The "Market Portfolio" Π M , which has no allocation in the risk-free asset, is also a solution of the mean/variance optimization problem (P) where there was no risk-free asset. So we can write Π M ∈ F

Demonstration:

Geometrically: it is obvious as otherwise there would be some portfolios more efficient (above) than those on the Capital Market Line.

Algebraically:

π M = 1 b-r 0 a Σ -1 (M -r 0 1 n ) = 1 b-r 0 a Σ -1 [(M -b a 1 n ) + ( b a 1 n -r 0 1 n )] = 1 b-r 0 a ( b a -r 0 )Σ -1 1 n + Σ -1 (M -b a 1 n ) = 1 a Σ -1 1 n + 1 b-r 0 a Σ -1 (M -b a 1 n ) = π a + λω a,b
which is the form of the portfolios of F. Q.E.D Until the end of this section we assume that b a > r 0

Tangent Portfolio

Theorem: Tangent Portfolio C(σ, m) is tangent to F(σ, m) at the point (σ Π M , m Π M )
Demonstration:

We know that Π M is on C and that Π M is on F.

Geometrically: If a line and an hyperbole have a contact point either they are tangent on this contact point or they cross each other. The situation where they cross each other is not possible here as it would imply that some portfolios of F are more efficient than any portfolios of C.

Remark:

Equivalently, we can say that the tangent to the Efficient Frontier F at the point Π M intersects the σ = 0 axis at the point 0 r 0

More Geometric Properties

Corollary: Geometry of the Efficient Frontier

For any risky efficient investment portfolio π = 1 b-ma Σ -1 (M -m1 n ) of F, the tangent to F(σ, m) at (σ π , m π ) intersects the {σ = 0} axis at the point 0 m .

For the risky efficient investment portfolio π = 1 a Σ -1 1 n of F, the tangent to F(σ, m) at (σ π , m π ) is parallel to the {σ = 0} axis. These results mean that there is a bijection between F -{ 1 a Σ -1 1 n } and R -{ b a } and that is it the tangents to the the F(σ, m) curve which establish the bijection between the portfolios and their parameters m.

Demonstration:

This result for a portfolio of parameter m corresponds to the result of the preceding corollary when taking r 0 = m. Q.E.D Security Market Line Definition:

The quantity ρ(R M , R P ) σ P σ M is noted β M (P) and is called the beta of the asset Π P (in respect to the Market Portfolio Π M ).

Remarks:

The equation:

m P -r 0 = (m M -r 0 )β M (P) (SML)
is valid for all investment portfolios and not only for efficient ones.

shows that only the risk correlated with the "Market Portfolio/Market Risk" is renumerated. is used in capital budgeting / CAPM to determine the price an asset should have based on its expected returns and beta with the sector. the beta can be estimated by statistical regression of the observed excess returns R P -r o over R M -r o over several periods. We can read on the graph, for any portfolio, the portion of the volatility correlated with the movements of the Market Portfolio.

Pierre Brugière (copyrights Pierre Brugière ) Portfolio Management -version 2 May 4, 2020

Security Market Line and "Arbitrage" Detections

Exercice: Show that if we consider an investment portfolio Π P with risk parameters (σ P , m P ) and if we call in the {(σ, m)} plane, (x, m P ) the intersection of the SML and the line {m = m P } then x = β M (P)σ M . Conclude that we can read in the {(σ, m)} plane the decomposition between systematic risk and idiosyncratic risk.

Remarks: An investor choosing a mutual fund to represent a large portion of his/her wealth should be concerned by the full risk of the fund and should look at the Sharpe Ratio.

Exercise 1: Show that the Sharpe Ratio is independent from the leverage Exercise 2: How do you read in a {(σ, m)} representation the Sharpe Ratio of a fund as the slope of a particular line?

Remarks:

A large pension fund which allocates money amongst many asset managers may assume that the idiosyncratic risk is going to be reduced/cancelled through diversification and in this case may be concerned only by the remuneration of the non diversifiable risk and by the Jensen Index of each Asset Manager's funds.

Exercise 1: Show that the Jensen Ratio is dependent on leverage

Exercise 2: How do you read in a {(β, m)} representation the Jensen Ratio of a fund as the distance above a particular line ?

Factor Model

Definition: K-factors model

For all assets i ∈ 1, N and for all instants t ∈ 1, T

r i (t) = a i + j=K j=1 β i j f j (t) + i (t) or matricially R(t) = A + BF (t) + E(t) with the assumptions that: ∀t ∈ 1, T , Var (F (t)) is invertible, E (E(t)) = 0 and Cov (F (t), E(t)) = 0 Remark 1: If (F (t), E(t)
) is assumed to be a Gaussian vector then R(t) is a Gaussian vector and we are still in the "Markowitz framework".

Factor Model

Remark 2:

R(t) =    r 1 (t) . . . r N (t)    A =    a 1 . . . a N    B =     β 1 1 . . . β 1 K . . . . . . . . . β N 1 . . . β N K     F (t) =    f 1 (t) . . . f K (t)    and E(t) =    1 (t) . . . N (t)   
Remark 3: additional assumptions are often made that E (F (t)) = 0 F (t) independent from F (t ) for t = t E(t) independent from E(t ) for t = t E(.) independent from F (.)

Var (E(t)) = diag (σ 2 i ) independent from t.

Factor Model

Remark 7: If we assume (F (t), E(t)) Gaussian, then a factor model is a "Markowitz model", where the vector of returns of the risky assets over the period [t -1, t] follows a Gaussian law of expectation A and variance-covariance matrix Σ = BΣ F B + diag (σ 2 i ) Remark 8: In the SML approach the tangent portfolio is a particular factor, explaining the expected remuneration, while in a factor analysis the focus is more on identifying the various sources of risks of the assets, which generate the correlations between them.

Factor Model -Example

Numerical Example: Consider the 3 factors model R = A + BF + E with 3 risky assets where:

E (R) =   5% 4% 6%   , B =   1 0 0 0 1 0 1 0 1   and P =   100% 0 50% 0 100% 0 50% 0 100%  
is the matrix of correlations for the factors and where:

σ(f 1 ) = 15%, σ(f 2 ) = 10%, σ(f 3 ) = 10%
Demonstration: A + BF (t) + E(t) = A + BVV F (t) + E(t) = A + BVdiag ( √ λ i )diag ( 1 √ λ i )V F (t) + E(t)) = A + BVdiag ( √ λ i ) diag ( 1 √ λ i )V F (t) + E(t) = A + DH(t) + E(t) with D = BVdiag ( √ λ i ) and H(t) = diag ( 1 √ λ i )V F (t) Now,Var [H(t)] = diag ( 1 √ λ i )Var [V F (t)]diag ( 1 √ λ i
) and

Var [V F (t)] = V Σ F V = diag (λ i ) so Var [H(t)] = diag ( 1 √ λ i )diag (λ i )diag ( 1 √ λ i ) = Id K . Q.E.D

APT Theorem

The APT conditions are satisfied if and only if:

∃λ 0 , λ 1 , λ 2 , • • • , λ K such that E (R) = λ 0    1 . . . 1    + B    λ 1 . . . λ K   
that we can also note as

E (R) = λ 0 1 N + Bλ (1)
Also, assuming that the decomposition (1) is valid then:

If it is possible to build a risk-free investment portfolio in the reduced model then, λ 0 is this risk-free rate If it is not possible to build a risk-free investment portfolio in the reduced model then, the decomposition (1) is not unique and we can take λ 0 = 0.

Factor Model -APT Theorem

Remark 1: If there is a risk free rate r 0 in the economy and a a risk free investment portfolio in the reduced model then usually it is expected that λ 0 = r 0 Remark 2:

As K < N it is possible to build in the reduced model a risk-free portfolio by choosing π = 0 such that π B = 0 a) If we can find such π such that π 1 N = 0 we are able to build a risk-free investment portfolio b) If for all such π, π 1 N = 0 we are only able to build risk-free self financing portfolios

Factor Model -APT Theorem (second implication)

We assume now that ∃λ 0 , λ such that E (R) = λ 0 1 N + Bλ and want to prove that the AOA conditions are satisfied in the reduced model. So, let π be a portfolio without risk in the reduced model

π investment portfolio without risk in the reduced model =⇒ π B = 0 =⇒ E (R π ) = π E (R) = λ 0 π 1 N = λ 0
so the first AOA condition that all risk free investment portfolios have the same return is satisfied

π self financing portfolio without risk in the reduced model =⇒ π B = 0 =⇒ E (R π ) = π E (R) = λ 0 π 1 N = 0
so the second AOA condition that all risk free self financing portfolios have a return of zero is satisfied This finishes the proof of the APT theorem

Factor Model -APT Theorem

Proposition

The two following propositions are equivalent there is no risk free investment portfolio in the reduced model

R = A + BF (1) Vect{b 1 , b 2 , • • • , b K } = Vect{1 N , b 1 , b 2 , • • • , b K } (2) Demonstration: (1) ⇐⇒ (x B = 0 =⇒ x 1 N = 0) ⇐⇒ Vect{b 1 , b 2 , • • • , b K } ⊥ ⊂ Vect{1 N } ⊥ ⇐⇒ Vect{1 N } ⊂ Vect{b 1 , b 2 , • • • , b K } ⇐⇒ (2) Q.E.D
Remark: the proposition above shows that when the APT conditions are satisfied and that there is no risk free investment portfolio in the reduced economy we can take in fact any value for λ 0 (and not only the value zero) Factor Model -APT Theorem

A + BF + E with E (R) = λ 0 1 N + Bλ implies that A + BE (F ) = λ 0 1 N + Bλ =⇒ R = λ 0 1 N + B(λ -E (F )) + BF + E =⇒ R = λ 0 1 N + B(λ -E (F ) + F ) + E Q.E.D.

Principal Components Analysis

Lemma:

Let x =   x 1 • • • x d   and y =   y 1 • • • y d   be in R d then x y = Tr (xy )
demonstration: trivial Proposition: Principal Components Analysis

Let Z =    Z 1 . . . Z d    be a random variable in R d then: E Z -E [Z ] 2 = Tr [Var (Z )] =
demonstration: E Z -E [Z ] 2 = E (E -E [Z ]) (Z -E [Z ]) = E Tr (E -E [Z ])(Z -E [Z ]) = Tr E (Z -E [Z ])(Z -E [Z ]) = Tr [Var (Z )] Q.E.D

Definition:

We call E Z -E [Z ] 2 the dispersion of Z and in dimension 1 this definition corresponds to the usual definition of variance.

Principal Components Analysis Proposition:

Let Z be a random variable in R d , Var (Z ) its matrix of variance covariance and λ 1 ≥ λ 2 ≥ • • • ≥ λ n ≥ 0 the eigenvalues of Var (Z ).

         sup (x i ) i∈[1,k] ∈B k i=k i=1
x i Var (Z )x i ∀i ∈ 1, k , x i x i = 1 ∀i, j ∈ 1, k , i = j =⇒ x i x j = 0 Pierre Brugière (copyrights Pierre Brugière )

Portfolio Management -version 2 May 4, 2020 116 / 127

Principal Components Analysis

The Lagrangian is i=k i=1

x i Var (Z )x i -i<j λ i,j x i x j -i=k i=1 λ i,i (x i x i -1)

∂L ∂x i = 0 ⇐⇒ 2x i Var (Z ) -2λ i,i x ij =i λ i,j x j = 0 ⇐⇒ 2Var (Z )x i = -2λ i,i x ij =i λ i,j x j .

As a consequence Var (Z )V ⊂ V with V = vect{x 1 , • • • , x k } and if we note Var (Z ) |V the restriction of Var (Z ) to V we have: x i (Z -EZ )x i and

p H k (Z -EZ ) 2 = i=k i=1 (x i (Z -EZ )) 2 = i=k i=1 x i (Z -EZ )(Z -EZ ) x i so E [ p H k (Z -EZ ) 2 ] = i=k i=1 x i E [(Z -EZ )(Z -EZ ) ]x i = i=k i=1
x i Var (Z )x i So the result follows from the previous proposition.

  is called the risk premium for X

  in a unique way the investment portfolio x 0 is the initial value of the portfolio i=n i=1 π i = 1 π i is the percentage of the value of the portfolio invested in asset i at time 0 and (π 1 , • • • , π n ) is called the allocation.

  derived from b) by taking expectations on both side of equality b). d) by definition (σ πa+λω a,b ) 2 = Var [R πa+λω a,b ] and according to b) this quantity equals Var [R πa + λR ω a,b ]. As the covariance between these two variables is zero, the expression is Var [R πa ] + λ 2 Var [R ω a,b ] and as seen previously Var [R ω a,b ] = 1. Q.E.D e) this results from d)

  Asset 1 leveraged, Asset 2 shorted) Note that Σ is not invertible here and a risk-free asset can be constructed Pierre Asset 1 leveraged, Asset 2 shorted) Note that Σ is invertible here and we obtain the "usual" Asset 1 leveraged, Asset 2 shorted) Note that Σ is not invertible here and a risk-free asset can be constructed Pierre

  and call it the Cone Inefficient Frontier All the assets and portfolios we can built have their risk parameters (σ, m) within the cone C(σ, m) and in particular F(σ, m) is included in C(σ, m).Demonstration:Straightforward when writing Π e = λΠ M + (1 -λ)Π 0 Exercise: Draw F and C in a model where there are two risky assets and a risk free asset and see what happens when you are changing the correlation between the two risky assets.



  are the standard deviations of the factors (F , E) is Gaussian with Cov (F , E) = 0 and Var (E) diagonal σ( 1 ) = 5%, σ( 2 ) = 5%, σ( 3 ) = 5% are the standard deviations of the specific risks there is a risk free asset with r 0 = 2% After calculating the law of R and applying Markowitz's results we find that:Factor Model -Example a) the investment portfolio of minimum variance π a verifies: E (R πa ) = 3.84% and σ(R πa ) = 10.69% b) the tangent portfolio verifies π T verifies: with E (R π T ) = 4.61% and σ(R π T ) = 12.74% c)R π T = 0.046 + 0.647F 1 + 0.353F 2 -0.037F 3 + 0.684 1 + 0.353 2 -0.037 3 d) from c)we can derive the β of the three risky assets to the tangent portfolio and find: β T (1) = 1.15, β T (2) = 0.77, β T (3) = 1.53 e) we can verify that for the three assets the SML is satisfied as: 5% = 2% + 1.15x(4.6% -2%) 4% = 2% + 0.77x(4.6% -2%) 6% = 2% + 1.53x(4.6% -2%)Factor Model -Example f) the residual e i of the returns of the tree risky assets in the SML modelr i = r 0 + β i (r T -r 0 ) + e i havea variance-covariance matrix of:   0.004 -0.007 0.001 -0.007 0.013 -0.002 0.001 -0.002 0.012   and verify: σ(e 1 ) = 5.98%, σ(e 2 ) = 11.39%, σ(e 3 ) = 10.92% As we can see in this example the 3 factors model enables a better explanation of the risks than the (one factor) SML model because: in the 3 factors model the residuals risks have lower variances than in the SML model in the 3 factors model the residuals are uncorrelated, proving that all common sources of risks have been identified. Factor Model -Standard Form Standard Form Theorem (normalization of the factors) We can re-write the factor model in the form: R(t) = A + DH(t) + E(t) with Var [H(t)] = Id K This form is called the Standard Form of the Factor Model

Remark 1 :

 1 Unicity of the decomposition of a K-factor model The equation R = A + BF + E where F is given with Σ F invertible E (E) = 0 and Cov (F , E) = 0 defines A and B in a unique way by B = Cov (R, F )Σ -1 F and A = E (R) -BE (F ) Proposition: Standard Form of an APT model According to the APT theorem any APT model can be written in the standard form: R = λ 0 1 N + BG + E where Var (G ) is invertible, E (E) = 0 and cov (G , E) = 0 Demonstration: R =

  Tr is the trace operator and the λ i are the eigenvalues of Var [Z ]

  For any k ≤ d let B k the set of all orthonormal families of k vectors of R d . Then for any k ∈ [1, d],sup(x i ) i∈[1,k]

( a )

 a Var (Z ) |V is symmetric definite positive (b) the eigenvalues of Var (Z ) |V are eigenvalues of Var (Z ) (c) i=k i=1x i Var (Z )x i = i=k i=1 x i Var (Z ) |V x i = Trace( i=k i=1 Var (Z ) |V x i x i ) = Trace(Var (Z ) |V Id |V ) =Trace(Var (Z ) |V ) and Trace(Var (Z ) |V ) is the sum of the k eigenvalues of Var (Z ) |V , which according to (b) is the sum of k eigenvalues of Var (Z ) and the sum is maximal if the eigenvalues are the k largest ones and in this case the sum equals i=k i=1 λ i . Q.E.D.

HΦ

  Let Φ be a scalar product of R d . We use the intrinsic notation Z ∼ N (0, Φ), where Φ is a quadatric form to mean that the Gaussian variable Z has a variance-covariance defined by Var (x Z ) = Φ(x, x). Let H be a sub-space of R d and Φ |H be the restriction of Φ to H. a) Show that:minλ Φ ≤ minλ Φ |H ≤ maxλ Φ |H ≤ maxλ Φ . b) Let Z ∼ N (0, Φ) be a Gaussian variable of R d . Show that p H (Z ) is a Gaussian variable of H which verifies p H (Z ) ∼ N (0, Φ |H ). c) Study the eigenvalues of Φ and Φ |H , when Mat(Φ) = Vect(e 1 + e 2 ). |H (x, x) = min x∈H, x 2 =1 Φ(x, x) ≥ min x∈R d , x 2 =1 Φ(x, x) b) Let Z ∼ N (0, Φ) in R d .As p H is a linear transformation, p H (Z ) is Gaussian andE (p H (Z )) = p H (E (Z )) = 0 ∀x ∈ H, Var (x p H (Z )) = Var (p H (x) Z ) = Var (x Z ) = Φ(x, x) = Φ |H (x, x)Therefore, p H (Z ) ∼ N (0, Φ |H ), in H. c) Var (Z ) has for eigenvalues 1 and 2 and Var (p H (Z )) has for value (eigenvalue) 3 2 and as expected 1 ≤ 3 2 ≤ 2.

  and call it the Inefficient Frontier

	Pierre Brugière (copyrights Pierre Brugière )	Portfolio Management -version 2	May 4, 2020	55 / 127

  Let H k denote a sub vector space of R d of dimension k (with k ≤ d) Let H k be the set of all vector spaces H k Let p H k be the orthogonal projection on H k then: maxH k ∈H k E [ p H k (Z -EZ ) 2 ] =

	Principal Components Analysis
	Corollary:
		i=k
		λ i
		i=1
	Pierre Brugière (copyrights Pierre Brugière )	Portfolio Management -version 2	May 4, 2020	118 / 127

Demonstration: Let (x i ) be an othornormal basis of H k then p H k (Z -EZ ) = i=k i=1

Pierre Brugière (copyrights Pierre Brugière )Portfolio Management -version 2 May 4, 2020

2 Pierre Brugière (copyrights Pierre Brugière ) Portfolio Management -version 2 May

4, 2020

May 4, 2020

Markowitz's Framework

We Note :

Proposition:

(H1) and (H2) =⇒ Σ is invertible Demonstration: If Σ was not invertible we could find π = 0 such that π Σπ = 0 and then there would be two cases: either π 1 n = 0 in which case we could find a self-financing portfolio made of the risky assets without risk or π 1 n = 0 in which case we could find an investment portfolio made of the risky assets without risk. in both cases this would be in contradiction with either (H1) or (H2).

Optimization without a risk-free asset Security Market Line Theorem: Security Market Line

Let Π M be the Market Portfolio as defined previously Let Π P be any investment portfolio composed of the risk-free and risky assets. Then:

m P -r 0 = (m M -r 0 )ρ(R P , R M ) σ P σ M (SML equation) and R P -r 0 = (R M -r 0 )ρ(R P , R M ) σ P σ M + with normal independant from R M and centered.

Demonstration:

Let Π P be an investment portfolio then:

If we apply the same calculation to Π M then:

From this we get:

Security Market Line

From which we get :

We now want to show a relationship for the r.v and not only for their expectations.

is Gaussian because it is an affine transformation of the vector of the returns of the risky assets which is assumed to be a Gaussian vector. Thus, to show that the first variable that we call is independent from the second one we just need to show that the covariance is zero.

Security Market Line

Definition:

| is called the systematic risk. "Economically", the fact that it is remunerated is explained by the fact that it cannot be "reduced by diversification".

σ is called the idiosyncratic risk or specific risk. "Economically", the fact that it is not remunerated is interpreted as the fact that a risk that can be "reduced by diversification" does not need to be remunerated.

Security Market Line and "Arbitrage" Detections

Remarks: The (β, m) of the assets we consider investing it according to the SML should be on a line.

The beta will usually be calculated in relation to a broader index to which these stocks belong

The expected returns will be based either on some historical estimates or some analysts predictions In practice, the points will not be perfectly aligned and a regression line will be calculated.

The assets over the line will look cheap

The assets below the line will look expensive In "pair-trading", strategies will be considered consisting in:

Selling assets lying below the line Buying assets lying above the line (usually in the same sector)

Pierre Brugière (copyrights Pierre Brugière )

Portfolio Management -version 2 May 4, 2020

Performance Indicators

Definition: Sharp Ratio

The Sharpe Ratio of an investment portfolio P is defined as: m P -r 0 σ P

Remarks:

Under the Markowitz's framework:

The Ratio should be maximal for portfolios belonging to the CML All the portfolios belonging to the CML have the same Ratio Wealth should be allocated:

First by determining a portfolio with the maximum Sharpe Ratio that can be built Then by allocating all the wealth between this portfolio and the risk free asset

The Sharpe Ratio is independent from the leverage has λΠ P + (1 -λ)Π 0 has the same Sharpe Ratio as Π P for any λ > 0. So the indicator is really "intrinsic to the fund".

The Sharpe Ratio is usually estimated by: mP -r 0

Performance Indicators

Definition: Jensen Index

The Jensen Index of an investment portfolio P is defined as:

Remarks:

Under the Markowitz's framework this quantity should be zero according to the SML In practice all the portfolios considered for investment are represented in the (β, m) plane where they should form a line (the SML) and where: the beta are estimated historically the expected returns are either historical estimates or analyst predictions Then a regression line is calculated and the portfolios above the line could be considered for addition to the investment portfolio as: their systematic risk is remunerated more than expected their idiosyncratic risk should disappear via diversification

Performance Indicators

Definition: Treynor Index

The Treynor Index of an investment portfolio P is defined as: m P -r 0 β M (P)

Remarks:

Under the Markowitz's framework the Treynor Index should be constant according to the SML. The Treynor Index is similar to the Jensen index in its objectives to detect funds for which there is an excess of remuneration of the systematic risk.

The excess is usually called the α ! Compared to the Jensen Index the advantage of the Treynor Index is that it does not dependent on leverage and thus is a more intrinsic measure.

Remarks:

Show that the Treynor Index does not depend on leverage.

Factor Model

We revisit here the SML equation for risky assets and investment portfolios:

in practice the i (t) and j (t) appear to be correlated and to represent a significant portion of the variance of the assets.

by exhibiting additional factors we aim at identifying better the common sources of risks (even when they may not be remunerated) and to end up with smaller non explained residual specific risks.

we want to determine the remunerations linked to all sources of risks through a non arbitrage argument.

we add here the time parameter t to show, in a times series analysis perspective, which parameters are assumed to be fixed and which ones are supposed to vary with time (on top of their randomness character when time is fixed).

Factor Model

Remark 4: Σ F is assumed to be symmetric definite positive. If this was not the case, we could find a vector u = 0 such that u Var [F (t)]u = 0 which would imply that Var [u F (t)] = 0 and u F (t) = Cte. This would imply that some of the factors would be redundant (could be "cointegrated").

Remark 5:

Factor Model

Remark 9: Factor models were introduced by Charles Spearman in 1904 in psychometrics.

Remark 10: In financial econometrics, the factors used are either:

Macroeconomics factors: ex GDP, inflation rate, unemployment rate..etc, in this case the F (t) are "exogene" i.e given and observable.

Fundamental factors: ex market cap, leverage, book/price ...etc which are as well exogene.

Statistical factors: in this case the F (t) are "endogene" / hidden factors of the model and the aim is to determine these F (t) as well as the corresponding sensibilities (i.e B). In this approach the factors can be interpreted as returns of some investments and self financing portfolios which are not correlated.

Factor Model -Standard Form

Reminder: Diagonalisation Theorem

Matricially, if we note V the matrix whose vectors columns are the V i then:

Here the λ i are the eigenvalues of Σ and the V i are the eigenvectors.

Pierre Brugière (copyrights Pierre Brugière ) Portfolio Management -version 2 May 4, 2020

Factor Model -Factor Decomposition

Exercise: Prove the previous result by writing Remark :

In fact,

•All self financing portfolios with no risk have a return of zero implies •All investment portfolios with no risk have the same return. So, the AOA conditions can be reduced to:

•All self financing portfolios with no risk have a return of zero

Factor Model -APT Theorem Definition: Asset Pricing Theory conditions

The APT conditions are said to be satisfied if and only if: There is no Arbitrage Opportunity in the reduced model R(t) = A + BF (t) (where the "diversifiable" risk E(t) is neglected)

If the APT conditions are satisfied we say that the model is an APT Model

Factor Model -APT Theorem Lemma 1

Let us define λ 0 by: λ 0 = 0 if there is no risk-free investment portfolio in the reduced model λ 0 is the risk-free rate of the risk-free investment portfolios if such portfolios exist in the reduced model Then, if there is no arbitrage in the reduced model we have:

Factor Model -APT Theorem therefore as there is no Arbitrage Opportunity in the reduced model we should have π A = λ 0 and thus π (A -λ 0 1 N ) = 0 and π (A -λ 0 1 N ) = 0 if π 1 N = 0 then π is a self-financing portfolio without risk which should therefore satisfy π A = 0 and also if λ 0 is defined

Lemma 2:

Factor Model -APT Theorem

Demonstration APT Theorem: (First implication) According to Lemma 1 and 2 the AOA in the reduced model implies that

Factor Model -APT Theorem

Numerical Example :

We consider the 2 factor model R = A + BF + E with 3 assets where:

Var (F ) invertible.

According to the Fundamental APT theorem there is no arbitrage in the reduced model i.i.f we can find λ 0 , λ 1 , λ 2 such that:

Indeed we find a unique solution λ 0 = 2%, λ 1 = 3%, λ 2 = 6% In the reduced Model π = (-1, -1, 3) is an investment portfolio without risk and when we calculate its expected return based on its components we find -5% -8% + 15% = 2% which is consistent with λ 0 = 2% We consider an economy with two risky assets whose returns r 1 and r 2 follow the following one factor model:

with f , e 1 and e 2 being independent of variance 1 and expectation zero.

We plot on the graph below the 800 simulations of

Principal Components Analysis -Numerical Example

We have the following results:

the green ellipse on the chart represents points for which the density function of the model is constant equal to the value derived for the point 1.20 1.44 .

the lengths of the axis (in purple) of the ellipse are proportional to the eigenvalues when we project the points on the red axis, the dispersion of the points projected is 3.06 which is what we expected, as we have projected on an axe corresponding to the (observed) eigenvalue 3.06.

Pierre Brugière (copyrights Pierre Brugière ) Portfolio Management -version 2 May 4, 2020