550.447 quantitative portfolio theory & performance ...daudley/447/jhuonly/550.447 sp13...

1

1.1

550.447Quantitative Portfolio Theory

& Performance Analysis

Week of March 4 & 11 (snow), 2013

Fast Algorithms, the Efficient Frontier & the Single-Index Model

1.2

Where we are

Chapters 1 - 3 of AL: Performance, Risk and MPT

Chapters 4, 5, and 11 of E&G The Opportunity Set, Efficient PFs and PF Selection

Chapters 5+ & 9 in Hull Closer Look & review of mean-variance calculations

for Risk, VaR, and Extreme Value Theory (EVT)

1.3

Assignment

For Week of March 4th

Read: A&L, Chapter 3 (Basic Elements of MPT) Read: E&G, Chapters 9 & 7 (Efficient Frontier and

Correlation Structure for Single-Index Model ) See Supplemental on Website

Problems: H5: 2, 5, 9, 11; 19 (Due Mar 4th) Problems: H6: 6 (Due Mar 4th) Problems: H9: 3, 6 (Due Mar 4th) Problems: EG7: 1, 2, 4 (Due Mar 13th) Problems: EG9: 1, 2 (Due Mar 13th)

1.4

Assignment

For Weeks of March 4th / March 11th

Read: A&L, Chapter 4 (Capital Asset Pricing Model and its Application to Performance Measurement)

Read: E&G, Chapters 9 & 7 (Efficient Frontier and Correlation Structure for Single-Index Model ) See Supplemental on Website

Problems: EG7: 1, 2, 4 (Due Mar 13th) Problems: EG9: 1, 2 (Due Mar 13th)

2

1.5

Assignment

Spring Break: March 18 – 23 Mid Term: NEW! April 3rd

Last Day of Class: Wednesday, May 1st

Final: Monday, May 13th ; 9:00am – Noon In the classroom: Whitehead 203

1.17

Fast Algorithms: Efficient Frontier

Markowitz and SharpeWe’ve looked at the formulations for the efficient

frontier, the complexity of the covariance structure, and the algorithmic solution

Sharpe provides an empirical simplification 1-factor model allows reduction of unknown

parameters in an appealing fashion Let’s view the algorithmic consequence of

Sharpe’s approach

1.18


Single-Index Model Suppose returns, Ri can be modeled as

Where Rm, is the return on the market index, and Where Rm , ei are r.v. with std dev σm , σei respectively ei is zero mean, and

ei is uncorrelated w/Rm : Estimates of from time series, regression

analysis – which provides an uncorrelated result (last) Finally, we assume, ei , ej are uncorrelated:

i i i m iR R e

cov( , ) 0i m i m me R E e R R 2, , and i i ei

0i jE e e 1.19


Single-Index Model Then we haveMean Return Variance of Return Covariance of returns

If the single index model holds, then for a PF The expected return is Variance

i i i mR R 2 2 2 2

i i m ei 2

ij i i m

1 1 1

N N N

P i i i i i i mi i i

R X R X X R

2 2 2

1 1 1

2 2 2 2 2 2

1 1 1 1

N N N

P i i i j iji i j

j i

N N N N

i i m i i j m i eii i j i

j i

X X X

X X X X

3

1.20


Single-Index Model So for the PF we can write

Giving a definition for PF alpha and beta

As for risk

Or

1 1

N N

P i i i i m P P mi i

R X X R R

1

N

P i ii

X

1

N

P i ii

X

2 2 2 2 2 2 2

1 1 1 1

2 2 2 2 2 2

1 1 1 1 1 1

N N N N

P i i m i i j m i eii i j i

j i

N N N N N N

i i j m i ei i i i i m i eii j i i i i

X X X X

X X X X X X

2 2 2 2 2

1

N

P P m i eii

X

1.21


Single-Index Model If we assume a large, equally weighted PF then

Which is 1/N times the average residual risk in the PF And as the PF becomes large, the importance of the

average residual risk diminishes to insignificant Thus So the measure of the contribution of a security to the

risk of a large PF is its beta – often used a measure of a securities risk

We will return to this shortly … but first …

2 2 2 2

1

1 1N

P P m eiiN N

1/22 2

1

N

P P m P m m i ii

X

1.22


Single-Index Model Key is the calculation of the Optimal Portfolio First we’ll go through the steps and then

rationalize/justify The steps are:1. Identify the universe of candidate securities2. Select those securities desirable to be included

in the optimal PF3. Calculate the weightings for those selected

1.23


Single-Index Model The main step is the selection process This is done using a measure of desirability – related

to the excess return to beta ratio This is a measure of additional return gained from a candidate

security per unit of non-diversifiable risk The additional return is relative to the risk-free rate

More formally, we use the ratio = the expected return on stock i = the return on the riskless asset = the expected change in the rate of return on stock i with

a 1% change in the market return

i F

i

R R

iFRiR

4

1.24


Single-Index Model The selection processWhen the measure is better than some

threshold, the security is included For convenience, rank the universe by excess return

over beta – from highest to lowest The optimum portfolio consists of all stocks for which

the excess return is greater than some cut-off, C* This far, the procedure is very simple were we to know

the cut-off Lets look at an example …

i F

i

R R

1.25


Single-Index Model The selection process Consider the 10 stocks and RF = 5%

If C* = 5, then only the 1st 5 are included in optimal PF

1.26


Single-Index Model The selection process So how do we find C* The cut-off is determined successively by assuming a

monotonically increasing-sized set including successively “lesser” securities

To do this we designate Ci as a candidate for C* Ci is calculated with i securities assumed in the optimal PF We know that we have found the optimum Ci , called C* , when

all securities used in the calculation of Ci have excess return to beta above Ci and all securities not used in the calculation have excess returns to beta below Ci

1.27


Single-Index Model The selection process Finding C* (result now; verification later) The stocks are ranked by excess return to risk from high to low For a PF of i stocks, Ci is given by

But we can show that this is just where:= exp change in ROR on stock w/ 1% change in opt PF= exp return on opt PF

22

1

22

21

1

ij F j

mj ej

i ij

mj ej

R R

C

iP P Fi

i

R RC

iP

PR

5

1.28


Single-Index Model The selection process Finding C* But we can show that this is just where:

= exp change in ROR on stock w/ 1% change in opt PF= exp return on opt PF

Of course we don’t know these until the opt PF is found, but we can still make use of the expression as …

We add securities to the candidate PF as long as Or from the above, as long as

iP P Fi

i

R RC

iP

PR

i Fi

i

R R C

iP P Fi Fi F iP P F

i i

R RR R R R R R

1.29


Single-Index Model The selection process Finding C* The rhs is just the expected excess return on a particular stock,

based solely on the exp performance of the opt PF The lhs is the analysts estimated excess return on the stock Thus, if analysis of a particular stock leads the PF manager to

believe it will perform better than would be expected, based on the opt PF, it should be in the PF

Now we look at how =>can be used in our framework to find C*

i F iP P FR R R R

22

1

22

21

1

ij F j

mj ej

i ij

mj ej

R R

C

1.30


Single-Index Model The selection process Finding C* – return to the Table (assume = 10%)

Which follows from …

22

1

22

21

1

ij F j

mj ej

i ij

mj ej

R R

C

2

m

1.31


Single-Index Model The 10 stocks and RF = 5%

6

1.32


Single-Index Model The selection process Finding C* The key point is Columns 5 & 6

are cumulative sums of columns 3 & 4, resp Column 7 is just

As soon as Col 7 reaches a maximum, or Col 7 is greater than Col 2 – because of our ordering scheme – we have found C*

2

2

column 51 column 6

mi

m

C

1.33


Single-Index Model The Optimal Portfolio Weightings First we recall the case where we allow short sales w/

riskless lending and borrowing (similar in other cases) The criteria is to maximize the objective function

Subject to Thus with We can write

P F

P

R R

11

N

ii

X

1 11

N N

F F i F i Fi i

R R X R X R

1

2 2

1 1 1

N

i i Fi

N N N

i i i j iji i j

j i

X R R

X X X

1.34


Single-Index Model The Optimal Portfolio Weightings

And finding the maximum involves solving the N equations

Which can be shown (Chapter 6) as equivalent to solving

for i = 1, … , N Where the optimum proportion to invest in each stock is

1

kNk

ii

ZXZ

1 1 2 2i F i i N iNR R Z Z Z

0 where 1, ,i

d i NdX

1.35



Lets look at: But for our single index Sharpe’s model we have

Or combining terms

And solving

Where we define

21 1 2 2

1

N

i F i i N iN i i j ijjj i

R R Z Z Z Z Z

2 2 2 2 2

1 1

N N

i F i i j ij i i m ei j i j mj jj i j i

R R Z Z Z Z

2 2

1

N

i F i ei j i j mj

R R Z Z

2

2 2 21

*N

i F i m i i Fi j j

jei ei ei i

R R R RZ Z C

2

1*

N

m j jj

C Z

7

1.36



As an aside, to get the earlier “daunting” expression for C* we must express

andIn terms that do not invoke

To do so, we multiply the left hand equation by beta and sum

We can solve for

2

2 21

Ni F i m

i j jjei ei

R RZ Z

2

1*

N

m j jj

C Z

1

N

j jj

Z

22

2 21 1 1 1

N N N Ni F i i

i i m j ji i i jei ei

R RZ Z

22

2 21 1 1

1N N N

i F i ji i m

i i jei ej

R RZ

1.37



Which, when substituted into the expression for C* gives

as before

22

122

1 22

1

*1

Ni F i

mNi ei

m j j Nj j

mj ej

R R

C Z

1.38



On slide 1.27 we said we could show that this optimal cutoff could be expressed as

where:= exp change in ROR on stock w/ 1% change in opt PF= exp return on opt PF

We have defined Also, Zi is proportional to the optimal fraction of the PF the

investor should hold in each stock, Xi

The proportionality constant multiplying Xi is the ratio of the excess return of the optimal PF to the variance of its return

iP P Fi

i

R RC

iP

PR2

1*

N

m j jj

C Z

1.39



The proportionality constant is the ratio of the excess return of the optimal PF to the variance of its return

Thus

From the single index model, we recognize the PF beta, so

22

1*

NP F

m j jj P

R RC X

2

2 22 2 2

1 1

*N N

mP F P Fm j j m j j P F P

j jP P P

R R R RC X X R R

2P F

P

R R

8

1.40



Multiplying by “1” ( = ) so

Where is the regression coefficient of the return on security i to the return on the PF

Which gives the other form for C* which we used in defining the selection process on 1.27

2 2

2 2

2

1*

1 cov( , ) 1

i m i P mP F P P F

i P i P

iP P FP F P F iP

i P i i

C R R R R

R Ri PR R R R

iP

i i

1.41


Single-Index Model The Optimal Portfolio WeightingsWe end the aside to formalize the approach and now

determine the optimal PF weightings The key equations are

and The second equation determines the relative investment in

each security, and The first scales the weights so they sum to 1: fully invested Note the significance that the residual variance, , in each

security plays in determining how much to invest in each

1

kNk

ii

ZXZ

2 *i i F

iei i

R RZ C

2ei

1.42


Single-Index Model Optimal Weightings For the example Our weightings are

Where the come from Col 4 of the first table, and The Zi : The Xi :

2

Col 4* Col 2 *i i Fi

ei i i

R RZ C C

i

1

kk N

ii

ZXZ

1

2

3

4

5

0.091 23.46%0.38750.0956 24.65%0.38750.0775 19.98%0.38750.110 28.36%0.38750.01375 3.54%0.3875

X

X

X

X

X

5.45 not 4.5 => 1.43


Single-Index Model Important to note that the previous result is exactly the

same as would have been found had the problem been solved using the established quadratic programming result Takes a fraction of the time Simple Calculations Easy to “debug”

The characteristics of a stock that make it attractive and relatively desirable are determined in advance or off-line Desirability of any stock is solely a function of its excess return to

beta ratio

9

1.45


Single-Index Model – With Short Sales Roughly the same technique as before Rank order via excess expected return over beta ratio Find C* and bifurcate set of stocks relative to C* Now, N = 10 , so go to last row (see next slide) for C10 = 4.52

Ratio higher than C* are longs, those lower are shortsWeightings use again, pos or neg Then % are found in two ways from either

Depending on definition of short sales (reinvest proceeds into longs (1st) or reinvest into risk-free rate, aka Lintner, (2nd ))

2 *i i Fi

ei i

R RZ C

1 1 or

N N

k k i k k ii i

X Z Z X Z Z

1.46


Single-Index Model – With Short Sales Returning to the example

1.47



10

11.379i

iZ

1.48



10

1.49


Single-Index Model Constant Correlation assumption and an

approach to selecting the optimum PF Utilizing the result that the efficient frontier can be

determined by solving the N simultaneous equations

But now we assume So Solving for Zi

2

1

N

i F i i j ijjj i

R R Z Z

2 2

1 1

N N

i F i i j i j i i i i j j i jj jj i

R R Z Z Z Z Z

ij i j

2

11

N

i i i F i j jj

Z R R Z

1.50


Single-Index Model Constant Correlation Solving for Zi

As before, to express C* in known terms, multiply by each and the and add up the N equations

Developing and solving for C* yields

Or

1

1 * where *1

Ni F

i j jji i

R RZ C C Z

i

1 1

**1 1

N Ni F

j jj j i

R R N CC Z

1 1

1* 1 *1 1 1 1

N Ni F i F

j ji i

R R R RNC CN

1*

1

Ni F

j i

R RCN

1.51


Single-Index Model Constant Correlation Approach Securities can be ranked according to their excess

return to standard deviation quotient

The cutoff, as used before, is found using C*

Lets look at an example …

i F

i

R R

11

ij F

ij j

R RC

i

1.52


Single-Index Model Constant Correlation Approach

11

1.53


Single-Index Model(let ρ = 0.5)

1.54


Single-Index Model Constant Correlation Approach To determine weightings in each of the 3 qualifiers The optimum to invest in each chosen stock is Xi where

And

1 *

1i F

ii i

R RZ C

1

kNk

ii

ZXZ

1.55


Single-Index Model Constant Correlation Approach The Example where C* = 21/4

1.56


Single-Index Model A final example

A final example

12

1.57


Single-Index Model A final example Using the earlier rule; ranking is 1, 3, 2, 5, 4

and C* = 11.82 so 1 & 3 are in

1.58


Single-Index Model The last thing for the single index model is

parameterization for each security, i The model The alpha and beta are determined by regression on

the returns series Beta

Alpha

i i i m iR R e

1

2 2

1

N

it it mt mtim t

i Nm

mt mtt

R R R R

R R

i it i mtR R

1.59


Single-Index Model The last thing for the single index model is

parameterization for each security, i The model Alpha and beta from regression on the returns series Model Statistics Estimate Error Coefficient of Determination – the square of the correlation

a measure of how much of the variation in a single stock is due to a variation in the market

Standard Error in beta for security i :

i i i m iR R e

2

2

1

1 N

ei it i i mtt

R RN

2

im i m mim i

i m i m i

i ei m 1.60


Single-Index Model How explanatory is this parameterization? How much association in beta one period to the next? Beta’s on large PF contain a great deal of information about

future betas – not so good for individual securities

Historical betas better predictors on PF than on stocks

13

1.61


Single-Index Model – How explanatory? Can the predictive ability be improved? Blume’s Results – Betas tend to one

Adjust prediction in next period Adds undesirable modification of average for a population of

stocks – this bias on individual betas should be removed Forecast improved by subtracting bias after adjusting

& under model 1.02 => 1.033 for `62-`68 For individual beta: subtract 1.033, add 1.02 for `62-`68

( 4̀8 5̀4) 1 ( 5̀5 6̀1) 1.02

2 10.343 0.677i i

1.62


Single-Index Model – How explanatory? Can the predictive ability be improved? Vasicek’s Results – adjust prediction: take a weighted average

of stock’s historical beta, , and the previous period average beta, , over a sample of stocks ML just takes average (equally weighted) Vasicaek:

= variance in distribution of historical period betas over a set of stocks

= variance in distribution of beta for security imeasured in the historical period,

This is a Bayesian Technique

221 1

2 1 12 2 2 21 11 1

ii i

i i

2

1

21i

1 1 1i ei m

1i1

1.63


Single-Index Model – How explanatory? Can the predictive ability be improved? Both Blume & Vasicek lead to better betas than not adjusting Is beta the right criteria? In PF analysis the key inputs are expected returns, variances

and correlations, so: Better to consider ability to give better correlations! This is

what matters to PF optimization Study of Correlation matrix forecast Use the historical correlation matrix itself Matrix from historical beta forecast Matrix from Blume-adjusted betas using 2 prior periods Matrix from Vasicek-adjusted Bayesian technique

2ij i j m

iji j i j

1.64


Single-Index Model – How explanatory? Can the predictive ability be improved? Historical correlation matrix was worst by far All single index models did better – surprising when it might be

though that these loose information in achieving simplification! Bayesian Technique is best if results are forced to have

stationary average correlation to the period where the model was fitted

More can be said about other factors that influence beta and how fundamentals can provide improvements to the forecast - later

550.447 quantitative portfolio theory & performance ...daudley/447/jhuonly/550.447 sp13...

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