7/2/2013copyright r. douglas martin1 5. transaction costs and mip 5.1 transaction costs constraints...

7/2/2013 Copyright R. Douglas Martin 1

5. TRANSACTION COSTS AND MIP5.1 Transaction costs constraints in MVO

5.2 Transaction costs penalty in quadratic utility

5.3 Discrete constraints via mixed integer programming

Reading: Scherer and Martin (2005), Chap. 3.3* Chincirini and Kim (2006), Chap. 10.1 – 10.5, 10A

* This is included in a copy of Chap. 3.2 & 3.3 posted to the class web site. You should ignore 3.2 except for optional casual reading at this point, and ignore all of the code in both sections as it was based on Rnuopt and this code will be replaced soon.


Turn-over constraints

– Assume costs are proportional and equal for all assets

Proportional costs

– Different for different assets

Fixed and proportional costs

– Add ticket costs independent of trade size

5.1 MVO Transaction Cost Constraints


Broker fees

– Percentage fees plus possible fixed cost– Small-caps: 20-33 bps (100 bps = 1%)– Large caps: 12-22 bps

Bid-ask spread

– Difference between “ask” price at which you can buy and “bid” price at which you can sell

– Becomes significant for thinly traded illiquid stocks

Market Impact– Price impact of large orders– Difficult to model and can be very significant

Overall Costs– Can range from 1% to 4% (but smaller broker deals are possible)

Why Transaction Costs?


Direct Constraints (this Section)

– Turnover– Proportional costs– Proportional costs plus fixed costs

Add Penalty to Quadratic Utility (next Section)

– Focus on proportional costs– Causes added nonlinearity– Can try approximate solution

Ways of Handling Transaction Costs?


initialiw

iw

iw

initiali i i iw w w w

1, 0, 0

n

i i i iiw w w w

Turnover constraints are implemented by practitioners to heuristicallysafeguard against transaction costs. If transaction costs are proportionaland equal across assets, it is sufficient to control turnover that is directlyrelated to transaction costs. So far we have not needed to know the initialholdings when constructing a portfolio, as we assumed no costs toturn our portfolio into cash and vice versa. Here, in addition to the vectorof initial holdings, we need two new sets of variables:

Turn-Over Constraints

assets bought

assets sold

bound on turn-over


1

min subject to

1

0

0

0.

i j iji j

i ii

ii

initiali i i i

n

i ii

i

i

i

w w

w

w

w w w w

w w

w

w

w


Example: Turnover Constraints

The next three long-only fully-invested efficient frontiers with turnover constraints are obtained using Scenario 7 of constraint.sets.test.R for the case of the first 10 midcap stocks in the data set crsp.short.Rdata, with initial equal weights of 1/10 and using the following three turnover constraints:

toc = .7toc = 1.2toc = 1.8

Note that the last choice above leads to the full efficient frontier for a long-only constraint.

Note: The weights bar-plot does not quite show this because the points

selected for the efficient frontier do not quite reach the max return stock.


0.00 0.05 0.10 0.15 0.20

0.00

00.

005

0.01

00.

015

0.02

00.

025

0.03

0

MV EFFICIENT FRONTIER

VOL

MAT

EMN

LEG

AAPL

UTRHB

BNK

APA

LNCR

BMET

toc = 0.6SRmax =0.318rf =0.003

0.05

410.

0542

0.05

450.

0551

0.05

600.

0571

0.05

850.

0601

0.06

20

0.06

510.

0832

BMETLNCRAPABNKHBUTRAAPLLEGEMNMAT

VOL

WE

IGH

TS

-1

0

1

2

3

4


0.00 0.05 0.10 0.15 0.20

0.00

00.

005

0.01

00.

015

0.02

00.

025

0.03

0


VOL

MAT

EMN

LEG

AAPL

UTRHB

BNK

APA

LNCR

BMET

toc = 1.2SRmax =0.332rf =0.003

0.05

260.

0528

0.05

320.

0539

0.05

480.

0560

0.05

740.

0590

0.06

080.

0628

0.06

500.

0673

0.06

980.

0723

0.07

500.

0778

0.09

27


VOL

WE

IGH

TS

-1

0

1

2

3

4


0.00 0.05 0.10 0.15 0.20

0.00

00.

005

0.01

00.

015

0.02

00.

025

0.03

0


VOL

MAT

EMN

LEG

AAPL

UTRHB

BNK

APA

LNCR

BMET

toc = 1.8SRmax =0.332rf =0.003

0.05

260.

0528

0.05

320.

0539

0.05

480.

0560

0.05

740.

0590

0.06

080.

0628

0.06

500.

0673

0.06

980.

0723

0.07

500.

0778

0.08

080.

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0.09

140.

1879


VOL

WE

IGH

TS

-1

0

1

2

3

4


1 1

min subject to

1 1

1

0

0

0.

i j iji j

i ii

n n

i i i i ii i

initiali i i i

i

i

i

w w

w

w tc w tc w

w w w w

w

w

w

1 10, 0, 0,

n n

i i i i i i i ii iw w tc w tc w w w

Proportional Cost Constraints

Have to pay for transaction costs out of proceeds:


Example: Proportional Costs Constraints


Adding Fixed Transaction Costs

1 1 1

max

max

min subject to

1 1

1

0, 0, 0

w

w

0,1 .

i j iji j

i ii

n n n

i i i i i i i ii i i

initiali i i i

i i i

i i

i i

i

w w

w

w f tc w tc w

w w w w

w w w

w

w

Copyright R. Douglas Martin 14

Chincarini & Kim (2006), Chap. 10.4

“before rebalancing”

“after rebalancing”

“before” $$ holdings

“after” $$ holdings

7/2/2013

biwaiw

0biV w

0aiV w

0 01

transaction valueTV = n

a bi iV w V w

5.2 Transaction Costs QU Penalty


If c = cost of rebalancing = fixed proportion of TV (idealization)

Then

Convenient to write as

(NOTE: can generalize to different costs for different assets, e.g. trading cost proportional to trading volume)

7/2/2013

01

TC = V ( )n

a bi i ic w w ic c

c

a bi iw wa bi iw w

0TC = V ( )a bw w c , !a bc w wdepends on

01

TC = transaction costn

a bi icV w w


Portfolio Value Growth

must subtract

Thus

7/2/2013

1 0 (1 )pV V

0 ( ) (1 )pV a bw w c

1 0 0(1 ) ( ) (1 )p pV V V a bw w c

1 0

0may neglect

( ) ( )p p p

V V

V a b a bw w c w w c


Transaction Cost Adjusted Quadratic Utility

This problem is highly non-linear since

C&K: Minimize

7/2/2013

12

12

( ) ( )

( )

U

a a a b a a

a a a

w w μ w w c w w

w μ c w w

( , ).c a bc w w

*

*

,

,

bi i

i bi i

c w wc

c w w

*w

12 a a aw μ w w


Then use the fixed to maximize

Will typically need to iterate the above two-step process to obtain a good final solution. No guarantee of convergence.

7/2/2013

12( ) a a aw μ c w w

c


Buy-in thresholds and cardinality constraints

– Portfolio managers and their clients hate small positions– You only want 25 out of 500 stocks with bounds on weights

Requires mixed integer programming (MIP)

Mathematical problem formulation

Example with pure cardinality constraint

– Best 2 out of 4 stocks– Non-convex efficient frontier

5.3 Discrete Constraints via MIP


Buy-In Thresholds

1 if asset i is selected

0 otherwisei

i

"large" enough number

0,1i iw

Can be 1 if no short-selling (long-only portfolio). Otherwise, you need to allow for weights larger than 1 due to short-selling


Cardinality Constrained Buy-In Thresholds

ilarge number, 0,1i iw

min maxi, 0,1i i i i iw w w

min large number, 0,1i i i i iw w

#iiassets

Type Formula

Either in or out

Either in or out of box

Either out or above

Cardinality constraint

This cardinality constraint is combined with one of the above “threshold” weight constraints.


Cardinality Plus Box Constraint

min max

min subject to

1

#

0,1

i j iji j

i ii

ii

i i i i i

ii

i

w w

w

w

w w w

assets


Best 2 Out of 4 Long-Only Example

4 uncorrelated assets with multivariate normal distribution having common variances of 20% and mean returns of 2%, 4%, 5%, 8%.

T = 50 sets of simulated returns

Example on next slide was produced with Rnuopt and will be replaced with example based on alternative R optimizer.


Risk

Re

turn

0.08 0.10 0.12 0.14 0.16 0.18

0.0

10

.02

0.0

30

.04

0.0

5

Mean - Variance Frontier with Cardinality Constraints

02

04

06

08

01

00

Frontier Portfolios

7/2/2013copyright r. douglas martin1 5. transaction costs and mip 5.1 transaction costs constraints...

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transaction costs constraints

transaction costs penalty

possiblewhy transaction

constraintsassume costs

costs toturn

discrete constraints

direct constraints