Transactions Costs

Important Topic

• Average manager underperforms by average amount of trading costs and fees.

• Portfolio management trades off expected returns against costs incurred with certainty.

• Traditionally, the importance of transactions costs has far exceeded the amount of attention paid.– The death of a thousand cuts.

Explicit and Implicit Costs

• Transactions costs have explicit and implicit components:

• Explicit components:– Commissions– Taxes

• Implicit components:– Spreads, market impact, opportunity costs– These show up in the prices. We don’t

separately pay them.

Defining and Measuring Costs• Implementation Shortfall Approach.• Treynor and Perold• Good overall framework for understanding

components of costs.• Basic idea:

– Compare real portfolio to a paper portfolio. Both portfolios start with same value. Real portfolio trades over interval, while paper portfolio does not. Compare final values.

– Implementation shortfall is difference in value between the two portfolios.

Implementation Shortfall

• Note: we will find it useful to work with numbers of shares, rather than our usual fractional portfolio holdings.

• Paper Portfolio– ni, pi(0), pi(T)

• Real Portfolio– mi(tj), pi(tj)– Trades occur at times {tj} between 0 and T.– Prices are net of commissions and taxes.

Self-financing Condition

• Trades are self-financing:

• Note that one of the assets can be cash.

1 shares of traded at i j i j i j jm t m t m t i t

1

0N

i j i ji

p t m t

Implementation Shortfall

• Paper portfolio performance over period:

• Real portfolio performance over period:

1

0N

P i i ii

Value n p T p

1

1

1

0 0

0 0 0

0 0 0

N

R i i i ii

N

i i i i i ii

N

i i i i i ii

Value m T p T m p

m T p T p p m m T

m T p T p p m T m

Shortfall Calculation

• We can rewrite trade term as:

• Hence, the shortfall becomes:

1 1 1

1 1 1 1

0 0 0

0

N N J

i i i i i ji i j

N J N J

i i j i j i ji j i j

p m T m p m t

p m t p t m t

1 1 1

0 0

opportunity cost+execution cost

N N J

P R i i i i i j i i ji i j

Value Value n T m T p T p p t p m t

Cost Components• Execution Cost

– Adds up shares traded times the difference between the traded price and the reference price (which is the t=0 price).

• Opportunity Cost– Adds up overall price moves times the discrepancy

between final shares in the real and paper portfolios.

• Possible set-up:– Receive new information at t=0 and immediately

rebalance paper portfolio. Allow real portfolio until t=T to implement new information. Compare using implementation shortfall methodology.

Transactions Costs Forecasting• Commissions and other explicit costs

– Easy to model– Typically small for institutional investors.

• Spreads– Also easy to observe and model.

• Market Impact– Extra cost of trading more than one share.– Liquidity provider assessment of counterparty:

• Sheep?• Wolf?• Probabilities vary with urgency of trading.

Market Impact Model

• Inventory Risk

• Liquidity provider receives offer to sell Vtrade.

• How long before the liquidity provider might receive similar orders (in aggregate) to buy Vtrade?

• Risk of holding the position over that time:

~ tradeclear

daily

V

V

250clear

inventory

Market Impact

• Liquidity provider demands price concession proportional to risk:

– Price impact proportional to square root of shares traded

• Market Impact cost proportional to shares traded times cost per share– Proportional to shares traded to power 3/2.

tradeinventory

daily

VPc c

P V

Example

• Trade 10,000 shares:– Move price from $50 to $50.25

– Price impact = 0.50%

– Market impact cost = 10,000 * ($0.25) = $2,500

• Trade 20,000 shares:– Price impact

– Move price to $50.35

– Market impact cost = 20,000 * ($0.35) = $7,000

0.50% 2 0.71%

3

27,000

2.8 2 2 22,500

Overall Cost Model

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

4.0%

4.5%

0 5 10 15 20 25

Order Size ($ millions)

On

e-w

ay T

ota

l C

ost

One-way Total Cost

Square Root Fit

trade

daily

VspreadCost commission c

P V

Thomas F. Loeb, “Trading Costs: The Critical Link between Investment Information and Results.” FAJ, 1983

Rule of Thumb Calibration

• It costs one day’s volatility to trade one day’s volume.– This sets coefficient c in model.– Calibration with trade data can refine this

further.

• Improvements on this simple model– Same framework, add structure.– Better forecasts of daily volume and volatility.– Elasticity: as prices move, how does this

change the mix of buy and sell orders?

Cost-aware Portfolio Construction

• We need to adjust our utility function:

• Notes:– Costs effectively lower portfolio .– Utility depends on initial position.– Cost is zero in absence of trading.

– h is the tcost amortization horizon

,IT TPA PA PA

h

CostU

h hh α h V h

Transaction Cost Amortization

• The quantities and are forecasts over time (i.e. flow variables).

• The quantity Cost forecasts costs at the moment of trading.

• We need to convert Cost for comparability with and .

• This number can have a large impact on portfolio construction, yet receives little attention.

Dynamics of Portfolio Management

• There is something fundamentally wrong with the prior approach.

• We are applying one-period technology to an inherently multi-period problem.

• The amortization horizon is but one symptom of that.

• The true dynamic problem optimizes expected utility over time, accounting for the evolution of alpha and risk over time, and naturally handling cost amortization.

Trading and Turnover

• We define turnover in terms of trading.

• Canonical example:– Go from 100% stock A to 100% stock B.– We count that as 100% turnover.

*

1

*

1

0,

0,

,

N

Pn

N

Sn

P S

TO Max h n h n

TO Max h n h n

TO Min TO TO

Turnover and Utility

• Trading costs lower turnover and impact our ability to add value through active management.

• We can gain some intuition for the extent of this problem by considering how much value we can add with limited turnover.

• We will start with an initial portfolio I, an optimal (ignoring costs) portfolio Q, and consider portfolios P which approach Q as we allow turnover to reach the required amount.

Turnover and Utility• We define P as:

• So P moves linearly from I to Q. We want to work out how utility changes as this happens. And, we will compare the improvement in utility over I, to the maximum available: the improvement of Q over I

1

1

P I Q I

Q Q I

Q T

w

w

w

h h h h

h h h

h h

Maximum Utility Improvement

• We achieve it with Q:

• And, using first order conditions for Q:

• These lead to additional useful relationship:

2 2max Q I Q I Q IU U U

22 , 2Q Q I QI

2 2max 2Q I QIU

Portfolio P Analysis• We calculate:

• We can simplify this to:

• Rule of thumb: we can achieve at least 75% of the added value with 50% of the turnover.

22 2 2

1

1 2 1

P I Q I Q I

Q T QT I

U U w

w w

2 2max

2max

1 2

2

P I Q I QIU U w U w w

w w U

Value Added/Turnover Frontier

0.00%

0.20%

0.40%

0.60%

0.80%

1.00%

0% 20% 40% 60% 80% 100%

Turnover

Ad

ded

Val

ue

Added Value

Optimal Trading

Trading As an Optimization Problem

• What is the trader’s challenge?– Move from initial portfolio to final portfolio while

keeping costs (market impact) low, risk relative to final portfolio low, and taking advantage of short term opportunities along the way.

• Utility Function

• What variables can trader control:– Shares traded each day

2trading short shortU MI

Continuous Single Stock Example

• We integrate utility along path

• Alpha and risk are straightforward:

• New component. We define market impact in proportion to speed of trading

,tradingU dt u h h

22 2

1

1

short

short

dt h t

dt h t

2MI dt c h

dhh

dt(where )

Calculus of Variations

• We can solve these general problems. The appendix shows that the optimal solution satisfies:

• In this case, the above implies:

0u d u

h dt h

22 1 2 0h c h

Single Stock Example

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time in Days

Market Impact DominatesRisk Aversion Dominates

Observations

• Commissions and spreads don’t matter.– We already know how many shares we want to

trade. Only issue is how fast.• Risk will motivate faster trading.• Market impact will motivate steady trading.• We need to specify the number of available

trading periods.• We usually impose monotonicity constraint.