Transaction Cost Analysis A-Z

Transaction Cost Analysis A-Z — November 2008

IV. Estimating Transaction Costs with Pre-Trade Analysis

This definition means that we need to consider cost and risk jointly. Accordingly, optimisation must be viewed as a means of determining the trading strategy that resolves the trader’s dilemma, which will find the suitable trade-off between cost and risk. Indeed, market impact and timing risk are conflicting terms; an optimal strategy results therefore in finding the proper trade- off. Broadly, optimisation in the context of trading can be formulated in three ways: • Minimise cost subject to a specified level of risk, that is: Min φ ( x k ) s.t. ℜ ( x k ) ≤ ℜ * ; ℜ * is the maximum risk exposure and φ ( x k ) the expected transaction cost. • Balance trade-off between cost and risk, that is: Min φ ( x k ) + λ . ℜ ( x k ) ; λ is the level of risk aversion or the marginal rate of substitution between cost and risk. • Maximise probability of price improvement, that is: Max Pr[ φ ( x k ) ≤ L*] ; L * is the highest acceptable cost. Specifically, we can illustrate the second formulation with the following proposal, which incorporates some real-world constraints, from Kissell and Glantz (2003): Objective function − D min ≤ x i ,k p i ⎡ ⎣ ⎢ ⎢ k ∑ i ∑

This constraint ensures that optimisation will provide a strategy that executes orders over the specified trading horizon. • Shrinking portfolio: i , j − 1 This constraint ensures that the portfolio is continuously decreasing in shares and prevents optimisation from making the position either longer or shorter than the initial position. r i , j ≤ r

x

i ,k

• Participation rate:

≤ α

x

+ v

i ,k

i ,k

This constraint places an upper bound on participation volume in each period.

• Cash balancing

⎡

0.95 x

i ∑

k ∑

i ,k

− D min ≤

x

p i

+ k . Δ p i

+

⎢

i ,k

X

( x

+

⎢

⎣

i

i ,k

⎤

0.95 x

I i

+ 0.05I i X i

i ,k

+ k . Δ p i

≤ D max

+

⎥

X

( x

+ 0.5v

)

⎥

⎦

i

i ,k

i ,k

x This constraint is usually used in the implementation of a trade list including buy and sell orders to ensure that the net cash position in any period is within a specified range. In the above approach, the objective function calculates the total cost by adding price appreciation and market impact plus λ units of timing risk. λ =1 indicates that investors are equally concerned with cost and risk. λ >1 (<1) refers to investors who are more (less) concerned with risk than cost and who thus prefer aggressive (passive) strategies. So, for a given risk aversion, the optimisation output includes the number of shares of each security to be traded in each period (trading schedule) as well as cost estimates for each component (cost profile). (2) Efficient trading frontier The approach described above is in i ,k + 0.5v i ,k ) + 0.05I i X i ⎤ ⎦ ⎥ ⎥ + λ . r k ' Cr k k = 1 n ∑

⎡

0.95I i

n ∑

m ∑

m ∑

n ∑

k .x

i ,k Δ p i

x

Min

+

⎢

i ,k

X

( x

⎢

⎣

k = 1

i = 1

i = 1

k = 1

i

i ,k

⎤

⎡

0.95I i

x

+ 0.05I i X i

n ∑

n ∑

m ∑

m ∑

n ∑

i ,k

+ λ .

r k

k .x

i ,k Δ p i

x

' Cr k

+

⎥

⎢

i ,k

X

( x

+ 0.5v

)

⎥

⎢

⎦

⎣

k = 1

k = 1

i = 1

i = 1

k = 1

i

i ,k

i ,k

Constraints • Completion:

n ∑

x

= X i

i ,k

k = 1

55 An EDHEC Risk and Asset Management Research Centre Publication