Transaction Cost Analysis A-Z

Transaction Cost Analysis A-Z — November 2008

IV. Estimating Transaction Costs with Pre-Trade Analysis

p 1

0

...

0

⎞

⎛

x

2

I

⎟

⎜

j ∑

j

K( x ) =

0

p 2

...

0

x

+ 0.5v j

X

⎟

⎜

D =

j

...

...

...

...

⎟

⎜

⎟

⎜

and then make two assumptions. First, that successive volumes are independent from period to period. Second, that the estimation error of the instantaneous market impact function I may be ignored. Hence, they obtain that liquidity risk is estimated as follows:

0

0

...

p m

⎠

⎝

From the above approximations for security volatility and covariance matrix, we can easily compute the total variance and risk for a static portfolio of m -securities. If X represents a vector of security positions where x i is the number of shares held in the i th security and C is the covariance matrix expressed in monetary units per share, we obtain the total risk of the portfolio as follows: σ 2 ( X ) = X ' CX and σ ( X ) = X ' CX . If we consider now a list of m -securities traded over n -periods, its price risk has to be derived from the changing portfolio approach. If x k is a vector where x i,k is the number of shares of security i to be traded in period k , r k is a vector where r i,k is the number of unexecuted shares of security i in period k and C the per-period covariance matrix expressed in monetary units per share, we compute the total risk in monetary units for the specified trading strategy as follows: σ 2 ( K ( x )) = σ 2 I X ( x j j ∑ ⎡ ⎣ ⎢ ⎢

2

2

x

⎡

⎤

I

I

⎛ ⎝⎜

⎞ ⎠⎟

j

j ∑

σ 2 ( K ( x )) = σ 2

=

⎢

⎥

X

X

( x

+ 0.5v j

)

⎣ ⎢

⎦ ⎥

j

j

2

2

2

x

x

⎛

⎞

⎤

I

⎛ ⎝⎜

⎞ ⎠⎟

j

j

j ∑

σ 2

=

⎥

⎜

⎟

X

+ 0.5v j

)

x

+ 0.5v j

⎝

⎠

⎦ ⎥

j

23 - For more detail, see Kissell and Glantz (2003), pp. 126-127.

Since the only random variable in this expression is v j , the authors solve it with the theorem below, which relies on the assumption that the function is at least twice differentiable. 23 THEOREM Let v be a random variable with E(v)=μ and σ ²(v)= σ ² . If Y=H(v) then, E(Y ) ≅ H( μ ) + H''( μ ) 2 σ 2 σ 2 (Y ) ≅ H'( μ ) [ ] 2 σ 2 If the expected value and variance of each period j is E(v j )=v j and σ ²(v j )= σ ² vj respectively, liquidity risk is computed as follows: H( v ) = x 2 ( x + 0.5v ) H'( v ) = − x 2 2( x + 0.5v ) 2 σ 2 (Y ) = x 4 σ 2 ( v ) 4( x + 0.5v ) 4 Hence, the liquidity risk of the execution schedule K x is given by:

n ∑

n ∑

σ 2 ( x k

) =

r k

' Cr k

σ ( x k

) =

r k

' Cr k

and

k = 1

k = 1

(b) Liquidity risk Liquidity risk is the risk of unexpected movements in market conditions. It must be assessed to account for the total risk involving price impact estimates.

To formulate liquidity risk, Kissell and Glantz (2003) start from the following total market impact expression:

52

An EDHEC Risk and Asset Management Research Centre Publication