Transaction Cost Analysis A-Z

Transaction Cost Analysis A-Z — November 2008

IV. Estimating Transaction Costs with Pre-Trade Analysis

4. hence, variance and volatility expressed in monetary units per share are easily obtained as follows: σ 2 ( Δ p t ) ≅ p 0 2 σ 2 ( r ) and σ ( Δ p t ) ≅ p 0 σ ( r ) . 5. therefore, the total variance and volatility in monetary units for an order is: σ 2 ( X ) ≅ X 2 p 0 2 σ 2 ( r ) and σ ( X ) ≅ Xp 0 σ ( r ) . Based on the variance of the security expressed inmonetary units, computing price risk for a specific trading strategy results in computing the risk of a one-security position that fluctuates per period. If x k is the number of shares traded in period k , r k is the number of unexecuted shares at the beginning of period k and σ ² is the per period variance for the security in monetary units/share, the total price risk of the trade schedule over n -periods is calculated as follows: σ 2 ( x k ) = r k 2 σ 2 k = 1 n ∑ n ∑ We can illustrate this approach with a small example. Suppose a trader has an order for 12,000 shares ( X ) and wants to execute 4,000 shares ( x i ) in each of the coming three periods. If the per period volatility of the security is estimated at € 0.04/share, the price risk of the given strategy is computed as follows: σ 2 ( x k Cov( i ,r j ) = E( r i r j ) − E( r i Cov( r i ,r j ) ≅ 1 p 0 ,i 1 p 0 , j Cov( Δ p i , Δ p j ) ≅ p 0 ,i Price risk of a specific trading strategy in a single security σ ( x k ) = r k 2 σ 2 k = 1

Price risk of a list of m -securities If we consider a list of m -securities traded over n -periods, the approach described above must be extended to a changing portfolio whose risk depends on both individual security volatility and the covariance of price movement across all pairs of securities. To address this issue, we need first to understand how to convert the covariance matrix into monetary units per share and how to compute the total portfolio variance and risk. Using the same approximation methodology as for individual security volatility, we can convert the covariance of returns of securities i and j as follows:

1

) ≅ 1 p i

Cov( r i

,r j

) = E( r i

r j

) − E( r i

)E( r j

E( Δ

⎡⎣

p

j

1

) ≅ 1 p i

1 ) − E( Δ p i ) − E( r i

1 i Δ p j ) = E( r i p 0 ,i

)E( r j

) ≅ E( Δ p

, Δ p j )E( Δ p j

)

,r j ⎡⎣

⎤⎦

1

) ) ≅ 1 p i

Cov( r i p

cov( Δ p i

j Cov( r i

,r j

p r j

)E( r j

E( Δ

⎡⎣

p

0 , j

j

cov( Δ p i

, Δ p j

)

1

1

Cov( r i

,r j

) ≅

cov( Δ p i

, Δ p j

)

p

p

0 ,i

0 , j

Hence, the covariance of security prices expressed in monetary units/share is: Cov( Δ p i , Δ p j ) ≅ p 0 ,i p 0 , j cov( r i ,r j ) ≅ p 0 ,i p 0 , j ρ i , j

p

cov( r i

,r j

) ≅ p

p

ρ i , j σ ( r i

) σ ( r j

) ,

0 , j

0 ,i

0 , j

where ρ i,j is the correlation coefficient, and the covariance in monetary units for two orders X i and X j is then: ) Expanding to a list of m -securities, the covariance matrix C in monetary units/ share is computed from the covariance matrix of return C r as follows: C = DC r D , where D is the diagonal matrix of the current price of the m -securities: Cov( X i Δ p i , X j Δ p j ) ≅ X i p 0 ,i X j p 0 , j cov( r i ,r j

) = 12000 2 × 0.04 2 + 8000 2 × 0.04 2 + 4000 2 × 0.04 2 = 358400

) = 358400 = 599

σ ( x k

2 ( x k ) = 12000 2 × 0.04 2 + 8000 2 × 0.04 2 + 4000 2 × 0.04 2 = 358400 x k ) = 358400 = 599 σ 2 ( x k ) = 358400 = 599 σ ( x k

) = 12000 2 × 0.04 2 + 8000 2 × 0.04 2 + 4000 2 × 0.04 2 = 358400

51 An EDHEC Risk and Asset Management Research Centre Publication

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