1. ©R. Schwartz Equity Markets: Trading and StructureSlide 1 Topic 3

2. ©R. Schwartz Equity Markets: Trading and StructureSlide 2 Returns Measurement

3. ©R. Schwartz Equity Markets: Trading and StructureSlide 3 Prices (P) P 0, P 1, P 2, …, P T P0P0 P1P1 P2P2 PTPT Time

4. ©R. Schwartz Equity Markets: Trading and StructureSlide 4 Arithmetic Returns (  P)  P 0,1 = P 1 – P 0  P 1,2 = P 2 – P 1  P T-1,T = P T – P T-1

5. ©R. Schwartz Equity Markets: Trading and StructureSlide 5 Percentage Returns (r) r 0,1 = ( P 1 – P 0 ) / P 0 =  P 0,1 / P 0 r 1,2 = ( P 2 – P 1 ) / P 1 =  P 1,2 / P 1 r T-1,T = ( P T – P T-1 ) / P T-1 =  P T-1,T / P T-1

6. ©R. Schwartz Equity Markets: Trading and StructureSlide 6 Price Relatives (PR) PR 0,1 = P 1 / P 0 = ( P 0 +  P 0,1 ) / P 0 = 1 + r 0,1 PR 1,2 = P 2 / P 1 = ( P 1 +  P 1,2 ) / P 1 = 1 + r 1,2 PR T-1,T = P T / P T-1 = ( P T-1 +  P T-1,T ) / P T-1 = 1 + r T-1,T PR 0,T = P T / P 0 = (P 1 / P 0 ) * (P 2 / P 1 ) *…* (P T / P T-1 )

7. ©R. Schwartz Equity Markets: Trading and StructureSlide 7 Log Returns (R) R 0,1 = ln(P 1 /P 0 ) = ln(P 1 ) – ln(P 0 ) = ln(1+ r 0,1 ) R 1,2 = ln(P 2 /P 1 ) = ln(P 2 ) – ln(P 1 ) = ln(1+ r 1,2 ) R T-1,T = ln(P T ) – ln(P T-1 ) = ln(1+ r T-1,T ) PR 0,T = P T / P 0 = (P 1 / P 0 ) * (P 2 / P 1 ) *…* (P T / P T-1 ) R 0,T = R 0,1 + R 1,2 +…+ R T-1,T =  R i-1,i = ln(P T ) – ln(P 0 ) R = ln(1+r) = ln(price relative)

8. ©R. Schwartz Equity Markets: Trading and StructureSlide 8 Question: Which of the following may be normally distributed? PP r PR R

9. ©R. Schwartz Equity Markets: Trading and StructureSlide 9 Two Period Log Returns P 2 = P 0 ( 1 + r 0,2 ) P 2 = P 0 ( 1 + r 0,1 ) ( 1 + r 1,2 ) 1 + r 0,2 = P 2 / P 0 = ( P 1 / P 0 ) * ( P 2 / P 1 ) = = ( 1 + r 0,1 ) ( 1 + r 1,2 ) R 0,2 = R 0,1 + R 1,2

10. ©R. Schwartz Equity Markets: Trading and StructureSlide 10 P* Returns in TraderEx When P* follows a random walk, P* returns are generated by draws from two distributions: 1.Poisson distribution (when does P* jump) 2.A lognormal distribution (how big is the jump) Ln(P* t ) = Ln(P* t-1 ) + R t the jump

11. ©R. Schwartz Equity Markets: Trading and StructureSlide 11 Means and Variances

12. ©R. Schwartz Equity Markets: Trading and StructureSlide 12 Log Returns: Two Period Mean Assume a constant Mean: E(R 0,1 ) = E(R 1,2 ) E(R 0,2 ) = E(R 0,1 ) + E(R 1,2 ) E(R 0,2 ) = 2E(R 0,1 )

13. ©R. Schwartz Equity Markets: Trading and StructureSlide 13 Log Returns: Two Period Variance Var(R 0,2 )=Var(R 0,1 )+Var(R 1,2 )+2Cov(R 0,1,R 1,2 ) Assume a constant Variance: Var(R 0,1 ) = Var(R 1,2 ) For Cov(R 0,1,R 1,2 ) = 0 Var (R 0,2 ) = 2 Var(R 0,1 ) What if Cov(R 0,1,R 1,2 ) < 0 ?

14. ©R. Schwartz Equity Markets: Trading and StructureSlide 14 Costs

15. ©R. Schwartz Equity Markets: Trading and StructureSlide 15 Trading Costs 1. Explicit costs commissions taxes etc.  Execution Costs (the implicit costs of trading) Bid-ask spread Market impact Delay/opportunity cost Implementation shortfall

16. ©R. Schwartz Equity Markets: Trading and StructureSlide 16 From Trading Costs to Volatility 1. The bid-ask spread 2. Market impact 3. Momentum trading 4. Imperfect price discovery Trading costs cause prices to bounce between higher and lower values

17. ©R. Schwartz Equity Markets: Trading and StructureSlide 17 Trading Costs & Volatility P* = Implicit transaction cost of buy or sell = Transaction price (triggered by buy order) = Transaction price (triggered by sell order) = Magnitude of C = Unobserved, costless trading price C P* Price Time

18. ©R. Schwartz Equity Markets: Trading and StructureSlide 18 Trading Costs & Volatility P* = Implicit transaction cost of buy or sell = Observed price of buy-triggered trade = Observed price of sell-triggered trade = C = Unobserved, costless trading price C P* Price Time

19. ©R. Schwartz Equity Markets: Trading and StructureSlide 19 Trading Costs & Returns Price Time PP TT

20. ©R. Schwartz Equity Markets: Trading and StructureSlide 20 Which is More Volatile? P* P* or the transaction price that we observe? Observed Transaction Price Time