Topic 3 R Schwartz Equity Markets Trading and

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

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

Prices (P) P 0 , P 1 , P 2 , … , PT Time P 0 ©R. Schwartz P 1 P 2 Equity Markets: Trading and Structure PT 3

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

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 = ( PT – PT-1 ) / PT-1 = PT-1, T / PT-1 ©R. Schwartz Equity Markets: Trading and Structure 5

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 PRT-1, T = PT / PT-1 = ( PT-1 + PT-1, T ) / PT-1 = 1 + r. T-1, T PR 0, T = PT / P 0 ©R. Schwartz = (P 1 / P 0) * (P 2 / P 1) *…* (PT / PT-1) Equity Markets: Trading and Structure 6

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

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

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 ©R. Schwartz Equity Markets: Trading and Structure 9

P* Returns in Trader. Ex 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 ) + Rt ©R. Schwartz the jump Equity Markets: Trading and Structure 10

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

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) = 2 E(R 0, 1) ©R. Schwartz Equity Markets: Trading and Structure 12

Log Returns: Two Period Variance Var(R 0, 2)=Var(R 0, 1)+Var(R 1, 2)+2 Cov(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 ? ©R. Schwartz Equity Markets: Trading and Structure 13

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

Trading Costs 1. Explicit costs • • • commissions taxes etc. 2. Execution Costs (the implicit costs of trading) • • Bid-ask spread Market impact Delay/opportunity cost Implementation shortfall ©R. Schwartz Equity Markets: Trading and Structure 15

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

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

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

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

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