In this assignment you will build a pairs trading strategy using the two instruments

provided, called, WHITE and BLACK. These have been picked to $$factor out$$ their

common factors such as industry, size, etc., so that we can assume their returns are driven by the same factors. We will focus on their $$residuals$$ as a basis for trading decisions, that is$,$ we will act when they $$deviate$$ from expectations, with a future expectation that they will revert.

I. Start by determining the suitability of the two instruments for pairs trading

using cointegration analysis. Specifically, using the Dickey$-$Fuller test,

analyze whether the difference in prices of the two series is stationary using

the data for the first $1,000$ days $($roughly $4$ years$).$ Remember, you are testing

not whether Black and White prices are stationary but whether their

DIFFERENCE is stationary. To do this, first construct the difference series $($of

prices$)$ and then calculate the daily deltas of this differenced series. As per the

handout of the Dickey$-$Fuller test, regress the deltas $($yt$)$ against the previous

values of the difference series $($Xt$-1)$ and interpret your result.

Does this look like a promising pair based on the stationarity test? Why or

why not?

II$.$ Independent of the result you get in part $1$ above, use the two instruments$$

data on the class website to build a pairs trading strategy that relies on their

$$spread$$ to be mean reverting. In our case we calculate three spreads based on

lookback windows of $5,10,$ and $20$ days. The spread will be a difference of

trailing N$-$day normalized returns between the two instruments. Do this as

follows:

a$.$ Compute the normalized N$-$day returns $($z$-$scores$)$ as usual for the two

instruments $($where you choose N to be say, $5,10,$ and $20$ days$)$ for each

stock based on $60$ trailing values.

b$.$ Compute the differences of the calculated normalized returns $($i$.$e$.,$ zdiff$5=$

zret$5\_($W$)$hite minus zret$5\_($B$)$lack$).$ These will be used to specify the entry

and exit levels for each trade. $($As stated above, use the trailing $60$ values

for normalizing returns, that is$,$ for calculating zPEP$5,$ zPEP$10,$ etc., and

then calculate the differences of the normalized returns between White and

Black, which are designated as zdiff$5,$ zdiff$10,$ and zdiff$20.$

c$.$ Apply the following decision rule: If zdiff$5($do the same for zdiff$10$ and

zdiff$20),$ exceeds some positive threshold, you $$short the spread,

$$ that is$,$short the outperformer and go long the underperformer with the

expectation that the spread will revert. Similarly, if the spread is below

some negative threshold, do the opposite, namely, go long the spread.

Use the spreadsheet template provided to do the assignment. $($Excel$)$

Calculate your forward one$-$day return by sizing the two positions in inverse proportion to their $20-$day trailing volatility, as in the previous assignments. That is$,$ you scale the forward return of the respective instruments on day T$+1$ by their relative weights calculated at the end of day T based on the trailing $20-$day volatility of their respective returns.

As parameters in your system, use the following five as defaults:

$*$ holding period $($i$.$e$.5$ days$)$ after which the trade is exited by default

$*$ long and short entry thresholds $($i$.$e$.-1$sigma for going long the spread and

$+1$sigma for going short the spread$)$

$*$ profit targets for long and short signals where you exit $($i$.$e$.+1$ sigma for a long

and $-1$ sigma for a short$)$

You can also extend the assignment by using a stop loss, but this isn$$t a requirement for this assignment.

Calculate the Sharpe ratios of the three trading strategies $-$ based on the $5,10$ and $20$ day spreads$)$ with a default holding period $($such as $5$ days$)$ and a profit target $($such as $1$ sigma$).$ Plot the equity graphs of the strategies.

Would you trade any of them? If so$,$ which one or which combo and why?