You are provided daily data for three ETFs: XLF$,$ XOP, XLK between January $2014$ and

January $2024.$ Your exercise is to combine their daily returns in two ways. In the first,

you average their returns, which means that your bets are $$equal sized$$ in terms of

dollars. In the second method, you will combine the three so that you take $$equal risk$$

which is expressed in terms of volatility. What this means is that you$$ll invest fewer

dollars if a series of more volatile and vice versa. This is commonly referred to as $$risk

parity.$$

Calculate the combination of returns by combining the ETFs in the two ways described

above. For the former, give the returns equal weights. For the latter, you will weight the

returns in inverse proportion to their $20-$day trailing volatility. So$,$ calculate the $20-$day

trailing window of daily returns for each instrument, and size $($or weight$)$ each instrument

it in inverse proportion to its $20$ day volatility. The three weights should add up to one.

Note that the returns are based on close to close prices and you are calculating the

weights at the end of each day $($sizing the position at the end of the day$),$ so that the

weights are applied the next day to the return. That is how you calculate your :volatilityweighted average returns$$ for each day. The intuition is that the higher the volatility, the

lower the dollars allocated and vice versa.

Requirements:

$$ Plot the two Equity Graphs for the equal and vol$-$weighted returns $($start them at

$100).$Which is better? Why?

$$ Plot the weights of the three ETFs over time. Do they look relatively smooth to

you? When would you expect sudden changes, like we see between $12/08/2014$

and $1/27/2015?$

If you feel like the challenge, calculate the worst drawdowns of the two strategies. A

drawdown is essentially a $$peak to trough$$ return. Which of the two exhibits better

drawdown behavior?