# Calculate weekly returns from daily stock prices?

If I have log returns for a specific stock, then the weekly log return is the log of Friday's closing price minus the log of Monday's closing price, i.e. $R_{weekly} = log(Price_{Friday}) - log(Price_{Monday})$.

However, can I also calculate the log return as the sum of the daily log returns? $R_{weekly}=[log(Price_{Monday}) - log(Price_{Sunday})] + [log(Price_{Tuesday}) - log(Price_{Monday})] + {...} + [log(Price_{Friday}) - log(Price_{Thursday})]$

• Both the methods should lead to the same answer under the assumption that the price changes are relatively small – Tim Mar 8 '16 at 21:07
• +log() and -log() terms in the second expression should cancel out, leaving exactly the first expression. – noob2 Mar 8 '16 at 22:44

Your second formula regarding the sum of day-to-day returns collapses as follows:

\begin{align} R_{weekly,2} &= \text{log}(Price_{Mon}) - \text{log}(Price_{Sun}) \\ &+ \text{log}(Price_{Tue}) - \text{log}(Price_{Mon}) \\ &+ \dotsc \\ &+ \text{log}(Price_{Fri}) - \text{log}(Price_{Thu}) \\ &= \text{log}(Price_{Fri}) - \text{log}(Price_{Sun}) \end{align}

Compared to the first formula,

$$R_{weekly,1} = \text{log}(Price_{Fri}) - \text{log}(Price_{Mon}),$$

the difference is

$$R_{weekly,2} - R_{weekly,1} = \text{log}(Price_{Mon}) - \text{log}(Price_{Sun}).$$

That is, you include one extra day-to-day return in the definition of $R_{weekly,2}$ as compared to the definition of $R_{weekly,1}$.

In my understanding, the relevant weekly return is the cumulative return over all seven days of the week (if there are no trades on, say, Sunday, then define the corresponding day-to-day return as zero), which collapses to

$$R_{weekly} = \text{log}(Price_{Fri,\ this \ week}) - \text{log}(Price_{Fri, \ last \ week})$$

(which is what @SimoneBortolato suggested before). However, depending on what you want to do with those weekly returns, other definitions could make sense as well.

I can't still comment a question, since I've still got low reputation, so I'm going to answer. I think you should calculate weekly returns from friday to friday (close of the week to close of the following week).