Stock Returns





Kerry Back

Close-to-close Returns

  • Let’s compute close-to-close daily stock returns. Let \(P_t\) denote the price at close on day \(t\).

  • If no dividends, the gain on day \(t\) is \(P_t - P_{t-1}\).

The return per $1 invested in a share at close on \(t-1\) is

\[(P_t - P_{t-1}) / P_{t-1}\]

What about dividends?  CVX in Fall 2021



Nasdaq’s statement

Chevron Corporation (CVX) will begin trading ex-dividend on August 18, 2021. A cash dividend payment of $1.34 per share is scheduled to be paid on September 10, 2021. Shareholders who purchased CVX prior to the ex-dividend date are eligible for the cash dividend payment.

Three Dates:

  • August 18: (begins trading ex-dividend)
  • August 19: (shareholders of record receive the dividend)
  • September 10: (dividend is paid)

Ex-dividend days

  • If you buy on Aug 18, you are not entitled to the dividend.

  • Purchase Aug 17 or before (and hold through Aug 17)

    • → will be shareholder of record on Aug 19
    • → receive dividend on Sept 10

  • Why Aug 17 → Aug 19?

    • T+2 settlement

  • For computing returns, the dividend goes on the ex day.
  • The close Aug 17 to close Aug 18 return for CVX is calculated as \((P_{\text{Aug18}} + 1.34 - P_{\text{Aug17}})/P_{\text{Aug17}}\).
  • We compound daily returns to get weekly, monthly, or annual returns:

\[ (1+r_1)(1+r_2)...(1+r_T)-1 \]

  • The weekly, etc. returns are as if the dividend was received on the ex day and reinvested in new shares.

Stock splits


If a company does an \(n\)-for-1 stock split, then each shareholder gets \(n\) new shares for each of her existing shares. Shares are worth roughly \(1/n\) as much.

Split adjusted prices


Data vendors routinely compute split-adjusted prices, scaling down old prices by the same factor for comparability to new prices.

Yahoo’s adjusted prices

  • finance.yahoo.com is a good source for data.

  • Yahoo’s adjusted closing prices are adjusted for splits and also adjusted for dividends on each ex date.

  • On Aug 18, 2021, the Aug 17 price was adjusted as

\[ {\hat{P}_{\text{Aug17}}}={P_{\text{Aug17}}}-1.34 \]

  • $1.34 of the value of at close on Aug 17 was the dividend. Adjusted price is the value without dividend.

  • So, Aug 17 price is scaled down by the factor

\[ {\hat{P}_{\text{Aug17}}}/{P_{\text{Aug17}}} \]

  • All prior prices are scaled down by the same factor.
  • Previous price ratios are therefore unchanged:

\[ \frac{\hat{P}_{t}}{\hat{P}_{t-1}}=\frac{P_{t}}{P_{t-1}}=1+r_t \]

Summary of Yahoo adjusted closing prices

  • On non ex-dividend days, percent change is the return \(P_t/P_{t-1} - 1\).
  • On ex-dividend days, percent change is

\[ \frac{P_{t}}{P_{t-1}-D_t} - 1 \]

CVX on Aug 18, 2021

The return calculated the usual way is:

\[ \frac{P_{\text{Aug18}}+1.34}{P_{\text{Aug17}}} - 1 =-0.0267 \]

The % change in Yahoo’s adjusted close is

\[ \frac{P_{\text{Aug18}}}{P_{\text{Aug17}}-1.34} - 1=-0.0271 \]

Getting Returns from Yahoo


Install pandas-datareader or, on colab, upgrade it.


!pip install --upgrade pandas-datareader

from pandas_datareader import DataReader as pdr

price = pdr('cvx', 'yahoo', start=2010)['Adj Close']
ret_daily = price.pct_change()

Compounding Yahoo returns


Can get monthly or annual return as % change in monthly or annual Yahoo-adjusted closing prices - equivalent to compounding Yahoo daily returns.


price = pdr('cvx', 'yahoo', start=2010)['Adj Close']
ret_monthly = price.resample('M').last().pct_change()
ret_annual = price.resample('Y').last().pct_change()

# change datetime to monthly or annual (optional)
ret_monthly.index = ret_monthly.index.to_period('M')
ret_annual.index = ret_annual.index.to_period('Y')