Returns


Kevin Crotty
BUSI 448: Investments

Where are we?

Last time:

  • Saving for retirement
  • Real and nominal cash flows and rates

Today:

  • Calculating returns
  • Fetching data
  • Summarizing returns

Calculating Returns

General definition

For an investment, a gross return is the value today scaled by the value in a prior period:

\[ \text{Gross Return}_t = \frac{\text{Value}_t}{\text{Value}_{t-1}} .\]

A net return is the change in value, scaled by the value in a prior period:

\[ \text{Net Return}_t = \frac{\Delta \text{Value}_t}{\text{Value}_{t-1}} .\]

We will usually use net returns in class.

Stock returns

For stocks, the value today is measured by the current price and any dividends \((D)\).

\[ R_t = \frac{P_t +D_t -P_{t-1}}{P_{t-1}} \]

  • Stock prices include future dividends until the ex-dividend date
  • For computing returns, the dividend goes on the ex day.

Stock splits

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

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

  • Example: 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.

Bond returns

For bonds, the value today is measured by the current price and any accrued interest \((AI)\) or coupon payments \((C)\).

\[ R_t = \frac{P_t + AI_t + C_t - (P_{t-1}+AI_{t-1})}{P_{t-1}+AI_{t-1}} \]

  • Prices quoted as clean prices, which exclude accrued interest.
  • Trades between coupon payments transact at dirty price

Compounding returns

  • We can 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 for stocks, or
    • interest payments were reinvested in the bond.

Fetching data

Yahoo Finance package


Install and import yfinance package

!pip install --upgrade yfinance
import yfinance as yf

Daily equity returns from Yahoo Finance


  • Yahoo provides daily data by default.
  • Adjusted closing prices are adjusted for splits and also adjusted for dividends on each ex date.


price = yf.download('AAPL', start='2000-01-01', end='2020-12-31', progress=False)['Adj Close']
ret_daily = price.pct_change()

Lower frequency returns from Yahoo


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


price = yf.download('AAPL', start='2000-01-01', end='2020-12-31', progress=False)['Adj Close']
ret_monthly = price.resample('M').last().pct_change()
ret_annual  = price.resample('Y').last().pct_change()

# change index from datetime to period (optional)
ret_monthly.index = ret_monthly.index.to_period('M')
ret_annual.index  = ret_annual.index.to_period('Y')

Pandas-datareader package


Install and import pandas-datareader package

!pip install --upgrade pandas-datareader
import pandas as pd
from pandas_datareader import DataReader as pdr

FRED data


Market and various portfolio returns + RF


Ken French Data Libary

  • Ken French is a Dartmouth finance prof and board member of investment firm Dimensional
  • Prolific researcher whose work we will be discussing later

Datasets include

  • benchmark returns including VW market return
  • portfolios sorted on characteristics
  • industry portfolios

Summarizing Returns

Arithmetic Average Returns

  • The arithmetic average return is sometimes used to estimate expected return: \[ \frac{1}{T}\sum_{t=1}^T r_t \]
  • Assumption: each realized return was a draw from a single distribution.
    • We will talk later in the class about what to do if we think this is a bad assumption.

Geometric Average Returns

  • We may also be interested in summarizing the realized performance of an investment over a time horizon.
  • The geometric average return is the constant return that would compound to the same compounded return experienced by an investor.
    • aka compound annual growth rate or time-weighted return.
  • The geometric average return is always less than the arithmetic average return.
    • The difference is larger when returns are more volatile.

Tesla

  • Tesla went down 50% between Nov 2021 and May 2022.
  • It then went up 50% between May 2022 and Aug 2022.
  • Were Tesla shareholders back to even?

  • For each $100 of Tesla stock, shareholders experienced 100 → 50

  • and then 50 → 75.

  • They lost 25%, even though the average return was zero.

  • So, lose 50% and make 50% → lose 25%. Suppose you

    • make 50% and then lose 50%?

    • lose 50% and then make 100%?

    • make 100% and then lose 50%?

Geometric Average Return

  • Given returns \(r_1, \cdots, r_n,\) the geometric average return is the number \(r\) such that

\[(1+r)^{n}=(1+r_1)\cdots(1+r_{n})\]

  • So earning \(r\) each period produces the same accumulation as the actual returns \(r_1, \cdots, r_n.\) We solve for \(r\) as

\[r=[(1+r_1)\cdots(1+r_n)]^{1/n}-1\]

Examples

  • make 50% and lose 50% → geometric average is

\[\sqrt{1.5 \times 0.5}-1=-0.134\]

  • make 100% and lose 50% → geometric average is

\[\sqrt{2 \times 0.5}-1=0\]

Some historical data

Average Returns in Python


Given a net return pandas data series ret, arithmetic average returns can be calculated:

ret.mean()


and the geometric averages can be calculated:

from scipy.stats import gmean
gmean(1+ret)-1

Variance and Standard Deviation

  • The dispersion in realized returns can be measured by either variance or standard deviation.

Sample standard deviation for sample mean \(m\):

\[ \sqrt{ \sum_{t=1}^T \frac{(r_{t}-m)^2}{T-1}}\]

To estimate standard deviation in python from a pandas data series ret

ret.std()

For next time: Returns of portfolios





BUSI 448