I decided to start a series of articles on the basics of mathematics in finance. It is a little technical however necessary for the future, especially when we will look into efficient frontier and optimal portfolios.

Today we talk about returns. What is it? How to compute them? What is the difference between daily, monthly and yearly returns? Why is it used in finance?

Key takeaways:

  • Stock prices are not comparable. Market capitalizations are.
  • Returns allow to compare stocks together, no matter their price or market capitalization.
  • Returns are flexible and can be annualized in order to compare different asset classes.

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Share price and market capitalization

The first characteristic of a stock people refers to (on TV, trading platforms and even at diners) is the share price (or stock price). It is the price of a share of a company:

Share price of different companies as a function of time. It does not tell much about stock performance.

It is completely artificial and contains little information about the underlying company. Every company is broken down in a different number of shares. Therefore you often can't compare the share prices of two different companies: a cheaper stock does not mean a cheaper company.

Example: Apple has a share price of $500 and a capitalization of about 1.9 Trillions. Whereas Amazon has a higher share price of $3000+ and a capitalization of 1.6 Trillions.

A more comparable amount is the market capitalization. It corresponds to the number of shares of a company times the share price. Safal Niveshak wrote an article about why you should think in market capitalizations, not stock prices

However market capitalization also has its caveats: as investors we are interested in making money. So when we buy a piece of a company we hope to sell it at a higher price in the future (i.e. at a higher market capitalization). And, as investors, we like to measure this growth as percentage.

Imagine a Company A is worth $10 Billions at time t1 and $11 Billions at time t2 and company B is worth $100 Billions at time t1 and $101 Billions at time t2. Intuitively, it's harder to go from $10B to $11B than it is to go from $100B to $101B (and the reward is larger). So what we are really interested in is not the market Capitalization in itself but its rate of growth. This is called returns. (Company A had a return of 10% during that period while Company B had only a return of 1%).

We are not really interested in the absolute value of a company but in its potential.

Returns

The main benefit of using returns, versus prices, is normalization: measuring a variable in a comparable metric. Using returns we can compare different companies of different sizes (and even different asset classes). Moreover you use returns to compute the money you are making or losing on a trade.

The return is the rate of growth of an asset or a portfolio. It is the money made or lost on an investment over some period of time. There are different way of representing returns:

  • Price return
  • Total return
  • Nominal return
  • Real return

More details on returns in Finance

The formula for the total stock return is the appreciation in the price p plus any dividends d paid divided by the original price of the stock.

\[ r = \frac{p_1 - p_0 + d}{p_0} \]

We will always refer to the total return because that what makes assets comparable with each other. For short period of time (a months to a few years) it is OK to use real return, or if the assets are quoted in the same currency. However if you start working with assets quoted in different currencies, real estate or gold, it is generally useful to take inflation and dividends into account.

Stock (daily) returns of Apple, Amazon and JP Morgan as a function of time

When you look at these graphs you want to tell (or even yell at) me: What are you talking about? These graphs are even more messier than before.

Indeed it is harder to read at first glance but it allows to compute many more things such as:

  • Volatility
  • Risk-return graphs
  • Efficient frontier and optimal portfolio

The properties (i.e. mean, variance, tails, kurtosis...) of the distribution of returns contains a lot of information that can be used to better understand the characteristics (and potential) of an asset.

Below you can see that the returns of JPM are much more centered to 0 than AAPL or AMZN. And the right-tail of the distribution of AMZN is larger.

It is in agreement with the price evolution of each of these stocks.

Distributions of daily returns of Apple, Amazon and JP Morgan. Notice how a small changes in the distribution of returns leads to large changes in the performance of the stocks.

Time horizon

Stock prices are available on a daily basis, so the most natural and easy way is to start with daily returns. However we've seen that daily returns can be hard to read for many reasons:

  • they are very volatile
  • it is hard to know what a 0.02% daily returns means on a monthly or yearly basis
  • it is not always possible to compare the returns of different asset classes especially if they don't quote every day

Therefore it is useful to be able to convert daily returns into monthly or yearly returns. Here is the procedure to annualize daily returns:

\[ A = {\prod_i (1+r_i)}^{T/t} \]

Where A is the annualized return, t is the number of observations (either on a daily, monthly or even quarterly basis), T is the factor for converting the observations to annual (e.g., 12 for converting monthly to annual). In effect, Π(1+r) is the cumulative return and T/t is the factor that annualizes the cumulative return. For example, for a manager with a 3-year record with monthly data, t = 36 and T = 12, and hence the annualization requires taking the 1/3 root or the cube root of the cumulative return.

Here is what it looks like for monthly and yearly returns:

Monthly and annualized returns of different companies as a function of time

That's all for today.

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Also, if you have any question or feedback: feel free to leave a comment :)

See you next week!

Kevin from Hook


Further readings:


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