# What Is a Security’s Rate of Return. Historical and Expected Return

Every investor’s main goal is to earn a good rate of return on his investment.

What do I mean by that?

Well, imagine you’ve bought one share of Apple’s equity at the beginning of the year. Back then, Apple shares were trading at $105. At the end of the year, their price increased to $116. So, apart from any transaction costs arising from the transaction and dividends paid by the company throughout the year, if you sell your one share today, you will receive $116, so you’ve made a profit of $11.

Is that a lot? Should you be happy with the investment you’ve made?

The only way to answer these questions is by comparing the investment in Apple shares with other investments, the amount you would have earned if you had bought other stocks.

## However, different stocks have different prices, right?

Therefore, we need a measure that ensures comparability between investments with different market prices.

And that’s done easily.

All we have to do is calculate a percentage rate of return of the investment. Once we do that, we can compare it to other investments.

The rate of return of the investment is given by the ending price minus the beginning price, and that is divided by the beginning price. So, in our case, it will be given by $116 minus $105 divided by $105. This gives us a 10.5% rate of return.

This computation of the rate of return is called simple rate of return.

It’s clear how we calculate a simple rate of return of an investment, right?

The amount we’ve made, minus the amount we’ve invested, divided by the amount we’ve invested.

If we assume Apple paid a $2 dividend at the end of the year, the rate of return calculation becomes $116 plus $2 minus $105 divided by $105. The simple rate of return becomes 12.4%.

We can calculate the logarithmic return of the investment by calculating the logarithm of $116 divided by $105, which is equal to log of $116 minus log of $105. The result we obtain by using the log formula is 10.0%.

Simple and log returns provide a slightly different result, so it is important to be consistent with the way we calculate returns. If we’ve chosen to calculate simple returns, we have to do it for all further financial calculations. And if we’ve decided to calculate log returns, we should use only log returns.

## Which one should you choose?

Well, there isn’t a general rule; however, most econometricians believe simple returns are preferable when you have to deal with multiple assets over the same timeframe, and log returns are preferable when you make calculations about a single asset over time.

There’s an important observation we have to make here. We should always remember the timeframe for which we’ve calculated a rate of return.

In the example we saw here, we calculated a yearly rate of return, because the Apple stock was held from the 1^{st} of January to the 31^{st} of December – one whole year.

We can use the formula to calculate rates of return for periods different than a year. But we should be careful as investments with different holding periods shouldn’t be compared. We should always remember the timeframe of the rates of return we are working with.

Typically, investors use daily, monthly, quarterly, or yearly returns. The most popular expression is annually.

In addition, we can easily convert daily, monthly, and quarterly returns yearly. We only have to use the following formula:

Annual~return = [{(daily~return+1)}^{365}]\times100-1

By applying it, we can easily adjust for daily, monthly, and quarterly returns.

Now that we know how to calculate an investment’s rate of return and how to convert rates of return with a different holding period in annual rates of return, we are almost ready.

We only have to explain the connection between historical and expected rates of return.

Everybody in Finance is interested in future rates of return. Given that, until now, scientists and researchers have not found a way to predict the future, we need to figure out a way to form an idea about future rates of return. We can’t predict them, but we can form a reasonable proxy and call that proxy expected rate of return.

The best idea we can have about a financial security’s expected rate of return is its past behavior. We’ve observed the security’s historical performance, and that information can help us build expectations about the future.