### In this post, I will explain:

##### 1) What is a cumulative return?

##### 2) How do you calculate a cumulative return using geometric linking?

##### 3) What is an annualized return?

##### 4) How do you calculate an annualized return?

### What Is A Cumulative Return For An Investment Portfolio?

##### A cumulative return measures, in percentage terms, how much your investments grew or declined over a specified period of time. In other words, how much wealth you *accumulated or lost due to your investments* over time.

##### For example, let’s say a portfolio starts with $100. Due to the investments held, five years later it is worth $130. The 5 year cumulative return is 30% (calculated as $30 gain / $100 beginning value, since there are no flows).

##### A cumulative return can be calculated for any period of time (e.g. a 5 day cumulative return is the return earned over 5 days, a 5 year cumulative return is the return over 5 years, a since inception cumulative return is the return since the portfolio incepted).

### How Do You Calculate A Cumulative Return?

#### Step 1: Gather the “Periodic Returns”

##### In the above example, I showed how you can calculate a cumulative return by taking the gain/loss over the beginning market value. We took the $30 gained over five years and divided it by the $100 starting value to get a 30% return.

##### However, using cumulative gains/losses can only be used to calculate a cumulative return when there are no flows (money added or withdrawn from the account). In the real world, people add or withdraw money from their accounts and so we have to use what are called “periodic returns” to calculate the “cumulative return.”

##### Periodic returns is a fancy term used to refer to the returns calculated for smaller time periods. For example, to calculate a 5 year cumulative return, you may have 5 annual returns, 60 monthly returns, or 1825 daily returns. The annual, monthly, and daily returns in these examples would be referred to as “periodic returns.”

##### The following is an example where we have 5 annual returns. These will be used to calculate cumulative and annualized returns.

#### Step 2: Geometrically Link the “Periodic Returns” to Calculate the “Cumulative Return”

##### Now that we have our periodic returns, how do we combine them to get the 5 year cumulative return? Can we just *add the 5 annual returns together*?

##### No. You need to use something called “geometric linking.” But don’t worry, it’s not as scary as it sounds.

##### You essentially add a 1 to each return, and then multiply them together, using the following formula:

##### This formula is used because money, including profits and losses, from the current period is reinvested into the next period. If you want to know all the details behind geometric linking and why you need to use it, please read this post on geometric linking.

##### In practice, many investment firms use software to calculate a geometrically linked return, but you can also calculate it in Excel using a function called PRODUCT. The PRODUCT function allows you to highlight the series of returns and it will multiply them together. This is what Performance Analysts typically use to geometrically link.

##### If we apply the product formula to our series of 5 returns we get 30%.

Note: the Excel document is locked but you can save a copy and then edit as you like.

## Geoff

An outstanding, lucid explanation that clarifies the different methods of calculating the returns on an investment or portfolio.

A must read for any investor who needs to understand the monthly reports from his portfolio manager

## Derek`

Hi Geoff,

How can I calculate for example 3 month return using a set of monthly returns say I need to calculate for 3 months and 6 months ?

## Krista

Hi,

Hi Derek,

You would calculate the 3 month cumulative return as (1 + monthly return1) * (1 + monthly return2) * (1 + monthly return3) – 1. Please check out my post below on geometric linking for a more detailed explanation or use the contact form to email me a specific question!

http://www.learninvestmentperformance.com/geometric-linking/

## Vijai Ananth

Dear Krista,

I must say that this is the best website to educate people on investments. It is so helpful and explains clearly the fundamentals.

I believe Annualized Return ( Annual Average Return ) is not the real world realized return. CAGR is the real world realized return for no flow, whereas IRR is the real world return for flow.

Kindly correct me if I’m wrong.

## krista

Yes, you are correct. When there are flows, the IRR is the actual realized return, not the TWR (time-weighted return). When there are no flows, the IRR or TWR will give the same results and both are the actual return.