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At some point or another mutual fund or stock investors would have encountered the acronym, CAGR – Compounded Annual Growth rate.

I would like to discuss what this stands for, its limitations and how to use it to interpret mutual fund or portfolio performance in a series of posts.

Imagine a fixed deposit that is subject to annual compounding at the rate of 10%. If I invest Rs. 100 in it, after one year I shall get 100 x (1+10%) = 110

After two years, 110 x (1+10%) = 100 x (1+10%)^{2} = 121

After three years, 100 x (1+10%)^{3} = 133.1

So we say the money compounds at the rate of 10%. Hence the compounded annual growth rate is 10%.

However, most fixed deposits do not compound in this simple fashion. They compound each quarter rather than each year.That is , we will need to use 100 x(1+10%/4)^4 = 110.38 for a FD of one year and

100 x(1+10%/4)^8 = 121.84 for a FD of 2 years (8 quarters) and so on,

So after three years, we have 100 x(1+10%/4)^12 = 134.48.

Notice that the corpus is a good Rs. 1 higher since the compounding frequency is higher.

Quarterly compounding is tough to understand and visualise. So I would like to know what is the rate at which the money compounds each year or, what is the CAGR?

I do not care about the frequency of actual compounding. I just need to know the CAGR. That is **I am assuming that the investment compounds annually **(note that this assumption is key to understanding CAGR)

To obtain the CAGR, I write

100 x (1+r)^{3} = 134.48

That is I would like to know what is *rate of return* which for annual compounding would give me a corpus identical to quarterly compounding. Thus *r* is the CAGR.

r = (134.48/100)^{(1/3) }-1 = 10.38%

This extra 0.38% originates in the frequency of compounding – quarterly in this case.

Let us use this example to define the CAGR,

r =(maturity value/investment)^(1/years)-1

It is important to recognise that *r* will be negative if the maturity value is lower than the investment!

**How does compounding work in a stock or mutual fund? **This is the most popular question I have received by email!

The reason for this is simple enough to understand. Most people are fed on ‘power of compounding’ propaganda. They are told that if you start a SIP in an equity fund, you money will compound at the rate of 14%.

A couple of years later, they see their investments in the red and being to wonder what went wrong.

Soon enough they recognise that compounding can be quite confounding!

It sure can, and it sure is, **if** the rate of return fluctuates like in a mutual fund or stock (or bond if … sold!)

So how does compounding work in a mutual fund or stock?

The simple and disappointing answer to this question is that mutual funds and stocks **do not compound wealth!**

The value of stocks and therefore the corresponding net asset value of (equity) mutual funds are influenced by a variety of logical and illogical factors : a scandal in a company, the taking over by a dynamic official or some good old profit booking. Just about anything that can affect the market will affect it sooner or later.

So there is no such thing as, you invest Rs. 100, and after an year I pay you interest. The next year I pay you interest on the investment plus the first year’s interest.

Stocks/mutual funds simply do not work that way.

**If the rate of return is fixed, the concept of a CAGR is useless**. I can take the relevant compounding formula (check out some of them in my fixed deposit calculators) and calculate the maturity value. That is good old compounding.

When something does not compound, it is hard for me to make sense out of it.* *

*Therefore, I choose to determine the net return assuming that mutual fund or stock behaves like instruments that compounds annually!*

The above statement is the key-takeaway from this post. Allow me re-state it:

I try to make sense of a stock/mutual funds performance by *pretending *that is compounds annually!

We pretend that

r =(maturity value/investment)^(1/years)-1

holds true for all instrument whether the interest rate is fixed or not. There is no justification for this choice. We try this out and see how it goes. Since the going is not terrible, we stick with it!

If I speak like that in a physics class, students accuse me of being unromantic! Perhaps so but that is the way math and science works: Assume and check how it fares!

So if I make a lump sum investment in a stock to determine CAGR, all I need to do is to use the above formula.

Since CAGR is the ‘average’ rate at which the investment has compounded (positively or negatively!) per year.

- It is the annual ‘average’ growth rate, it makes sense only for periods above 1 year – that is, since the instrument is volatile, the period should be reasonably long enough so that a long term trend can be inferred.
- The ‘average’ here is a geometric average and not an arithmetic average.

To understand this idea, consider our fixed deposit offering annual offering 10% per annum.

After 3 years the maturity amount is given by

100 x (1+ 10%) x (1+10%) x (1 +10%) = 100 x (1+10%)^{3} = 100 x (1+r)^{3}

Therefore I write,

^{ }(1+r)^{3} =(1+ 10%) x (1+10%) x (1 +10%)

r = [(1+ 10%) x (1+10%) x (1 +10%)]^(1/3) -1

This is known as the geometric average. Of course *r * is the CAGR. So,

r =(maturity value/investment)^(1/years)-1

is still valid. Notice the similarity in structure between the two definitions.

I now define CAGR as

r = [(1+ r_{1}) x (1+r_{2}) x (1 +r_{3}) …. x (1+r_{n})]^(1/years) -1 = (maturity value/investment)^(1/years)-1

Here r_{1}, r_{2,} r_{3},… r_{n} are the rates of annual interest.

For a fixed deposit, this formula is an over-kill since r_{1} = r_{2 }=r_{3}, …, = r_{n (= 10% for the above example). }

However for a stock or a mutual funds the annual interest can be very different! For a detailed analysis of annual stock market returns, see this:**Understanding the Nature of Stock Market Returns**

Therefore, for a lump sum investment made in a stock or a mutual fund 3 years ago, there two ways in which I can calculate CAGR.

1. Simply note the value as on date (say after 3 years).

r =( value today /investment)^(1/3)-1

2. I tabulate the annual returns (365 days from the date of investment).

r1 = return after one year (from date of investment = +10%

r2 = return after two years = -25%

r3 = return after three years = +3%

value today = Investment x (1+ _{10%}) x (1-25%) x (1 +_{3%}) = Investment x (1+ r)^{3}

The CAGR, r = [(1+ _{10%}) x (1-25%) x (1 +_{3%})]^(1/3) -1

To summarise, CAGR is the average annual growth rate of an instrument assuming annual compounding.

The CAGR completely washes out the intermediate volatility. It is just worried about the end points – the investment and final value.

The stock could have increased steadily in this period or could have see-sawed several times in this period. The CAGR completely ignores this.

So CAGR by itself does not tell you the complete picture. You will need to worry about the standard deviation. That is the average deviation from the arithmetic mean ( [r_{1} +r_{2} +r_{3}]/3).

Excuse me for suddenly dropping the arithmetic mean from nowhere! Unfrotunately that is how it works. I will be happy to explain as to why this is necessary, if you leave a comment

The standard deviation of a mutual fund is the single most important parameter that investor should worry about. Read more: **How to select mutual fund categories suitable for your financial goals?**

**What about a SIP?** **How does CAGR work then? **

**To be continued ….**

Feel free to share your views on this post.

If you are familiar with the CAGR, check this out, **Visualizing Mutual Fund Volatility Measures**

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