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Many of us have trouble understanding how the growth of volatile instruments like equity, bonds gold or even real estate can be quantified. We come across many terms like CAGR, IRR and XIRR in this context.

In this post, the similarity and differences between the CAGR (compounded annualised growth rate) and the IRR (internal rate of return) are examined with the help of an illustration.

First, it is important to understand why compound interest formulae are used for volatile instruments where there is no actual compounding.

We take a risk by investing in a volatile security. So we must be compensated for it. Ideally more than a fixed income product where there is no risk. This is known as a risk** premium**.

The only way I can calculate risk premium is by using the same compounding formula used for the fixed income product to the volatile security. This allows a comparison of returns from the two instruments.

This is why we talk about CAGR and XIRR for stocks.

Let us consider:

- an imaginary instrument which offers a
**constant**return of 20% and - a highly volatile instrument

## I Lump sum investment

**Case 1: Constant returns**

First, let us consider a one-time investment of 12,000 and let it grow for 15 years.

In this case, the above sheet is an over-kill as all I need to do to find the maturity amount is to calculate

12000 x (1+20%)^15 = 1,84,884 (rounded off)

The CAGR in this case obviously is 20% as the ‘average’ rate at which the money compounds = 20% as there is no variation in yearly returns.

The average that CAGR refers to, is *not the arithmetic average* that we usually use but the *geometric average. *

This is given by,

(1+20%) x (1+20%)x (1+20%) ….. x(1+20%) = (1+CAGR)^15

On the left, (1+20%) is multiplied 15 times as that is the investment duration.

So,

1+ CAGR =**[** (1+20%) x (1+20%)x (1+20%)x(1+20%) x (1+20%)x (1+20%)x(1+20%) x (1+20%)x (1+20%)x (1+20%) x (1+20%)x (1+20%)(1+20%) x (1+20%)x (1+20%)**]**** **

From which the CAGR can be calculated. Of course, when all the annual returns are the same the above expression is trivial to solve:

I expanded it explicitly for us to recognize that when the annual returns fluctuate, we need to plug in each year’s return in the above expression.

Notice the last column in the above table. Those are the inputs for IRR calculation in Excel. Investments are entered as a negative number (-12,000 in this case) and the final maturity value is positive. It is important to enter a zero in the intervening period.

To calculate IRR, all we need to do is to type

=IRR(F2:F17)

in any cell we want. The answer is 20%.

Notice that

**CAGR = IRR (lump sum investment and constant returns)**

**Case 2: Fluctuating returns**

Next we will consider a one-time investment in an instrument with highly volatile returns for 15 years.

Notice that the returns fluctuate wildly. However, the maturity value is identical to the above case in which annual returns were fixed at 20%.

This fascinating occurrence is an illustration of volatile growth.

If we set out to calculate the CAGR, we have

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1+ CAGR =**[** (1+20%) x (1-20%)x (1+123%)x(1-10%) x (1+5%)x (1+10%)x(1-15%) x (1+15%)x (1+125%)x (1+30%) x (1+40%)x (1-30%)(1+80%) x (1+25%)x (1+10%)**] **(returns are rounded off)

Which gives CAGR =20%

Since the input entries in the IRR column are the same as above, so is the answer, 20%.

So we conclude,

**CAGR = IRR (lump sum investment and variable returns)**

## II Periodic investments

Now we move on to when periodic investments are made.

**Case 1: Constant returns**

We will assume that each year Rs. 12,000 is invested. Typically the same math used if Rs. 1000 is invested per month.

An example of this type of investment is the recurring deposit.

Again, the annual return is a constant 20%.

In this case, the formula to calculate the maturity value is a bit involved

12000 x (1+20%) x[(1+20%)^15-1]/20% = 10,37,306 (rounded off)

Notice the entries in the IRR entry column.

**CAGR = IRR (periodic investment and constant returns)**

**Case 2: Fluctuating returns**

Now for periodic investments in a volatile instrument. This corresponds to SIPs in equity and debt mutual funds or gold ETFs.

Notice that the maturity amount is the same as when returns were constant. However, **the IRR is **20%** but the CAGR is lower.**

If we adjust the returns so that the CAGR is 20%, we get

Notice now that the corpus is much higher (by about 18%) and so is the IRR.

The point is, the notion of a CAGR cannot be used when returns fluctuate **and **when periodic investments are made.

What is important is to recognise that IRR represents the annualized compounded growth rate (CAGR) if

- a lump sum investment is made in an instrument with constant returns (fixed deposit, bonds)
- a lump sum investment is made in an instrument with variable returns (stocks, mutual funds)
- periodic investments are made in an instrument with constant returns (recurring deposit)

IRR does not represent the CAGR if

- periodic investments are made in an instrument with variable returns (mutual fund SIPs)

In such a case, IRR is a measure of growth but it cannot be equated to CAGR! (many, including me, make this mistake).

IRR cannot be used when investments are not periodic. A monthly SIP is not exactly periodic. For obvious reasons, they are not exactly spaced 30 days apart. So one will have to take into account the date of investments and receipts along with the amounts involved. This is done using the Excel function, XIRR.

It is important to recognise that both IRR and XIRR are approximation techniques and can be troublesome to use when someone loses a lot of money! See an example **here**

See here to understand how **XIRR for a mutual fund SIP is calculated.**

Many goal planners (including mine) while calculating corpuses use CAGR and not IRR. As shown above this is incorrect and will end up overestimating the required corpus and hence the monthly investment.

I will redo the goal planners to compute the corpus via IRR. Hopefully this will reduce the stress associated with using a retirement calculator 🙂

Here is the Excel used to generate above scenarios: **Compounding with volatile returns**

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M. Pattabiraman(PhD) is the founder, managing editor and primary author of freefincal. He is an associate professor at the Indian Institute of Technology, Madras. since Aug 2006. Connect with him via Twitter or Linkedin
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Thanks for this clarification. Pls send the excel file.

Thanks for this clarification. Pls send the excel file.

Thank you. I have provided the link at the end of the post.

Great explanation in simple terms of a fairly complex topic. For SIP estimation amount IRR is the relevant measure. For regular investments in PPF etc IRR and CAGR will be equivalent if you assume the interest rate to be uniform. However, even here IRR is suitable as the interest rates are unlikely to be the same over a 15 year period.

Great explanation in simple terms of a fairly complex topic. For SIP estimation amount IRR is the relevant measure. For regular investments in PPF etc IRR and CAGR will be equivalent if you assume the interest rate to be uniform. However, even here IRR is suitable as the interest rates are unlikely to be the same over a 15 year period.

This is what I've wanted to know for ages! This is why, if a lump sum is in hand, it is better to invest it at once (provided the duration is very long) rather than staggering it over the entire (v.long) investment period. Unfortunately, the way SIPs are elevated to be the BEST, not many people realize this (including me). It still remains the best for a salaried individual though..

This is what I've wanted to know for ages! This is why, if a lump sum is in hand, it is better to invest it at once (provided the duration is very long) rather than staggering it over the entire (v.long) investment period. Unfortunately, the way SIPs are elevated to be the BEST, not many people realize this (including me). It still remains the best for a salaried individual though..

This is what I've wanted to know for ages! This is why, if a lump sum is in hand, it is better to invest it at once (provided the duration is very long) rather than staggering it over the entire (v.long) investment period. Unfortunately, the way SIPs are elevated to be the BEST, not many people realize this (including me). It still remains the best for a salaried individual though..

Thanks for clarification with an excel example.

My question is what if the investments are blends of many instruments,

1. MF regular/irregular SIPS.

2. MF regular/irregular SWP

3. MF regular /irergular STP

4. Lumpsum invetments

5. Equity

May be plug fitting into above excel sheet might be used for above scenario? Please share excel sheet.

XIRR can used in general for all scenarios.

I have now provided the link at the end of the post.

I really do not know how generous your personality is ! Inspite of your extremely tight schedule, not forgetting into to response to every comment/post in sweetest manner, I pray and wish such quality must own/inbibe into me.

Thanks a lot.

Thank you. Responding to people who make the effort to comment is the very least that I could do.

Thanks for clarification with an excel example.

My question is what if the investments are blends of many instruments,

1. MF regular/irregular SIPS.

2. MF regular/irregular SWP

3. MF regular /irergular STP

4. Lumpsum invetments

5. Equity

May be plug fitting into above excel sheet might be used for above scenario? Please share excel sheet.

One point worth noting is that when using IRR to compare different instruments, overall period must be same/similar. Otherwise, unaccounted re-investment risk may provide misleading picture. Consider for example IRR for following 2 instruments:

1. Lump Sum 1000 (30% tax benefit, net investment 700) at beginning and 8% annually compounded for next 5 years

2. Lump Sum 1000 (30% tax benefit, net investment 700) at beginning and 8% annually compounded for next 10 years

Naturally, 2nd instrument continues to give return after 6th year onwards just with first investment, yet IRR of first will be higher.

Yes, quite true.

Pattu Sir,

I liked the points on IRR vs CAGR mentioned in the last few paragraphs of the post.

Sometimes it is better to ‘overestimate the required corpus.’ What do you say sir?

True Sreekanth, Agree!

Thanks for simplified CAGR vs IRR,Easy to understand and use.Yes I need the Excel calculator kindly share

As usual, again a stellar explanation of something that is perceived to be complex! Thank you Pattu, for all your efforts!

Sir, thanks for covering these fundamental concepts. I guess for all practical purposes, XIRR is more useful since the exact dates can be mentioned. I went through the video that was linked in your other article (about the drawbacks of XIRR). Is it safe to use XIRR as long as only investments are made for a longer duration? Say, if I’m investing in mutual funds for the long term (10+ years) and rarely make withdrawals, I’m assuming its safer to use XIRR in this scenario.

Yes only XIRR can be used for all situations. However if there is a huge loss, XIRR cannot be computed.

Hi Pattu,

Pardon my poor analytical skills.But i have a query. As I understand the only way to increase our annual SIP (under DIrect schemes from AMC ONLINE) is to cancel the existing SIP and start a new SIP of increased amount for the remaining period under the same folio. This seems to be the case with at-least ICICI and HDFC,i believe. So, my question is, whether,doing such an exercise every year affect the compounding towards our final goal in anyway??

Increasing the investment amount is a great way to create wealth. Either you will need to cancel current sip and start a new one with the bigger amount or start a second SIP. Best is to manually invest the additional amt instead of opting for SIP route.

Thank you very much. 🙂

Thank you. Yes that is true. IRR or XIRR can be used universally

Thank you. I have provided the link at the end of the post.

Yes I agree it is better to invest a lump sum asap at best within a few weeks.

Dear Pattu,

I enjoyed reading about the difference between CAGR, and IRR / XIRR. I always got confused when to use which in the calculations, as well as how one is different from the other. This post definitely clarifies that.

On a separate note, I heard in one of the classes, that “In finance all answers are wrong”. The reason being, that in the end whatever estimations are made will not be correct. You will either over estimate or under estimate. What matters is how much is the error margin for our estimations. With CAGR, Or IRR/XIRR I believe that the margin of error will not be significant over a long period of time. Hence either of the two measures are fine to use. Would that be a right way at looking at the nerdy excel stuff ?

Thank you. I try not to think too much about it and i do not try to predict what cannot be predicted.

Thanks Pattu, this post arrived just the right time.

Btw, would you know what VR guys are using?

Dear Pattu sir,

Thank you for clearing the doubt i was searching for. So what is a use of CAGR then, where it should be used. should we ever worry about CAGR? does CAGR talks anything about volatility?

If you noticed, the CAGR is not dependent on the principal invested (i.e. our SIP monthly or annual installments), it depends only on each year’s rate of interest and total no. of years. Hence it is used as a marketing tool for the Mutual Fund AMCs and also for websites such as Money control and ValueResearch which compares the 5-yr or 10-yr CAGR of several Mutual Funds for our simple understanding. However the IRR/XIRR result will vary from individual to individual based on how much he/she invested in the fund each year, whether it was lump sum or SIP, etc. Hence, in short, while CAGR is relevant for the MF company, the IRR/XIRR is relevant for the individual.

thanks prashanth

thank you Pattu sir.. Since MBA days IRR was haunting :).

thank you.

Thanks a lot

Sir,

Unable to access graphs/chart. This is true for most of the articles.

Please give an access.

Thank you.

Fixed it now. Thank you. Let me know what articles you want fixed first and I will get to it.

Dear Pattu sir,

This article was published again on 16th instant. I read it first time regarding IRR and CAGR..

You observed ” IRR does not represent the CAGR if periodic investments are made in an instrument with variable returns”.

So the concept ” sequence of returns” place bigger part .

Yes of course.

Dear Pattu, Thanks a lot for this article. I have two scenarios – 1) Calculate growth of net worth (across multiple asset classes) and 2) Calculate growth of a single asset/mutual fund.

For (2), XIRR is best as per your article. But for overall networth growth, is it still XIRR? I’ve been using CAGR (just using starting and ending values for each asset class) as entering the in/out values for XIRR becomes tedious when calculating it for the entire portfolio across real estate, loans, savings accounts, fixed deposits, PPF, EPF, NPS, mutual funds, bonds, gold and direct equity.

What is your view? I am not really concerned with the absolute value, just want to make sure that the networth has a relatively consistent growth year-on-year.

If there is more than one investment, it is always XIRR.

Hi,

Just wanted to understand what is the way one should look at a 1 month return for the following exmaple. I invest Rs 100 on 1st July 2016 and it becomes 102 on 31st July 2016. A simple return is 2%. An XIRR gives me 27% and an annualised return gives me 24.3%. How should one look at this

For one month or less than 1Y periods, annualized returns do not mean anything.

Excellent explanation and coverage on the topic of CAGR Vs IRR. I am looking for CAGR caclulator with different annual rates during holding period, unable to find on freecal. any pointers will be helpful….thanks

Really excellent explanation. Every one should understand CAGR, IRR AND XIRR. Sin Q / Cos Q.

Thank you