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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

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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