Understanding Returns: Absolute return, CAGR, IRR etc

inShare
We make investments to earn returns. Investment return is the change in
value of an investment over a given period of time. For example

Mr Sharma invested Rs 10,000  in a mutual fund and 2 years of holding

it became Rs 20,000
Mr. Patil  invested Rs 50,000  in Gold for 7 years and it became Rs
4,00,000
Who earned more? How to calculate returns so that we do not end up
comparing apple to oranges! There are many different measures such as,
absolute return, simple annualised return, compounded annual growth,
among others. In this article we shall see different ways to compare
returns.

What is Return or Return on Investment?

Return is the gain or loss in the value of an asset in a particular period.It
is usually quoted as a percentage. The general rule is that the more risk
you take, the greater the potential for higher return – and loss. Return on
investment has two basic components.

 Interest and/or dividends, the income generated by the

underlying investment.
 Appreciation(Depreciation), an increase(decrease) in the value
of the investment.
Why finding return on investment is important:

 Determining return on investment is an important part

of investment review to know whether your investments are on track
and make appropriate adjustments.
 Estimating a return on investment also helps in choosing among
investment options – Should one invest in Gold/Real
Estate/Stocks/Mutual Funds/Fixed Deposits ?
Let’s learn about various kind of returns!
Absolute return or Point to Point Returns : Absolute return is the
increase or decrease  that an investment achieves over a given period of
time expressed in percentage terms. It’s calculated as follows:
Absolute returns = 100* (Selling Price – Cost Price)/ (Cost Price)

For example you invested in  asset in January 2005 at a price of  Rs

12000.  And you sold the investment in January 2012 at the cost of Rs
3200. Absolute returns in this case will be:

Absolute returns        = 100* (32000 – 12000)/12000

                   = 100 * 20000/12000
=  166.67%
This measurement of return is the simplest and it does not consider time
period. Most times it produces a large number so people are impressed!

Simple Annualized Return: The increase in value of an investment,

expressed as a percentage per year. expressed as
Simple Annualized Return= Absolute Returns/Time period. 
Suppose investment of Rs 1,00,000  becomes 1,24,000 over three years.

Absolute return = 100* ( 12400 – 100000/100000 ) =24 %.

Simple annualized return = 24/3  =  8 %
Average Annual Return (AAR)
Average annual return (AAR) is the arithmetic mean of a series of rates
of return. The formula for AAR is:

AAR = (Return in Period 1 + Return in Period 2 + Return in Period 3 +

…Return in Period N) / Number of Periods or N
Let’s look at an example. Assume that an investment  XYZ records the
following annual returns:

Year Annual Return

2005 20%
2006 25%
2007 22%
2008 -10%
AAR for the period from 2005 to 2008: = (20% + 25% + 22% -10%) / 4
= 57%/4 = 14.25%

AAR is somewhat useful for determining trends. However, because

returns compound (they generally not add) AAR is typically not
regarded as a correct form of return measurement and thus it is not a
common formula for analysis. In addition, one or a few particularly high
or low data points (“outliers”) can skew the average and provide
misleading results. ( Ref: InvestingAnswers : Definition of AAR)
Compound Annual Growth Rate or CAGR
CAGR is the year-over-year growth rate of an investment over a
specified period of time. It’s an imaginary number that describes the rate
at which an investment would have grown if it grew at a steady rate.
Let’s assume you invested Rs 10,000 in Apr 2010 and by Apr 2011 your
investment became Rs 30,000, by Apr 2012 it became Rs 15,000.  What
was the return on your investment for the period?

Initial
Amount Profit/Loss
 Profit/Loss(in for the Percentage(in the
 Description the period) period  period)
Return in year 100*( 20,000/10,000
2010-11 20,000  10,000 ) = 200%
100 *(-
Return in year 15,000/30,000) =
2011-12 -15,000  30,000 -50%
Absolute
Returns from 15,000-10,000 100*(5,000/10,000)
Apr 2010-2012 =5000  10,000 = 50%
Simple 5000 in 2 years  10,000  100*(5000/10,000 *
 Annualized
Return
from Apr 2010-
2012 2)= 25%
CAGR is a way to smoothen out the returns, it determines an annual
growth rate on an investment whose value has fluctuated from one
period to the next as shown in picture below. In that sense CAGR isn’t
the actual return in reality. This is similar to saying that you went on a
trip and averaged 60 km/hr. Whole time you did not actually travel 60
km/hr  Sometimes you were traveling slower, other times faster.

CAGR graph
The formula to calculate CAGR is :

CAGR Formula
So CAGR for above example is :

= ((15,000/10,000) ^ (1/2)) -1
= 22.47%
If the investment states that it had an 8% annualized return over ten
years, that means if you invested on Apr 1, and sold your investment on
 Mar 31 exactly ten years later, you earned the equivalent of 8% a year.
However, during those ten years, one year the investment may have
gone up 20% and another year it may have gone down 10%. In the
example if the investment Rs 10,000 would have grown at the rate of
22.47% every year and at end of two years it would be Rs 15,000 as
shown in calculation below.

Year  Initial Value   Growth Final Value

 1  10,000  2,247  12,247
 2  12,247  2752  15,000
 

Good and Bad of CAGR: CAGR is the best formula for evaluating

how different investments have performed over time. Investors can
compare the CAGR in order to evaluate how well one stock/mutual fund
has performed against other stocks in a peer group or against a market
index. The CAGR can also be used to compare the historical returns of
stocks to bonds or a savings account. But the bad points of CAGR are:
 CAGR does not reflect investment risk, and you must use the
same time periods.
 CAGR does not reflect volatility. Investment returns are
volatile, they can vary significantly from one year to another. CAGR
give the illusion that there is a steady growth rate even when the value
of the underlying investment can vary significantly.
For  example ,investment in XYX had the following price trend over
three years:

Year Price
0 5,000
1 22,000
2 5,000
This could be viewed as a great investment if you were smart enough to
buy  at 5,000 and one year later sell it at 22,000 for a 340% gain. But if
 one more year later the price was 5 ,000 and you still have it in  you
would be even. If you bought XYZ in year 1 at 22,000 and still had it in
Year 2, you would have lost 77% of your value (from 22,000 to 5,000).
Ads can tout a fund’s 20% CAGR in bold type, but the time period used
may be from the peak of the last bubble, which has no bearing on the
most recent performance.  More Details about CAGR
at Investopedia: Compound Annual Growth Rate: What You Should
Know
Rolling Returns
Returns that we have looked at so far have used the value of investment
at beginning and at end of the period. But does that tell the full
story. Suppose there two investments, A and B, both purchased for 12
and both sold seven years later for 32 as shown in figure below.

Absolute Returns and CAGR same

 Point-to-point returns : Both have generated an absolute return
of 100* (32-12)/32 =166.67%.
 CAGR :CAGR calculation would also yield identical results,
that is, 10.41%
What’s different between the two funds is the consistency of their
respective performance-A is consistently rising whereas B, after
showing a robust growth for the first four years, declines consistently
after 2009. Clearly, A is a better choice, which you’d never know by
relying on the point-to-point/CAGR return measures.
Ref EconomicTimes:How to calculate returns from a mutual fund
Rolling returns paint quite a different picture in this example. Rather
than a single return spanning the entire time frame, rolling returns divide
the period into equal but smaller periods. The rolling return is the
average of the smaller period returns. For example, consider rolling 12-
month periods over the same five years. To get this, you’d start by
calculating the single point return over the first twelve months. Next,
you’d calculate the single point return for months 2-13. You’d continue
the process by adding the next month and dropping the oldest month
until you had single point results for all 12-month periods in the seven
years. The average of these results is the rolling 12-month
return. HDFCFund:Sensex rolling return calculator can calculate rolling
returns of Sensex from any date between 03 Apr 1979 to 30 Aug 2011.
Using the calculator Rolling returns from 1st Apr 1997 are as shown in
picture below:
15 years Rolling returns Calculation by HDFC MF Fund Calculator
But rolling period statistics have their limitations, too. By their very
nature, they tend to put more weight on returns in the middle of the
measurement period and less on those at either end.  The KleinPost:
Rolling Returns vs Fixed Period Returns has more details.
BenchMark
Mr Sharma earned 12% returns in mutual fund. Is it good or bad? It’s
like saying a child has got 92% in exams? Questions asked is compared
to what? If Mr Sharma’s mutual fund gave 12% when Nifty gave a
return of 15%! 12% is not so good but if Mr Sharma’s mutual fund gave
12% when Nifty gave 7% then 12% is good. When evaluating the
performance of any investment, it’s important to compare it against an
appropriate benchmark.
A benchmark is usually an index of the same or similar class  against
which the performance of a stock, bond, mutual fund can be measured.
For example, bond yields are generally compared to benchmark yields
on Government securities of similar maturity. Stocks, Mutual fund
performance is often compared to Sensex or Nifty . Comparing a
portfolio to an inappropriate benchmark could yield misleading
information. For example, the BSE 200 may be an appropriate
benchmark for a portfolio investing exclusively in large and mid-cap
domestic stocks, but it may be inappropriate for a portfolio investing in
bonds or only large cap fund . Benchmarking lies at the heart of the
controversy between passive and active management trying to answer
the question “What value was added by the manager’s decisions”.
Benchmark helps to answer question Should I invest in HDFC Top 200
Fund or Exchange Traded Fund based on Nifty?
Relative Return
Relative returns enable us to know the true return earned by the fund
over and above the benchmark.It determines how the return of a given
stock or fund compares to that of benchmark. This can be useful in
making investment decisions.For example, if the stock you are holding
achieves a return of 20 per cent, while the benchmark index say Nifty
 managed 15.58 per cent, then the stock has achieved a relative return of
a +4.42 per cent. A stock that falls less than the benchmark in a falling
market is considered to have done well, as it manages to contain losses
for the investor.
Peer Returns
Suppose you invest in State Bank of India(SBI)  and the stock increases
in a year by 7%. Other banking stocks rise  like HDFC Bank by
9% ICICI Bank by 6.5%, Punjab National Bank (PNB) by 5% and
Banking Index by 6%. It means SBI has given better returns as
compared to Banking Index  and among its peers better than PNB but
less than HDFC Bank and ICICI bank. Peer return helps in selection of
investment within a particular sector or sub-group of an asset class such
as banking stock in above example.
Risk Adjusted Returns
Is performance just a matter of high returns, or should you also be
concerned about risk? For example, two investment options have the
same 7% annual return. One is 100% invested in Govt. Bonds, and the
other is 100% invested in mid-capitalization stocks. With the Govt
bonds, there is very little volatility in returns. while with the mid-
capitalization investment, there is extreme volatility. Risk matters
and  the Risk-adjusted return measures how much risk is involved in
producing that return. It is generally expressed as a number or
rating. The concept is very popular while comparing the returns
from equity mutual fund schemes.  There are five principal risk
measures: alpha, beta, r-squared, standard deviation and the Sharpe
ratio. Each risk measure is unique in how it measures risk. Quoting
from Investopedia : 5 Ways To Measure Mutual Fund Risk
 Alpha: It takes the volatility (price risk) of a security or fund
portfolio and compares its risk-adjusted performance to a benchmark
index. The excess return of the investment relative to the return of the
benchmark index is its alpha. A positive alpha of 1.0 means the fund
has outperformed its benchmark index by 1%. Correspondingly, a
similar negative alpha would indicate an underperformance of 1%.
For investors, the more positive an alpha is, the better it is.
 Beta: Beta is calculated using regression analysis, and you can
think of beta as the tendency of a security’s returns to respond to
swings in the market. A beta of 1 indicates that the security’s price
will move with the market. A beta of less than 1 means that the
security will be less volatile than the market. A beta of greater than 1
indicates that the security’s price will be more volatile than the
market.  For example, if a fund portfolio’s beta is 1.2, it’s theoretically
20% more volatile than the market.
 R-Squared: R-Squared is a statistical measure that represents
the percentage of a fund portfolio’s or security’s movements that can
be explained by movements in a benchmark index. R-squared values
range from 0 to 100.Mutual fund investors should avoid actively
managed funds with high R-squared ratios, which are generally
criticized by analysts as being “closet” index funds. According to
Morningstar, a mutual fund with an R-squared value between 85 and
100 has a performance record that is closely correlated to the index. In
these cases, why pay the higher fees for so-called professional
management when you can get the same or better results from an
index fund? 
 Standard Deviation: Standard deviation measures the
dispersion of data from its mean i.e the more that data is spread apart,
the higher the difference is from the norm. In finance, standard
deviation is applied to the annual rate of return of an investment to
measure its volatility (risk). A volatile stock would have a high
standard deviation. With mutual funds, the standard deviation tells us
how much the return on a fund is deviating from the expected returns
based on its historical performance.
 Sharpe Ratio :The Sharpe ratio tells investors whether an
investment’s returns are due to smart investment decisions or the
result of excess risk. It is calculated by subtracting the risk-free rate of
return (Treasury Bond) from the rate of return for an investment and
dividing the result by the investment’s standard deviation of its
return. It was developed by Nobel laureate economist William
Sharpe. This measurement is very useful because although one
portfolio or security can reap higher returns than its peers, it is only a
good investment if those higher returns do not come with too much
additional risk. The greater an investment’s Sharpe ratio, the better its
risk-adjusted performance.
So far we focused on investing only one time – we had beginning value
and end value. But How do you calculate your returns when you every
year/period you invest different amount and at the end you receive your
Money back or you get some dividend ?

Internal rate of Return(IRR) and  Extended Internal Rate of Return

(XIRR)
Suppose your invest 5,000 , 10,000 , 6,000 , 4,000 and 6,500 in 5 yrs and
Get 53,000 at the end of 5 yrs , what is your Return ? Or you invested Rs
10,000 in the stock which gave 10% annual dividends. At end of 5 years
you sell stock for 12,000 Rs. The return that is used in such cases is IRR
or Internal Rate of Return. Its used to calculate the returns given some
amount at a fixed interval . For example after every 3 months or after
every 1 yr . The only thing which matters is that there should be equal
distance between two installments. When you invest you need to -ve as
in example beginning we invested Rs 10,000 which is written as
-10,000. While when we get something out of investment like dividend
we need to show it as positive quantity. IRR for investment in stock
comes out to be 13.08% as shown in picture below.
Year Amount
0 -10,000
1   1,000
2   1,000
3   1,000
4   1,000
5   1,000+12,000
IRR 13.08%
 But what when  the payments are at Irregular interval, in that case we
use Extended Internal Rate of Return or XIRR . Excel Function IRR
and XIRR are used for calculating,   Jagoinvestor : What is IRR and
XIRR and how to Calculate it covers the concept in
details. Onemint:What is IRR and how is it calculated?  explains about
IRR.
Note:The article is for educational and informational use and do not
construe this as professional financial advice. Check out our Disclaimer.
Example of Returns Reported
A very good question raised by our reader Suhas in his comment So
what type of return does MF use when they state returns in their fund
card or popular websites?
Typically a simple point-to-point return is preferred when the holding
period is less than one year and CAGR is ideal for longer holding
periods. For example the returns of Birla SunLife Equity Fund are
reported as follows

Sample Mutual Fund Returns

Related Articles:
 Calculator for Calculating Returns (Absolute, CAGR)
 Time Value of Money
 Finding Info on NFOs of FMP,MFs,FD and Saving Interest
Rates
Investment returns over a particular period of time can be stated as
absolute return Or, they might be stated as an annualised return . For
multiple investments over period of time one needs to use IRR or XIRR.
For Investors should understand how investment returns are calculated
and which return to consider for making investment decisions.
Awareness about calculating the returns from investment is essential to
be a smart investor. Which return do you use for reviewing or choosing
investments?