251) How Total, Annual & Annualized Returns are Related

251) How Total, Annual & Annualized Returns are Related

How many of you are confused between the different Returns terms used?

a) Total Return
b) Annual Return
c) Average Total Returns
d) Annualized Return

Have you wondered how the Returns are related?

The graph above shows the relationships between the different returns terms.

The Black fluctuating line is the investment performance.
The 3 Green straight joining lines are the 3 Annual Returns.
The Red line is the Total Return.
The Blue sloping upward exponential curve is the Annualized Return. It is sloping because the effect of compounding. The value increases faster as over the years.

Let us look at the relationships below and understand the differences.

FV = PV(1 + i)^n
FV = PV((1+i1)(1+i2)(1+i3)…(1+in))

(1+TR) = (1 + i)^n
FV = PV(1 + TR)
TR = (FV – PV)/PV
where
FV = Future Value (End Value)
PV = Present Value (Begin Value)
i = Annualized Return
in = Annual Return for year n
n = No of years
TR = Total Return for n years

A) Total Returns (TR)

From the basic formula of investment:

Total Return = (End Value - Begin Value)/Begin Value
TR = (FV - PV)/PV

For Total Return, the period can consist of 1 year or many years. It calculates from the beginning to end of the investment.

If the period is only 1 year, then it is the same as Annual Return.

B) Annual Return (in)

Annual Return basically means the investment return for only 1 year period. Each year will have its own Annual Return.

Example: 

i1 = Annual Returns for 1 Jan 1st year to 31 Dec 1st year.

i2 = Annual Returns for 1 Jan 2nd year to 31 Dec 2nd year. 

i2019 = Annual Returns for 2019 means the investment return from 1 Jan to 31 Dec 2019. 

i2020 = Annual Returns for 2020 means the investment return from 1 Jan to 31 Dec 2020.

It is the same as Total Return for 1 year period.

TR1 = i1 = (FV1 - PV1)/PV1

Annual Return for 2019  = (Value in 31 Dec 2019 - Value in 1 Jan 2019)/Value in 1 Jan 2019

i2019 = (FV in 31 Dec 2019 - PV in 1 Jan 2019)/PV in 1 Jan 2019

The Total Returns for many years are the compounded value of many Annual Returns. 

Total Return for n years:

1+TR = (1+i1)(1+i2)(1+i3)...(1+in)

C) Average Total Returns (ATR)

Average Total Return is simply the Total Return divided by the number of years of the investment.

ATR = TR/n

D) Annualized Returns (i)

Annualized Returns is the Total Return that is re-scaled to a period of 1 year. 

This means that if you keep compounding the same Annual Return value, you will get the Total Return.

(1+i)^n = (1+i1)(1+i2)(1+i3)…(1+in)  = (1+TR) 

Let us apply an example to the formulas.

You invested $1,000 on 1 Jan 2017.

In 2017, the Annual Return = 5%

In 2018, the Annual Return = 8%

In 2019, the Annual Return = -4%

a) What is the Total Return rate, TR?

b) What is the Average Total Return, ATR?

c) What is the Annualized Return rate, i?

d) What is the Investment Value after 3 years on 31 Dec 2019, FV?

PV = $1,000
i1 = 5%
i2 = 8%
i3 = -4%
n = 3 years

a) What is the Total Return rate, TR?

Total Return for 3 years:

1+TR = (1+i1)(1+i2)(1+i3)

1+TR = (1+5%)(1+8%)(1-4%)

1+TR = (1.05)(1+08)(0.96)

1+TR = 1.08864

TR     = 0.08864 = 8.864%

b) What is the Average Total Return, ATR?

ATR = TR/n

ATR = 8.864%/3

ATR = 2.955%

c) What is the Annualized Return rate, i?

(1+i)^3 = (1+i1)(1+i2)(1+i3)

(1+i)^3 = (1+5%)(1+8%)(1-4%)

(1+i)^3 = (1.05)(1+08)(0.96)

(1+i)^3 = 1.08864

(1+i)     = Root 3 of 1.08864

(1+i)     = 1.028714

i            = 0.028714

i            = 2.8714%

Note: Annualized Return is not the same Average Total Return. 

There are many who confuse between ATR and AR.

d) What is the Investment Value after 3 years on 31 Dec 2019, FV?

At the end of the investment period, you want to know what is the $ value you have.

You can get the Future Value, FV from any one of the formulas below:

FV = PV(1 + i)^n
FV = PV((1+i1)(1+i2)(1+i3)…(1+in))

FV = PV(1 + TR)

Formula 1:

FV = PV(1+i)^n

FV = 1000(1+2.8714%)^3

FV = 1000(1.028714)^3

FV = 1000(1.08864)

FV = 1088.64

Formula 2:

FV = PV(1+i1)(1+i2)(1+i3)

FV = 1000(1+5%)(1+8%)(1-4%)

FV = 1000(1.05)(1+08)(0.96)

FV = 1000(1.08864)

FV = 1088.64

Formula 3:

FV = PV(1 + TR)

FV = 1000(1+8.864%)

FV = 1000(1.08864)

FV = 1088.64

Hope the formula and calculation example above helps you to understand the relationships between the different returns terms used in investment.

For more explanation, go to these links:

179) Unit Price and Number of Units Growth Rates Effect on Investment Values.

https://highlevelrules.blogspot.com/2018/06/unit-price-and-number-of-units-effect.html

209) Expanded Formula for Investment

https://highlevelrules.blogspot.com/2018/09/expanded-formula-for-invesment.html