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What is Geometric Average return and Arithmetic return? How to calculate it?

 Geometric average return: It is an average rate of return that includes the compounding effects in multiple periods.

Formula: GAR = (((1+R1)* (1+R2)*(1+R3)*(1+R4)....(1+Rn))^1/n) -1

Arithmetic return: It is an average of set of data of different periods. It is calculated by adding the return of some period and then dividing the number with the number of data.

Formula: AR = (R1+R2+R3+R4....Rn)/n

AR =Arithmetic return

R1+R2+R3+R4....Rn = Rate of return

N = Number of observations

Example: Find out the Average return of 4 years rate of return in a given data:

Year

Rate of return

1

3.5%

2

-2.9%

3

6%

4

8.5%

 

Solution: The average return of 4 years rate of return:

AR = (R1+R2+R3+R4)/n

= (3.5+ (-2.9)+6+8.5)/4

= 3.775%

AR = Arithmetic return

N =Number of periods

Example: There are two stock are given Stock A and Stock B with their rate of return for 6 years. Find out the arithmetic return of stock A and stock B.

Year

Stock A

Stock B

1

-6.8%

4.9%

2

5.4%

8.4%

3

3.8%

6.4%

4

-2.5%

7.2%

5

3.6%

-5.2%

6

4.8%

-3.5%

 Solution: The arithmetic return of stock A:

AR = (R1+R2+R3+R4+R5+R6)/n

= ((-6.8) + 5.4+3.8 + (-2.5)+3.6+4.8)/6

= 8.30/6

= 1.38%

The arithmetic return of stock A:

AR = (R1+R2+R3+R4+R5+R6)/n

= (4.9 + 8.4+6.4 +7.2+(-5.2)+ (-3.5))/6

= 18.20/6

= 3.03%

The arithmetic return of stock A and B is 1.38% and 3.03% respectively.

Example: The rate of return of stocks of clothing company PQR and company LMN is given below. Find out the geometric average return of both the companies stocks.

Year

PQR company stock

LMN company stock

1

0.07

0.095

2

0.05

0.015

3

0.03

-0.032

4

0.025

0.06

5

-0.08

0.045

 

Solution: The geometric average return of PQR company stock is:

GAR = (((1+R1)* (1+R2)*(1+R3)*(1+R4)*(1+R5))^1/5) -1

= (((1+0.07)* (1+0.05)*(1+0.03)*(1+0.025)*(1+ (-0.08)))^1/5) -1

= ((1.07*1.05*1.03*1.025*0.92) ^0.2)-1

= (1.091244^0.2) -1

=1.017617-1

= 0.017617

=1.76%

The geometric average return of LMN company stock is:

GAR = (((1+R1)* (1+R2)*(1+R3)*(1+R4)*(1+R5))^1/5) -1

= (((1+0.095)* (1+0.015)*(1+ (-0.032))*(1+0.06)*(1+0.045))^1/5) -1

= ((1.095*1.015*0.968*1.06*1.045) ^0.2)-1

= 1.191729 -1

= 0.191729

= 19.17%

Where,

GAR =Geometric Average return

Example: Find out the Arithmetic return and geometric return of Stock P and Stock Q. The rate of return is given in a table.

Year

Stock P

Stock Q

1

0.075

0.025

2

0.052

-0.015

3

-0.013

-0.032

4

0.015

0.055

5

-0.08

0.065

 

Solution: The arithmetic return of stock P:

AR = (R1+R2+R3+R4+R5)/n

= ((0.075+0.052+ (-0.013) +0.015+ (-0.08))/5

= 0.049/5

=0.0098

=0.98%

The arithmetic return of stock Q:

AR = (R1+R2+R3+R4+R5)/n

= (0.025 + (-0.015) + (-0.032) + 0.055 + 0.065) /5

= 0.098/5

=0.0196

=1.96%

The geometric return of stock P:

GAR = (((1+R1)* (1+R2)*(1+R3)*(1+R4)*(1+R5)) ^1/5) -1

= ((1+ 0.075)*(1+0.052)*(1+ (-0.013))*(1+ 0.015)*(1+ (-0.08)) ^0.2) -1

= (1.075 * 1.052 * 0.987*1.015*0.92) ^0.2) -1

= (1.042306 ^0.2) -1

=1.008322-1

=0.008322

=0.83%

The geometric return of stock Q:

GAR = (((1+R1)* (1+R2)*(1+R3)*(1+R4)*(1+R5)) ^1/5) -1

= ((1+ 0.025)*(1+ (-0.015))*(1+ (-0.032))*(1+ 0.055)*(1+ 0.065) ^0.2) -1

= (1.025 * 0.985*0.968*1.055*1.065) ^0.2) -1

= (1.098089 ^0.2) -1

=1.01889-1

=0.01889

=1.89%

 

 

 

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