Question

An investment of $1,000,000 yields $1,100,000 in exactly one year. Calculate the annual rate of return on this investment with:

annual compounding

semiannual compounding

Monthly compounding

Continuous compounding.

Answer 1

Annual Compounding:

FV = Deposit ( 1+ r)^n

where r is int rate per period and n is no. of periods

FV = Deposit ( 1+ r)^n

$1,100,000 = $1,000,000 ( 1 + r )^1

1 + r = $ 1,100,000 / 1,000,000

= 1.1

r = 1.1 -1 = 0.10 i.e 10%

Semi Annual Compounding:

FV = Deposit ( 1+ r)^n

$1,100,000 = $1,000,000 ( 1 + r )^2

(1 + r)^ 2 = $ 1,100,000 / 1,000,000

= 1.1

1 + r = (1.1)^ (1/2)  

= 1.0488

r = 1.0488 - 1

= 0.0488 i.e 4.88% per six months

Annual Rate = 4.88% * 12/6

= 9.76%

Monthly Annual Compounding:

FV = Deposit ( 1+ r)^n

$1,100,000 = $1,000,000 ( 1 + r )^12

(1 + r)^ 12 = $ 1,100,000 / 1,000,000

= 1.1

1 + r = (1.1)^ (1/12)  

= 1.0080

r = 1.0080 - 1

= 0.0080 i.e 0.8% per month

Annual Rate = 0.8% * 12/1

= 9.57%

Contineous Compounding:

FV = Deposit * e^rn

where r is int rate per anum

e^r = 1100000 / 1000000

= 1.1

r = Ln (1.1)

r = 0.0953 i.e 9.53%