Question

At what annual rate would the following have to be invested?

$7,054, to grow to $86,066, in 32 years.

Answer 1

Solution:
	Annual rate is
	r=8.13% in two decimal
	r= 8.13092% in 5 decimal
	Use as per requirement of question , but remember when you take 5 or 6 decimal then you will get exact results not in two decimal as the question is decimal sensitive.
Working Notes:
	Annual rate of interest is the rate at which 7054 is compounded to future value in 32 years to 86066 , Hence we have to used concept future value of present value of deposit to get the annual rate.
	C0 = Deposit= Present value = $7054
	FV=Future value of the deposit after 32 years = 86066
	r=Annual rate of interest = r=??
	n = no. Of periods = 32 year
	Using future value formula
	FV= C0 x (1+r)^n
	86066 = 7054 (1+ r)^32
	(86066/7054) = (1+r)^32
	Taking Log on both side
	Log(86066/7054) = Log(1+r)^32
	Log(12.2010207) = Log(1+r)^32
	using relation loga^b = b x Log a
	Log(12.2010207) = 32 x Log(1+r)	Log(12.2010207) = 1.086396164
	(1.086396164/32) = Log (1+r)
	(1+r) = Antilog (1.086396164/32)
	(1+r) = Antilog (0.03394988)	We got from online financial calculator
	(1+r) = 1.081309155	antilog(0.03394988) =1.081309155
	r= 1.081309155 - 1
	r= 0.081309155
	r= 8.1309155%
	r=8.13% in two decimal
	r= 8.13092% in 5 decimal
Lets Check is our computed rate is correct.
	Using future value formula
	FV= C0 x (1+r)^n
	86066 = 7054 (1+ r)^32
	86066 = 7054 (1+ 8.1309155%)^32
	86066 = 86065.99923
	86066 = 86066.00
	Hence above computed rate is correct only you uses 5 decimal or above.
Please feel free to ask if anything about above solution in comment section of the question.