Question

Find the real interest rate (when the compound interest is quarterly) when a capital of EUR 29,000 gives a final value of EUR 33,000 in 6 years. WITHOUT Excel please.

Answer 1

Solution:
	Real interest rate is
	Real rate of interest = 2.15934%	in 5 decimal
	Real rate of interest = 2.16%	in 2 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:
	Real rate of interest is the rate at which EUR 29000 is compounded quarterly to future value in 6 years to EUR 33,000 , Hence we have to used concept future value of present value of deposit to get the annual rate.
	C0 = Capital= Present value = $29000
	FV=Future value of the deposit after 6 years =$33000
	r= rate per period=Quarterly interest rate = r=(Real rate of interest /4)=??
	n = no. Of periods = No. of quarter in a year x no. of years = 4 x 6 =24
	Using future value formula
	FV= C0 x (1+r)^n
	33000 = 29000 (1+ r)^24
	(33000/29000) = (1+r)^24
	Taking Log on both side
	Log(33000/29000) = Log(1+r)^24
	Log(1.13793103448) = Log(1+r)^24
	using relation loga^b = b x Log a
	Log(1.13793103448) = 24 x Log(1+r)	Log(1.13793103448) = 0.056115942
	(0.056115942) =24 x Log (1+r)
	(0.056115942/24) = Log (1+r)
	(0.002338164249) = Log (1+r)
	now we move log from right side to left side then it becomes antilog
	antilog(0.002338164249) = (1+r)
	(1+r) = antilog (0.002338164249)
	(1+r) = antilog (0.002338164249)	We got from online financial calculator
	(1+r) = 1.0053983409590392	antilog(0.002338164249) =1.0053983409590392
	r= 1.0053983409590392- 1
	r= 0.0053983409590392
	r= 0.53983409590%
	r= Real rate of interest/4 =0.53983409590%
	Real rate of interest = 0.53983409590% x 4
	Real rate of interest = 2.1593363836%
	Real rate of interest = 2.15934%	in 5 decimal
	Real rate of interest = 2.16%	in 2 decimal
Lets Check is our computed rate is correct.
	Using future value formula
	FV= C0 x (1+r)^n
	33000 = 29000 x (1+ r)^24
	33000 = 29000 x (1+(2.15934%/4))^24
	33000 = 33000.00712
	33000 =33000
	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.