Question

Find the present values of these ordinary annuities. Discounting occurs once a year. Round your answers to the nearest cent. $500 per year for 10 years at 12%. $ $250 per year for 5 years at 6%. $ $400 per year for 16 years at 0%. $ Rework previous parts assuming that they are annuities due. Round your answers to the nearest cent. $500 per year for 10 years at 12%. $ $250 per year for 5 years at 6%. $ $400 per year for 16 years at 0%. $

Answer 1

Solution:
a.	present values of these ordinary annuities
	$500 per year for 10 years at 12%.	$2,825.11
	$250 per year for 5 years at 6%	$1,053.09
	$400 per year for 16 years at 0%	$6,400.00
Working Notes:
	Present value of annuity = P x (1-(1/(1+i)^n))/i
	P= annual cash flow
	i=interest rate
	n= no. Of years
	$500 per year for 10 years at 12%.
	Present value of annuity = P x (1-(1/(1+i)^n))/i
	P= annual cash flow = $500
	i=interest rate =12%
	n= no. Of years = 10 years
	Present value of annuity = P x (1-(1/(1+i)^n))/i
	= 500 x (1-(1/(1+.12)^10))/.12
	=2825.111514
	=$2,825.11
	$250 per year for 5 years at 6%
	Present value of annuity = P x (1-(1/(1+i)^n))/i
	P= annual cash flow = $250
	i=interest rate =6%
	n= no. Of years = 5 years
	Present value of annuity = P x (1-(1/(1+i)^n))/i
	= 250 x (1-(1/(1+.06)^5))/.06
	=1,053.090946
	=$1,053.09
	$400 per year for 16 years at 0%
	P= annual cash flow = $400
	i=interest rate =0%
	n= no. Of years = 16 years
	Present value of annuity = P x n
	= 400 x 16
	=6,400.00
	these can also be calculated using excel formula
	=PV(rate,nper,pmt,(FV),(type))
For	$500 per year for 10 years at 12%.
	rate=12%
	nper=10
	pmt=-$500
	fv=0
	type= 0   for ordinary annuity
	=PV(rate,nper,pmt,(FV),(type))
	=PV(12%,10,-500,0,0)
	2825.111514
For	$250 per year for 5 years at 6%
	rate=6%
	nper=5
	pmt=-$250
	fv=0
	type= 0   for ordinary annuity
	=PV(rate,nper,pmt,(FV),(type))
	=PV(6%,5,-250,0,0)
	1053.090946
For	$400 per year for 16 years at 0%
	rate=0%
	nper=16
	pmt=-$400
	fv=0
	type= 0   for ordinary annuity
	=PV(rate,nper,pmt,(FV),(type))
	=PV(0%,16,-400,0,0)
	6400
b.	present values of these ordinary annuities due
	$500 per year for 10 years at 12%.	$3,164.12
	$250 per year for 5 years at 6%	$1,116.28
	$400 per year for 16 years at 0%	$6,400.00
Working Notes:
	Present value of annuity due = P x (1+i) (1-(1/(1+i)^n))/i
	P= annual cash flow
	i=interest rate
	n= no. Of years
	$500 per year for 10 years at 12%.
	Present value of annuity due = P x (1+i) (1-(1/(1+i)^n))/i
	P= annual cash flow = $500
	i=interest rate =12%
	n= no. Of years = 10 years
	Present value of annuity due = P x (1+i) (1-(1/(1+i)^n))/i
	= 500 x (1+.12) (1-(1/(1+.12)^10))/.12
	=3,164.124896
	=$3,164.12
	$250 per year for 5 years at 6%
	Present value of annuity due = P x (1+i) (1-(1/(1+i)^n))/i
	P= annual cash flow = $250
	i=interest rate =6%
	n= no. Of years = 5 years
	Present value of annuity due = P x (1+i) (1-(1/(1+i)^n))/i
	= 250 x (1+0.06) (1-(1/(1+.06)^5))/.06
	=1,116.2764
	=$1,116.28
	$400 per year for 16 years at 0%
	P= annual cash flow = $400
	i=interest rate =0%
	n= no. Of years = 16 years
	Present value of annuity = P x16
	= 400 x 16
	=6,400.00
	these can also be calculated using excel formula
	=PV(rate,nper,pmt,(FV),(type))
For	$500 per year for 10 years at 12%.
	rate=12%
	nper=10
	pmt=-$500
	fv=0
	type= 1   for annuity DUE
	=PV(rate,nper,pmt,(FV),(type))
	=PV(12%,10,-500,0,1)
	3164.124896
	$250 per year for 5 years at 6%
	rate=6%
	nper=5
	pmt=-$250
	fv=0
	type= 1   for annuity DUE
	=PV(rate,nper,pmt,(FV),(type))
	=PV(6%,5,-250,0,1)
	1116.276403
	$400 per year for 16 years at 0%
	rate=0%
	nper=16
	pmt=-$400
	fv=0
	type= 1   for annuity DUE
	=PV(rate,nper,pmt,(FV),(type))
	=PV(0%,16,-400,0,1)
	6400
Notes:	When there is 0% rate of interest present value will be just total amount deposited .
Please feel free to ask if anything about above solution in comment section of the question.