Question

10 How many uniform annual receipts would be required from now that an initial investment of $1,000,000 would be recovered? Note that the annual receipts are $131,000 and starts three years from now at an interest rate of 12% per year.

               

Answer 1

Annual receipts	$                     131,000.00
Interest rate	12%
Initial Investment	$                 1,000,000.00
Annual payment stream start date	3 years from now
Value of initial investment after 3 years	1000000*(1+12%)^3
Value of initial investment after 3 years	$                 1,404,928.00
PV of future cash flows should be equal to 1,404,928
PV of annuity for making payment
P = PMT x (((1-(1 + r) ^- n)) / r)
Where:
P = the present value of an annuity stream
PMT = the dollar amount of each annuity payment
r = the effective interest rate (also known as the discount rate)
n = the number of periods in which payments will be made
$                                                                                         1,404,928.00	= 131000* (((1-(1 + 12%) ^- n)) / 12%)
1404928/131000	=(((1-(1 + 12%) ^- n)) / 12%)
10.72	=(((1-(1 + 12%) ^- n)) / 12%)
10.7246*12%	=(1-(1 + 12%) ^- n
1.28695	=(1-(1 + 12%) ^- n
1.286952-1	=-(1 + 12%) ^- n
0.28695	=-(1 + 12%) ^- n
Using log both sides
Log 0.28695	n log (1.12)
Using anti log tables
-0.54219377	n*0.04921802
N=	-11.0161638
So here period is coming as negative, which represents that initial amount will never be recovered.
let's check it another way.
If the amount is received for infinity, then PV of this stream will be=	Annual payment/rate
If the amount is received for infinity, then PV of this stream will be=	131000/12%
If the amount is received for infinity, then PV of this stream will be=	$                 1,091,666.67
As we can see if this stream is received for infinity, even then PV comes to 1,091,666.67 which is less than 1,404,928