Question

The general management of Electric Motors has considered having $ 75,000 available within 10 years, to complete the expansion of its current administrative office. For such purpose have designed a savings plan with the intention of reaching the established goal, taking advantage of the promotion of the Bank "El Buen Amigo" that offers them a fixed rate of 6% annual capitalized semi-annually, for the entire time that deposits are being made savings. The plan is to start depositing a certain amount next year, which could increase at the rate of $ 1,500 per year up to the fifth deposit. After this fifth deposit could increase your savings at a rate of 10% per year up to the tenth year. If the general management is convenient, this savings program will provide the necessary funds, what should be the amount to be deposited in the first year?.

Answer 1

i=6%

n=2 for semi annual capitalization

Effective interest rate,r =(1+i/n)^n-1=(1+6%/2)^2-1=(1+3%)^2-1=1.03^2-1=1.0609-1=0.0609 or 6.09%

Let first deposit be A and gradient,G=$1500

FV of A for five years=A*((1+r)^n-1)/r

=A*((1+6.09%)^5-1)/6.09%

=A*(1.0609^5-1)/0.0609

=A*(1.3439-1)/0.0609

=A*0.3439/0.0609

=5.6472A

FV of gradient= G*((1+r)^n-r*n-1)/r^2

=1500*((1+6.09%)^5-6.09%*5-1)/6.09%^2

=1500*(1.0609^5-0.0609*5-1)/0.0609^2

=1500*(1.3439-0.3045-1)/0.00371

=1500*0.0394/0.00371

=$15929.92

Total FV at end of 5 years= 5.6472A+15929.92

Total FV of depsoits till 5 year= (5.6472A+15929.92)*(1+6.09%)^5=(5.6472A+15929.92)*1.0609^5=(5.6472A+15929.92)*1.3439= 7.5893A+21408.22

Deposit of 5th year= A+1500*4=6000+A

Deposit at 6th year=(6000+A)*(1+10%)=(6000+A)*1.1=6600+1.1A

FV of deposits from 6 to 10 years =A*((1+r)^n-(1+g)^n)/(r-g)

=(6600+1.1A)*((1+6.09%)^5-(1+10%)^5)/(6.09%-10%)

=(6600+1.1A)*(1.0609^5-1.1^5)/(-3.91%)

=(6600+1.1A)*(1.3439-1.6105)/(-0 0391)

=(6600+1.1A)*(-0.2666)/(-0.0391)

=(6600+1.1A)*6.8187

=45003.22+7.5A

Total FV=75000=7.5893A+21408.22+45003.22+7.5A

=66411.44+15.0893A

Or, 15.0893A=75000-66411.44

Or,A=8588.56/15.0893=$569.18