Question

How much must be deposited in an account to have annual withdrawals of $ 2607 forever? Interest is compounded quarterly and the interest rate is 9% for the first 13 years and 15 % thereafter.

The answer is close to

Answer 1

Solution:-

Given,

Amount of annuals withdrawals is $2607 per year forever.

Interest rate 9% per year for the first 13 Year and 15% thereafter compounded quarterly.

First Calculate the effective interest rate for both the time periods then use that rate to

Calculate amount of deposit required to withdraw $2607 per year forever.

Interest rate calculation is as follows.

Effective interest rate=[1+(nominal interest rate/compounding period in year)]^compounding period in year-1

=[(1+9%/4)^4-1]

=[(1+.09/4)^4-1]

=[(1+0.0225)^4-1]

=[(1.0225)^4-1]

=1.093083-1

=0.093083=9.3083%

There after 15% compounded quarterly

=[(1+15%/4)^4-1]

=[(1+.15/4)^4-1]

=[(1+0.0375)^4-1]

=[(1.0375)^4-1]

=1.158650-1

=0.1586504=15.8650%

Thus, the effective interest rate are 9.3083% and 15.8650%

Following the way to calculate the amount of initial deposite.

PV=$2607(P/A, 9.3083%,13)+$2607/ 15.8650%(P/F,9.3083,13)

    =$2607[(1+i)^n-1/i(1+i)^n]+$2607*100/15.8650[1/(1+i)^n]

    =$2607[(1+0.093083)^13-1/0.093083(1+0.093083)^13]+$2607*100/15.8650[1/(1+0.093083)^13]

    =$2607[(1.093083)^13-1/0.093083(1.093083)^13]+260700/15.8650[1/(1.093083)^13]

   =$2607[3.18046-1/0.093083(3.18046)]+16432.39836(1/3.18046)

   =$2607(2.18046/0.29605)+16432.39836(0.3144)

   =$2607(7.36517)+5166.346

   =19200.9819+5166.346

   =$24367.344

Hence, the amount of initial deposit must be $24367.344