Question

1.) FINDING THE REQUIRED INTEREST RATEYour parents will retire in 20 years. They currently have $260,000 saved, and they think they will need $900,000 at retirement. What annual interest rate must they earn to reach their goal, assuming they don't save any additional funds? Round to TWO decimal places.  

2.) TIME TO REACH A FINANCIAL GOAL You have $51,238.94 in a brokerage account, and you plan to deposit an additional $4,000 at the end of every future year until your account totals $250,000. You expect to earn 13% annually on the account. How many years will it take to reach your goal?  Round UP to the nearest year. (Example 5.01 years = 6 years) Your answer should include numerical value only.

Answer 1

1.We use the formula:
A=P(1+r/100)^n
where
A=future value
P=present value
r=rate of interest
n=time period.

900,000=260,000*(1+r/100)^20

(900,000/260,000)^(1/20)=(1+r/100)

(1+r/100)=1.0641

r=(1.0641-1)*100

=6.41%(Approx).

2.

We use the formula:
A=P(1+r/100)^n
where
A=future value
P=present value
r=rate of interest
n=time period.

Hence A for 51,238.94=51,238.94*(1.13)^n

Also Future value of annuity=Annuity[(1+rate)^time period-1]/rate

=$4000[(1.13)^n-1]/0.13

Hence

250,000=51,238.94*(1.13)^n+$4000[(1.13)^n-1]/0.13

250,000=51,238.94*(1.13)^n+$30769.23077[(1.13)^n-1]

250,000=51,238.94*(1.13)^n+$30769.23077(1.13)^n-$30769.23077

(250,000+$30769.23077)=(1.13)^n[51238.94+$30769.23077]

(250,000+$30769.23077)/[51238.94+$30769.23077]=(1.13)^n

(1.13)^n=3.423673862

Taking log on both sides;

n*log 1.13=log3.423673862

n=log 3.423673862/log 1.13

10 years(Approx).