Question

When Ricky is 22 years old, she begins steadily investing $200/month (K=$2400) for 10 years at 5% compounded continuously.

1. What is the investment worth at the end of 10 years (when she is 32 years old)?

2. Ricky stops monthly contributions and lets her investment grow for the next 40 years at 5% compounded continuously?

3. When she is 72, how much did her contributions grow? (hint: subtract 24000 from value at age 72)

Answer 1

Answer to first question :

1) Here interest is compounded continously so I supposed that interest is compounded on monthly basis.

Formulae to calculate the invesment worth after a certain period is as follows :

Amount at the end of certain period =

PMT * (((1 + r/n)^(nt) - 1)) / (r/n))) * (1+r/n)

where PMT stands for monthly contributions

r stands for rate of interest

n stands for number of times interest is compounded

t stands for the total time

Here PMT = $200 , r =5/100 = 0.05 , n= 12 (because 1 year is 12 months) and t = 10 years

Applying the formulae,

Investment worth at the end of 10 years = 200 ((( 1+0.05/12)^(12*10)-1))/(0.05/12)))*(1+0.05/12)

= 200 ((1+0.0042)^120-1))/0.0042*(1+0.0042)

= 200 ((1.0042)^120-1))/0.0042(1.0042)

= 200((1.6536)-1)/0.0042 (here 0.0042*1.0042 = 0.0042)

=200(0.6536)/0.0042

=130.72/0.0042

=31123.81

The above calculations taken upto 4 decimal places and the last calculated amount 31123.81 is uptill 2 places of decimal. Computations done to calculate to the power of 120 and then amount is put .

So the answer is investment worth at the end of 10 years = $31123.81 (when she is 32 years old)

2nd Part to calculate the amount after she stops the monthly investment and consolidated amount after 40 years . Here intial investment is $31123.81

Now , the formulae to calculate is P(1+r/n)^(nt)

Where P is the intial investment = $31123.81

r=rate of interest = 5/100= 0.05

n= number of times interest is compunded per unit time period = 12

t = total time period = 40 years

Contribution to grow after 40 years when she is 72 years = 31123.81(1+0.05/12)^(12*40)

=31123.81 (1+0.0042)^480

=31123.81 (1.0042) ^480

=31123.81 (7.4766)

=232700.27

So when she is 72 years old , contribution will grow to $232700.27