Question

Hi please help me on this compounding interest question :D

Answer 1

hints:
1. vital formula for this question:
For a monthly deposit of A over n years at an annual interest rate of i, the monthly interest is then r=i/12, and 12n is the number of months.
The future value after n years is given by
\(\Large FV=\frac{A((1+r)^{12n}-1)}{r}\)
2. For parts (a) and (b), use above formula to work out 5 years at a time, and don't forget to double the deposit (A) every five years as instructed. Remember to ADD the additional contribution from every 5 years to the previously accumulated to calculate the next period.
3. For part (c),
1. first calculate the future value (at 60) resulting from deposits during the 45-50 period, call this amount F. You need to do this in three steps, FV at 50, 55 and 60.
2. Subtract 1.1 millions from the amount obtained in (b), call this T. This is the excess, or what he needs to keep his minimum future value at 1.1 million.
3. If T>F, the excess is greater than what he earned between 45 and 50, therefore he does not have to deposit anything during 45-50.
If T<F, F is what he earned from the deposits during 45-50, and F-T is what he needs to contribute. In this case, prorate the amount of monthly deposit (during 45-50) according to the ratio (F-T)/F. The prorated deposit is the answer required.

Answer 2

Note: this is a problem to test your understanding of compound interest. Convince yourself that you understand WHY you do each step. Please do not just blindly follow the steps, or worse, accept an answer from someone else without doing any calculations by yourself.

Answer 3

@mathmate Is there a way that I could use the calculator for example, come up with the N, I/Y, PMT, PV and FV values? I did key in with the formula used and the calculator, the answers were not the same. eg. for a. N=60 months, I/Y=4/12, PMT=-500, PV=0, FV=33149.49

While using th formula, i get a very huge no. ~4.70x10^10

Answer 4

Using the formula, I get for A=500, i=0.04, r=0.04/12, n=5*12,
FV=33149.489091
Perhaps you're not using the formula right. Read the definition of the parameters,
n=number YEARS
r=annual interest divided by 12, i.e. 0.04/12
A=500 (per month)
The above formula is made for monthly payments, but the interest and period are in years.
Else you may have a problem putting the parentheses correctly on the calculator.
BTW, what model of calculator do you use?

Answer 5

* actually n=5, the factor 12 is built-in the formula.

Answer 6

Ahh, i got my mistake, i key in r= 4/12 instead of r=0.04/12 , thank you!

Answer 7

You're welcome! :)

Answer 8

@mathmate can i check with you, for year 40-45, is it \[FV=\frac{ 1000(1+\frac{ 0.04 }{ 12 })^{60} }{ \frac{ 0.04 }{ 12 } } \] or\[FV=\frac{ 1000(1+\frac{ 0.04 }{ 12 })^{120} }{ \frac{ 0.04 }{ 12 } } \] plus the 33149.49 ?

Answer 9

im quite confused on this matter, but if i could confirm with you, i shd be able to do the rest :)

Answer 10

For year 40-45, you would have two components,
One of the components is the 33149.48909 at 4% p.a. over the next 5 years.
(I use more decimal places because over the next 20 years, it makes a difference of 33 cents if you rounded the amount to cents.)

The other component is given in your first expression, which incidentally is just double that of 35-40, since the monthly deposit is also doubled, but the period remains 5 years.

The total should be just over $100,000 (at the end of 40-45).

Answer 11

@mathmate just over $100,000 like $99,448.46727 to be precise? but thats not over $100,000 though., am i doing it correcty? I did the first expression and added the 33k.

Answer 12

Show your work if you want help to spot the problem. The final number does not tell me much! lol
Are you using the calculator functions? If so, it is important to make sure they work the same way we expect. Please give me at least the two components separately as a start.

Answer 13

\[A=33149.48909(1+\frac{ 0.04 }{ 12 })^{60}=40475.41327\]

\[FV= \frac{ 1000((1+\frac{ 0.04 }{ 12 })^{60}-1) }{ \frac{ 0.04 }{12 } }= 66298.97818\]

adding them both together gives 106.774.3915

i went to redo hopefully i amended my mistake :D

Answer 14

@mathmate

Answer 15

Yes, I have exactly the same number for 40-45,
106774.3914540556

Answer 16

yay! great! now i understand! thanks!

Answer 17

Very good, keep working at it and we can soon compare notes for 55-60!
I have a touch less than 1.25 million.

Answer 18

@mathmate yay! i have got FV for age 60 at 1246238.097! but fr part c, we have to use the FVs calculated at b right?
45-50 ; FV=132597.9564
50-55; FV= 265195.9127
55-60; FV= 530391.8255
if subtract 1.1M then all of them would be T<F?

Answer 19

Yes, I have the same number at age 60.
I also have the value of the 45-50 investment at 50 as 132597.9564.
However, we need to grow this amount for the next 10 years independent of the additional investments after 50, which means to isolate the 45-50 investment's value at 60.
What would you do to achieve that?

Answer 20

you mean make the 45-50 investment as PV of 60?

Answer 21

Exactly!

Answer 22

Can you then get this value at 60? By subtracting, we will know how much we're short of reaching 1.1 million if he did not invest during this period at all.

Answer 23

Hm, which formula do we use? the FV=PV(1+i)^n?

Answer 24

Exactly, for the next 10 years.

Answer 25

\[FV=132597.9564(1+0.04)^{10}= 196277.3672\]

Answer 26

Almost, but don't forget the investment is still compounded each month!

Answer 27

\[FV=132597.9564(1+\frac{ 0.04 }{ 12 })^{120}= 197681.367\]

Answer 28

Exactly!
So how much of this amount do we need to make just 1.1 million instead of 1.25 million?

Answer 29

1.1M-197681.367= 902318.633

Answer 30

Sorry, that's not what I meant.
Perhaps a diagram helps.

From here, we need to find the monthly investment during 45-50.