I request a Financial God! (1 Viewer)

Wonderlay · Jul 10, 2017

A person pays $2000 into an investment fund every six months, and it earns interest at a
rate of 6%pa, compounded monthly. How much is the fund worth at the end of ten years?

This question is under Rates and Finance (Sequence and Series)
Thank you!

The answer is $55 586.38 btw!

InteGrand · Jul 10, 2017

InteGrand · Jul 10, 2017

$\noindent Here's some explanations for the main steps. The reason that$

$A_{n} = \left(1 + \frac{0.06}{12}\right)^{6}A_{n-1} + 2000$

$\noindent is true is that the value just after the $n$-th deposit is the value just after the previous ($(n-1)$-st) deposit (which is $A_{n-1}$), multiplied by $\left(1 + \frac{0.06}{12}\right)^{6}$ (since this is the factor by which the $A_{n-1}$ grows over six months using monthly compounding at a rate of $6\%$ p.a.), plus the amount we then finally deposit (which is $2000$).$

$\noindent Once we have established that $A_{n} = aA_{n-1} + b$, we can use this recurrence relation multiple times (or use induction) to show that we have $A_{n} = a^{n-1}A_{1} + b\sum_{k = 0}^{n-2}a^{k}$.$

$\noindent Once we have $A_{20}$ (the value of the fund just after the deposit at time $9.5$ years), we just multiply this by $ \left(1 + \frac{0.06}{12}\right)^{6}$ (the compound interest factor) to get the value of the fund at the end of $10$ years, since we are assuming there are no more deposits after the one at time $9.5$ years (so no need to add on $2000$).$

BenHowe · Jul 10, 2017

It is poorly worded because you don't know whether the payments are made in advance or arrears

Note what integrand is explaining is given in actuarial notation as $i^{12}=6\%$

the power is the number of the number of times it is compounded and the percentage is the nominal amount. The above rate can also be expressed as 6% p/a payable/compounded/charged/convertible monthly. Note this can be misleading in the real work context for people because 6% p/a compounded monthly is not the same value as the 6% p/a compounded quarterly etc. The formula is also given above by integrand

These questions are best visualised by drawing out a time line and marking the total period 12 months in a year times the 10 years and then each increment being bi-annually. Then you can write down the geometric progression clearly. Always do your measure of time in terms of the compounding period of the interest rate!

Wonderlay · Jul 10, 2017

Thank you both!

I request a Financial God! (1 Viewer)

Wonderlay

New Member

InteGrand

Well-Known Member

InteGrand

Well-Known Member

BenHowe

Active Member

Wonderlay

New Member

Users Who Are Viewing This Thread (Users: 0, Guests: 1)