# Which is the final capital after 10 years?

We want to calculate the final capital after 10 years for a savings plan with the following data:

Monthly deposit at the beginning of the month 200 Euro; Interest rate coupon annually; Interest rate 4.5% p.a.



I have done the following:

The intereset rate is $q=1+\frac{p}{100}=1+\frac{4,5}{100}=1+0,045=1,045$.

Since we are paying every month for $10$ years, we are paying for $12\cdot 10=120$ months.

Using the formula \begin{equation*}\overline{R_m}=\frac{r\cdot q_m\cdot \left (q_m^m-1\right )}{q_m-1}\end{equation*}

with $m=120$, $r=200$ und $q_m=\sqrt[12]{q}=\sqrt[12]{1,045}=1.003675$,

we get the following

\begin{align*}\overline{R_m}&=\frac{200\cdot 1.003675\cdot \left (1.003675^{120}-1\right )}{1.003675-1}\approx 30206.10\end{align*}



My result is not the same as the answer of the book. What have I done wrong? Do we not use that formula?

• It looks like you've used some sort of annuity formula but not all terms are defined. How often is the compounding? Have you checked that you have used the correct annuity formula (first payment immediate)? Apr 20 '17 at 12:53
• Which terms are not defined? This formula is for the case that we pay at the beginning of a period, or not? @Any Apr 20 '17 at 12:59
• That looks fine if the compounding is annual. Apr 20 '17 at 13:10
• It is annual. @Any Apr 20 '17 at 13:15
• As in the anseer, the question is whether the rate is effective annual or something else Apr 20 '17 at 13:20

To get the required result in the book you first have to calculate the amount of the equivalent annual payment. The formula for payments at the beginning of every month is

$C_1=12\cdot r+\frac{\color{blue}{13}\cdot r\cdot i}{2}$

In case of payments at the end of every month $\color{blue}{13}$ has to be replaced by $11$.

$C_1=12\cdot 200+\frac{\color{blue}{13}\cdot 200\cdot 0.045}{2}=2458.5$

To get the Future value after 10 years we use the formula for annual payments.

$C_{10}=2458.5\cdot \frac{1-1.045^{10}}{1-1.045}=30210.56$

But in general I wouldn´t say that your method is worse then the method above. Your result differs from my result about $0.015\%$ only.

• So, if we want to use the formula that I used, we have to use $m=13\cdot 130$ instead of $12\cdot 10=120$ ? Or can we not use that formula at all? If yes, why can we not use it? And how can we know when we can use it and when not? Apr 20 '17 at 18:32
• No I wouldn´t say that. For me your thoughts and your calculation are correct. Both results are almost equal. If you want the exact result of the book you do my way. But if you want only a correct result you can choose between the two ways of calculation. Note that $0.015\%$ variation is really small. Apr 20 '17 at 18:47
• Ok!! I have also an other question. When we have a monthly deposit and an interest rate $x\%$ p.a., we have to know the interest rate of a month, right? We have that $q=1+x$. This factor for a month is $q_m=\sqrt[12]{q}$, or not? Is the interest rate of a month always $\frac{x}{12} \%$ ? Apr 20 '17 at 19:01
• @MaryStar I think I have posted a nice answer to your question at the following link. Short answer: The equivalent interest factor is $q_m=\sqrt[12]{q}$. The interest factor $1+\frac{i}{12}$ is a good approximation. Link: math.stackexchange.com/a/2206483/144421 Apr 20 '17 at 19:31
• Ok!! Thank you very much!! :-) Apr 20 '17 at 23:08

I'm not sure about the formula you are using, but this is how I'd do it. Say you start of by putting in $c$ the first month and then you have a monthly interest on that at $4.5$%. In the start your capital is $y_0=c$, one, two and $n$ months later, respectively, right after the deposit, you have $$\begin{array}{lcl} y_1 &=& 1.003675c+c = (1.003675+1)c \\ y_2&=& 1.003675(1.003675c+c) = (1.003675^2+1.003675+1)c \\ &\vdots \\ y_n&=& (1.003675^n+1.003675^{n-1}+...+1)c= \frac{(1.003675^{n+1}-1)c}{1.003675-1} & \end{array}$$

By the geometric sum. Now plug in $n=120$ and $c=200$.

I have some doubts about your question, but I am assuming that there is a new deposit made every month for 10 years, so 120 deposits of 200 Eur each. This means that the 1st deposit will earn interest for 120 periods (months) while the last, just for one period.

If the rate is 4.5% p.a nominal, than the real is just 4.5% / 12 = 0.375% Interest Rate Calculator

This being said, the total amount should be 30353.01329 Eur

• But in my book the answer is $30210,56$ Eur. So, do we have to use an other formula? Apr 20 '17 at 13:19