11. You deposit $200 each month into an account earning 3% interest compounded monthly. a. How much will you have in the account in 30 years

.

a)200*1.03^30 = $485.45

b)

c)  200*1.03^30 minus $200

11. You deposit $200 each month into an account earning 3%(per annum) interest compounded monthly.

I am assuming that the money goes in at the beginning of the month and the interest is paid at the end of the month.

a. How much will you have in the account in 30 years?

b. How much total money will you put into the account?

c. How much total interest will you earn?

a)    You can also do this with the future value of an ordinary annuity formula 

C=200     n=30*12=360      i=0.03/12 = 0.0025

Amount after 30 years = $116547.38

$${\mathtt{200}}{\mathtt{\,\times\,}}\left({\frac{\left({{\mathtt{1.002\: \!5}}}^{{\mathtt{360}}}{\mathtt{\,-\,}}{\mathtt{1}}\right)}{{\mathtt{0.002\: \!5}}}}\right) = {\mathtt{116\,547.376\: \!919\: \!658\: \!203\: \!63}}$$

b)  $${\mathtt{200}}{\mathtt{\,\times\,}}{\mathtt{360}} = {\mathtt{72\,000}}$$

You  have put $72,000 into the account.

c)   Interest = $116547.38 - $72,000 = $44547.38

$${\mathtt{116\,547.38}}{\mathtt{\,-\,}}{\mathtt{72\,000}} = {\mathtt{44\,547.38}}$$

The small difference between my result and Melody's is that I've assumed the the total includes the interest paid at the end of the last month; whereas Melody's assumes that in the last month only the $200 deposited at the beginning of the month is added in.

.

For this type of investment, the “Annuity Due” formula returns the correct future value for standard depository account interest payments, correlating to the beginning of the month deposit and end of the month interest payment.  This returns the same value as Alan’s result.

$$\\
\noindent \text {Annuity Due: }{FV = D \dfrac{(1 + r)^{n} - 1} {r}(1 + r)}\hspace{20pt} |\hspace{10pt} \text {\small D=Deposit per interval}\\

\\ 200*\dfrac{(1.0025^{360}-1)}{0.0025}*(1.0025)\;=\;116838.75$$

In practice, financial institutions use the “average daily balance” to calculate the interest on deposits. The result returns a value with limits between the two formulas.

$$\text {FV(annuity ordinary) \leq $ Total Interest $ \leq $ FV(annuity due)}$$

Yes, sorry Alan I did not realize that our answers were different. 

Alan is totally corect.

This is the formula for the the future value of an ordinary annuity.  This is where the money is put into the account at the END of the time period instead of at the beginning.

It is easy to adjust it for this question.  

Here we have $200 invested at the very beginning so we must ADD 200 plus the interest it will accrue for the whole 30 years  that will be    C(1+i)^n = 200*(1.0025^360)

but NO $200 is  invested at the very end SO we must subtract C = $200 at the end

so we get 

$$\\FV=\left[\frac{(1+i)^n-1}{i}\right]+C(1+i)^n-C\\\\
$If you rearrange this you will find that it is identical to Nauseated's formula.$\\\
FV=$my original answer$+C(1+i)^n-C\\\\
FV=116547.38+200(1.0025)^{360}-200\\\\$$

$${\mathtt{116\,547.38}}{\mathtt{\,\small\textbf+\,}}{\mathtt{200}}{\mathtt{\,\times\,}}{\left({\mathtt{1.002\: \!5}}\right)}^{\left({\mathtt{360}}\right)}{\mathtt{\,-\,}}{\mathtt{200}} = {\mathtt{116\,838.748\: \!442\: \!299\: \!145\: \!509\: \!1}}$$

FV = $ 116838.75  

This is identical to Alan's answer.  And Nauseated also agreed that it is correct.

 ------------------------------------------

Nauseated has gone one more step with this.

He has said:

"In practice, financial institutions use the “average daily balance” to calculate the interest on deposits."

Yes  this is correct  

Nauseated has then stated that:

"The result returns a value with limits between the two formulas. "

Yes I can see how this could possibly be justified..   but I would like Nauseated to justify/discuss this statement. :)

I was in finance for years. I never heard of treating saving account like any kind of annuity. This looks like a ton of BS to me.

It’s not that you never heard of it, you simply forgot. That can happen when you are stoned in class. BTW, playing Monopoly is not the same as being in finance.

In any case, a managed depository savings account can match the return of either type of annuity. I will demonstrate this in the next post.

My goodness, that is excessively polite for you Nauseated !

Are you unwell ?

The following data sets shows first 12 months and the last 12 months of a hypothetical account of 360 interest cycles.

The first column displays the $200 deposit; the second column displays the end of month interest payment on the previous balance and the $200 first day of month deposit.

Column 3 displays the new balance.

Column 4 is the multiplier that weights the deposit for the purpose of interest calculation. The multiplier is from 1 to 0, representing the average daily deposit of the $200, where 1 is a deposit on the first day and 0 is a deposit on the last day. (No withdraws are made from this account and only the deposit is weighted).

Column 5 is the average daily of the deposit.

Column 6 is the interest.

Column 7 is the balance.

Note that the final balance in column 3 matches the annuity due, while the final balance in column 7 matches the annuity ordinary.

$$\displaystyle

\noindent \small {First data set. Months 1-12}\\

\begin{tabular}{lllllll}
1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline

200.00 & 0.50 & 200.50 & 0.00 & 0.00 & 0.00 & 200.00 \\
200.00 & 1.00 & 401.50 & 0.00 & 0.00 & 0.50 & 400.50 \\
200.00 & 1.50 & 603.01 & 0.00 & 0.00 & 1.00 & 601.50 \\
200.00 & 2.01 & 805.01 & 0.00 & 0.00 & 1.50 & 803.01 \\
200.00 & 2.51 & 1,007.53 & 0.00 & 0.00 & 2.01 & 1,005.01 \\
200.00 & 3.02 & 1,210.54 & 0.00 & 0.00 & 2.51 & 1,207.53 \\
200.00 & 3.53 & 1,414.07 & 0.00 & 0.00 & 3.02 & 1,410.54 \\
200.00 & 4.04 & 1,618.11 & 0.00 & 0.00 & 3.53 & 1,614.07 \\
200.00 & 4.55 & 1,822.65 & 0.00 & 0.00 & 4.04 & 1,818.11 \\
200.00 & 5.06 & 2,027.71 & 0.00 & 0.00 & 4.55 & 2,022.65 \\
200.00 & 5.57 & 2,233.28 & 0.00 & 0.00 & 5.06 & 2,227.71 \\
200.00 & 6.08 & 2,439.36 & 0.00 & 0.00 & 5.57 & 2,433.28
\end{tabular}$$

$$\displaystyle

\noindent \small {First data set. Months 349-360}\\

\begin{tabular}{lllllll}
1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline
200.00 & 278.06 & 111500.59 & 0.00 & 0.00 & 276.86 & 111222.53 \\
200.00 & 279.25 & 111979.84 & 0.00 & 0.00 & 278.06 & 111700.59 \\
200.00 & 280.45 & 112460.29 & 0.00 & 0.00 & 279.25 & 112179.84 \\
200.00 & 281.65 & 112941.94 & 0.00 & 0.00 & 280.45 & 112660.29 \\
200.00 & 282.85 & 113424.79 & 0.00 & 0.00 & 281.65 & 113141.94 \\
200.00 & 284.06 & 113908.85 & 0.00 & 0.00 & 282.85 & 113624.79 \\
200.00 & 285.27 & 114394.13 & 0.00 & 0.00 & 284.06 & 114108.85 \\
200.00 & 286.49 & 114880.61 & 0.00 & 0.00 & 285.27 & 114594.13 \\
200.00 & 287.70 & 115368.31 & 0.00 & 0.00 & 286.49 & 115080.61 \\
200.00 & 288.92 & 115857.23 & 0.00 & 0.00 & 287.70 & 115568.31 \\
200.00 & 290.14 & 116347.38 & 0.00 & 0.00 & 288.92 & 116057.23 \\
200.00 & 291.37 & 116838.75 & 0.00 & 0.00 & 290.14 & 116547.38
\end{tabular}$$

The second data sets are the same as the first, except the multiplier is set to 0.75. This corresponds to a depositor making two deposits of $100 each on the first and 15 of each 30-day month.

$$\displaystyle

\noindent \small {Sceond data set. Months 1-12}\\

\begin{tabular}{lllllll}
1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline
200.00 & 0.50 & 200.50 & 0.75 & 150.00 & 0.38 & 200.38 \\
200.00 & 1.00 & 401.50 & 0.75 & 150.00 & 0.88 & 401.25 \\
200.00 & 1.50 & 603.01 & 0.75 & 150.00 & 1.38 & 602.63 \\
200.00 & 2.01 & 805.01 & 0.75 & 150.00 & 1.88 & 804.51 \\
200.00 & 2.51 & 1007.53 & 0.75 & 150.00 & 2.39 & 1006.90 \\
200.00 & 3.02 & 1210.54 & 0.75 & 150.00 & 2.89 & 1209.79 \\
200.00 & 3.53 & 1414.07 & 0.75 & 150.00 & 3.40 & 1413.19 \\
200.00 & 4.04 & 1618.11 & 0.75 & 150.00 & 3.91 & 1617.10 \\
200.00 & 4.55 & 1822.65 & 0.75 & 150.00 & 4.42 & 1821.51 \\
200.00 & 5.06 & 2027.71 & 0.75 & 150.00 & 4.93 & 2026.44 \\
200.00 & 5.57 & 2233.28 & 0.75 & 150.00 & 5.44 & 2231.88 \\
200.00 & 6.08 & 2439.36 & 0.75 & 150.00 & 5.95 & 2437.84
\end{tabular}$$

$$\displaystyle

\noindent \small {Sceond data set. Months 349-360}\\

\begin{tabular}{lllllll}
1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline
200.00 & 278.06 & 111500.59 & 0.75 & 150.00 & 277.76 & 111431.07 \\
200.00 & 279.25 & 111979.84 & 0.75 & 150.00 & 278.95 & 111910.02 \\
200.00 & 280.45 & 112460.29 & 0.75 & 150.00 & 280.15 & 112390.17 \\
200.00 & 281.65 & 112941.94 & 0.75 & 150.00 & 281.35 & 112871.52 \\
200.00 & 282.85 & 113424.79 & 0.75 & 150.00 & 282.55 & 113354.08 \\
200.00 & 284.06 & 113908.85 & 0.75 & 150.00 & 283.76 & 113837.84 \\
200.00 & 285.27 & 114394.13 & 0.75 & 150.00 & 284.97 & 114322.81 \\
200.00 & 286.49 & 114880.61 & 0.75 & 150.00 & 286.18 & 114808.99 \\
200.00 & 287.70 & 115368.31 & 0.75 & 150.00 & 287.40 & 115296.39 \\
200.00 & 288.92 & 115857.23 & 0.75 & 150.00 & 288.62 & 115785.00 \\
200.00 & 290.14 & 116347.38 & 0.75 & 150.00 & 289.84 & 116274.84 \\
200.00 & 291.37 & 116838.75 & 0.75 & 150.00 & 291.06 & 116765.90
\end{tabular}$$

...

My goodness, that is excessively polite for you Nauseated !

Overt politeness is one of my faults. I try, but I am not always successful.

Are you unwell ?

 Yes! I am unwell. I am Nauseated.

It seems worse than usual.  . . . Maybe it's sympathetic morning sickness. :)

Is Mrs Nauseated pregnant?    Congratulations !

The name you have inflicted upon her will suit her well for a while :/

ok Nauseated you have a lot of figures there but I am going to be so bold as to summarize your reasoning.

The question does not state WHEN the money was deposited into the account.

IF it is deposited at the very beginning of the month then it is an annuitiy due question and yours and Alan figures are correct (a tiny bit too big)

If it is deposited at the very end of the month then it is an ordinary annuity question and my original answer was in fact correct. (a tiny bit to small)

HOWEVER

If the $200 deposit is made at another time or times during the month then this will make the Future value lie between  the 2 extremes.

SEE Nauseated - that was not so hard.  You really did not need all those tables of values.

I am quite positive you will correct me if I have misinterpreted your logic.

-----------------------------------

The reason I have added "a tiny bit too big" and a "tiny bit too small" is because there is always going to be one day at the end, or the beginning, of the month which has the old value at the beginning and the new value at the end so effectively this day has interest paid on the new deposit of only $100  (average of 0 and 200) [Not 0 (ordinary annuity) or 200 (annuity due)  as used for the development of the formula]