# Thread: I have a question of an annuity calculation..

1. ## I have a question of an annuity calculation..

My math professor gave me this question to solve:

If Tom contributes $4000 at the end of each semiannual period (semiannually) into a retirement account with 8% interest compounded quarterly, and he continues these semiannual payments for 10 years, I know that he will have$119,607.89 in his account, as opposed to the $119,112.31 that he should have if interest were compounded semiannually. Suppose that Tom contributes$R at the end of each period for y years and that there are m payments per year. If the interest rate r is compounded c times in a year (for any c > m), write the future value of this financial scheme as an algebraic function of R, m, y, r and c.
You’ll know that you have the right answer if you can use your formula using the information in the scenario above, with R = 4000, r = .08, m = 2, y = 10 and c = 4.
If someone can help me with the solution it will be greatly appreciated. I can't seem to grasp this whole annuity thing. He gave me 2 hints saying that:

1. Push forward the first few payments to the end of term value using the compound interest formula.
2. Know what a monotonic polynomial is.

2. Are you 100% certain that these are the correct answers?
$119,607.89 ,$119,112.31

3. Originally Posted by westside7
Suppose that Tom contributes \$R at the end of each period for y years and that there are m payments per year. If the interest rate r is compounded c times in a year (for any c > m), write the future value of this financial scheme as an algebraic function of R, m, y, r and c.
You must have meant:
for any
$\left\{ \begin{array}{l} c < m \\ c = m \\ c > m \\ \end{array} \right.
$

If so, then

$S = R \cdot \frac{{\left( {1 + {\textstyle{r \over c}}} \right)^{yc} - 1}}{{\left( {1 + {\textstyle{r \over c}}} \right)^{{\textstyle{c \over m}}} - 1}} = 4,000\frac{{\left( {1 + {\textstyle{{.08} \over 4}}} \right)^{10 \times 4} - 1}}{{\left( {1 + {\textstyle{{.08} \over 4}}} \right)^{{\textstyle{4 \over 2}}} - 1}} \approx \ 119,607.8875...
$

If $c \to \infty$, then

$S = R \cdot \frac{{e^{ry} - 1}}{{e^{{\textstyle{r \over m}}} - 1}}$

4. What would the formula be if the deposits are made at the beginning of each period?

Originally Posted by jonah
You must have meant:
for any
$\left\{ \begin{array}{l} c < m \\ c = m \\ c > m \\ \end{array} \right.
$

If so, then

$S = R \cdot \frac{{\left( {1 + {\textstyle{r \over c}}} \right)^{yc} - 1}}{{\left( {1 + {\textstyle{r \over c}}} \right)^{{\textstyle{c \over m}}} - 1}} = 4,000\frac{{\left( {1 + {\textstyle{{.08} \over 4}}} \right)^{10 \times 4} - 1}}{{\left( {1 + {\textstyle{{.08} \over 4}}} \right)^{{\textstyle{4 \over 2}}} - 1}} \approx \ 119,607.8875...
$

If $c \to \infty$, then

$S = R \cdot \frac{{e^{ry} - 1}}{{e^{{\textstyle{r \over m}}} - 1}}$

5. Originally Posted by vaionline
What would the formula be if the deposits are made at the beginning of each period?
If the deposits were made at the beginning of each period, then
$\ddot S = R \cdot \frac{{\left( {1 + {\textstyle{r \over c}}} \right)^{yc} - 1}}{{\left( {1 + {\textstyle{r \over c}}} \right)^{{\textstyle{c \over m}}} - 1}} \cdot \left( {1 + {\textstyle{r \over c}}} \right)^{{\textstyle{c \over m}}}
$

or
$\ddot S = R \cdot \frac{{\left( {1 + {\textstyle{r \over c}}} \right)^{yc + {\textstyle{c \over m}}} - 1}}{{\left( {1 + {\textstyle{r \over c}}} \right)^{{\textstyle{c \over m}}} - 1}} - R
$

If $c \to \infty$ then
$\ddot S = R \cdot \frac{{e^{ry} - 1}}{{e^{{\textstyle{r \over m}}} - 1}} \cdot e^{{\textstyle{r \over m}}}
$

or
$\ddot S = R \cdot \frac{{e^{ry + {\textstyle{r \over m}}} - 1}}{{e^{{\textstyle{r \over m}}} - 1}} - R
$