Suppose T contributes $R at end of each period for y years.

[westside7]

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.

 

[stapel]

Suppose that Tom contributes $R at the end of each period for y years and that there are m payments per year....

Duplicate post.

 

[Denis]

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.

SO, as example, with monthly payments (m = 12), r needs to compound at least 13 times in a year; correct?
If weekly payments (m=52), then c>52; correct?
If weekly payments, payment is assumed every 1/52nd of the year, regardless of the number of days in year; correct?
What if payment is daily: 360 days used? 365.25 ?

Shouldn't your professor restrict the payment frequency to something reasonable;
like, shouldn't stuff like every 7 months be eliminated? Like frequency => 2 ?

 

[Denis]

Well, whatever...

The formula for Future Value of periodic deposit is:
D[(1 + i)^n - 1] / i, where:
D = deposit amount
n = number of periods
i = periodic interest factor

Your semi-annual example: 4000(1.04^20 - 1) / .04 = 119112.31

When the deposit dates and interest compounding dates don't match,
we need to calculate what interest rate factor achieves same results.
This is done this way:
i = (1 + r/c)^(c/f) - 1, where:
i = interest rate factor
r = annual rate
c = compounding frequency
f = frequency of deposits

The ball is now in your court!
You said: "My math professor gave me this question to solve" ; that's YOU, not us :idea:

 

[jonah]

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
\(\displaystyle \left\{ \begin{array}{l} c < m \\ c = m \\ c > m \\ \end{array} \right.\)
If so, then
\(\displaystyle S = R \cdot \frac{{\left( {1 + {\textstyle{r \over c}}} \right)^{yc} - 1}}{{\left( {1 + {\textstyle{r \over c}}} \right)^{{\textstyle{c \over m}}} - 1}} = 4,000\frac{{\left( {1 + {\textstyle{{.08} \over 4}}} \right)^{10 \times 4} - 1}}{{\left( {1 + {\textstyle{{.08} \over 4}}} \right)^{{\textstyle{4 \over 2}}} - 1}} \approx \$ 119,607.8875...\)

If \(\displaystyle c \to \infty\), then

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

 

[vaionline]

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

 

[Denis]

All that means is that you get an extra period's interest, so use regular formula and add a period's interest:
(regular formula) * (1+i)

 

[jonah]

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
\(\displaystyle \ddot S = R \cdot \frac{{\left( {1 + {\textstyle{r \over c}}} \right)^{yc} - 1}}{{\left( {1 + {\textstyle{r \over c}}} \right)^{{\textstyle{c \over m}}} - 1}} \cdot \left( {1 + {\textstyle{r \over c}}} \right)^{{\textstyle{c \over m}}}\)
(just as Sir Denis indicated)
or
\(\displaystyle \ddot S = R \cdot \frac{{\left( {1 + {\textstyle{r \over c}}} \right)^{yc + {\textstyle{c \over m}}} - 1}}{{\left( {1 + {\textstyle{r \over c}}} \right)^{{\textstyle{c \over m}}} - 1}} - R\)
If \(\displaystyle c \to \infty\) then
\(\displaystyle \ddot S = R \cdot \frac{{e^{ry} - 1}}{{e^{{\textstyle{r \over m}}} - 1}} \cdot e^{{\textstyle{r \over m}}}\)
or
\(\displaystyle \ddot S = R \cdot \frac{{e^{ry + {\textstyle{r \over m}}} - 1}}{{e^{{\textstyle{r \over m}}} - 1}} - R\)