Accrued interest

Question

I know how to calculate accrued interest over time on an initial amount. However, my assignment has me artificially adding additional sums intermittently. I'm curious if there is a formula to do that. The specific task is to calculate: $\$7000$ is invested at the beginning of each year for $25$ years with an annual interest of $7$%. Find the sum of the payments over the entire investment period.

Answer 1

This may be handled simply with a geometric series. Let $P$ be the amount invested every year, $i$ be the interest rate, and $N$ be the number of years of the investment period. Then treat each investment at the beginning of the year separately. Thus, the sum $S$ is

$$S = P\left (1+i\right)^n + P\left (1+i\right)^{n-1} + \cdots P\left (1+i\right)$$

This assumes that we are evaluating the sum at the end of a year. Using the formula for a geometric series, I get

$$S=P \left(1+i\right) \frac{\left(1+i\right)^n-1}{i} $$

Plug in $P=\$7000$, $i=0.07$, $n=25$. I get $S \approx \$ 473735$.

Answer 2

Indeed, one can calculate the present value of a set of periodic payments, discounted at that period interest rate.

This is explained in section "13.1.3 A Closed Form for the Annuity Value" section of the "July 25, 2013 draft of the complete class text" of the MIT course "Mathematics for Computer Science" (6.042/18.062J Spring 2013)

That class materials page is currently available at http://courses.csail.mit.edu/6.042/spring13/class-material.shtml and the direct link PDF file (a 9.5 Megabytes file, with a "Creative Commons" license) is available at http://courses.csail.mit.edu/6.042/spring13/mcs.pdf

Quoting from the page 468 of that book:

13.1.3 A Closed Form for the Annuity Value

Using equation 13.2, we can derive a simple formula for V, the value of an annuity that pays m dollars at the start of each year for n years.

$$V = m\bigg(\frac{1 - x^n}{1-x}\bigg)$$ $$=m \bigg(\frac{1 + p - (1/(1 +p))^{n - 1}}{p}\bigg)$$

For the above formula, in the section "13.1.1 The Future Value of Money" (on pages 466-467), it is explained that:

• $m$ stands for the periodic payment value (7,000 USD in the OP example)
• $p$ is the interest rate (7% in this case)
• $x$ stands for 1/(1 + $p$)
• $n$ stands for the number of periods (25 in this case).

So, applying the formula to the OP example we would have:

$$V =7000 \bigg(\frac{1 + 0.07 - (1/(1 +0.07))^{25 - 1}}{0.07}\bigg) \approx \$87,285.34 $$

EDIT #2:

The above \$87,285.34 value is the Present Value (as the Wikipedia article for "Present Value" says, it's the "future amount of money that has been discounted to reflect its current value, as if it existed today").

If we wanted the Future Value instead (quoting Wikipedia again, "the nominal future sum of money that a given sum of money is "worth" at a specified time in the future assuming a certain interest rate"), then we could simply do $ \$87,285.34 \times (1 + 0.07)^{25}$ and we would reach the \$473,735 value that Ron Gordon has arrived to, in his answer. :)

EDIT #1: The book gives an example of calculating V (Present Value) :

Equation 9.5 is much easier to use than a summation with dozens of terms. For example, what is the real value of a winning lottery ticket that pays \$50,000 per year for 20 years? Plugging in m = \$50,000, n = 20, and p = 0.08 gives V = \$530,180. So because payments are deferred, the million dollar lottery is really only worth about a half million dollars!