PLEASE USE EXCEL!

Part $2$

Find the future value of an increasing annuity.

Imagine you want to accumulate certain amount, or future value, by making the same payments for

certain number of periods. How much should your payments be$?$

Strategy for solution:

To develop the general formula let's first figure out the future value of $n$ payments of $$1.$

The future value of the last payment made at time $n$ is just $$1.$

The future value of the next to last payment made at time $n-1$ is just $$(1+i),$ the accumulated value

of a payment of $$1$ over one period.

The future value of the payment made at time $n-2$ is $$(1+i{)}^2,$ the accumulated value of a payment

of $$1$ over $2$ periods.

The future value of the payment made at time $1$ is $$(1+i{)}^{n-1},$ the accumulated value of a payment of

$$1$ over $n-1$ periods.

The future value of all payments is the sum

$F=1+(1+i)+(1+i{)}^2+(1+i{)}^3+$cdots$+(1+i{)}^{n-1}={\displaystyle \underset{k=0}{\overset{n-1}{}}}(1+i{)}^k$

In order to find a general formula for this sum, we need to review the sum of geometric progression.

A geometric series is always going to look like

$1+a+a^2+a^3+$cdots$+a^n$ for some number $a.$

Now, what does this have to do with finance? Let's look at an example.

Let $S=1+r+r^2+$cdots$+r^{n-1}={\displaystyle \underset{k=0}{\overset{n-1}{}}}{r}^k$

$S=1+r+r^2+$cdots$+r^{n-1}$

$rS=r+r^2+$cdots$+r^{n-1}+r^n$

The difference is:

$S-rS=1-r^n$

$S(1-r)=1-r^{n}S={\displaystyle \underset{k=0}{\overset{n-1}{}}}{r}^k=\frac{r^n-1}{r-1}$

In our example $r=1+i,$ thus we have $F=\frac{(1+i{)}^n-1}{i}.$

The future value $F$ when the payment is equal to $R$ is:

$F=R\frac{(1+i{)}^n-1}{i}$

Questions to answer:

Imagine that you want to invest into an increasing annuity to accumulate a certain amount of money

by the time you graduate. To make it more realistic, think of the set amount of money you can put away

every month.

a$.$ Using Excel, calculate how much will you have by the time you graduate from college if you make a set

payment into this annuity every month?

b$.$ Using Excel, calculate how much will you have after you make $12$ payments?

c$.$ Confirm result you received in part b$.$ by populating the following table and using the same logic as

was used to develop the formula for future value of increasing annuity.