If you have an investment or savings plan through which you earn compound interest and regularly make additional payments or deposits, you may wish to model and analyse various investment options to choose the best for you.
To do this, we can extend our knowledge from modelling compound interest.
Analysing our investment with a recurrence relation
Let's examine this using an example.
example 1
We'll invest $\$5000$$5000 at $6%$6% per annum interest compounded monthly. At the end of each month, after the interest has been added, we'll made an additional deposit of $\$100$$100.
Firstly we need to change our interest rate from annually to monthly by dividing by $12$12, so we get $0.5%$0.5%.
The balance at the end of the first few months can be calculated as follows.
| End of month 1 | $1.005\times5000+100$1.005×5000+100 | $=$= | $\$5125$$5125 |
|---|---|---|---|
| End of month 2 | $1.005\times5125+100$1.005×5125+100 | $=$= | $\$5250.63$$5250.63 |
| End of month 3 | $1.005\times5250.63+100$1.005×5250.63+100 | $=$= | $\$5376.88$$5376.88 |
Notice for our purposes here we round each balance to $2$2 decimal places and then use the rounded amount in our summary.
To be able to examine the balance in a few years time, we don't really want to manually calculate the values in the table. Instead, we'll write a recurrence relation and then analyse it our investment through the sequence facility of our calculator.
$B_{n+1}$Bn+1, the balance at the end of month $n$n, can be defined as:
$B_{n+1}=1.005B_n+100$Bn+1=1.005Bn+100, where $B_0=5000$B0=5000
Notice I've used our initial investment amount and provided $B_0=5000$B0=5000 rather than using $B_1=5125$B1=5125. We can use either of these.
To determine the value of our investment after $2$2 years, we'll use the sequence facility on our calculator.
We can see we'll have a total of $\$8178.99$$8178.99 after $24$24 months.
Analysing our investment with a spreadsheet
We can investigate what our investment would look like with a spreadsheet.
You can change the original investment amount, the annual interest rate, how often the interest is compounded and the regular deposit amount.
Note that here our extra deposits always match the number of compounding periods each year.
Sometimes we are given a portion of a spreadsheet and asked to find certain values or write various formulae used. Let's take a look at another example.
example 2
(a) What is the quarterly interest rate for this investment?
Think: We need to use one of the interest rate values in the table to help us work it out
Do: $10450r=209$10450r=209, so $r=0.02$r=0.02
The quarterly interest rate is $2%$2% or $8%$8% per annum.
(b) Write a formula for the balance at the end of the first year
Think: The balance at the end of the first year is contained in cell $E5$E5
Do: $E5=1.02\times B5+D5$E5=1.02×B5+D5
(c) How much was originally invested?
Think: We need to write a formula here to solve for $X$X in the table.
Do: $1.02X+250=10450$1.02X+250=10450
Solving we get $X=10000$X=10000
(d) Write a recurrence relation that gives the balance of the investment at the end of each quarter.
Think: We need to put everything together from any row in our spreadsheet.
Do: $B_{n+1}=1.02B_n+250$Bn+1=1.02Bn+250, where $B_0=10000$B0=10000
Analysing our investment with the CAS finance application
Let's go back to our original investment of $\$5000$$5000 invested at $6%$6% per annum compounded monthly and ask a few questions that are too time consuming to answer with a recurrence relation or a spreadsheet.
1. How long does it take for our original investment to triple in value?
We could answer this by scrolling through the table of our sequence facility on our calculator, but it will be much faster to use the finance facility on our calculator.
r
We want to solve for $N$N, the number of installment periods it will take to triple our investment.
I is our annual interest rate of $6%$6%.
PV is the present value of our investment and it is $-5000$−5000 because from our point of view we have taken $\$5000$$5000 away from ourselves to put in an investment
PMT is the payment of $\$100$$100 we'll make each month, and again we express this as $-100$−100
FV is the future value of our investment and we're interested in when this amount will be $\$15000$$15000
P/Y is the number of payments or deposits we'll make each year, which is $12$12
C/Y is the number of compounding periods each year, which is also $12$12.
Solving we get:
So it will take us $68$68months to triple our money.
2. How much should we deposit each month if we want our original investment to double at the end of two years?
Check each of the fields in the calculator to make sure you know how each has been filled in.
Solving we get:
So we would need to make deposits of $\$171.60$$171.60 each month to double our money in $2$2 years.