# High School Math Tip 12: Tackling Superannuation and Repayment with Time Value of Money

Suppose you are asked to find the total amount at the end of 20 years if \$500 was invested at the beginning of every year at 6% p.a..

The usual method requires us to find the total value after 1 year, after 2 years, and so on, until we find a general pattern and then calculate the total amount after 20 years by a considering it as a geometric series. This method works fine and indeed you are encouraged to apply this method when the question mandates it.

However, a more efficient method is to consider the time value of money. What would the very first \$500 be worth at the end of the 20 years?

Since it was invested at the start of the first year, it has 20 full years to accumulate (compound) interest. Hence its value at the end of the 20 years is

Now what about the second \$500? Since it was invested exactly 1 year later, it has one year less to accumulate (compound) interest. Hence its value at the end of the 20 years is

This pattern continues until the very last investment. Since the last \$500 only has 1 year to accumulate interest, its value at the end of the 20 years is

Hence the total value of the investments at the end of the 20 years is

Note that there are necessarily 20 terms in the summation since each term corresponds to the one of the 20 investments made over the 20 years.

A quick reasonableness check is also recommended. A total of \$500 x 20 = \$10 000 has been invested. Considering the effect of interest, nearly doubling is not unreasonable.

Consider another example:

\$2000 was borrowed at 12% p.a. to be repaid in equal monthly instalments over 10 years. Find the value of each monthly instalment.

Since the instalments are monthly, the compounding frequency can also be assumed to be monthly, with a rate of

Let the value of each monthly instalment be M.

At the end of the 10 years, the loan is worth

since it has the full 120 months to accumulate (compound) interest.

At the end of the 10 years, the first repayment is worth

since it has 119 months to accumulate (compound) interest.

We can assume that repayments are made at the end of each time period, because otherwise we could have just borrowed \$M less.

At the end of the 10 years, the second repayment is worth

since, occurring exactly 1 month after the first repayment, it has 118 months to accumulate (compound) interest.

Continuing this pattern as we did before, at the end of the 10 years, the last repayment is worth just

since it has no time to accumulate any interest whatsoever.

To repay fully, the total value of the repayments has to equal the value of the loan at the end of the 10 years. We can thus work out M:

Again, a reasonableness check is recommended. A total of about \$30 x 120 = \$3600 was repaid. Considering the effect of interest, nearly double the amount borrowed is not unreasonable.

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