(a) In your own words, describe the steps required to calculate the balance owing on a reducing balance loan after n years.
(b) Take a reducing balance loan problem whose solution you know and use the steps in part (a) to reconstruct the solution for the balance owing.
(c) If the answer you obtained using your description doesn't match the known answer, edit your description and try again. Repeat with other problems until you are happy that your description is correct.
(a) Using the process outlined in the lesson, derive a recursive formula for the value of an annuity after n years using simple interest.
You will need to know that
\[ \begin{align}
1+2+3+ . . . +n &= \frac{n(n+1)}{2}
\end{align} \]
(b) A perpetuity is an annuity that pays an amount A periodically in perpetuity. The value of a perpetuity is
\[ \begin{align}
V &= \frac{A}{r}
\end{align} \]
where r is a rate expressed as a real number.
This is often used to value the dividend stream from a company. Look up the latest dividend from CBA and value the dividend stream using an interest rate of 5%.
You can also use perpetuities to measure relative risk. A higher rate implies higher risk.
Look up the latest dividends from CBA and BHP and adjust the rate in the perpetuity equation so the value matches the share price in each case. Which is riskier - lending money for housing or digging commodities out of the ground?