This lecture (which is shorter than usual because of the quiz) deals with a small but potentially confusing detail which we skimmed over earlier in the semester.
We have discussed the difference between discrete and continuous compounding of interest. However, in both cases we have considered cash flows to occur at the end-points of discrete time intervals. There are many situations where this isn't true; for example, suppose an investment earns money continuously, and we put the weekly earnings in the bank every Friday, so over a year our bank balance grows from zero to $1,000. How does the bank calculate the interest it owes us?
We first introduce some notation: if an amount $F is paid in many regular, uniform installments over a year, we denote it by `F bar' [an F with a bar over the top, which I haven't yet figured out how to write in HTML.].
Compounding
Discrete
Continuous
Cash Flow
Discrete
(Familiar formulae or tables in Appendix A)
(Convert i=er OR use tables in Appendix B)
Continuous
Up to the bank
`Funds Flow Conversion Factor' OR use tables in Appendix C
If the bank uses discrete compounding, it can choose any one of several conventions to determine the principal on which it calculates interest. For example, it can take the sum in the account at the end of the period; or the average sum in the account; or (this is what my bank does) the minimum amount in the account over the period.
One way of taking an average is to use the `mid-period convention', according to which all of the funds deposited over the year are considered to be deposited halfway through the year. If the interest rate is i, the interest on a principle P is then calculated from
F = P(1+i)0.5
If the bank uses continuous compounding, there is only one formula that can be used for calculating interest, the funds flow conversion formula. (which you should not memorise.)
This formula gives the amount A you'll have in the bank at the end of the year, given that the bank continuously compounds at a nominal rate of r% per year, and that the total amount you pay into the bank over the year in regular installments is A bar.
A = A bar ((er-1)/r)
This is the formula used to generate the tables in Appendix C of the textbook.
(Neither the quizzes, mid-term nor final exam in this course will require
you to know this formula.)