To calculate the interest payable on a loan, we need to know the principal, the interest rate, and the interval at which interest is compounded. The typical case is where interest is compounded annually, that is, every year we calculate the interest accumulated over the past year, add it on to the amount already owed, and charge interest on the whole sum over the next year. But other compounding intervals are possible. For example, suppose you are quoted an interest rate of i per month, compounded monthly. What annual interest rate does this correspond to? We can answer this in two stages: firstly, i per month, compounded monthly, corresponds to 12i per year, compounded monthly. This is the nominal interest rate. However, if we want to know what annual interest rate compounded annually corresponds to i, we need the effective annual interest rate, which is given by j=(1+i)12-1.
Consider two extreme cases:
If we divide the year into M intervals, the interest rate for each interval will be r/M. The effective annual interest rate will then be
j=(1+r/M)M-1
=((1+r/M)M/r)r-1
=er-1
( Or, you can just as easily convert the continuous interest rate to an equivalent effective annual rate, via the above formula.)
The cash-flow diagram shows receipts by an arrow above the horizontal axis, and payouts by an arrow below the axis. The horizontal axis is divided into time periods. The interest rate should be clearly shown. It's useful to keep the arrow heights roughly to scale -- this will give you an idea of which effects are negligible -- but there's no point in making them exactly to scale, since you're not going to be using the diagram for any kind of geometrical construction.
All these methods compare alternative strategies available to you or your company. In addition to the financial consequences of each alternative strategy, there may be other relationships between the strategies. In particular, they may be independent, exclusive, or contingent. The most permissive of these three cases is where the strategies are independent; in this case, we can implement all of them, none of them, or any combination of them. The least permissive is the case where the strategies are exclusive -- we can implement at most one of them. In the third case, causal connections between the strategies may only permit us to implement them in certain combinations.
It is important to note that which of these categories we're dealing with must be determined before the economic analysis; it doesn't emerge from the economic analysis. Whichever case we have, we can re-define it as an example of the second case, by exhaustively listing all the legitimate combinations of strategies.
Suppose we have a sum of money P to invest. If we don't choose any of the alternative strategies, we could just put the money in the bank. What is the present worth of this `do nothing' alternative?
Putting the money in the bank is worth -P, since we're paying P to the bank. If we leave the money in the bank for N years, it will then be worth F=P(F/P,i,N). The present worth of F is then F(P/F,i,N)=P(F/P,i,N)(P/F,i,N)=P. So, adding the present worth of expenditures and receipts, we get 0. This shows that any alternative with a positive present worth is preferable to doing nothing.
Some strategies may guarantee us an infinite series of payments, or commit us to an infinite series of payouts. (For example, hiring Methuselah to a tenure-track position). What is the present worth of such a series?
The present worth of such a series is referred to as its capitalized cost, and can be calculated as follows. If A is one payment in the series, then
P = A (P/A,i,N) as N tends to infinity
= A((1+i)^N-1/(i(1+i)^N)
= A/i
A company has been using manual drafting methods for thirty years. It currently employs 10 drafters at $800/week each. The head of the drafting department is considering two alternatives:
(i) The department can buy 8 low-end workstations at $2,000 each. Two of the drafters can be given twelve months notice; at the end of the twelve months they will get $5,000 severance pay each. The remaining 8 can be trained in AutoCAD; the first training course is available in twelve months, and costs $2,000 for each participant. After completing this course, each drafter gets a $100/week raise.
(ii)The department can buy five high-end workstations at $5,000 each. All of the current drafters will be given a year's notice, and five new graduates hired at $1,200/week. These new graduates will be trained in Pro-Engineer; to keep current with this package, they will need a $5,000 retraining session every six months.
The department is currently able to perform its assigned drafting services for the rest of the company, and either of the two alternatives would allow it to continue that performance.