At the beginning of the recent film,
Sense and Sensibility
, Mr John Dashwood and his avaricious wife, Mrs
John Dashwood, are debating how he can most cheaply discharge his obligations towards his half-sister, the
recently widowed Mrs Henry Dashwood and her three daughters, Elinor, Marianne and Margaret. He first
considers giving her a lump sum of $1,500; then, since $1,500 seems a lot to part with in one lump, he
considers paying her an annual sum of $100 for as long as she lives. But, objects his wife, although Mrs
Dashwood is old, such an arrangement will encourage her to cling to life for an unreasonable time; and,
should she survive for more than 15 years, her half-brother will lose money by the arrangement. [Actually,
Mrs Dashwood is only 40, but Jane Austen evidently considers this to be near senility; Mrs Dashwood
herself says at one point in the novel that she can scarcely expect to survive another fifteen years.]
It is clear from this episode that Mrs John Dashwood cannot have studied engineering economics; for in fact,
Mr Dashwood
would
save money by the latter alternative, even if Mrs Henry Dashwood were to live
considerably longer than 15 years. Mrs John Dashwood's error lies in comparing future and present sums of
money. To make such a comparison valid, we must take into account the potential of money to earn interest.
$100 in hand will always be worth more than $100 a year from now, by the amount of interest that the sum
will earn over that year.
To make sensible comparisons between present and future sums, then, we must bring all the sums being
compared into the same moment of time. This moment can be either present or future. The value of present
money increases as we move it into the future, while the value of future income falls as we move it back to
the present.
The central notion here is that of
equivalence
. A present and future sums of money are equivalent if a rational
person would be indifferent as to which he or she received. Thus, for example, if the best available interest
rate is 10%, I should be indifferent between receiving $100 now or $110 this time next year. (In this, and in
the remainder of this lecture, we will assume an inflation rate of zero.)
The calculations needed to establish equivalence are relatively straightforward, but, since these calculations
will often be steps in more complex calculations, it is useful to establish a standard notation for the quantities
involved. The four most needed concepts are
P
, the present worth;
F
, future worth (or `compound amount');
i
, the interest rate per period; and
N
, the number of time periods we are considering.
These quantities are related by the simple formula
P=F(P/F,i%,N)
and its inverse,
F=P(F/P,i%,N)
The conversion factor can be calculated from a simple formula, which you can readily deduce, but it is
usually more convenient to look it up in compound interest tables, such as those found at the back of most
engineering economics textbooks.
We also need to consider the common case in which a series of equal payments are made at regular intervals.