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Now, let’s make a few changes to the scenario described above. Suppose that instead
of borrowing the $8,000 from his uncle, Jon borrowed the money from the same credit
union where he set up his sinking fund. Would this change how we worked the problem?
Of course not! It didn’t matter in the slightest whether Jon was borrowing the money from
his uncle, his credit union, or anyone else for that matter. Jon’s monthly sinking fund pay-
ment is exactly the same regardless of who loaned him the $8,000.
But from a practical point of view, this does sound kind of strange. After all, the idea was
that Jon wouldn’t have to make any payments to his uncle until the very end. But if the lender
is the same credit union to which he is making his deposits anyway, it would seem awfully
awkward to keep the loan and the sinking fund separate given that both are deals between
the same players: Jon and his credit union. There is no reason why the two deals couldn’t be
kept separate, but it would be far simpler to just have Jon’s monthly payments go directly
toward paying off his loan.
With that, we’ve made an important leap. The key to dealing with annuity present values
is recognizing thatJon and His Uncle
To Jon: $8,000 To his uncle:To credit union: $236.90Jon and His Credit Union
To Jon: $8,000is the same as:To credit union: $236.90Jon and His Credit Union
To Jon: $8,000This realization enables us to develop formulas for annuity present values.Formulas for the Present Value of an Annuity
Suppose that we know the payments, term, and interest rate for an annuity, and want to
determine its present value. We can find the future value that this annuity could accumulate
to by using:FV PMT s _n (^) |i
As we’ve just seen, we can relate this future value to the present value by means of the
compound interest formula
FV PV(1 i)n
In both cases, the FV we are talking about is the same amount of money. So then
PV(1 i)n PMT s _n (^) |i
since both of these are equal to the FV.
To get the present value by itself, we can divide both sides of this equation by (1 i)n
to get:
PV
PMT s _n (^) |i
___(1 i)n
Which we can adjust slightly to get:
PV PMT
s _n (^) |i
___(1 i) (^) n
Notice that this formula is essentially the ordinary annuity present value formula from the
start of this section, except that s _n (^) |i /(1 i)n is sitting where a _n (^) |i sat. Since both formulas give
us the present value, it must be that we have found our formula for a _n (^) |i!
4.4 Present Values of Annuities 171