The Solution of a Difference
Equation for a Compound
Interest Account
Basic Compound Interest Formula
Recall that the new balance of an
account that earns compound interest
can be found by using the formula
Bnew = (1 + i) Bprevious .
Recall the following:
• Bnew represents the new balance (which can also be thought of as the next balance)
• Bprevious represents the previous balance
• i represents the interest rate per compound period (i.e. i = r/m)
Formula for a Future Balance of a
Compound Interest Account
A future balance of a compound interest account
can be found by using the formula
F = (1 + i)nP .
Recall the following:
• F represents the future balance
• n represents the number of compound periods
• i represents the interest rate per compound period (i.e. i = r/m)
• P represents the principal deposited into the account
• The above formula allows us to find a future balance without having to find any of the previous
balances.
The formula, F = (1 + i)nP , is a solution
of the difference equation Bnew = (1 + i) Bprevious
.
The following slides are a proof of the above
statement.
Recall the General Form of a
Difference Equation
In chapter 11, a generalized difference
equation (with an initial value) is written as:
yn = a∙yn-1 + b
, y0
where
• yn represents the next value in the list
• yn-1 represents the previous value in the list
• a is some value that is the coefficient of yn-1
• b is some constant value
• y0 is the initial value
The formula Bnew = (1 + i)Bprevious is a
difference equation since it is in the form of
yn = a∙yn-1 + b .
The next slide will show this, but first we will
rewrite Bnew = (1 + i)Bprevious as
Bnew = (1 + i)∙Bprevious + 0.
Bnew = (1 + i) ∙ Bprevious + 0
yn
=
where
•
Bnew is equivalent to the yn
•
Bprevious is equivalent to the yn-1
•
a = (1 + i)
•
b=0
•
Note that b/(1 – a) = 0 [Proof:
a ∙ yn-1
+ b
b/(1 – a) = 0/(1 – (1 + i)) = 0 / (– i) = 0. ]
The Initial Value
The initial value, y0 , for a account that earns
compound interest is the principal, P (which is
the initial deposit). Thus y0 = P.
Recall When Solving a Difference
Equation
Since a ≠ 1 (the reason will be, or was, discussed
in the lecture), then the solution of the difference
equation can be found using
yn = b/(1 – a) + ( y0 – b/(1 – a) ) an .
Now substituting the chapter 10 notations into
the above formula.
The Substitutions
Substituting 0 for b/(1 – a), P for y0 , and (1+i) for
a, the solution of
yn = b/(1 – a) + ( y0 – b/(1 – a) ) an
becomes
yn = 0 + ( P – 0) (1+i)n .
Note that we did not change the notation of yn ; we will address this later.
Simplification Using Algebra
The formula
yn = 0 + ( P – 0) (1+i)n
can be simplified to
yn= P (1 + i)n ;
which is equivalent to
yn= (1 + i)n P
.
We used the additive property of 0 and the commutative property of multiplication.
The yn Notation Change
Now since yn , in yn= (1 + i)n P , represents a
specified value in the list (mainly the nth value
in the list after the initial value) without using
any of the previous values (with the exception
of the initial value P), the yn can be changed
to an F to represent the future value.
So yn= (1 + i)n P is changed to F = (1 + i)n P .
Therefore
F = (1 + i)n P
is the solution of the difference equation
Bnew = (1 + i)Bprevious .
Q.E.D.