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 B new 1 i B ID: 264676
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Slide1
The Solution of a Difference Equation for a Compound
Interest AccountSlide2
Basic Compound Interest Formula
Recall that the new balance of an account that earns compound interest can be found by using the formula
B
new
=
(1 +
i
)
B
previous
.
Recall the following:
B
new
represents the new balance (which can also be thought of as the next balance)
B
previous
represents the previous balance
i
represents the interest rate per compound period (i.e.
i
=
r
/
m
)Slide3
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
)
n
P
.
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.Slide4
The formula,
F
=
(1
+
i
)
n
P
,
is a solution
of the
difference equation
B
new
=
(1 +
i
)
B
previous
.
The following slides are a proof of the above statement.Slide5
Recall the General Form of a Difference Equation
In chapter 11, a generalized difference equation (with an initial value) is written as:
y
n
=
a∙y
n-1
+
b
,
y
0
where
y
n
represents the next value in the list
y
n-1
represents the previous value in the list
a
is some value that is the coefficient of
y
n-1
b
is some constant value
y
0
is the initial valueSlide6
The formula
B
new
=
(1 +
i
)
B
previous
is a
difference equation since it is in the form of
y
n
=
a∙y
n-1
+
b .
The next slide will show this, but first we will
rewrite
B
new
=
(1 +
i
)
B
previous
as
B
new
=
(1 +
i
)
∙B
previous
+
0
.Slide7
B
new
= (1 +
i
)
∙
Bprevious + 0 yn = a ∙ yn-1 + bwhereBnew is equivalent to the yn Bprevious is equivalent to the yn-1 a = (1 + i)b = 0Note that b/(1 – a) = 0 [Proof: b/(1 – a) = 0/(1 – (1 + i)) = 0 / (– i) = 0. ]Slide8
The Initial Value
The initial value,
y
0
, for a account that earns compound interest is the principal,
P
(which is the initial deposit). Thus
y
0 = P.Slide9
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
y
n
= b/(1 – a) + ( y0 – b/(1 – a) ) an . Now substituting the chapter 10 notations into the above formula.Slide10
The Substitutions
Substituting 0 for
b
/(1 –
a
),
P
for
y
0 , 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. Slide11
Simplification Using Algebra
The formula
y
n
= 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. Slide12
The
y
n
Notation Change
Now since
y
n
, 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 .Slide13
Therefore
F
= (1
+
i)n P is the solution of the difference equation Bnew = (1 + i)Bprevious . Q.E.D.