08-25-2018, 02:32 PM
This program is a slightly shorter version of the 2nd solution that uses the function NPV:
Example: 201410 → 37368
10 STO PMT
8 STO FV
2014 R/S
Formulas Used
Net Present Value
NPV = net present value of a discounted cash flow.
CFj = cash flow at period j.
\(NPV=CF_0+\frac{CF_1}{(1+i)^1}+\frac{CF_2}{(1+i)^2}+\cdots+\frac{CF_n}{(1+i)^n}\)
Percentage
\(\Delta\%=100\ \frac{x-y}{y}\)
Explanation
We can calculate the remainders by dividing the number continuously by the base we want to transform to (here stored in FV):
These remainders have to be multiplied by powers of the base we transform from (here stored in PMT):
And then when we add up all the terms we get:
To calculate the polynomial we use NPV.
For this to work we have to solve for \(i\) in:
\(1+\frac{i}{100}=\frac{1}{b}\)
where \(b\) is the base we transform from.
Thus
\(\begin{align*}
i &= 100\ (\frac{1}{b}-1)\\
&= 100\ \frac{1-b}{b}
\end{align*}\)
That's why we can use the \(\Delta\%\) function to calculate \(i\):
I was a bit surprised that ENTER is needed here:
But it appears that STO PV disables the stack lift.
From the manual:
Code:
01 - 44 13 STO PV
02 - 36 ENTER
03 - 45 15 RCL FV
04 - 10 ÷
05 - 43 25 INTG
06 - 43 13 CFo
07 - 45 15 RCL FV
08 - 10 ÷
09 - 43 25 INTG
10 - 43 35 x=0
11 - 43,33 14 GTO 14
12 - 43 14 CFj
13 - 43,33 07 GTO 07
14 - 45 14 RCL PMT
15 - 1 1
16 - 24 Δ%
17 - 44 12 STO i
18 - 45 13 RCL PV
19 - 42 13 NPV
20 - 45 14 RCL PMT
21 - 45 15 RCL FV
22 - 30 −
23 - 20 ×
24 - 40 +
Example: 201410 → 37368
10 STO PMT
8 STO FV
2014 R/S
Formulas Used
Net Present Value
NPV = net present value of a discounted cash flow.
CFj = cash flow at period j.
\(NPV=CF_0+\frac{CF_1}{(1+i)^1}+\frac{CF_2}{(1+i)^2}+\cdots+\frac{CF_n}{(1+i)^n}\)
Percentage
\(\Delta\%=100\ \frac{x-y}{y}\)
Explanation
We can calculate the remainders by dividing the number continuously by the base we want to transform to (here stored in FV):
Code:
2014 ÷ 8 = 251 → 6 = 2014 - 8 × 251
251 ÷ 8 = 31 → 3 = 251 - 8 × 31
31 ÷ 8 = 3 → 7 = 31 - 8 × 3
3 ÷ 8 = 0 → 3 = 3 - 8 × 0
These remainders have to be multiplied by powers of the base we transform from (here stored in PMT):
Code:
6 = 1 × 2014 - 8 × 1 × 251
30 = 10 × 251 - 8 × 10 × 31
700 = 100 × 31 - 8 × 100 × 3
3000 = 1000 × 3 - 8 × 1000 × 0
And then when we add up all the terms we get:
Code:
3736 = 2014 + 2 × (251 + 10 × 31 + 100 × 3)
To calculate the polynomial we use NPV.
For this to work we have to solve for \(i\) in:
\(1+\frac{i}{100}=\frac{1}{b}\)
where \(b\) is the base we transform from.
Thus
\(\begin{align*}
i &= 100\ (\frac{1}{b}-1)\\
&= 100\ \frac{1-b}{b}
\end{align*}\)
That's why we can use the \(\Delta\%\) function to calculate \(i\):
Code:
14 - 45 14 RCL PMT
15 - 1 1
16 - 24 Δ%
17 - 44 12 STO i
I was a bit surprised that ENTER is needed here:
Code:
01 - 44 13 STO PV
02 - 36 ENTER
03 - 45 15 RCL FV
But it appears that STO PV disables the stack lift.
From the manual:
Quote:In addition, the stack does not lift when a number is entered if the last operation performed was storing a number into a financial register.