08-25-2018, 02:32 PM

This program is a slightly shorter version of the 2nd solution that uses the function NPV:

Example: 2014

10 STO PMT

8 STO FV

2014 R/S

Formulas Used

Net Present Value

NPV = net present value of a discounted cash flow.

CF

\(NPV=CF_0+\frac{CF_1}{(1+i)^1}+\frac{CF_2}{(1+i)^2}+\cdots+\frac{CF_n}{(1+i)^n}\)

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: 2014

_{10}→ 3736_{8}10 STO PMT

8 STO FV

2014 R/S

Formulas Used

Net Present Value

NPV = net present value of a discounted cash flow.

CF

_{j}= 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.