Finance problems (TVM) on hp 49G+ hp 50G Periods and compounds per year are different

[Karinne]

hi guys.

I have an HP 49g+ calculator, which belonged to my older brother, I have had it for 6 months and I always use it in RPN, although I am still a beginner.

I am now using it for my finance classes. But I have a problem. In finance mode, when I want to solve TVM problems, my calculator does not have the option for payments per year (P/Yr) and compounding per year (C/Yr), separately.

And I see that other students have Casio or Texas financial calculators, and they do it right away. Or even others have Texas programmable calculators, and they have the appropriate finance menu.

Do you know if there is a program that can help me with this, or a hidden menu in the calculator?

I don't want to buy a financial calculator.

[Massimo Gnerucci]

(05-13-2024 04:00 AM)Steve Simpkin Wrote:  I may be missing something as I am not that familiar with the Finance App on the HP 49g+/HP 50g. When I choose APP, select Finance, choose TVM and press Num, I see the following:



It appears to have the P/Yr and C/Yr fields you are looking for. What am I missing?

But that's a Prime...

On the 50g C/Yr and Group Size are missing.

[Steve Simpkin]

(05-13-2024 07:03 AM)Massimo Gnerucci Wrote:  

(05-13-2024 04:00 AM)Steve Simpkin Wrote:  I may be missing something as I am not that familiar with the Finance App on the HP 49g+/HP 50g. When I choose APP, select Finance, choose TVM and press Num, I see the following:



It appears to have the P/Yr and C/Yr fields you are looking for. What am I missing?

But that's a Prime...

On the 50g C/Yr and Group Size are missing.

Well I knew I was probably missing something. I had both emulators running and appear to have pulled up the wrong one. Hopefully someone can help Karinne find a workaround.

[fhub]

(05-13-2024 01:35 AM)Karinne Wrote:  Do you know if there is a program that can help me with this, or a hidden menu in the calculator?

Well, you could use my program 'TVM-CalcPro_2015' from my website:
https://fhub.jimdofree.com/

It has C/Yr and P/Yr, and a lot of other features, too.

Regards,
Franz

[mfleming]

(05-13-2024 01:35 AM)Karinne Wrote:  I am now using it for my finance classes. But I have a problem. In finance mode, when I want to solve TVM problems, my calculator does not have the option for payments per year (P/Yr) and compounding per year (C/Yr), separately.

Go to hpcalc.org and use the Search field in the upper right corner with the text "TVM". That will return several hits including this one

https://www.hpcalc.org/details/9392

The web site hpcalc.org is a great resource for beginning users!

[Karinne]

(05-13-2024 11:05 AM)mfleming Wrote:  

(05-13-2024 01:35 AM)Karinne Wrote:  I am now using it for my finance classes. But I have a problem. In finance mode, when I want to solve TVM problems, my calculator does not have the option for payments per year (P/Yr) and compounding per year (C/Yr), separately.

Go to hpcalc.org and use the Search field in the upper right corner with the text "TVM". That will return several hits including this one

https://www.hpcalc.org/details/9392
The web site hpcalc.org is a great resource for beginning users!

edit[/b][/i]
Thank you, so much.

It's perfect.

[Albert Chan]

(05-13-2024 01:35 AM)Karinne Wrote:  In finance mode, when I want to solve TVM problems, my calculator does not have the option for payments per year (P/Yr) and compounding per year (C/Yr), separately.

Found the relation of (I/Yr, P/Yr, C/Yr) for per period effective interest rate.

i = (1 + (I/Yr)/(C/Yr)) ^ ((C/Yr)/(P/Yr)) - 1

If (P/Yr) = (C/Yr), it reduced to simple ratio: i = (I/Yr) / (P/Yr)

We can use log1p/expm1 for better accuracy.

i = expm1(log1p((I/Yr)/(C/Yr)) * (C/Yr)/(P/Yr))
(I/Yr) = expm1(log1p(i) * (P/Yr)/(C/Yr)) * (C/Yr)

Example:

lua> ipy, ppy, cpy = .05, 12, 3 -- (I/Yr), (P/Yr), (I/Yr)
lua> i = expm1(log1p(ipy/cpy) * cpy/ppy)
lua> i
0.004140875317607807
lua> expm1(log1p(i) * ppy/cpy) * cpy -- == ipy
0.05

lua> n, pv, fv = 60, 30000, 0
lua> tvm(n, i, pv, nil, fv) -- solve for pmt
-565.7117263797815

HP Prime TVM app:
N=60, PV=30000, FV=0, I%Yr=5.00, P/Yr=12, C/Yr = 3, END=1      → PMT = -565.71

[Albert Chan]

(05-13-2024 07:07 PM)Albert Chan Wrote:  lua> ipy, ppy, cpy = .05, 12, 3 -- (I/Yr), (P/Yr), (I/Yr)
lua> i = expm1(log1p(ipy/cpy) * cpy/ppy)
lua> i
0.004140875317607807

It may be easier to use APR, EFF functions for i (if we have them)

lua> APR = fn'i,n: expm1(log1p(i)/n)*n'
lua> EFF = fn'i,n: expm1(log1p(i/n)*n)'
lua> APR(EFF(ipy, cpy), ppy) / ppy
0.004140875317607807

TI BA-35 Solar

5 [►EFF] 3 =    → 5.083796293
[►APR] 12 =     → 4.969050367
[÷12]               → 0.414087531

[Gil]

Solution with the HP50G:

1) Use the finance application of the calculator, with the explained example of Albert Chan
(the summary of the 3 steps/calculations below for nominal I%YR are in bold)
2) However, instead of the initial nominal interest of 5% to be normally put in I%YR, calculate the corresponding nominal interest x, as follows:
a) Effective/real interest, relative to Compounding, for the fractional period:
5/100 / (C/YR) = 0.05/3 —> effective/real interest for 4 months = 1.667%
b) Now, calculate, always relative to the Compounding, the effective yearly interest:
(0.05/ C/YR +1)^(C/YR) = (0.05/ 3 +1)^ 3 =1.050837 —> effective real interest = 5.0837%
c) Then calculate the corresponding effective/real interest for the fractional period, relative to the Payments = 1/(P/yr) = 1/12 (ie here for 1 month):
1.050837^(1/ P/YR)=1.050837^(1/ 12)= 1.004140875 —> effective/real interest for 1 month Payment = 0.4140875%
d) Now, put in I%YR the corresponding value x (nominal yearly interest) instead of the initial nominal yearly 5%:
x=(1.004140875-1)* P/YR *100=(1.004140875-1)*12*100 = 4.969050384 for I%YR
3) Complete 60 for N, copy if necessary the above value x = 4.969... into I%YR, 30000 for PV, 12 for P/YR (that is available here!)
4) And go to PMT line and press SOLVE
to get -565.711726419.

[Albert Chan]

(05-13-2024 07:07 PM)Albert Chan Wrote:  HP Prime TVM app:
N=60, PV=30000, FV=0, I%Yr=5.00, P/Yr=12, C/Yr = 3, END=1      → PMT = -565.71

HP-12C, mode=END

5 Enter 3 ÷ [I] 0 [PMT] 1 [FV]
3 [N] [PV] 12 [N] [I]              --> 0.414087532
60 [N] 3E4 [PV] 0 [FV] [PMT] --> -565.7117265

[Albert Chan]

There is nothing special about a Year, we could go directly for i

I%/Yr = 5, C/Yr = 3                            → 12/3 = 4 months, $3 will grow to $3.05

Less keys to type, faster to calculate, and more accurate!

HP-12C, mode=END

4 [N] -3 [PV] 0 [PMT] 3.05 [FV] [I]      → 0.414087532
60 [N] 3E4 [PV] 0 [FV] [PMT]              → -565.7117264, Error = 0 ULP

Note for HP-12C: above work only if N is positive integer (too bad...)

[Gil]

A TVM short program for HP48/49/50G including both C:YR & P:YR, and using the solver.

Payments can be at the End (variable END10=1) or at the beginning of the period (variable END10=0).

Code
\<<
\<< I%YR 100 / C\166YR / 1 + C\166YR ^ P\166YR INV ^ DUP SWAP 1 - \-> ri ii
\<< NofP DUP - PV ri NofP ^ * + FV - ri NofP ^ 1 - ii / END10 1 == 1 ri IFTE * PMT * +
\>>
\>> STEQ 30 MENU
\>>


Working example
For 5%,
5 and A/F1-key, ie 5 and I%YR

For 3 Compounding in a year,
3 and B/F2-key , ie 3 and C¦YR

For 12 Payments in a year,
12 and C/F3-key, ie 12 and P¦YR

For number of payments (Total)
= P¦YR × number of years =12 * 5 = 60,
60 and D/F4-key, ie 60 NofP

30000 and E/F5-key, ie 30000 PV

0 and F/F4-key, ie 0 FV

Then press NXT-key (3rd line, 3rd column)

Then payments at the end? Yes =1
—> 1 A/F1-key, ie 1 END10

Then give any value to PMT, for instance zero
0 B/F2-key, ie 0 PMT

And now use the solver as follows:
Shift-Left (arrow left to key/figure/number 4)
& then press B/F2-key (PMT)

You should get the following result:
PMT: - 565.71.

[Albert Chan]

PM exchange between Gil and me, for the "real" meaning of C/yr
Comments welcome.

Gil Wrote:Let's say I%YR=5%
compounds/YR = 3
and payments/YR = 12.

When so doing, does the bank really gives a monthly interest of .414087532% for each monthly payment (for the first, 2nd and 3rd payment?) I understand the calculation, but I have my doubts about the effective bank practice. Or does the bank instead wait for the 4 payments to be on the account to then start to calculate/"offer" the interest?
In that latter case we should of course never have 12 payments in a year, but have only 3.

Albert Chan Wrote:I was also confused of how to generate a mortgage table, when (P/Yr) ≠ (C/Yr)
Do you get a saw-tooth shaped table?

From HP Prime amortization table, answer is No!

N=60, PV=30000, FV=0, I%Yr=5.00, P/Yr=12, C/Yr = 3, END=1 → PMT = -565.71

Code:

1    −441.48    −124.23    29558.52
2    −443.31    −122.40    29115.21
3    −445.15    −120.56    28670.06
...

1st payment, rate = 124.23 / 30000 = 0.4141%
2nd payment, rate = 122.40 / 29558.52 = 0.4141%
3rd payment, rate = 120.56 / 29115.21 = 0.4141%
...

The way I see it, (C/Yr > 1) is just a fancy way to boost-up rate, without saying so.
It is a way to confuse people. Why do we care how the bank does thing internally?

Banking shopping, ask for APY!

(I%YR = 5, C/Yr = 3) --> APY = 5.0838%
(I%YR = 5, C/Yr =∞) --> APY = 5.1271%

Gil Wrote:The HP50G has the amortisation tables that once I checked to be correct from what I learnt, but it does not provide the option C/YR (the calculation for PMT is made with C/YR = P/YR).

Not sure that the bank employees know what's going on with their interest calculations.

They use programs, as the car sellers also do, that show sometimes small unexplained discrepancies with accurate, correct calculation.

Apparently, they know how to make profits in a different way.

You can publish my and your answers, Albert.

Regards

[Gil]

For the same example
N=60
I%nominal=5%
PV=30000
FV=0
C/YR=3
P/YR=12

Suppose that you don't have the options C/YR and P/YR
on your calculator.

Then, instead of 5%, put the corresponding value x% calculated as follows:
0.05/3
0.05/3 +1
(0.05/3+1)^3
((0.05/3+1)^3)^(1/12)=1.00414087532
1.00414087532-1
x%=(1.00414087532-1)*100=0.414087532%
And you should get PMT=-565.711726.

Suppose, now, that you don't have any available program,
then procede as follows to calculate PMT:
PV*(... )+PMT*[... ]=FV=0

PV*(1.00414087532^60)
+PMT*[(1.00414087532^60-1)/0.00414087532)=0

30000*1.2813824439
+PMT*[(1.2813824439-1)/0.00414087532)=0

38441.473317
+PMT*[67.9524067] =0

PMT(at the end) =-38441.473317/[67.9524067] =565.711726
PMT (at the beginning) =PMT(at the end) / 1.00414087532
PMT (at the beginning) =-565.711726 /1.00414087532
PMT (at the beginning) = -563.378845

Balance after one full year = 12 months
(payments at the end of each month)

PV*(...)
+PMT*[...]

PV*(1.00414087532^12)
+PMT*[(1.00414087532^12-1)/0.00414087532]

PV*(1.05083796299)
+PMT*[0.05083796299/0.00414087532]

PV*(1.05083796299)
+PMT*[12.277105]

30000*(1.05083796299)
+-565.711726*[12.277105]

=24579.836

Within the first year, the debt went down from 30000 to 24579.836.
—> "Reduction" of 5420.164.
But we paid 12*PMT=12*-565.711726=6788.541.
—> Interests paid during the first year (12 full months)
=6788.541 - 5420.164 = 1368.377

Balance after exactly 12 payments
(but payments at the beginning of each month,
the 12th payment being made after exactly 11 months
[and not here after a full year=12 months]
—> balance "a few seconds" after the 11th month)

PV*(...)
+PMT*[...]/1.00414087532

PV*(1.00414087532^ 11 )
+PMT*[(1.00414087532^12-1)/(0.00414087532/1.00414087532)] /1.00414087532

PV*(1.00414087532^ 11 )
+PMT*[(1.00414087532^12-1)/0.00414087532]


PV*(1.0465045)
+PMT*[0.05083796299/0.00414087532]

PV*(1.0465045)
+PMT*[12.277105]

30000*(1.0465045)
+-563.3788*[12.277105]

31'395.1355-6916.6615
=24478.474

[Gil]

Suppose a slightly different problem
as suggested by SlideRule in another post (17B/19B GPM: initial payment)

Assume, for simplication here, that C/YR=P/YR=12 (ie, for the time being, C/YR≠3 as in the previous case).
Let PV=60000, total of years=30, I%YR=12.
The first 12 payments during a full year are, as before, all equal to PMT.
Then, the rules change, with the next 5 years the payments being increased as follows:
the second year, as you are supposed to dispose of a higher income, they increase by 7.5%, so the next 12 payments of the 2nd year will be equal to TVM×1.075;
the 3rd year, the 12 payments will be again increased by 7.5% and then be all equal to (TVM×1.075)×1.075=TVM×1.075²;
the 4th year, (TVM×1.075²)×1.075=TVM×1.075³;
the 5th year, TVM×1.075⁴;
finally, the 6th year (last and 5th increase), TVM × 1.075^5.
Then, up to the end of the reimbursement, all payments remain equal to that last amount TVM × 1.075^5.


Mathematically

PV+
{PMT×1.075^0 × [(v^12-1)/i /(1.01^12)^0)] +
PMT×1.075^1 × [(v^12-1)/i /((1.01^12)^1)] +
PMT×1.075^2 × [(v^12-1)/i /(1.01^12)^2)] +
PMT×1.075^3 × [(v^12-1)/i /(1.01^12)^3)] +
PMT×1.075^4 × [(v^12-1)/i /(1.01^12)^4)] +
PMT×1.075^5 × [(v^(360-12*5)-1)/i /(1.01^12)^5)]
} =0

PV+
{PMT×(v^12-1)/i ×[Sum, (1.075/1.01)^j, j=0..(5-1)] +
PMT×[v^(360-12×5)-1)/i]×(1.075/1.01^12)^5
}=0,
with v=1/1.01, i=0.01.

Then,
PV+PMT ×
{(v^12-1)/i ×[Sum, (1.075/1.01)^j, j=0..(5-1)] +
[v^(360-12×5)-1)/i]×(1.075/1.01^12)^5
}=0,
with v=1/1.01, i=0.01


Then,
PMT = -PV /
{(v^12-1)/i ×[Sum, (1.075/1.01)^j, j=0..(5-1)] +
[v^(360-12×5)-1)/i]×(1.075/1.01^12)^5
},
with v=1/1.01, i=0.01,
which gives PMT=-474.825.


Transposition to a more general TVM-Growth HP48-50G Solver program

I transposed the corresponding program, letting it have equally the more usual case without growing payments (I.Growing%=0 or number of Years for the Growing payments YG=0).

To check the case/example of Albert Chan and Karinne in the above post, I included the possibility of having the number of compoundings/year, C.Y, ≠ number of payments/year, P.Y.

Besides, I added here the choice of the payments being made at the end of each period (variable END10=1) or at the beginning of each period (variable END10=0).

Finally, I gave also the possibility of having a remaining global amount FV to be still paid at the very end of the process, ie at the end of YTOT.


Use

Launch the Sover program by pressing its name TVM.G.

Enter, in the Solver, your values followed by the corresponding variable names as follows: 30 YTOT 12 I%Y 12 P.Y 12 C.Y 7.5 IG%Y NEXT-key 60000 PV 0 FV 1 END10.

Then press, always in the Solver, LeftShift P1 to get the value of the payments during the first year.

The following years, the payments will be increased (or not) according to your indications (the several calculations are not included in the result here) until reaching (or not) a constant value P.CST (LeftShift P.CST or RightShift P.CST).


Suppose now that, on the first year, you could afford a payment of maximum 450/month —> PMT =-450

What would be the monthly amounts to be paid on the 2nd year and from year 6/to end of year 30?

Press -450 P1
Press NXT
Press LeftShift IG%Y

You get IG%Y = 9.04416%.
To be paid back monthly in the 2nd year: 450×1.0904416=490.6987
To be paid monthly from year 6 till end year 30:
450×1.0904416^5=693.7845
(or RightShift P.CST and you get also directly, -693.7845, with negative sign indicating that it's a sum to be reimbursed [every month]).

[Gil]

Just changed the case when no payment increase/Growth

(IG%Y=0 or #Yrs of payments Growth, YG, =0).

Instead of writing
IF
THEN YTOT 'YG' STO 1. 'rg' STO
END

I put now
IF
THEN 1 'YG' STO 1. 'rg' STO 0 'IG%Y' STO
END

Code:


\<<
  \<< YTOT I%Y + C.Y + P.Y + YG + IG%Y + PV + FV + END10 + DUP - I%Y 100. / C.Y / 1. + C.Y ^ P.Y INV \->NUM ^ DUP INV IG%Y 100. / 1. + \-> r v rg
    \<< YG IG%Y * 0. ==
      IF
      THEN 1 'YG' STO 0 'IG%Y' STO 1. 'rg' STO
      END PV + FV v P.Y ^ YTOT ^ * + 1. v P.Y ^ - r 1. - / 0. 0. YG 1. -
      FOR j rg r P.Y ^ / j ^ +
      NEXT * 1. v YTOT YG - P.Y * ^ - r 1. - / rg r P.Y ^ / YG ^ * + END10 1. == 1. r IFTE * P1 * + P1 rg YG ^ * 'P.CST' STO
    \>>
  \>> STEQ 30. MENU
\>>

[Karinne]

(05-14-2024 12:14 AM)Albert Chan Wrote:  

(05-13-2024 07:07 PM)Albert Chan Wrote:  HP Prime TVM app:
N=60, PV=30000, FV=0, I%Yr=5.00, P/Yr=12, C/Yr = 3, END=1      → PMT = -565.71

HP-12C, mode=END

5 Enter 3 ÷ [I] 0 [PMT] 1 [FV]
3 [N] [PV] 12 [N] [I]              --> 0.414087532
60 [N] 3E4 [PV] 0 [FV] [PMT] --> -565.7117265

If i have to solve [i] on hp 12C , for N=60, PV=30000, FV=0, PMT= 565.711 P/Yr=12, C/Yr = 3 I=?

How?

[Werner]

in case C/YR divides P/YR (which is the case here):
60 [N] 3e4 [PV] -565.7117265 [PMT] 0 [FV] [END] [i] -> 0.414087533
-3 * [PMT] 0 [PV] 4 [N] [FV] -> 5.000
( [RCL] [i] CYR [CHS] [*] [PMT] 0 [PV] PYR [ENTER] CYR [/] [N] [FV] -> I%YR )

Cheers, Werner

explanation: (corrected.. thanks, Albert!)

             I%YR/CYR  CYR/PYR
i% = [ ( 1 + -------- )       - 1 ]*100
               100


now we must find I%YR !

               i%  PYR/CYR
I%YR = [ ( 1 + --- )        - 1 ]*CYR*100
               100
                           (1 + i%/100)^N - 1
now, remember USFV(i%,N) = -----------------
                                i%/100

and PV*(1+i%/100)^N + PMT*USFV(i%,N) + FV = 0

so if we take N=PYR/CYR, PV=0 and PMT = -i%*CYR, FV will give us I%YR

[Albert Chan]

(06-07-2024 07:31 AM)Werner Wrote:  PV*(1+i%/100)^N + PMT*USFV(i%,N) + FV = 0

so if we take N=PYR/CYR, PV=0 and PMT = -i%*CYR, FV will give us I%YR

Too bad HP-12C have different meaning for non-integer N. Why?
Yes, I know N is period, but we can use TVM functionality for other things! Like this trick!

Just to be clear, I%YR here refer to APR (R=CYR)

I would probably multiply by CYR after TVM calculation.
That way, we get I2% = effective rate per (1/CYR year) for free.

I% = 0.414087533 --> I2% = 1.66666667 --> APR(R=CYR) = I2%*CYR = 5.00000001

---

Here is another trick, (theoretical) final payment with a 4-banger!

After we solve TVM for PMT, its value likely have fractional cents.
We can't paid fractional cents, and likely use another number, say PMT2

PV/USPV(I%,N) + FV/USFV(i%,N) + PMT = 0

I like to let c = 1/USPV(I%,N), d = 1/USFV(I%,N), because c - d = i

c*PV + d*0 + PMT = 0               // original loan
c*PV + d*FV + PMT2 = 0           // we are paying PMT2 instead, with balance FV
FV = (PMT-PMT2) / d                 // subtract, solve for balance

d = c - i = -PMT/PV - i

Final payment = PMT2 + FV = PMT2 + (PMT2-PMT)/(PMT/PV + i)

[Karinne]

(06-07-2024 07:31 AM)Werner Wrote:  in case C/YR divides P/YR (which is the case here):
60 [N] 3e4 [PV] -565.7117265 [PMT] 0 [FV] [END] [i] -> 0.414087533
-3 * [PMT] 0 [PV] 4 [N] [FV] -> 5.000
( [RCL] [i] CYR [CHS] [*] [PMT] 0 [PV] PYR [ENTER] CYR [/] [N] [FV] -> I%YR )

Cheers, Werner

Thanks!!!!!

There are many steps