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Full Version: 12c TVM problems where N is less than 1?
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Hello,

I have a problem that I can solve for PV on other financial calculators without any difficulty but which just returns 0 on the 12c. I am looking for the PV of \$115,000 per annum for 6 months at an effective annual rate of 7.65%.

Because this is an effective rate with compounding built in already, and the 12c doesn't convert effective rates to nominal rates, I can't just divide that rate by 12 to get a monthly rate; it would be inaccurate. So my approach would be to set N at 0.5, PMT at 115000 and I at 7.65. This works quite well on other calculators, but the 12c returns a result of 0 for PV.

Can the 12c not solve for PV where N is less than 1? I can get the right result on the 12c using the actual formula for the PV of an annuity but, wow, I might as well go and use a pen and paper at that rate!

Can you show us what would the (monthly) cash flow look like?

Hi.

Just for the sake of comparison: which other calculator did you use, how did you perform the calculation and what results did you get?

Cheers.

Luiz (Brazil)

Did you mean to put 115000 in PMT? Maybe FV would be more appropriate, as otherwise the payment happens at the end of the year (and since we only care about six months, that means it didn't happen at all).

First of all: I do not own a 12C and I never did. But while the 12C usually expects an integer number of periods, it seems to be able to handle compound interest rates as well. Here's a quote from the manual, p. 58:

Quote:
At your option, the calculations of i, PV, PMT, and FV can be performed with either simple interest or compound interest accruing during the odd period. If the C status indicator in the display is not lit, simple interest is used. To specify compound interest, turn the C indicator on by pressing [STO] [EEX]. Pressing [STO] [EEX] again turns the C indicator off, and calculations will then be performed using simple interest for the odd period.

Does this work for you?

Dieter

Halo, Dieter;

I also took this into consideration, but when compound interest is performed this way, the integer part is supposed to be different of zero and it accepts non-integer values for 'n' (number of periods). The expression to calculate compound interest has a factor that is computed with

`(1+i)^(n)`
In this factor, 'n' is the period of time (accepts non-integer values) and 'i' is the absolute value for interest rate (the percentage value divided by 100). If 'n' is zero, this factor is always '1' ('i' is supposed to be positive, non-zero).

I tried with payment in the beginning and in the end of the compound period, having the odd (fractional) period being considered as compound or single interest (C indicator on and off) but it seems to me that there is some major consideration in the way the problem is handled with the HP12C, so I asked him to show us the way he did with the other models.

As a teacher, I'm much interested on knowing what is going on, too.

Cheers.

Luiz (Brazil)

Edited: 11 Nov 2012, 9:30 a.m.

James,

tested on HP12c+ as:

```f CLEAR FIN
100 PV
107.65 CHS FV
i? => 0.62 ...
g BEG
0 FV
.5 g n (=> 6)
115000 PMT
PV? => -679,521,81```

Same results on TI BAII+ Professional, HP-17bII, BWK 7.02

Best regards

PS: Corrected earlier post because you've requested calculating with an effective rate of 7.65% p.a.

Edited: 11 Nov 2012, 10:11 a.m.

In your example, one period equals one month. So you're calculating the PV for a monthly payment of 115.000,- ...

Dieter

Hi, Peter.

As James mentioned:

Quote:
(...) the 12c doesn't convert effective rates to nominal rates, I can't just divide that rate by 12 to get a monthly rate; it would be inaccurate.
I can only guess this is the compound calculation that he was trying to avoid, thought.

Cheers.

Luiz (Brazil)

In this case a closer look at the used formulas is helpful - cf. 12C manual, page 186. As far as I can see, the desired result would be returned if the standard formula (without odd period) would be used even if n is not an integer. The two other formulas that handle odd periods - while INT(n) = 0 - can be simplified to

```  0  =  PV * (1 + 0,5 i) + FV
```
resp.
```  0  =  PV * (1+i)0,5 + FV
```
which means that for FV = 0 the solution for PV is simply zero as well, just as stated in the original post.

A possible solution here may be the following approach: simply force the 12C to use the standard formula, i.e. make sure n is an integer, for instance 1. Since FV is zero, this can be accomplished by using a modified interest rate:

```  0,0765 ENTER ENTER 1 + 0,5 CHS y^x  CHS 1 + / 1 - 100 x
```
Of course this may also be done with a short user program.

This calculation returns an interest rate of 111,4045 percent. Now use this for [i] and enter 1 as [n], and you should get PV = -54398,08. Which is the result the 34s TVM solver returns for the original data. ;-)

Dieter

Indeed, this is a much better way of looking at the problem and exploring the solution.

Luiz (Brazil)

@Dieter,

Thx.

Seems I was misled by James using PMT for entering the 115.000 ... X=(

Calculating again:

```f CLEAR FIN
100 PV
107.65 CHS FV
2 n
i? => 3,7545 ... (which satifies condition 7.65% p.a.)
0 fv
115000 pmt
1 n (which satisfies condition .5 year)
pv? => -110.838,55
```

@Luiz,

not using compound calculation can be controlled via German Savingsbook-Method (Sparbuchmethode) of the European HP-17bII+
This method cumulates "normal" interest until the years end.

There we have:

```
#J   =         0.50 (# of years)
I%J  =         7.65 (i% p.a.)
BARW =  -110 763.30 (pv)
RATE =   115 000.00 (pmt)
ENDW =         0.00
T    =       180.00 (1st pmt after 180 of 360 days)
#RJ  =         2.00 (pmts per year)
#I%J =         1.00 (compounding periods per year)
```

Because my 12c+ is a pure RPN unit, I'm not using it very often - compared to the 17bII. So I have to investigate the 12c issue a bit more now.

Best regards

Edited: 11 Nov 2012, 1:31 p.m.

Quote:
-110 763.30 (pv)

That's what I get on both the HP-12C and HP-12 Prestige with default settings:

```f CLEAR FIN
115000 FV
.5 n
7.65 i
PV    -->    -110,763.30
```

I had tried to change the settings, but I wasn't able to make them return PV = 0 on this example.

Regards,

Gerson.

Edited: 11 Nov 2012, 3:07 p.m.

Dieter: If I follow those keystrokes that you've provided, I get a nominal rate of -96.1 percent, which doesn't look very appealing. I think the keystrokes you want are

1,0765 ENTER 0,5 y^x 1 - 100 *

to get the semiannual effective rate.

EDIT: Trying it again, I do get the 111 you got above. Don't know what I did the first time, sorry. And I think I see where you're going with this, which isn't where I was going.

Edited: 12 Nov 2012, 2:55 a.m.

Hello everyone,

Wow, lots of feedback, what a great bunch. Sorry for my lack of response - been a busy past couple of days. To be clear, I am comparing the HP 12c to a Sharp EL-738 which does return the "correct" result when using an N of less than 1. By "correct", I mean the result I am expecting and the one that matches the result of manually calculating the PV using the formula for an annuity I learned back at university. I will write out the actual key presses at the end of the post.

If the 7.65% in the problem I described were a simple interest rate, then I would just divide by 12 using g,I on the 12c, but it is actually compound interest - the annual effective rate. On the EL-738, I can easily convert between effective rates (EFF) and nominal rates with monthly rest periods (APR) by keying in 12(x,y)7.65 2nd F ->APR. Once I have the APR with monthly rests then I have a rate that can be divided by 12. In other words, where the annual effective rate is 7.65%, the annual APR with monthly rests is 7.39%, which can then be divided into a monthly rate of 0.62%. Just dividing the original effective rate by 12 gives 0.64% per month.

Small difference, I know, but it adds up.

So, the fact that I cannot easily convert the 7.65% effective annual rate to a nominal rate with monthly rests, I cannot solve the main TVM problem with months as the period unit. I am stuck with years. Obviously this would be resolved if there were some way to convert the effective rate to a nominal rate automatically on the 12c, but there isn't according to this link:

Now, I remember looking this up a while ago and thinking it was far too hard to remember on the spot, but taking a second look at it now, it doesn't seem TOO bad. However, far more complex than the process on the EL-738 (I have a BAII+ sitting in the drawer - maybe I should start using that?).

Finally, just to be clear, the context of the problem is valuing a 5-year lease where there is a rent-free period of 6 months at the start and the annual rent is \$115,000, with a discount rate of 7.65% p.a. effective. I need the PV of that rent free period, so I can then deduct it from the overall value of the lease. Also, solving it from the other direction (just valuing 4.5 years' income and forgetting about the 6 months) is no good - the rent-free period needs to be broken down explicitly.

I should add that I don't necessarily disagree with the 12c's logic, that is to say that after only half a period, no actual cash flow has occurred so the result SHOULD be zero, but unfortunately the 12c leaves me no easy way to solve this on a monthly basis due to the effective/nominal rates issue. Hopefully this makes sense.

KEY PRESSES TO SOLVE ON EL-738

115000 PMT

0.5 N

7.65 I/Y

COMP PV => -54,398.08

KEY PRESSES ON 12c

115000 PMT

0.5 n

7.65 i

PV => 0.00

SOLVING USING MY FORMULA FOR THE PV OF AN ANNUITY

(apologies in advance, I have no idea how to format this properly)

PV factor: [(1+i)^n-1]/i x 1/(1+i)^n

Therefore PV factor: [(1.0765)^0.5-1]/0.0765 x 1/(1.0765)^0.5

Reduces to: 0.49 x 0.96 = 0.47

now

PV: PV factor x PMT

Therefore PV: 0.47 x 115,000 = \$54,398.08

Hi Peter,

0.62 is actually the i figure I need. I am really curious to know how you got the result of 0.62. I see your keystrokes, but how can you solve for i with just a PV and FV? What are you putting in n?

Best,
James

I suspect he's putting in 12 for n. (That's what matches what I did anyway.)

It's true that there isn't a button for interest conversions, but the link you provided is the mechanism for doing the conversion on the 12c and is what Peter did.

If we're looking for a technique that only requires the "big five" buttons without a lot of formulas, I think this is the sort of thing you're looking for:

```g BEGIN
1 PV
1.0765 CHS FV
2 n
i
0 PV
115000 FV
PMT
```

The basic idea would seem to be that if you are looking for how much six rent-free months "cost" you, then work out what the equivalent rent payment would be if you charged rent by a six-month period instead of by the year.
This first four lines compute the periodic interest rate (I chose semiannual since you are looking at six-month periods, but you could do 12 for monthly etc), then used that to see what the equivalent payment would be for that six-month period.

Hi.

For the records, I have a TI MBA (70's) and it matches the EL-38 with the same keystroke sequence (with a few label changes to match the keys):

```115000 [PMT]
0.5  [N]
7.65 [%i]
[CPT] [PV] => -54,398.08051```
Go figure...

Hi.

This is the fastest way I know (so far) do convert nominal to effective interest rate. In this case, n must be 12 (year -> 12 months). You use the initial amount of 100 (PV) and add your interest rate to it to compute the final amount, say, 7.65 [+] if you wish (FV). Select the number of months (12) for 'n' and compute the effective rate for this nominal annual increase with [i]. Please, try this:

```[f]CLEAR[FIN]
12 [n]
100 [PV]
112 [CHS][FV]  (alternatively: [+][CHS][FV] adds interest rate in Y-register, which coincides with # of periods in this example)
[i]  ->   0.95 (the effective monthly rate for a 12% nominal year rate)```
This sequence computes an effective monthly rate for a nominal interest rate of 12% a year. Not 1% a month, as we'd have by simply dividing the interest rate by 12, or using 12 [g][12÷].

Cheers.

Luiz (Brazil)

Edited: 12 Nov 2012, 8:36 a.m.

Yes, the HP 12C does not convert interest rates from one number of compounding periods per year to another. And, James, a lease is an example of an annuity due where the payments are made at the beginning of the period. So you need to do two things. Set the 12C to BEGIN mode. Next convert your effective interest rate to one of semi-annual compounding. This can be done with the financial keys, but I've created a program which I've use for years. It can convert between different compounding periods for interest/interest rates, discount/discount rates, discount/interest rates, interest/discount rates, and continuous rates to any of the others.

I'll present the program listing below, but first your problem. With your calculator in BEGIN mode, calculate the PV of 5 lease payments. I get \$498,880.95. Next you need to convert the annual lease payment into a semi-annual payment. For this you need to convert the annual rate of 7.65% to a semi-annual rate. My program gives you 3.753517..... With the calculator still in BEGIN mode, enter N=2, I=3.753517..., PV=115,000 (since the payments are made at the beginning of the period), and FV=0. Compute PMT. This is the payment that must be made at the beginning of the lease and the six-month period to equal the payment of \$115,000 at the beginning of the period. The PMT I get is \$58,559.53. Subtract this from the PV of the 5 lease payments and your answer for the PV of the lease should be \$440,321.42.

My program for interest rate conversions:

```1.    STO 3
2.    Rv
3.    STO 2
4.    Rv
5.    1
6.    0
7.    0
8.    /
9.    STO 1
10.   RCL 2
11.   X=0
12.   GTO 21
13.   RCL 1
14.   X<>Y
15.   /
16.   1
17.   +
18.   RCL 2
19.   Y^X
20.   GTO 23
21.   RCL 1
22.   e^X
23.   RCL 3
24.   x=0
25.   GTO 31
26.   1/X
27.   Y^X
28.   1
29.   -
30.   GTO 33
31.   Rv
32.   LN
33.   1
34.   0
35.   0
36.   *
37.   0
38.   X<>Y
39.   X<=Y
40.   CHS
41.   STO i
42.   RCL i
43.   RCL 3
44.   X=0
45.   GTO 52
46.   *
47.   0
48.   X<>Y
49.   X<=Y
50.   CHS
51.   GTO 00
52.   Rv
/ = division
* = multiplication
- = subtraction
```
This program takes 3 arguments. The interest or discount rate to be converted. The number of its compounding periods (a negative value if it is a discount rate). And the number of compounding periods it is to be converted to (a negative value if it is a discount rate). If either rate is a continuous rate, enter 0 for its compounding periods.

The converted interest rate divided by its new number of compounding periods is automatically stored in the i register. The new nominal rate is displayed.

For example: Convert 5.5% compounded monthly to a rate compounded quarterly.

5.5 ENTER
12 ENTER
4 R/S
5.5252 is displayed
1.3813 is stored in the i register

Note: Many times interest rates are referred to as discount rates. A true discount rate is used when interest is paid in advance, not in arrears as with interest rates on loans and mortgages.

Edited: 25 Nov 2012, 3:35 p.m. after one or more responses were posted

Hi, MacDonald;

I hope you forgive me but I took the liberty of giving your listing a different look. Please, take this as a respectful act.

Consider these symbols and meanings:

/ = division
* = multiplication
- = subtraction
x<>y = X exchange Y
Rv = roll down

```1.    STO 3
2.    Rv
3.    STO 2
4.    Rv
5.    1
6.    0
7.    0
8.    /
9.    STO 1
10.   RCL 2
11.   X=0
12.   GTO 21
13.   RCL 1
14.   x<>y
15.   /
16.   1
17.   +
18.   RCL 2
19.   Y^X
20.   GTO 23
21.   RCL 1
22.   e^X
23.   RCL 3
24.   x=0
25.   GTO 31
26.   1/X
27.   Y^X
28.   1
29.   -
30.   GTO 33
31.   Rv
32.   LN
33.   1
34.   0
35.   0
36.   *
37.   0
38.   x<>y
39.   X<=Y
40.   Rv
41.   STO i
42.   RCL i
43.   RCL 3
44.   X=0
45.   GTO 52
46.   *
47.   0
48.   x<>y
49.   X<=Y
50.   CHS
51.   GTO 00
52.   Rv```
Hopefully this listing reads a little better, and it is totally based on your original one. As a matter of fact, I copied your original listing and pasted it here prior to edit it.

Best regards.

Luiz (Brazil)

Edited: 12 Nov 2012, 10:17 p.m.

Luis,

That's the way I wrote it, but when I posted it it was screwed up. Thanks!

Hi, MacDonald.

My pleasure!

There is a way to keep whatever you type in when writing the post exactly the way it is. You just need to add a [pre] - as in pre-formated - in the top of it and a [/pre] after the last line you wish to look this way. So, if you key in:

[pre]

```1  STO 00
2  2
3  *
4  RCL 00
5  -```
[/pre]

it will be shown like this:

```1  STO 00
2  2
3  *
4  RCL 00
5  -```

Best regards.

Luiz (Brazil)

Quote:
The PMT I get is \$58,559.53.

Thank you for this -- I had originally gotten this answer but it didn't match the answer from the formula as posted, so I had been confused. I went back to double-check the formula, and I think I see the problem, as that is the formula for a plain annuity, not an annuity due. For an annuity due, you should get [(1+i)^n-1]/i * (1+i)/(1+i)^n.

[NOTE to the OP: If you put your other calculators in BEGIN mode, you should also get this answer from your n=0.5 setup, or at least the calculators I have also give that.]

```I have recently acquired a 12-C 30th AE, so I decided to have a go at what you're trying to do with this.  If I understand you correctly, there's a five year term on this lease and only five payments:
Pmt    When
---    ----
1     6 mos
2     beg of year 2
3     beg of year 3
4     beg of year 4
5     beg of year 5
There is no payment at the beginning of year 1 (hence the "free rent for 6 mos").
I'm not a financial guru, so I may be a bit fuzzy on the terminology and rules of interpretation here.  If I understand the concepts correctly it seems to me that this problem can be broken down into three steps:
1) Calculate the PV of the part of the lease that has the 115K payments (relative to the start of the second year)
2) Calculate the partial payment at 6 mos
3) Determine the true PV for both of the above amounts and add them together
Here's how I did the above:
1) PV of 115K payments at beginning of 2nd year
CLEAR FIN
g BEG
4  n
7.65  i
115000  PMT
PV     =>     -413,247.84   (STO 1 for later use)
2) Partial payment at 6mos
CLEAR FIN
g BEG
2  n
100 CHS   PV
107.65  FV
i     =>     3.75  (STO 2 for later use)
0  PV
115000  FV
PMT     =>     -54,398.08
3) Move amounts to beginning of lease and sum
(6mos PMT is already in X register on stack)
CHS  FV
1  n
RCL 2  i     (3.75 [6mos eff. int rate])
0  PMT
PV     =>     -52,429.60  (STO 3 for later use)
CLEAR FIN
1  n
7.65  i
RCL 1  FV     (PV of 4 115K payments)
PV     =>     383,880.95
x<>Y CHS
+     =>     436,310.55
Granted, my TVM experience is limited to a university-level financial math class in the early '80s.  Have I misinterpreted something important for this situation?
```

The things that stick out are:
In your step 2 the interest rate is right, but since we are dealing with pre-payments we should put the 115000 in PV and the 0 in FV.

And in your step 3b, you shouldn't change the interest rate since you only want to move back six months and not a full year. And if you're going to do that, you don't want to add your part two answer to this answer; after all you're not getting any money for those free six months so there's nothing to add. (That should give you \$398,293.83.)

EDIT: No wait sorry, for 3b I was thinking of four yearly payments and a free six months, not six months out of five full years. Sorry. You'll need to fix step 2 still.

Edited: 25 Nov 2012, 2:18 a.m.

Luiz,

0.95% is the effective monthly rate for an effective annual rate of 12%.

For a nominal annual rate of 12%, the effective monthly rate is 1% and the effective annual rate is 12.68%

Cheers, Werner

I haven't quite had time to wrap my head around this yet, but it feels like just changing the PV<->FV values in step 2 isn't all that would need to change. Doesn't that introduce an inconsistency with regards to the treatment of the other "moves"?

Quote:
I haven't quite had time to wrap my head around this yet, but it feels like just changing the PV<->FV values in step 2 isn't all that would need to change. Doesn't that introduce an inconsistency with regards to the treatment of the other "moves"?

Since the other payments are in the PMT register, putting the calculator in BEGIN mode will make sure all the other payments are at the beginning. Since we are only using PV/FV here, the BEGIN/END setting won't do anything for us.