Solving the TVM equation for the interest rate

[ttw]

I'd like to add that getting these interest rates is more important than most people realize. Of course, most math is more important i everyday life than most people understand for that matter.

When comparing two investments or two mortgages or something similar, one should always choose the investment with the best rate or mortgage with the least (if the payments are affordable.) It' isn't (as the radio and TV pseudo financial pseudo advisors sometime say) that one compares the "total of interest payments" or worrying about monthly payments for a purchase no longer owned (or even paying a ballplayer who's been traded.) It's only the interest rate (AKA "rate of return.") A short term mortgage mortgage (15 vs 30) years may have bigger payments but lower interest rates; if one can afford it, the lower rate always give bigger bang for the buck. The same with investments; get the best rate of return. A lower rate on car payments may make longer terms favorable; one may be paying for car (or ball player or house) already traded or sold to achieve a low effective interest rate.

Likewise, higher interest means faster retirement fund growth.

[Albert Chan]

(02-12-2020 05:25 PM)Albert Chan Wrote:  This formula is based on Pade[1,1], I centered 0, of NFV = F + P + ((1+I)^N-1)*(P+M/I)

Solving Pade[1,1] approximated NFV = 0, for 1/I :

\( \Large {1\over I} ≈
{\binom{N}{3}M + \binom{N}{2}P \over \binom{N}{2}M + \binom{N}{1}P} -
{\binom{N}{2}M + \binom{N}{1}P \over F + P + M N}\)

Doing the same example, N=11, F=40000, P=0, M=-2564

1/I ≈ (-423060/-141020) - (-141020/11796) = 44102/2949

I ≈ 2949/44102 ≈ 6.687%, which under-estimated true rate (6.780%) by tiny 0.093%

We can reuse the formula, getting a closer estimate.

With above I=6.687%, calculated F = 39804.11, error = 40000 - 39804.11 = 195.89
Tried the formula again, with F = 40000 + 195.89, to compensate this error

I ≈ 1/(3 - (-141020/(11796 + 195.89))) ≈ 6.775%

Again, using this new I, calculated F = 39989.83, error = 40000 - 39989.83 = 10.17

Interpolate for 0 error, I = 6.775% - (6.687% - 6.775%) * 10.17/(195.89 - 10.17) = 6.7798%

[Erwin]

(04-15-2018 08:48 PM)Carsen Wrote:  Problem #1: n=32 PV=-6,000 FV=10,000 PMT=0 i=???
Answers: 12C=1.609139492% 15C=1.609139501% Prime=1.60913949196%

Problem #2: n=36 PV=13,000 PMT=-372.53 FV=0 i=???
Answers: 12C=0.1692576094% 15C=0.1692569782% Prime=0.169257604168%

Problem #3: n=36 PV=5,750 PMT=-176.89 FV=0 i=???
Answers: 12C=0.56266017270% 15C=0.5626013555% Prime=0.562601726101%

Problem #4: n=360 PV=75,000 PMT=-425.84 FV-0 i=???
Answers: 12C=0.458302324% 15C=0.4583302039% Prime=0.458330232637%

The 12C results is from my 12C bought in 2016. The 15C results is your (Dieter's) program. The Prime's results is using the TVM formula in the 12C manual using your initial estimate formula. Looking at the 15C and the Prime's results, I would say that the estimate works really well. I wonder if there is a even better way to produce a estimate.

Hello,

cause I was curious I tried the above examples with the finance pac from the HP-71b with the following solutions:

Problem #1: i= 1.609139492% same as the 12c
Problem #2: i= 0.169257426%
Problem #3: i= 0.562601665%
Problem #4: i= 0.458330232%

So it looks like, that the HP-71b routine for small interest rates use a different method compared to the other calculators. But no idea what calculation method the HP-71b use, there are no hints in the user manual.

regards Erwin

[Gamo]

Albert Chan

Thanks for the comment, I found that in the HP-55 Mathematics Programs

On the Direct Reduction Loan Interest Rate the program used the

Harmonic Mean approach for the Suggested Guess solution but

the book doesn't show the formula on this I'll find out about this

approach later.

Thanks

Gamo

[Gamo]

From HP-55 Mathematics Programs

The Suggested Guess for

1. Direct Reduction Loan Interest Rate with known [n, PV and PMT]
2. Sinking Fund Interest Rate with known [n, FV and PMT]

To find the guess the HP-55 programs book used these formulas.

1. Direct Reduction Loan Interest Rate

[1 / (PV / PMT)] - [(1 / n^2) x (PV / PMT)]

2. Sinking Fund Interest Rate

[((FV/PMT) - n) x 2] ÷ [(n - 1)^2 + (FV/PMT)]

Gamo 2/2020

[Csaba Tizedes]

(04-15-2018 05:30 PM)Carsen Wrote:  

(04-15-2018 08:39 AM)Dieter Wrote:  So the result seems to depend on previous input ?!

I believe you are right Dieter. Try this on your HP-12C emulator.

CLEAR FIN
n = 10
PMT = -1,000
FV = 10,000
i = 0.001

If you solve for i, the answer is 2.152976E-11, rather than 0 when you leave the value of i at 0. This proves that since we changed the initial value in i from 0 to 0.001, it changed the result. It's almost like it uses the value in the i register as a initial guess. Just as you concluded before.

IMHO there is a test for result=0 AND if the test is false the algorithm checks the abs(ch% of result)<=eps%. It seems a more possible explanation.

Csaba

[Albert Chan]

(04-15-2018 08:48 PM)Carsen Wrote:  Problem #1: n=32 PV=-6,000 FV=10,000 PMT=0 i=???
Answers: 12C=1.609139492% 15C=1.609139501% Prime=1.60913949196%

Problem #2: n=36 PV=13,000 PMT=-372.53 FV=0 i=???
Answers: 12C=0.1692576094% 15C=0.1692569782% Prime=0.169257604168%

Problem #3: n=36 PV=5,750 PMT=-176.89 FV=0 i=???
Answers: 12C=0.56266017270% 15C=0.5626013555% Prime=0.562601726101%

Problem #4: n=360 PV=75,000 PMT=-425.84 FV-0 i=???
Answers: 12C=0.458302324% 15C=0.4583302039% Prime=0.458330232637%

I made function guess_i(n, pv, pmt, fv), fitting TVM of i as quadratics

lua> guess_i(32,-6000,0,10000) * 100
1.609567691201156
lua> guess_i(36,13000,-372.53,0) * 100
0.16925628259544864
lua> guess_i(36,5750,-176.89,0) * 100
0.5625600411596099
lua> guess_i(360,75000,-425.84,0) * 100
0.4601141074876585