Home |
Search |
Today's Posts |
#1
Posted to microsoft.public.excel.worksheet.functions
|
|||
|
|||
Interest Functions
Hi Guys I have a problem, and am not too good in excel ... and am not a financial wizard so please be patient :-) The file I am working with is available 'here' (http://www.jacqdar.com.au/bits/AMORTIZE2.xls) (1) I need to work out the value of 2 years worth of interest on a loan - details would be Loan = 30,180.00 interest = 14.90% pa (calculated daily) term of loan would be 60 months I am currently using an ammortisation speadsheet which calculates the interest per month etc ... at this time I simply highlight and calculate the interest which would be two years worth ... the problem is this calculation varies from the bank interest calculations (although the repayments are the same and the total interest payable amounts are the same) The banks calculations are said to be done on the basis that the minimum monthly repayment is met and there are no additional fees thrown in ... so I cant understand the difference ... (2) In addition to the above question, but working on the same loan, traditionally amortisation tables calculate the amount of the repayments, the interest component and the principal component ... some include a column where you can factor in additional repayments on a row by row basis (like the one I am referring to) the table assumes that the payment is made on the due date ... what I would like to be able to do is (a) insert the actual payment dates (b) if no payment is recorded for that period, automatically insert an overdue fee Any and all help greatly appreciated. MTIA Darrin -- dwe ------------------------------------------------------------------------ dwe's Profile: http://www.excelforum.com/member.php...o&userid=27914 View this thread: http://www.excelforum.com/showthread...hreadid=490382 |
#2
Posted to microsoft.public.excel.worksheet.functions
|
|||
|
|||
Interest Functions
"dwe" wrote:
The file I am working with is available 'here' (http://www.jacqdar.com.au/bits/AMORTIZE2.xls) [....] Loan = 30,180.00 interest = 14.90% pa (calculated daily) term of loan would be 60 months I am currently using an ammortisation speadsheet which calculates the interest per month etc ... [....] the problem is this calculation varies from the bank interest calculations That does not surprise me. The bank probably compounds interest daily, whereas the spreadsheet you point to compounds monthly. (although the repayments are the same and the total interest payable amounts are the same) The latter would surprise me, unless the bank's statement of the total interest is only an estimate. If 14.9% is the nominal annual rate, the daily rate is 14.9%/365. Since you have the payment date in the spreadsheet, the monthly interest can be computed as: (PreviousBalance)*(1 + 14.9%/365)^(DueDate - PreviousDueDate) - (PreviousBalance) Actually, you should compute =ROUND(...,2), where "..." is the expression above. And you need to add the loan date in the row with the initial balance. Does that come a little closer to the bank's computation? PS: If 14.9% is the APR, other adjustments are needed to match the bank's computation. |
#3
Posted to microsoft.public.excel.worksheet.functions
|
|||
|
|||
Interest Functions
Postscript and errata ....
I wrote: the monthly interest can be computed as: (PreviousBalance)*(1 + 14.9%/365)^(DueDate - PreviousDueDate) - (PreviousBalance) The following alternative might seem more clear: ROUND(FV(14.9%/365, DueDate - PreviousDueDate,, -PreviousBalance), 2) - PreviousBalance Actually, you should compute =ROUND(...,2), where "..." is the expression above. With the previous formula, "..." should be just the "(...)*(...)^(...)" part. That is, round before subtracting the previous balance in the end. |
#4
Posted to microsoft.public.excel.worksheet.functions
|
|||
|
|||
Interest Functions
"dwe" wrote in message
... Hi Guys I have a problem, and am not too good in excel ... and am not a financial wizard so please be patient :-) The file I am working with is available 'here' (http://www.jacqdar.com.au/bits/AMORTIZE2.xls) (1) I need to work out the value of 2 years worth of interest on a loan - details would be Loan = 30,180.00 interest = 14.90% pa (calculated daily) term of loan would be 60 months 1 - Banks don't know mathematics 2 - Then the often make mistakes in calculations 3 - When they make mistakes the loss is always for client So, these are the exact parameters for your plan (as is results from your file): Loan = 30 180.00 Montly interest = 1.2416768% Terms = 60 months A monthly interst rate of 1.2416768% corresponds to 15.961% annual rate. You get this with the formula: (1 + 0.012414768)^12 - 1 = 0.15961 = 15.961% From 15.961% per year you get daily interest rate with: (1 + 0.15961)^(1/365) - 1 = 0.000405791 = 0.0405791% Then the overdue fee for 23 day-delay in montly payment is: 716.40 * ((1 + 0.000405791)^23 -1) = 6.71623 Anything less then this is a gift compared with mathematical results, anything more is a robbery. I suppose the second is the case. Ciao Bruno |
#5
Posted to microsoft.public.excel.worksheet.functions
|
|||
|
|||
Interest Functions
Hi Bruno and Joe - many thanks for the responses ... Wow, well I am lost somewhat ... but will keep going ... Bruno - how do you arrive at 15.961% annual rate - when the interest said to be charged by the bank is 14.9% ... MTIA Darrin -- dwe ------------------------------------------------------------------------ dwe's Profile: http://www.excelforum.com/member.php...o&userid=27914 View this thread: http://www.excelforum.com/showthread...hreadid=490382 |
#6
Posted to microsoft.public.excel.worksheet.functions
|
|||
|
|||
Interest Functions
To calculate an effective annual rate, ask the question "if I borrowed $100 at
the start of the year, made no payments, how much would I owe at the end of the year?". If interest is compounded monthly, the answer is: =FV(14.9%/12,12,0,-100) = 115.961 You now know that 14.9% compounded monthly is the same as 15.961% compounded annually. 14.9% compounded daily is an effective annual rate of 16.06%. As you can see, the compounding period has a significant effect on the actual interest being charged. -- Regards, Fred "dwe" wrote in message ... Hi Bruno and Joe - many thanks for the responses ... Wow, well I am lost somewhat ... but will keep going ... Bruno - how do you arrive at 15.961% annual rate - when the interest said to be charged by the bank is 14.9% ... MTIA Darrin -- dwe ------------------------------------------------------------------------ dwe's Profile: http://www.excelforum.com/member.php...o&userid=27914 View this thread: http://www.excelforum.com/showthread...hreadid=490382 |
#7
Posted to microsoft.public.excel.worksheet.functions
|
|||
|
|||
Interest Functions
"Bruno Campanini" wrote:
"dwe" [...] wrote: Loan = 30,180.00 interest = 14.90% pa (calculated daily) term of loan would be 60 months [....] So, these are the exact parameters for your plan [...]: Loan = 30 180.00 Montly interest = 1.2416768% Terms = 60 months A monthly interst rate of 1.2416768% corresponds to 15.961% annual rate. You get this with the formula: (1 + 0.012414768)^12 - 1 = 0.15961 = 15.961% I think you intended to write 1.2416667% (14.9%/12). I suspect that you had correctly computed (1 + 14.9%/12)^12 - 1 = 15.960861%. But after you rounded that to 15.961%, you tried to reverse-engineer the monthly rate with: (1 + 0.15961)^(1/12) -1 = 1.2416768%. From 15.961% per year you get daily interest rate with: (1 + 0.15961)^(1/365) - 1 = 0.000405791 = 0.0405791% I concur that this computation would be consistent with the usual way to compute the monthly payment. But I would use accurate numbers to minimize numerical error, namely: (1 + (1 + 14.9%/12)^12 - 1)^(1/365) - 1 = (1 + 14.9%/12)^(12/365) - 1 = 0.0405788%. To be honest, I am not sure that lenders do not do just that, instead of compounding 14.9%/365, as I had said. Your formula would explain why the OP's numbers match the lender's computation of the payment and the total interest, even though interest for individual months did not match. I am trying to get ahold of my daughter's loan records to vet all of this for modern (US) loans. Then the overdue fee for 23 day-delay in montly payment is: 716.40 * ((1 + 0.000405791)^23 -1) = 6.71623 That might be one way for the lendor to compute the late fee plus interest. But there is no "mathematical" right or wrong here. In fact, the late fee could be totally unrelated, as is the case with my 30-year-old loan. The late fee was 8% per month times the monthly payment, even though the nominal annual rate was 8.75%. Anything less then this is a gift compared with mathematical results, anything more is a robbery. Perhaps. But arguably the lender chooses a high late fee partly to discourage you from being late and partly due to uncertain opportunistic costs that the lender might incur because late payments might mean that the lender has less capital to work with. |
#8
Posted to microsoft.public.excel.worksheet.functions
|
|||
|
|||
Interest Functions
" wrote
in message ... "Bruno Campanini" wrote: "dwe" [...] wrote: Loan = 30,180.00 interest = 14.90% pa (calculated daily) term of loan would be 60 months [....] So, these are the exact parameters for your plan [...]: Loan = 30 180.00 Montly interest = 1.2416768% Terms = 60 months A monthly interst rate of 1.2416768% corresponds to 15.961% annual rate. You get this with the formula: (1 + 0.012414768)^12 - 1 = 0.15961 = 15.961% I think you intended to write 1.2416667% (14.9%/12). I suspect that you had correctly computed (1 + 14.9%/12)^12 - 1 = 15.960861%. But after you rounded that to 15.961%, you tried to reverse-engineer the monthly rate with: (1 + 0.15961)^(1/12) -1 = 1.2416768%. No. I calculated effective montlhy rate of interest from: 1 - principal 30 180.00 2 - terms = 60 month 3 - montly payment = 716.40 Say Im the monthly interest Iy the yearly interest Id the daily interest Then from (1 + Iy) = (1 + Im)^12 I get Iy = (1 + Im)^12 - 1 From (1 + Iy) = (1 + Id)^365 I get Id. = (1 + Iy)^(1/365) This is the only mathematically-correct way to make such calculations. For sure the parts may agree on a more practical way of making such calculations. But a "mora practical way" translates in a little "more profit" for one part and a little "more loss" for the other. No doubt on this point. That might be one way for the lendor to compute the late fee plus interest. But there is no "mathematical" right or wrong here. In fact, the late fee could be totally unrelated, as is the case with my 30-year-old loan. The late fee was 8% per month times the monthly payment, even though the nominal annual rate was 8.75%. Perhaps. But arguably the lender chooses a high late fee partly to discourage you from being late and partly due to uncertain opportunistic costs that the lender might incur because late payments might mean that the lender has less capital to work with. I agree with you on this point. Normally the bank states a higher interest rate for late overdue fee, the client accepts and they undersign an act. Then the "act" is the "law" between them. You can only tell a mathematical judgment, nothing more. Ciao Bruno |
#9
Posted to microsoft.public.excel.worksheet.functions
|
|||
|
|||
Interest Functions
Hi. Just two cents:
=Effect(14.9%,12) returns 15.961% We note the equation for Effective monthly rate from above... =(1 + 14.9%/12)^12 - 1 We note that as the time period tends towards infinity, the above equation reduces to: =EXP(14.9%)-1 16.067% -- Dana DeLouis Win XP & Office 2003 "Fred Smith" wrote in message ... To calculate an effective annual rate, ask the question "if I borrowed $100 at the start of the year, made no payments, how much would I owe at the end of the year?". If interest is compounded monthly, the answer is: =FV(14.9%/12,12,0,-100) = 115.961 You now know that 14.9% compounded monthly is the same as 15.961% compounded annually. 14.9% compounded daily is an effective annual rate of 16.06%. As you can see, the compounding period has a significant effect on the actual interest being charged. -- Regards, Fred "dwe" wrote in message ... Hi Bruno and Joe - many thanks for the responses ... Wow, well I am lost somewhat ... but will keep going ... Bruno - how do you arrive at 15.961% annual rate - when the interest said to be charged by the bank is 14.9% ... MTIA Darrin -- dwe ------------------------------------------------------------------------ dwe's Profile: http://www.excelforum.com/member.php...o&userid=27914 View this thread: http://www.excelforum.com/showthread...hreadid=490382 |
#10
Posted to microsoft.public.excel.worksheet.functions
|
|||
|
|||
Interest Functions
I agree that EFFECT and NOMINAL are simple ways of converting to/from effective
interest rates, but I find people understand much more quickly when I use my future value example. As soon as they calculate the future value of $100, they say "now I understand how compound interest works." -- Regards, Fred "Dana DeLouis" wrote in message ... Hi. Just two cents: =Effect(14.9%,12) returns 15.961% We note the equation for Effective monthly rate from above... =(1 + 14.9%/12)^12 - 1 We note that as the time period tends towards infinity, the above equation reduces to: =EXP(14.9%)-1 16.067% -- Dana DeLouis Win XP & Office 2003 "Fred Smith" wrote in message ... To calculate an effective annual rate, ask the question "if I borrowed $100 at the start of the year, made no payments, how much would I owe at the end of the year?". If interest is compounded monthly, the answer is: =FV(14.9%/12,12,0,-100) = 115.961 You now know that 14.9% compounded monthly is the same as 15.961% compounded annually. 14.9% compounded daily is an effective annual rate of 16.06%. As you can see, the compounding period has a significant effect on the actual interest being charged. -- Regards, Fred "dwe" wrote in message ... Hi Bruno and Joe - many thanks for the responses ... Wow, well I am lost somewhat ... but will keep going ... Bruno - how do you arrive at 15.961% annual rate - when the interest said to be charged by the bank is 14.9% ... MTIA Darrin -- dwe ------------------------------------------------------------------------ dwe's Profile: http://www.excelforum.com/member.php...o&userid=27914 View this thread: http://www.excelforum.com/showthread...hreadid=490382 |
#11
Posted to microsoft.public.excel.worksheet.functions
|
|||
|
|||
Interest Functions
Hey guys this is great and I really appreciate the assistance ... One more thing I need to work out is the rate of total interest payable on a worst case scenario .. 1 - principal = 30, 180.00 2 - interest is 14.90% pa calculated daily 3 - terms = 60 month 4 - montly payment = 716.40 5 - late payment fee = 85 6 - interest on arrears is calculated at 24% pa on a daily basis example: the borrower loans $30,180.00 - his monthly repayments are due on say the 1st of every month following the date of funding - he fails to pay his first instalment - but pays it 14 days late - he would then be liable for the payment of $714 + $85.00 + 24% on $714 for 14 days How do I equate this to a % on a pa basis ... such as 14.90% equates to 15.961% pa Hoping I make sense ... Regards DWE -- dwe ------------------------------------------------------------------------ dwe's Profile: http://www.excelforum.com/member.php...o&userid=27914 View this thread: http://www.excelforum.com/showthread...hreadid=490382 |
#12
Posted to microsoft.public.excel.worksheet.functions
|
|||
|
|||
Interest Functions
Fred - I have to say your $100 example assisted me greatly in understanding the effective interest rate ... I am truly greatful for simplified answers because as I said in my opening post - I am no financial wizard :-) DWE -- dwe ------------------------------------------------------------------------ dwe's Profile: http://www.excelforum.com/member.php...o&userid=27914 View this thread: http://www.excelforum.com/showthread...hreadid=490382 |
#13
Posted to microsoft.public.excel.worksheet.functions
|
|||
|
|||
Interest Functions
dwe wrote:
6 - interest on arrears is calculated at 24% pa on a daily basis [....] How do I equate this to a % on a pa basis ... such as 14.90% equates to 15.961% pa The correct way to ask the question is: what is the effective annual rate of the nominal annual rate of 24% compounded daily. The answer is: =FV(24%/365,365,,-1) - 1 which is 27.11%. That is the same as (1 + 24%/365)^365 - 1 and EFFECT(24%,365). But I hope the FV() formulation helps you understand the concept. |
#14
Posted to microsoft.public.excel.worksheet.functions
|
|||
|
|||
Interest Functions
"Bruno Campanini" wrote:
" [...] wrote: So, these are the exact parameters for your plan [...]: Loan = 30 180.00 Montly interest = 1.2416768% Terms = 60 months [....] I think you intended to write 1.2416667% (14.9%/12). I suspect that [...] you rounded that to 15.961%, you tried to reverse-engineer the monthly rate with: (1 + 0.15961)^(1/12) - 1 = 1.2416768%. [....] No. I calculated effective montlhy rate of interest from: 1 - principal 30 180.00 2 - terms = 60 month 3 - montly payment = 716.40 Then I compute 1.24168311% = RATE(60,716.40,-30180). If you continue to disagree, please provide the exact formula or method and numbers that you use to compute 1.2416768%. I would be very interested -- no pun intended ;-). This is the only mathematically-correct way to make such calculations. No need to pontificate. You are preaching to the choir. I was not questioning your methods, merely your number. If by "mathematically correct", you mean a monthly interest rate that reduces the principal to zero with 60 payments all of which are $716.40, then RATE(60,716.40,-30180) -- approx 1.24168311% -- is the correct rate, not 1.2416768%. This is self-evident if you build an amortization table and dispense with any rounding. With 1.2416768%, the last payment must be $716.29, whereas with RATE(60,716.40,-30180), the last payment is "exactly" $716.40, formatted to 6 decimal places. Alternatively, you can use the following formula: FV = (-30180 + 716.40*(1 - (1+i)^(-60))/i)/((1+i)^(-60)) When i = 1.24168311%, FV is nearly zero (-7E-6); FV is -2E-8 when i = RATE(60,716.40,-30180). When i = 1.2416768%, FV is 0.11, indicating that that 60 payments of $716.40 is too much. $0.11 is consistent with requiring the last payment to be $716.29 ($716.40 - $.11) in the aforementioned amoritzation table. No matter! I think you make the important point that lenders might use (1 + 14.9%/12)^(12/365), not 14.9%/365, to determine the effective daily rate. The first formula yields the same effective annual rate as the nominal monthly rate (14.9%/12). Arguably, that does seem more "mathematically consistent". I cannot say for sure at this time which formula lenders use -- or even if they all make the same choice. |
#15
Posted to microsoft.public.excel.worksheet.functions
|
|||
|
|||
Interest Functions
" wrote
in message ... [...] Then I compute 1.24168311% = RATE(60,716.40,-30180). If you continue to disagree, please provide the exact formula or method and numbers that you use to compute 1.2416768%. I would be very interested -- no pun intended ;-). No pun at all, only a little of mathematics. Given: Pm = monthly payment n = terms = 60 months Im = monthly interst = 0.01246768 C = principal = 30180 Im Pm = C * ----------------------------- = 716.3987949... = 716.40 1 1 - ( ----------------) ^ n 1 + Im But I don't have Im, just I must calculate it from the above formula. I can do this in a lot of ways. I chose the iteration method so: Pm 1 Im = ------- * 1 - ( ----------------) ^ n C 1 + Im Put Im = your "guess" (say 0.0125) into the second member. You get Im = 0.012472421 Put again this Im into the second member You get Im = 0.012454076 Put again this Im into the second member You get Im = 0.012441771 You see Im is decreasing every iteration, starting from the 6th figure, the first 5 figures being stable. Do until you have the first 10 or 15 or what you want figures stable. No need to pontificate. You are preaching to the choir. I was not questioning your methods, merely your number. If by "mathematically correct", you mean a monthly interest rate that reduces the principal to zero with 60 payments all of which are $716.40, then RATE(60,716.40,-30180) -- approx 1.24168311% -- is the correct rate, not 1.2416768%. This is self-evident if you build an amortization table and dispense with any rounding. With 1.2416768%, the last payment must be $716.29, whereas with RATE(60,716.40,-30180), the last payment is "exactly" $716.40, formatted to 6 decimal places. I can't continue any longer discussing of mathematics while you are discussing of Excel formulas. Here is my amortization table made in Excel sheet using mathematical, not Excel's, formulas. A = 30 180.00 Tassi relativi a Durate in periodi Rate/anno = 12 Anni 1/12 di anno di 1/12 di anno i(1) = 15.96% 2 0.012416768 24 i(2) = 15.96% 2 0.012416768 24 i(3) = 15.96% 1 0.012416768 12 iv = 6.00% 0.004867551 60 716.40 42 983.93 12 803.93 30 180.00 Rata QI QC DE DR 30 180.00 1 716.40 374.74 341.66 341.66 29 838.34 2 716.40 370.50 345.90 687.56 29 492.44 3 716.40 366.20 350.20 1 037.76 29 142.24 4 716.40 361.85 354.55 1 392.31 28 787.69 5 716.40 357.45 358.95 1 751.26 28 428.74 .................................................. .................................................. ................ 58 716.40 26.04 690.36 28 773.45 1 406.55 59 716.40 17.46 698.93 29 472.39 707.61 60 716.3987949 8.786260412 707.6125345 30180 -2.17824E-10 For your convenience I've formatted the last row with full decimals. No need of adjusting any elements to get the exact balance. Only format cells to two decimals. Excel is an extraordinary good program, allowing everybody to deal with complex mathematical problems, but it uses often algorithms and approximation methods which sometimes give results slightly different from the ones you can get with proper math formulas. I think this is one case. As from my opinion, the difference laying mainly in that 14.90/12 which is only allowed if 14.90 is a "nominal yearly interest rate monthly convertible". Ciao Bruno |
#16
Posted to microsoft.public.excel.worksheet.functions
|
|||
|
|||
Interest Functions
" wrote in message ...
[...] Then I compute 1.24168311% = RATE(60,716.40,-30180). If you continue to disagree, please provide the exact formula or method and numbers that you use to compute 1.2416768%. I would be very interested -- no pun intended ;-). No pun at all, only a little of mathematics. Given: Pm = monthly payment n = terms = 60 months Im = monthly interst = 0.01246768 C = principal = 30180 Im Pm = C * ----------------------------- = 716.3987949... = 716.40 1 1 - ( ----------------) ^ n 1 + Im But I don't have Im, just I must calculate it from the above formula. I can do this in a lot of ways. I chose the iteration method so: Pm 1 Im = ------- * 1 - ( ----------------) ^ n C 1 + Im Put Im = your "guess" (say 0.0125) into the second member. You get Im = 0.012472421 Put again this Im into the second member You get Im = 0.012454076 Put again this Im into the second member You get Im = 0.012441771 You see Im is decreasing every iteration, starting from the 6th figure, the first 5 figures being stable. Do until you have the first 10 or 15 or what you want figures stable. No need to pontificate. You are preaching to the choir. I was not questioning your methods, merely your number. If by "mathematically correct", you mean a monthly interest rate that reduces the principal to zero with 60 payments all of which are $716.40, then RATE(60,716.40,-30180) -- approx 1.24168311% -- is the correct rate, not 1.2416768%. This is self-evident if you build an amortization table and dispense with any rounding. With 1.2416768%, the last payment must be $716.29, whereas with RATE(60,716.40,-30180), the last payment is "exactly" $716.40, formatted to 6 decimal places. I can't continue any longer discussing of mathematics while you are discussing of Excel formulas. Here is my amortization table made in Excel sheet using mathematical, not Excel's, formulas. A = 30 180.00 Tassi relativi a Durate in periodi Rate/anno = 12 Anni 1/12 di anno di 1/12 di anno i(1) = 15.96% 2 0.012416768 24 i(2) = 15.96% 2 0.012416768 24 i(3) = 15.96% 1 0.012416768 12 iv = 6.00% 0.004867551 60 716.40 42 983.93 12 803.93 30 180.00 Rata QI QC DE DR 30 180.00 1 716.40 374.74 341.66 341.66 29 838.34 2 716.40 370.50 345.90 687.56 29 492.44 3 716.40 366.20 350.20 1 037.76 29 142.24 4 716.40 361.85 354.55 1 392.31 28 787.69 5 716.40 357.45 358.95 1 751.26 28 428.74 .................................................. ........ 56 716.40 42.87 673.53 27 401.20 2 778.80 57 716.40 34.50 681.90 28 083.09 2 096.91 58 716.40 26.04 690.36 28 773.45 1 406.55 59 716.40 17.46 698.93 29 472.39 707.61 60 716.3987949 8.786260412 707.6125345 30180 -2.17824E-10 For your convenience I've formatted the last row with full decimals. No need of adjusting any elements to get the exact balance. Only format cells to two decimals. Excel is an extraordinary good program, allowing everybody to deal with complex mathematical problems, but it uses often algorithms and approximation methods which sometimes give results slightly different from the ones you can get with proper math formulas. I think this is one case. As from my opinion, the difference laying mainly in that 14.90/12 which is only allowed if 14.90 is a "nominal yearly interest rate monthly convertible". Ciao Bruno |
#17
Posted to microsoft.public.excel.worksheet.functions
|
|||
|
|||
Interest Functions
"Bruno Campanini" wrote:
I can't continue any longer discussing of mathematics while you are discussing of Excel formulas. And I cannot continue any discussion with you as long as you suffer from the delusion that anything I have said has anything to do with the Excel implementation of financial functions. Although I might use Excel functions to express the exact value of something, I have verified my results independently using "mathematics" as you put it -- by which I presume you mean algebraic formulas involving simple arithmetic operators (+,-,*,/,^) -- and using "mathmetics" with another calculating device (HP 12C). (HP 12C results differ slightly from PC computation after 9 significant digits. I conjecture that the HP 12C, released in 1981, might not use the IEEE-754 double-precision format internally, first release as a standard in 1985. Just a guess.) I will just tidy up some loose ends and be done with this discussion. I chose the iteration method so: Pm 1 Im = ------- * 1 - ( ----------------) ^ n C 1 + Im [....] Do until you have the first 10 or 15 or what you want figures stable. When I do this, I get the result that I expected, namely 1.24168310958368% -- approx 1.24168311% -- after 58-60 iterations, which matches the RATE() results, I might add. I cannot say why you get different results. There can be many reasons, none of which have to do with "mathematical correctness" (approach). No matter! Your method of determining the interest rate based on the (rounded?) payment is fundamentally flawed, at least for the purpose of this discussion. The OP asked why his/her computation of the interest amount during a period did not match the lender's. In effect, the OP is asking how lenders determine the interest rate and the interest amount. Lenders do not determine the interest rate from the payment amount, as you attempt to, nor do they have to in order to be "mathematically correct". Instead, they compute the payment based on the nominal interest rate, which they set based on business requirements, along with other terms of the loan (loan amount, term of the loan, repayment frequency, and interest charge frequency). There is nothing fundamentally with letting the payment depend on the interest. To summarize .... If the interest is compounded monthly, the mortgage payment can be computed by: PMT = PV * r / (1 - 1/((1 + r)^n)) where PV is the loan amount, n is the length of the loan in months, and r is i/12, where i is the nominal annual interest rate (not the APR). In Excel, it is simply PMT(i/12, n,, -PV). Both methods yield the same result for the OP's loan parameters, namely $716.396878002440. The lender should always round __up__ PMT at least to cents to ensure that the last payment is no more than the others -- unless the lender chooses to disclose the difference. In my experience, lenders do round up. If the interest is compounded daily (as in the OP's case), you correctly point out that the mortgage payment __should__ be computed as above, but r is (1 + i/365)^(365/12) - 1. In Excel: PMT((1 + i/365)^(365/12) - 1, n,, -PV). In Excel, r computed using FV(i/365, 365/12,, -1) - 1. All of theses approaches yield the same result, namely r = 1.24915081949526% and PMT = $717.818668999167 -- approx $717.82. However, I cannot say if any lenders use either of those formulas to determine the payment when interest is compounded daily. Instead, the payment might still be determined by the compounded-monthly formula, and the daily interest rate might be as simple as i/365. Either or both can result in a larger last payment, which I believe the lender would have to disclose in the Reg Z statement (for US loans). That certainly seems to be that case for the OP. I believe that is the "mathetical incorrectness" that you intended to identify. And I agree. As an aside, I hasten to point out that even if the lender uses the more accurate compounded-daily formula, the last payment is likely to be slightly higher -- at least in amortization tables that I have created. This is because months contain 30, 31, 28 and 29 days, not 365/12 days. So there will always be some "mathematical incorrectness" due to numerical analytical realities, albeit this disparity is small in the OP's amortization table. (In the OP's case, equal payments of $718.06 with a final payment of $717.28 would work.) In any case, I do not recall seeing disclosure of a balloon payment in the conventional US mortgages that I have been involved in directly or indirectly. So I wonder if the lender is prepared to simply eat the extra interest in the last period (suprise!). Many mortgages are paid off before the last scheduled payment anyway. Nonetheless, if the payment is based on the monthly compounding rate, but interest is compounded daily, I believe that results in an acceleration of interest -- and indeed an excess of interest paid if the loan lasts long enough. Since I sincerely doubt the latter can be the case for US loans at least, I have some doubts about this analysis. Unfortunately, I do not have any (US) Reg Z statements close at hand to look at. So I cannot check the analysis against reality -- not until next week. But I hope this summary serves to put the discussion back on track. PS: One remaining question in my mind is whether lenders round the interest computation each month, or if they carry the numbers (interest and balance) to whatever precision their computer hardware/software allows. |
#18
Posted to microsoft.public.excel.worksheet.functions
|
|||
|
|||
Interest Functions
" wrote
in message ... It seems to me that's clear enough we both know financial mathematics to correctly deal with an amortization plan. May be I don't know enough your language to go deeper end deeper into the matter. Then let me summarize the origin of my objection. It was that 14.90% simply declared as "Annual interest rate" instead of "Annual nominal interst rate, monthly convertible". In fact the table was built up using a monthly effective interest rate of 14.9%/12, which leads to (1+14.9%/12)^12-1 =15.96...% as annual effective interest rate. Then that "Annual interest rate = 14.90%" in not correct or very ambiguous. (HP 12C results differ slightly from PC computation after 9 significant digits. I conjecture that the HP 12C, released in 1981, might not use the IEEE-754 double-precision format internally, first release as a standard in 1985. Just a guess.) My HP12C doen't work any more; as a handy calculator I use its brother HP11C or HP28S. But I made the calculations on this matter only using Excel worksheet elementary formulas, not those infamous Single/Double VBA data types. Do until you have the first 10 or 15 or what you want figures stable. When I do this, I get the result that I expected, namely 1.24168310958368% -- approx 1.24168311% -- after 58-60 iterations, which matches the RATE() results, I might add. I cannot say why you get different results. There can be many reasons, none of which have to do with "mathematical correctness" (approach). I depends mainly to the fact of considering the monthly payment equal to 716.40 rounded to 2 decimals, or 716.40 as a format made up 15 decimals. To summarize .... If the interest is compounded monthly, the mortgage payment can be computed by: PMT = PV * r / (1 - 1/((1 + r)^n)) where PV is the loan amount, n is the length of the loan in months, and r is i/12, where i is the nominal annual interest rate Which correctly leads only to an effective monthly interest rate of i/12. And you can't use i/365 to get a daily effective interest rate. But you should get (1+i%/12)^12-1 = Iy% = annual effective unterest rate; from (1+Iy%)=(1+Id%)^365 you get Id% = (1+Iy%)^(1/365)-1 as effective daily interest rate and 365*Id% as Annual nominal interst rate, daily convertible. I think these points are clear enough to both of us. Ciao Bruno |
Reply |
Thread Tools | Search this Thread |
Display Modes | |
|
|
Similar Threads | ||||
Thread | Forum | |||
User-defined functions created in Excel 2000 fail in Excel 2003 | Excel Discussion (Misc queries) | |||
Database functions should use criteria in formula, as 1-2-3 does | Excel Worksheet Functions | |||
How do I calc interest when using multiple dates within a given ye | Excel Worksheet Functions | |||
Function help with Calculating Interest for two different investme | Excel Worksheet Functions | |||
Are financial functions calculated based on compound interest? | Excel Worksheet Functions |