Comparing the present value of total payment today and partial payments over 3 months
Its kind of a dumb question because no one believes that you can earn 8% in the short term in the market, but for arguments sake the math is painfully easy.  Keep in mind I am an engineer not a finance guy. So the first payment will earn you one month at 8%, the second, two.  In effect three months at 8% on 997.  You can do it that way because the payments are equal: 997 * (.08 /12) *3 = earnings ~= 20 So with the second method you pay: 997 * 3 - 20 =  2971 "I got $3394.83 The first problem with this is that it is backwards.  The NPV (Net Present Value) of three future payments of $997 has to be less than the nominal value.  The nominal value is simple:  $2991.   First step, convert the 8% annual return from the stock market to a monthly return.  Everyone else assumed that the 8% is a monthly return, but that is clearly absurd.  The correct way to do this would be to solve for m in  But we often approximate this by dividing 8% by 12, which would be .67%.  Either way, you divide each payment by the number of months of compounding.   Sum those up using m equal to about .64% (I left the calculated value in memory and used that rather than the rounded value) and you get about $2952.92 which is smaller than $2991.   Obviously $2952.92 is much larger than $2495 and you should not do this.  If the three payments were $842.39 instead, then it would about break even.   Note that this neglects risk.  In a three month period, the stock market is as likely to fall short of an annualized 8% return as to beat it.  This would make more sense if your alternative was to pay off some of your mortgage immediately and take the payments or yp pay a lump sum now and increase future mortgage payments.  Then your return would be safer.   Someone noted in a comment that we would normally base the NPV on the interest rate of the payments.  That's for calculating the NPV to the one making the loan.  Here, we want to calculate the NPV for the borrower.  So the question is what the borrower would do with the money if making payments and not the lump sum.   The question assumes that the borrower would invest in the stock market, which is a risky option and not normally advisable.  I suggest a mortgage based alternative.  If the borrower is going to stuff the money under the mattress until needed, then the answer is simple.  The nominal value of $2991 is also the NPV, as mattresses don't pay interest.  Similarly, many banks don't pay interest on checking these days.  So for someone facing a real decision like this, I'd almost always recommend paying the lump sum and getting it over with.  Even if the payments are ""same as cash"" with no premium charged." What's the present value of using the payment plan? In all common sense the present value of a loan is the value that you can pay in the present to avoid taking a loan, which in this case is the lump sum payment of $2495.  That rather supposes the question is a trick, providing irrelevant information about the stock market. However, if some strange interpretation is required which ignores the lump sum and wants to know how much you need in the present to pay the loan while being able to make 8% on the stock market that can be done. I will initially assume that since the lender's APR works out about 9.6% per month that the 8% from the stock market is also per month, but will also calculate for 8% annual effective and an 8% annual nominal rate. The calculation If you have $x in hand (present value) and it is exactly enough to take the loan while investing in the stock market, the value in successive months is $x plus the market return less the loan payment.  In the third month the loan is paid down so the balance is zero. I.e. So the present value of using the payment plan while investing is $2569.37. You would need $2569.37 to cover the loan while investing, which is more than the $2495 lump sum payment requires.  Therefore, it would be advisable to make the lump sum payment because it is less expensive: If you have $2569.37 in hand it would be best to pay the lump sum and invest the remaining $74.37 in the stock market.  Otherwise you invest $2569.37 (initially), pay the loan and end up with $0 in three months. One might ask, what rate of return would the stock market need to yield to make it worth taking the loan? The APR proposed by the loan can be calculated.   The present value of a loan is equal to the sum of the payments discounted to present value.  I.e. with  ∴ by induction  So by comparing the $2495 lump sum payment with $997 over 3 x monthly instalments the interest rate implied by the loan can be found. Solving for r If you could obtain 9.64431% per month on the stock market the $x cash in hand required would be calculated by This is equal to the lump sum payment, so the calculated interest is comparable to the stock market rate of return.  If you could gain more than 9.64431% per month on the stock market it would be better to invest and take the loan. Recurrence Form Solving the recurrence form shows the calculation is equivalent to the loan formula, e.g. becomes v[m + 1] = (1 + y) v[m] - p where v[0] = pv where In the final month v[final] = 0, i.e. when m = 3 Compare with the earlier loan formula: s = (d - d (1 + r)^-n) / r They are exactly equivalent, which is quite interesting, (because it wasn't immediately obvious to me that what the lender charges is the mirror opposite of what you gain by investing). The present value can be now be calculated using the formula. Still assuming the 8% stock market return is per month. If the stock market yield is 8% per annum effective rate and if it is given as a nominal annual yield, 8% compounded monthly
From the description, you have a post-1998 income contingent loan. The interest rate on those is currently 1.5% but it has varied quite a bit in the last few years due to the formula used to calculate it, which is either the inflation rate (RPI), or 1% + the highest base rate across a group of banks - whichever is smaller. This is indeed really cheap credit compared to any commercial loan you could get, though whether you should indeed just repay the minimum depends on making a proper comparison with the return on any spare money you could get after tax elsewhere. There is a table of previous interest rates. From your description I think you've had the loan for about 4 years - your final year of uni, one year of working without repayments and then two years of repayments. A very rough estimate is that you would have been charged about £300 of interest over that period. So there's still an apparent mismatch, though since both you and I made rough calculations it may be that a more precise check resolves it. But the other thing is that you should check what the date on the statement is. Once you start repaying, statements are sent for a period ending 5th April of each year. So you may well not be seeing the effect of several months of repayments since April on the statement. Finally, there's apparently an online facility you can use to get an up to date balance, though the administration of the loans repaid via PAYE is notoriously inefficient so there may well be a significant lag between a payment being made and it being reflected in your balance, though the effect should still be backdated to when you actually made it. It's easiest to get your payment from the PMT function in Excel or Google Sheets.  So a $100,000 30 year mortgage at 3% looks like this: The basic calculation is pretty simple.  You take the annual interest rate, say 3%, divided by 12, times the existing principal balance: The idea is that borrowers would like to have a predictable payment.  The earlier payments are proportionally more interest than principal than later payments are but that's because there is much more principal outstanding on month 1 than on month 200. Easier to copy paste than type this out. Credit: www.financeformulas.net Note that the present value would be the initial loan amount, which is likely the sale price you noted minus a down payment.   The loan payment formula is used to calculate the payments on a loan.   The formula used to calculate loan payments is exactly the same as the   formula used to calculate payments on an ordinary annuity. A loan, by   definition, is an annuity, in that it consists of a series of future   periodic payments. The PV, or present value, portion of the loan payment formula uses the   original loan amount. The original loan amount is essentially the   present value of the future payments on the loan, much like the   present value of an annuity. It is important to keep the rate per period and number of periods   consistent with one another in the formula. If the loan payments are   made monthly, then the rate per period needs to be adjusted to the   monthly rate and the number of periods would be the number of months   on the loan. If payments are quarterly, the terms of the loan payment   formula would be adjusted accordingly. I like to let loan calculators do the heavy lifting for me. This particular calculator lets you choose a weekly pay back scheme. http://www.calculator.net/loan-calculator.html My theory, if you must be in debit, own it at the least expense possible. The interest you will pay by the end, combined with the future value of money. Example: The Future value of $3000 at an effective interest rate of 5% after 3 years =$3472.88 Present value of $3000 at 5% over 3 years =$2591.51 you will need more money in the future to pay for the same item The reason it's broken out is very specific: this is showing you how much interest accrued during the month.  It is the only place that's shown, typically. Each month's (minimum) payment is the sum of [the interest accrued during that month] and [some principal], say M=I+P, and B is your total loan balance.  That I is fixed at the amount of interest that accrued that month - you always must pay off the accrued interest.  It changes each month as some of the principal is reduced; if you have a 3% daily interest rate, you owe (0.03*B*31) approximately (plus a bit as the interest on the interest accrues) each month (or *30 or *28).  Since B is going down constantly as principal is paid off, I is also going down. The P is most commonly calculated based on an amortization table, such that you have a fixed payment amount each month and pay the loan off after a certain period of time.  That's why P changes each month - because it's easier for people to have a constant monthly payment M, than to have a fixed P and variable I for a variable M. As such, it's important to show you the I amount, both so you can verify that the loan is being correctly charged/paid, and for your tax purposes. Your calculations are correct. It is likely that the bank's software has a rounding error.  In effect, either your bank is overstating your interest by eight cents per month, or your bank is insisting that you prepay your principal by eight cents per month.  If the bank's ongoing interest calculations are correct, your final payment will be slightly smaller (because of the prepaid principal, and because of compound interest on those prepayments).  I have performed similar calculations for my mortgages over the years, and except upon early payoff in the middle of the month, I have always matched my banks' calculations to the penny.  Ironically, this means that my banks' formulas are a bit weird: After making these adjustments, even my calculations for the mid-month payoff matched to the penny. From here The formula is M = P * ( J / (1 - (1 + J)^ -N)). M: monthly payment RESULT = 980.441... P: principal or amount of loan 63963 (71070 - 10% down * 71070) J: monthly interest; annual interest divided by 100, then divided by 12. .00275 (3.3% / 12) N: number of months of amortization, determined by length in years of loan. 72 months See this wikipedia page for the derivation of the formula What you want is the distribution yield, which is 2.65. You can see the yield on FT as well, which is listed as 2.64. The difference between the 2 values is likely to be due to different dates of updates. http://funds.ft.com/uk/Tearsheet/Summary?s=CORP:LSE:USD "By the sounds of things, you're not asking for a single formula but how to do the analysis... And for the record you're focusing on the wrong thing. You should be focusing on how much it costs to own your car during that time period, not your total equity.  Formulas:  I'm not sure how well you understand the nuts and bolts of the finance behind your question, (you may just be a pro and really want a consolidated equation to do this in one go.) So at the risk of over-specifying, I'll err on the side of starting at the very beginning. Any financial loan analysis is built on 5 items: (1) # of periods, (2) Present Value, (3) Future Value, (4) Payments, and (5) interest rate.  These are usually referred to in spreadsheet software as NPER, PV, FV, PMT, and Rate. Each one has its own Excel/google docs function where you can calculate one as a function of the other 4. I'll use those going forward and spare you the 'real math' equations.  Layout: If I were trying to solve your problem I would start by setting up the spreadsheet  up with column A as ""Period"". I would put this label in cell A2 and then starting from cell A3 as ""0"" and going to ""N"". 5 year loans will give you the highest purchase value w lowest payments, so n=60 months... but you also said 48 months so do whatever you want.  Then I would set up two tables side-by-side with 7 columns each. (Yes, seven.) Starting in C2, label the cells/columns as: ""Rate"", ""Car Value"", ""Loan Balance"", ""Payment"", ""Paid to Interest"", ""Principal"", and ""Accumulated Equity"". Then select and copy cells C2:I2 as the next set of column headers beginning in K2. (I usually skip a column to leave space because I'm OCD like that :) ) Numbers: Now you need to set up your initial set of numbers for each table. We'll do the older car in the left hand table and the newer one on the right. Let's say your rate is 5% APR. Put that in cell C1 (not C3). Then in cell C3 type =C$1/12. Car Value $12,000 in Cell D3. Then type ""Down Payment"" in cell E1 and put 10% in cell D1. And last, in cell E3 put the formula =D3*(1-D$1). This should leave you with a value for the first month in the Rate, Car Value, and Loan Balance columns. Now select C1:E3 and paste those to the right hand table. The only thing you will need to change is the ""Car Value"" to $20,000. As a check, you should have .0042 / 12,000 / 10,800 on the left and then .0042 / 20,000 / 18,000 on the right.  Formulas again: This is where spreadsheets become amazing. If we set up the right formulas, you can copy and paste them and do this very complicated analysis very quickly.  Payment The excel formula for Payment is =PMT(Rate, NPER, PV, FV). FV is usually zero. So in cell F3, type the formula =PMT(C3, 60, E3, 0). Obviously if you're really doing a 48 month (4 year) loan then you'll need to change the 60 to 48. You should be able to copy the result from cell F3 to N3 and the formula will update itself.  For the 60 months, I'm showing the 12K car/10.8K loan has a pmt of $203.81. The 20K/18K loan has a pmt of 339.68.  Interest The easiest way to calculate the interest is as =E3*C3. That's (Outstanding Loan Balance) x (Periodic Interest Rate). Put this in cell G4, since you don't actually owe any interest at Period 0.  Principal If you pay PMT each month and X goes to interest, then the amount to principal is ""PMT - X"". So in H4 type =-F3 - G3. The 'minus' in front of F3 is because excel's PMT function returns a negative amount. If you want to, feel free to type ""=-PMT(...)"" for the formula that's actually in F3. It's your call.  I get 159 for the amount to principal in period 1.  Accumulated Equity As I mentioned in the comment, your ""Equity"" comes from your initial Loan-to-Value and the accumulated principal payments. So the formula in this cell should reflect that. There are a variety of ways to do this... the easiest is just to compare your car's expected value to your loan balance every time. In cell I3, type =(D3-E3). That's your initial equity in the car before making any payments. Copy that cell and paste it to I4. You'll see it updates to =(D4-E3) automatically. (Right now that is zero... those cells are empty, but we're getting there) The important thing is that as JB King pointed out, your equity is a function of accumulated principal AND equity, which depreciates. This approach handles those both.  Finishing up the copy-and-paste formulas I know this is long, but we're almost done.  Rate // Period 1 In cell C4 type =C3.  Payment // Period 1 In cell F4 type =F3.  Loan Balance // Period 1 In cell E4 type =E3-H4. Your loan balance at the end of period is reduced by the principal you paid. I get 10,641.  Car Value // Period 1 This will vary depending on how you want to handle depreciation. If you ignore it, you're making a major error and it's not worth doing this entire analysis...  just buy the prettiest car and move on with life. But you also don't have to get it scientifically accurate. Go to someplace like edmunds.com and look up a ballpark. I'm using 4% depreciation per year for the old (12K) car and 7% for the newer car. However, I pulled those out of my ass so figure out what's a better ballpark.  In G1 type ""Depreciation"" and then put 4% in H1. In O1 type ""Depreciation"" and then 7% in P1. Now, in cell D4, put the formula =D3 * (1-(H$1/12)).  Paste formulas to flesh out table As a check, your row 4 should read 1 / .0042 / 11,960 / 10,641 / 203.81 / 45 / 159 / 1,319. If so, you're great. Copy cells C4:I4 and paste them into K4:Q4.  These will update to be .0042 / 19,883 / 17,735 / 339.68 / 75 / 265 / 2,148.  If you've got that, then copy C4:Q4 and paste it to C5:C63. You've built a full amortization table for your two hypothetical loans. Congratulations.  Making your decision I'm not going to tell you what to decide, but I'll give you a better idea of what to look at. I would personally make the decision based on total cost to own during that time period, plus a bit of ""x-factor"" for which car I really liked.  Look at Period 24, in columns I and Q. These are your 'equities' in each car. If you built the sheet using my made-up numbers, then you get ""Old Car Equity"" as 4,276. ""New Car Equity"" is 6,046. If you're only looking at most equity, you might make a poor financial decision.  The real value you should consider is the cost to own the car (not necessarily operate it) during that time... Total Cost = (Ending Equity) - (Payment x 24) - (Upfront Cash).  For your 'old' car, that's (4,276) - (203.81 * 24) - (1,200) = -1,815.75 For the 'new' car, that's (6,046) - (339.68 * 24) - (2,000) = -4,106.07.  Is one good or bad? Up to you to decide. There are excel formulas like ""CUMPRINC"" that can consolidate some of the table mechanics, but I assumed that if you're here asking you would have gotten stuck running some of those.  Here's the spreadsheet: https://docs.google.com/spreadsheet/ccc?key=0Ah0weE0QX65vdHpCNVpwUzlfYjlTY2VrNllXOS1CWUE#gid=1" What you want to do is figure out how much you're paying in interest, solely (ie, the interest part of each payment), add that up over 48 months, then figure out the net value of the cash inflow/outflow for the points over 48 months (ie, 3.5% annual return on the positive or negative value).  Sum those two.  Then you can see your P&L, and your total cash outflow (up to you if you add a % to your negative initial outflow, and how exactly you consider your $2k closing costs; I agree with JoeTaxpayer about adding at least closing costs to the loan amount.  If you have money to pay the points that would otherwise be earning money, you could alternately consider it a negative cash (ie, instead of accruing 3.5% it's a negative balance accruing that). In excel I'd do something like: Then track changes in H and I when you change columns B and C and G. "You'd have to check the terms of your contract. On most installment loans, I think, they calculate interest monthly, not daily. That is, if you make 3 payments of $96 over the course of the month instead of one payment of $288 at the end of the month (but before the due date), it makes absolutely zero difference to their interest calculation. They just total up your payments for the month. That's how my mortgage works and how some past loans I've had worked. All you'd accomplish is to cost yourself some time, postage if you're mailing payments, and waste the bank's time processing multiple payments. If the loan allows you to make pre-payments -- which I think most loans today do -- then what DOES work is to make an extra payment or an overpayment. If you have a few hundred extra dollars, make an extra payment. This reduces your principle and reduces the amount of interest you pay every month for the remainder of the loan. And if you're paying $1 less in interest, then that extra dollar goes against principle, which further reduces the amount you pay in interest the next month. This snowballs and can save you a lot in the long run. Better still, instead of paying $288 each month, pay, say, $300. Then every month you're nibbling away at the principle faster and faster. For example, I calculate that if you're paying $288 per month, you'll pay the loan off in 72 months and pay a total of $6062 in interest. Pay $300 per month and you'll pay it off in 67 months with a total of $6031 interest. Okay, not a huge deal. Pay $350 per month and you pay it off in 55 months with $5449 interest. (I just did quick calculations with a spreadsheet, not accurate to the penny, but close enough for comparison.) PS This is different from ""revolving credit"", like credit cards, where interest is calculated on the ""average daily balance"". With a credit card, making multiple payments would indeed reduce your interest. But not by much. If you pay $100 every 10 days instead of $300 at the end, then you're saving the interest on 20 days x $100 + 10 days x $100, so 12.5% = 0.03% per day, so 0.03% x ($2000+$1000) = 90 cents. If you're mailing your payments, the postage is 49 cents x 2 extra payments = 98 cents. You're losing 8 cents per month by doing this." The MWRR that you showed in your post is calculated incorrectly. The formula that you use... ($15,750 - $15,000 - $4,000) / ($15,000 + 0.5 x $4,000) Translates into a form of the DIETZ formula of (EMV-BMV-C)/(BMV + .5 x C) The BMV is the STARTING balance. And as a matter of fact, the starting balance was NOT 15,000. It was IN FACT 11,000. See, the starting value for a month MUST BE the ending value of the prior month. So the BMV of 11,000 would give you the correct answer. Because if you added 4,000 at the start of the month (on day 1), it would have to have been ADDED to the 11,000 of the PRIOR month's ENDING value. Make sense? That would also mean that the addition of 4000 to the 11000 would imply that you started day 1 with 11,000. Make sense? Summary: When doing the calculations, you may use the ending value on the last day of the month to get your EMV. BUT YOU MAY NOT take the ending value on day 1 to get the BMV. That simply can not make sense since you already added a bunch of money during the day. Think about it. Davie Using the following loan equations where  and With the balance b[n] in period n given by Applying the OP's figures Check & demonstration Switching to $96 payment every 10 days, with 365.2422 days per year  Paying $96 every 10 days saves $326.85 and pays the loan down 2.68 months quicker. "Well, the first one is based on the ""Pert"" formula for continuously-compounded present value, while the second one is the periodically-compounded variant.  Typically, the continuously-compounded models represent the ideal; as the compounding period of time-valued money shrinks towards zero, and the discount rate (or interest rate if positive) stays constant over the time period examined, the periodic equation's results approach that of the continuously-compounded equation. Those two assumptions (a constant rate and continuous balance adjustment from interest) that allow simplification to the continuous form are usually incorrect in real-world finance; virtually all financial institutions accrue interest monthly, for a variety of reasons including simpler bookkeeping and less money paid or owed in interest. They also, unless prohibited by contract, accrue this interest based on a rate that can change daily or even more granularly based on what financial markets are doing. Most often, the calculation is periodic based on the ""average daily balance"" and an agreed rate that, if variable, is based on the ""average daily rate"" over the previous observed period. So, you should use the first form for fast calculation of a rough value based on estimated variables. You should use the second form when you have accurate periodic information on the variables involved. Stated alternately, use the first form to predict the future, use the second form in retrospect to the past." "A real simple definition or analogy of present Value would be the ""Principal"" or ""Loan Amount"" being lent and the future value as being returning the Principal along with cost of borrowing The (1+i)^n is the interest you earn on present value The (i+i)^-n is the interest you pay on future value The first one is the FVIF or future value of a $1 The second one is the PVIF or present value of a $1 Both these interest factors assume interest is paid annually, if the interest payment is made more often within the payment year then interest factors look this way m is the frequency of interest payment, the higher this frequency the more of interest you pay or earn and you pay or earn the most interest when compounding occurs on each small fraction of time This entails here e is the Euler's e  Thus the interest factors turn to this The preceeding examples only considered a single repayment at future date. Now if you were obliged to make periodic loan repayments say in amount of $1 for n number of periods. Then the present value of all such periodic payment is the ""Principal"" or amount you borrowed. This is the sum of discounted periodic payments as  if we replace 1/1+i with x then this turns out to be geometric series of the form This simplifies to  replacing (1/1+i) for x we get which is the present value of periodic payment in amount of $1 The future value of periodic payments in amount of $1 can be arrived at multiplying the PVIFA by (1+i)^n giving Once again the interest is compounded per annum and for intra-year compounding you would have to at first find the annual effective yield AEY to use as the effect rate is the PVIFA and FVIFA calculation for continuous compounding All the calculation discussed thus far did not take inflation into consideration, if we were to adjust the amounts for a growth of g% then the present value of a $1 would be as follow Once again you would have to use AEY if compounding frequency of interest is intra-year Now assume that each loan repayment increases or decreases by an extra amount Q per period. To find the present value of series of payments P that increase or decrease per period by an amount Q we would do the following calculations Here and All of these calculations have been available in tadXL add-in for finance and incrementally being offered as JavaScript financial functions library tadJS. Please note that the tad series of the financial functions library for various environments such as Excel, JavaScript, PHP, Ruby, Microsoft.net and others are property of the author writing this post. All of these libraries except one for Excel are available for FREE for public use. And the future value of such payments with increments may be found by multiplying the PV by (1+i)^n as follows Here"