Calculating Future and Present value into mortgage comparisons
Keep in mind the number of months or years before you break even. You pay money to lower the interest rate, and lower the monthly cost. But it takes a number of months, using your numbers $7,000 to save $160 a month will take ~43 months. That is before figuring in the future or present value. If you sell or refinance the mortgage, the initial points to lower the rate is gone. You mentioned 15-20 years in your comment on mhoran_psprep's response. This is the most important factor to consider in the points vs. rate question. With a horizon that long it sounds like the points are probably a better option for you. There is a neat comparison tool at The Mortgage Professor's website that may help you build your spreadsheet or simply check the numbers you are getting. "Using the fact that you'd save $160/mo by spending $7000, I'd look at it this way -  If I were to lend you the $7000 at 12%/yr, $160 would pay it off in 58 months. At 18%/yr, 72 months or just 6 years. You can run spreadsheets to get breakeven scenarios, and mhoran is on track with his answer, but breakeven is just one point to consider. Beyond that date, it's free money.  My approach is to look at it with a question - ""How much interest could I afford to pay to make that monthly savings worthwhile?"""
"Sometimes an assumptions is so fundamentally flawed that it essentially destroys the relevance and validity of any modelling outputs.  ""Obviously, we're assuming the company can pay it back""  Is one of those assumptions.  The person gets a notes stating that they will get $525 'IN ONE YEAR' You need to divide $525/(1+Cost of Capital)^n n being the number of periods to find out what the note will be worth today.  Google 'Present Value of an Annuity' to deal with debt that is more complex than you have $500 now and give me $525 in a year..." First, assuming you are making payments for a savings deposit.  The present value of the deposit is the sum of the all the payments discounted to present value.  In this case they would be discounted by the rate of inflation: £100 deposited next year is worth less than £100 today because it will be eroded by inflation.  With a higher rate of inflation the payments are discounted more heavily, so the present value decreases.  A deposit, or annuity due (see Calculating the Present Value of an Annuity Due), can be expressed mathematically like so:-  ∴ by induction  So for example, the following annuity has a present value of £1,136.76 The total amount that will be paid for the annuity is 12 x £100 = £1,200.  With a higher rate of inflation, say 2% per month, and with the same 12 x £100 payments, the present value of the annuity decreases. In Excel (£1,078.68) A similar case is that for a loan, or ordinary annuity (see Calculating the Present Value of an Ordinary Annuity), except the discounting factor is the loan interest rate rather than inflation and repayments are made at the end of each period rather than at the start. The present value of a loan is the value of the all the future repayments discounted to present value.  With a higher interest rate the payments are discounted more heavily, so the present value decreases.  A loan can be expressed mathematically like so:-  ∴ by induction  So for example, the following values fulfill a loan worth £1,125.51 The total amount that will be paid for the loan is 12 x £100 = £1,200.  With a higher rate of interest, say 2% per month, and the same 12 x £100 repayments, the present value of the loan that can be obtained decreases. In Excel (£1,057.53) Basically you need to use a time-value-of-money equation to discount the cashflows back to today. The Wikipedia formula will likely work fine for you, then you just need to pick an effective interest rate to use in the calculation. Run each of your amounts and dates though the formula (there are various on-line calculators to pick from, and sum up the values. You did not mention your location or jurisdiction, but a useful proxy for the interest rate would be the average between the same duration mortgage rate and fixed-deposit rate at your bank; it should be close enough for your purposes - although if an actual lawsuit is involved and the sums high enough to have lawyers, it might be worth engaging an accountant as well to defend the veracity of both the calculation and the interest rates chosen. EDIT: After reading one of the comments on the original question, I realized that there is a much more intuitive way to think about this. If you look at it as a standard PV calculation and hold each of the cashflows constant. Really what's happening is that because of inflation the discount rate isn't the full value of the interest rate. Really the discount rate is only the portion of the interest rate above the inflation rate. Hence in the standard perpetuity PV equation PV = A / r r becomes the interest rate less the inflation rate which gives you PV = A / (i - g). That seems like a much better way to get to the answer than all the machinations I was originally trying. Original Answer: I think I finally figured this out. The general term for this type of system in which the payments increase over time is a gradient series annuity. In this specific example since the payment is increasing by a percentage each period (not a constant rate) this would be considered a geometric gradient series. According to this link the formula for the present value of a geometric gradient series of payments is: Where P is the present value of this series of cashflows.   A_1 is the initial payment for period 1 (i.e. the amount you want to withdraw adjusted for inflation).   g is the gradient or growth rate of the periodic payment (in this case this is the inflation rate)   i is the interest rate   n is the number of payments This is almost exactly what I was looking for in my original question. The only problem is this is for a fixed amount of time (i.e. n periods). In order to figure out the formula for a perpetuity we need to find the limit of the right side of this equation as the number of periods (n) approaches infinity. Luckily in this equation n is already well isolated to a single term: (1 + g)^n/(1 + i)^-n}. And since we know that the interest rate, i, has to be greater than the inflation rate, g, the limit of that factor is 0. So after replacing that term with 0 our equation simplifies to the following: Note: I don't do this stuff for a living and honestly don't have a fantastic finance IQ. It's been a while since I've done any calculus or even this much algebra so I may have made an error in the math. "I was wondering how ""future cash flows of the asset"" are predicted? Are they also predicted using fundamental and/or technical analysis? There are a many ways to forecast the future cash flows of assets. For example, for companies: It seems like calculating expected/required rate using CAPM does not belong to either fundamental or technical analysis, does it? I would qualify the CAPM as quantitative analysis because it's mathematics and statistics. It's not really fundamental since its does not relies on economical data (except the prices). And as for technical analysis, the term is often used as a synonym for graphical analysis or chartism, but quantitative analysis can also be referred as technical analysis. the present value of future cash flows [...] (called intrinsic price/value, if I am correct?) Yes you are correct. I wonder when deciding whether an asset is over/fair/under-valued, ususally what kind of price is compared to what other kind of price? If it's only to compare with the price, usually, the Net asset value (which is the book value), the Discount Cash flows (the intrinsic value) and the price of comparable companies and the CAPM are used in comparison to current market price of the asset that you are studying. Why is it in the quote to compare the first two kinds of prices, instead of comparing the current real price on the markets to any of the other three kinds? Actually the last line of the quote says that the comparison is done on the observed price which is the market price (the other prices can't really be observed). But, think that the part: an asset is correctly priced when its estimated price is the same as the present value of future cash flows of the asset means that, since the CAPM gives you an expected rate of return, by using this rate to compute the present value of future cash flows of the asset, you should have the same predicted price. I wrote this post explaining some valuation strategies. Maybe you can find some more information by reading it." i will not calculate it for you but just calculate the discounted cash flow (by dividing with 1.1 / 1.1^2 / 1.1^3   ...)of each single exercise as stated and deduct the 12.000 of the above sum.  in the end compare which has the highest npv "(Ignoring any ethical considerations of defaulting on a mortgage obligation) If your credit score fell, future costs of borrowing would be greater, if you could borrow money at all. The true financial cost of these penalties would depend on your own intentions and circumstances. The value of a credit score cannot be quantified in any absolute sense because of these circumstances which will vary from person to person. Suppose a decision would result in your score falling from 700 to 600. As a result, a future car loan would cost something like 2% more in interest per year. Let's also suppose you would not be able to find a bank to give you money to buy a new home.  The difficulties caused by this situation would depend on the person. Some people have no intention of taking out another mortgage or buying another car, and to them, a lowered credit score might make no difference. To others, it would be desirable to avoid these penalties. The value of credit score would be equal to how much you would be willing to pay in the present to avoid these future penalties.  If you intend to borrow money in the near future, and somehow know how much more interest you will pay as a result of your lower score, then you could approximate the value of the credit drop by summing all of the additional interest costs discounted into the present. Something a little like this: Year's Additional Cost in Interest / (1 + Your Personal Opportunity Cost of Funds) ^ (Years in the future this extra interest will be paid).  See time value of money explanation: Can you explain ""time value of money"" and ""compound interest"" and provide examples of each? It could also be the case that you miss out on valuable financial opportunities as a result which could be added to the present value of the credit score drop.  For example, if the drop made you totally unable to purchase an investment property that would net you $10,000 in cash flows /year (which would yield cash flows immediately). The value of that investment would have been $10,000 / (1 + Your Personal Opportunity Cost of Funds). Since you would not be able to make that investment (in this example) as a result of the decreased score, the present value of the investment would be an opportunity cost of the lower score. There are other hard-to-quantify costs of low credit to consider as well, ranging from housing to employment to bragging rights.  Hopefully this has helped :)" You can explore the scenarios in which it is better to rent or to buy using this application: http://demonstrations.wolfram.com/BuyOrRentInvestmentReturnCalculator/ In the possibly unlikely scenario shown below, at the term of the mortgage (20 years) the tenant and the buyer have practically the same return on investment.  At this point the tenant's savings would be sufficient to buy a house equivalent to the buyer's, and this would be the advisable course of action (based on the figures alone). First, let me fill in the gaps on your situation, based on the numbers you've given so far. I estimate that your student loan balance (principal) is $21,600. With the variable rate loan option that you've presented, the maximum interest rate you could be charged would be 11.5%, which would bring your monthly payment up to that $382 number you gave in the comments. Your thoughts are correct about the advantage to paying this loan off sooner.  If you are planning on paying off this loan sooner, the interest rate on the variable rate loan has less opportunity to climb. One thing to be cautious of with the comparison, though: The $1200 difference between the two options is only valid if your rate does not increase.  If the rate does increase, of course, the difference would be less, or it could even go the other way.  So keep in mind that the $1200 savings is only a theoretical maximum; you won't actually see that much savings with the variable rate option. Before making a decision, you need to find out more about the terms of this variable rate loan: How often can your rate go up? What is the loan rate based on? I'm not as familiar with student loan variable rate loans, but there are other variable rate loans I am familiar with: With a typical adjustable rate home mortgage, the rate is locked for a certain number of years (perhaps 5 years).  After that, the bank might be allowed to raise the rate once every period of months (perhaps once every year).  There will be a limit to how much the rate can rise on each increase (perhaps 1.0%), and there will be a maximum rate that could be charged over the life of the loan (perhaps 12%). The interest rate on your mortgage can adjust up, inside of those parameters.  (The actual formula used to adjust will be found in the fine print of your mortgage contract.)  However, the bank knows that if they let your rate get too high above the current market rates, you will refinance to a different bank.  So the mortgage is typically structured so that it will raise your rate somewhat, but it won't usually get too far above the market rate.  If you knew ahead of time that you would have the house paid off in 5 years, or that you would be selling the house before the 5 years is over, you could confidently take the adjustable rate mortgage. Credit cards, on the other hand, also typically have variable rates.  These rates can change every month, but they are usually calculated on some formula determined ahead of time.  For example, on my credit card, the interest rate is the published Prime Rate plus 13.65%.  On my last statement, it said the rate was 17.15%.  (Of course, because I pay my balance in full each month, I don't pay any interest.  The rate could go up to 50%, for all I care.) As I said, I don't know what determines the rate on your variable rate student loan option, and I don't know what the limits are.  If it climbs up to 11.5%, that is obviously ridiculously high.  I recommend that you try to pay off this student loan as soon as you possibly can; however, if you are not planning on paying off this student loan early, you need to try to determine how likely the rate is to climb if you want to pick the variable rate option. Line one shows your 1M, a return with a given rate, and year end withdrawal starting at 25,000.  So Line 2 starts with that balance, applies the rate again, and shows the higher withdrawal, by 3%/yr.  In Column one, I show the cumulative effect of the 3% inflation, and the last number in this column is the final balance (903K) but divided by the cumulative inflation.  To summarize - if you simply get the return of inflation, and start by spending just that amount, you'll find that after 20 years, you have half your real value.  The 1.029 is a trial and error method, as I don't know how a finance calculator would handle such a payment flow. I can load the sheet somewhere if you'd like.  Note: This is not exactly what the OP was looking for. If the concept is useful, I'll let it stand. If not, downvotes are welcome and I'll delete. Illustrating with a shorter example:  Suppose I deposit 1,000 USD. Every year I deposit another 100 USD. I want to know how much money will be on that savings account in 4 years.  The long-hand calculation is Expressed with a summation  And using the formula derived from the summation (as shown by DJohnM) So for 20 years Note in year 20 (or year 4 in the shorter example) the final $100 deposit does not have any time to accrue interest before the valuation of the account. Question is, what is this number 0.01140924 13.69/12=0.01140924 In addition, how does one come out with the EIR as 13.69% pa? When calculating payments, PV = 9800, N=36 (months), PMT=333.47, results in a rate of 1.140924% per period, and rate of 13.69%/yr. No idea how they claim 7.5% In Excel, type =RATE(36,333.47,-9800,0,0) And you will get 1.141% as the result.  36 = #payments, 333.47 = payment per period, -9800 is the principal (negative, remember this) And the zeros are to say the payments are month end, second zero is the guess.  Edit - I saw the loan is from a Singapore bank. It appears they have different rules on the rates they quote. As quid's answer showed the math, here's the bank's offer page -  The EIR is the rate that we, not just US, but most board members, are used to. I thought I'd offer an example using a 30 year mortgage.   Yo can see above, a 6% fixed rate somehow morphs into a 3.86% AR. No offense to the Singapore bankers, but I see little value in this number. What surprises me most, is that I've not seen this before.  What's baffling is when I change a 15yr term the AP drops to less than half. It's still a 6% loan and there's nothing about it that's 2 percent-ish, in my opinion.  Now we know. In short, your scenario could work in theory, but is not realistic... Generally speaking, you can borrow up to some percentage of the value of the property, usually 80-90% though it can vary based on many factors.  So if your property currently has a value of $100k, you could theoretically borrow a total of $80-90k against it.  So how much you can get at any given time depends on the current value as compared to how much you owe.  A simple way to ballpark it would be to use this formula: (CurrentValue * PercentageAllowed) - CurrentMortgageBalance = EquityAvailable.     If your available equity allowed you to borrow what you wanted, and you then applied it to additions/renovations, your base property value would (hopefully) increase.  However as other people mentioned, you very rarely get a value increase that is near what you put into the improvements, and it is not uncommon for improvements to have no significant impact on the overall value.  Just because you like something about your improvements doesn't mean the market will agree. Just for the sake of argument though, lets say you find the magic combination of improvements that increases the property value in line with their cost.  If such a feat were accomplished, your $40k improvement on a $100k property would mean it is now worth $140k.  Let us further stipulate that your $40k loan to fund the improvements put you at a 90% loan to value ratio.  So prior to starting the improvements you owed $90k on a $100k property.  After completing the work you would owe $90k on what is now a $140k property, putting you at a loan to value ratio of ~64%.  Meaning you theoretically have 26% equity available to borrow against to get back to the 90% level, or roughly $36k.  Note that this is 10% less than the increase in the property value.  Meaning that you are in the realm of diminishing returns and each iteration through this process would net you less working capital. The real picture is actually a fair amount worse than outlined in the above ideal scenario as we have yet to account for any of the costs involved in obtaining the financing or the decreases in your credit score which would likely accompany such a pattern.  Each time you go back to the bank asking for more money, they are going to charge you for new appraisals and all of the other fees that come out at closing.  Also each time you ask them for more money they are going to rerun your credit, and see the additional inquires and associated debt stacking up, which in turn drops your score, which prompts the banks to offer higher interest rates and/or charge higher fees... Also, when a bank loans against a property that is already securing another debt, they are generally putting themselves at the back of the line in terms of their claim on the property in case of default.  In my experience it is very rare to find a lender that is willing to put themselves third in line, much less any farther back.  Generally if you were to ask for such a loan, the bank would insist that the prior commitments be paid off before they would lend to you.  Meaning the bank that you ask for the $36k noted above would likely respond by saying they will loan you $70k provided that $40k of it goes directly to paying off the previous equity line. I would use neither method.  Taking a short example first, with just three compounding periods, with interest rate 10%.  The start value y0 is 1. So after three years the value is 1.331, the same as y0 (1 + 0.1)^3. Depreciating (like inflation) by 10% (to demonstrate) gets us back to y0 = 1 Appreciating and depreciating by 10% cancels out: Appreciating by 10% interest and depreciating by 3% inflation: This is the same as y0 (1 + 0.1)^3 (1 + 0.03)^-3 = 1.21805 So for 50 years the result is y0 (1 + 0.1)^50 (1 + 0.03)^-50 = 26.7777 Note You can of course use subtraction but the not using the inflation figure directly.  E.g. (edit: This appears to be the Fisher equation.) 2nd Note Further to comments, here is a chart to illustrate how much the relative performance improves when inflation is accounted for.  The first fund's return is 6% and the second fund's return varies from 3% to 6%.  Inflation is 3%. What is the formula for calculating the total cost of a loan with extra payments towards the principal? The formula you require is the standard one for calculating the time to repay.  With larger repayments the time to completely repay the loan is reduced. Where The total cost of the loan is then n * d. Explanation & Calculation The formula for a loan is derived from the sum of the cash flows discounted to present value being equal to the principal.  For further info see the section here titled: Calculating the Present Value of an Ordinary Annuity The summation can be reduced to a closed form by induction:  Rearranging for d and n With the OP's figures The original monthly repayment is $1,006.96 Adding $200 each month ... With the higher repayment the loan is repaid in 257.36 months instead of 360.  (Of course a bank would simply take a reduced payment in month 258, but the amounts work out the same.) The saving is $51,882.37 Addendum If the repayments increase was made part-way through the term of the loan the summation and formula would be  Where Then For example, if for the first ten years the payments are $1,006.96 and for the remaining time the repayments are $1,206.96 The loan is completely repaid in 302.528 months. The saving is Plotting over a range of m months: