Calculating the cost of waiting longer for money
This looks correct to me, for simple interest. If you are dealing with compound interest, the formula would be: So, A = 500000(1+0.036/365)^(30), or 501,481.57, or an interest of 1481.57, assuming the 3.6% is the annual nominal interest rate and it is compounded daily. Note that you are ignoring the depreciation and also ignoring the percentage of customers who will forfeit their debt in the 30 - 60 day period. The cost of an extra 30 days is $1459.80
You want the net expense of the surcharge minus the rewards to be no more than the interest that you would pay otherwise.   Where t is the compounding period for the rate D expressed as a fraction of the overall period for D.  So if D is an annual rate (not the APR, the simple rate), it would be expressed as something like 1/365 if compounded daily.  That is the number of years in the compounding period.  If a monthly rate or weekly compounding, that would change.   And p is the number of such time periods in the grace period.  So if the grace period were one month, this might be 30.   Other variables are as used in the question, all expressed as percentages (which is why I'm dividing by 100).  The D rate should be the simple rate, like 6% not the APR of 6.24% or whatever.   Note that I'm saying <=.  When equal, there is no financial advantage or disadvantage.  You could choose either method for the same cost.  Now, one method may be more annoying to implement, in which case you might add a fee for it on one side or the other of the equation.  Or simply change the less than or equal to be just less than.   I may be missing something that you should consider but I don't know.  The problem is generic enough that pertinent details might be hidden.  But hopefully this at least gives you a framework under which to consider it. Edited to incorporate the comments elsewhere of @Atkins Assuming, (apparently incorrectly) that duration is time to maturity: First, note that the question does not mention the coupon rate, the size of the regular payments that the bond holder will get each year.  So let's calculate that. Consider the cash flow described.  You pay out 1015 at the start of Year #1, to buy the bond.  At the end of Years #1 to #5, you receive a coupon payment of X.  Also at the end of Year #5, you receive the face value of the bond, 1000.  And you are told that the pay out equals the money received, using a time value of money of 4.69% So, if we use the date of maturity of the bond as our valuation date, we have the equation: Maturity + Future Value of coupons = Future value of Bond Purchase price 1000 + X *( (1 + .0469)^5-1)/0.0469 = 1015 * 1.0469^5 Solving this for X, we obtain 50.33;  the coupon rate is 5.033%.  You will receive 50.33 at the end of each of the five years.   Now, we can take this fixed schedule of payments, and apply the new yield rate to the same formula above;  only now, the unknown is the price paid for the bond, Y. 1000 + 50.33 * ((1 + 0.0487)^5 - 1) / .0487 = Y * 1.0487^5 Solving this equation for Y, we obtain: Y = 1007.08 "I'm just trying to visualize the costs of trading. Say I set up an account to trade something (forex, stock, even bitcoin) and I was going to let a random generator determine when I should buy or sell it. If I do this, I would assume I have an equal probability to make a profit or a loss. Your question is what a mathematician would call an ""ill-posed problem.""  It makes it a challenge to answer.  The short answer is ""no.""  We will have to consider three broad cases for types of assets and two time intervals. Let us start with a very short time interval.  The bid-ask spread covers the anticipated cost to the market maker of holding an asset bought in the market equal to the opportunity costs over the half-life of the holding period.  A consequence of this is that you are nearly guaranteed to lose money if your time interval between trades is less than the half-life of the actual portfolio of the market maker.  To use a dice analogy, imagine having to pay a fee per roll before you can gamble.  You can win, but it will be biased toward losing. Now let us go to the extreme opposite time period, which is that you will buy now and sell one minute before you die.  For stocks, you would have received the dividends plus any stocks you sold from mergers.  Conversely, you would have had to pay the dividends on your short sales and received a gain on every short stock that went bankrupt.  Because you have to pay interest on short sales and dividends passed, you will lose money on a net basis to the market maker.  Maybe you are seeing a pattern here.  The phrase ""market maker"" will come up a lot. Now let us look at currencies.  In the long run, if the current fiat money policy regime holds, you will lose a lot of money.  Deflation is not a big deal under a commodity money regime, but it is a problem under fiat money, so central banks avoid it.  So your long currency holdings will depreciate.  Your short would appreciate, except you have to pay interest on them at a rate greater than the rate of inflation to the market maker. Finally, for commodities, no one will allow perpetual holding of short positions in commodities because people want them delivered.  Because insider knowledge is presumed under the commodities trading laws, a random investor would be at a giant disadvantage similar to what a chess player who played randomly would face against a grand master chess player.  There is a very strong information asymmetry in commodity contracts.  There are people who actually do know how much cotton there is in the world, how much is planted in the ground, and what the demand will be and that knowledge is not shared with the world at large.  You would be fleeced. Can I also assume that probabilistically speaking, a trader cannot do   worst than random? Say, if I had to guess the roll of a dice, my   chance of being correct can't be less than 16.667%. A physicist, a con man, a magician and a statistician would tell you that dice rolls and coin tosses are not random.  While we teach ""fair"" coins and ""fair"" dice in introductory college classes to simplify many complex ideas, they also do not exist.  If you want to see a funny version of the dice roll game, watch the 1962 Japanese movie Zatoichi.  It is an action movie, but it begins with a dice game.   Consider adopting a Bayesian perspective on probability as it would be a healthier perspective based on how you are thinking about this problem.  A ""frequency"" approach always assumes the null model is true, which is what you are doing.  Had you tried this will real money, your model would have been falsified, but you still wouldn't know the true model. Yes, you can do much worse than 1/6th of the time.  Even if you are trying to be ""fair,"" you have not accounted for the variance. Extending that logic, then for an inexperienced trader, is it right to   say then that it's equally difficult to purposely make a loss then it   is to purposely make a profit? Because if I can purposely make a loss,   I would purposely just do the opposite of what I'm doing to make a   profit. So in the dice example, if I can somehow lower my chances of   winning below 16.6667%, it means I would simply need to bet on the   other 5 numbers to give myself a better than 83% chance of winning. If the game were ""fair,"" but for things like forex the rules of the game are purposefully changed by the market maker to maximize long-run profitability.  Under US law, forex is not regulated by anything other than common law.  As a result, the market maker can state any price, including prices far from the market, with the intent to make a system used by actors losing systems, such as to trigger margin calls.  The prices quoted by forex dealers in the US move loosely with the global rates, but vary enough that only the dealer should make money systematically.  A fixed strategy would promote loss. You are assuming that only you know the odds and they would let you profit from your 83.33 percentage chance of winning. So then, is the costs of trading from a purely probabilistic point of   view simply the transaction costs? No matter what, my chances cannot   be worse than random and if my trading system has an edge that is   greater than the percentage of the transaction that is transaction   cost, then I am probabilistically likely to make a profit? No, the cost of trading is the opportunity cost of the money.  The transaction costs are explicit costs, but you have ignored the implicit costs of foregone interest and foregone happiness using the money for other things. You will want to be careful here in understanding probability because the distribution of returns for all of these assets lack a first moment and so there cannot be a ""mean return.""  A modal return would be an intellectually more consistent perspective, implying you should use an ""all-or-nothing"" cost function to evaluate your methodology." "For 60 days I got $2,958.  What you have is how much it would cost you over a 3 month period ASSUMING that ALL of your receivables were paid at 30 days rather than 60. But I'm confused by ""our return is 3.6%"", is that the interest you're charging the customer for paying late? Would the invoice be 3.6/30 n-60? I'm not sure" The periodic rate (here, the interest charged per month), as you would enter into a finance calculator is 9.05%. Multiply by 12 to get 108.6% or calculate APR at 182.8%. Either way it's far more than 68%.  If the $1680 were paid after 365 days, it would be simple interest of 68%. For the fact that payment are made along the way, the numbers change.  Edit - A finance calculator has 5 buttons to cover the calculations:  N = number of periods or payments %i = the interest per period PV = present value PMT = Payment per period FV= Future value In your example, you've given us the number of periods, 12, present value, $1000, future value, 0, and payment, $140. The calculator tells me this is a monthly rate of 9%. As Dilip noted, you can compound as you wish, depending on what you are looking for, but the 9% isn't an opinion, it's the math. TI BA-35 Solar. Discontinued, but available on eBay. Worth every cent.  Per mhoran's comment, I'll add the spreadsheet version.  I literally copied and pasted his text into a open cell, and after entering the cell shows,  which I rounded to 9.05%. Note, the $1000 is negative, it starts as an amount owed.  And for Dilip - 1.0905^12 = 2.8281 or 182.8% effective rate. If I am the loanshark lending this money, charging 9% per month, my $1000 investment returns $2828 by the end of the year, assuming, of course, that the payment is reinvested immediately. The 108 >> 182 seems disturbing, but for lower numbers, even 12% per year, the monthly compounding only results in 12.68% 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. 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. Mostly ditto to Dillip Sarwate. Let me just add: I don't know how you're making your payments, whether through the biller's web site, your bank's web site, by mail, in person, etc. But whatever the mechanism, if there is a chance that waiting until the due date to pay may mean that you will miss the due date: don't. The cost of a late payment charge is likely to far exceed any interest you would collect on your savings. Bear in mind that we are talking pennies here. I don't know how much the monthly bills that we are discussing here come to. Say it's $3,000. I think that would be a lot for most people. You say you're getting 3.6% on your savings. So if, on the average, you pay a bill 2 weeks later than you might have, you're getting an extra 2 / 52 x 3.6% x $3,000 in interest, or $4 per month. I think the last time I paid a late fee on a credit card it was $35, so if you make one mistake every 8 months and end up getting a late fee it will outweigh any savings.  Personally, I pay most of my bills through either my bank's web site or the biller's web site. I schedule all payments when I get a paycheck, and I generally try to schedule them for 1 week before the due date, so there's plenty of breathing room. This is of course using 'new math'.  Namely, if I lend you $100 and you keep it for a month, I've lent you $100.  In fact, if I loan you $100 for a year I've loaned you $100.  But if I lend you $100 overnight, you pay me back in the morning, I lend you $100 overnight, you pay me back in the morning and we repeat that for a month, I've supposedly lent you $3,000.    That's some interesting math for sure. "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." Interest rates are always given annually, to make them comparable. If you prefer to calculate the rate or the total interest for the complete time, like 10 years or 15 years or 30 years, it is simple math, and it tells you the total you will pay, but it is not helpful for picking the better or even the right offer for your situation. Compare it to your car's gas mileage- what sense does it make to provide the information that a car will use 5000 gallons of gas over its lifetime? Is that better than a car that uses 6000 gallons (but may live 2 years longer?) I think they're compounding the interest daily. That means you have to look at the number of days between payments to judge how much the interest charge is. From February 3 to March 2 is 28 days (2012 was a leap year). From March 2 to April 3 is 32 days. That's an increase of about 14% in number of days between payments, which accounts reasonably well to the ~$18 difference in interest charge.  Daily compounding also explains the minor fluctuations in the other interest charges. I think if you compute interest/day for each month, you'll find that it is, indeed, decreasing over time. Given that a mutual fund manager knows, at the end of the day, precisely how many shares/units/whatever of each investment (stock, equity, etc.) they own, plus their bank balance,  It is calculating this given. There are multiple orders that a fund manager requests for execution, some get settled [i.e. get converted into trade], the shares itself don't get into account immediately, but next day or 2 days later depending on the exchange. Similarly he would have sold quite a few shares and that would still show shares in his account. The bank balance itself will not show the funds to pay as the fund manager has purchased something ... or the funds received as the fund manager has sold something. So in general they roughly know the value ... but they don't exactly know the value and would have to factor the above variables. That's not a simple task when you are talking about multiple trades across multiple shares. Nathan's answer was a +1 from me. The answer is not always simple. Having the money available is surely the first step. Using Pete's process aligns with this.  Another thought is depending where you are in your finances, delay by a day for every $100 in cost. e.g. For a $1000 purchase, sleep on it for 10 days. Adjust the number for your circumstance. What matters is the combination of the loan amount and the time (hours/days/whatever) you've loaned out the money.  $100 * 10 hours * 30 times = 30K dollar-hours  The other loan is $100 * 24 hours * 30 days = 72K dollar-hours  But using your math the original loan is 30X the amount of the second.