Calculating required rate of return for an income-generating savings account
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.
You are looking for the Internal Rate of Return. If you have a spreadsheet like Microsoft Excel you can simply put in a list of the transactions (every time money went in or out) and their dates, and the spreadsheet's XIRR function will calculate a percentage rate of return. Here's a simple example. Investment 1 was 100,000 which is now worth 104,930 so it's made about 5% per year. Investment 2 is much more complicated, money was going in and out, but the internal rate of return was 7% so money in that investment, on average, grew faster than money in the first investment. "For 3X, it's about 114, and for 4X, 144, which naturally, is twice 72.  These are close, back of napkin, results. With smart phone apps offering scientific calculators, you should get comfortable just taking the nth root of a number for a more precise answer.  Update in response to Brick's comment. The rule of 72 says that (n)(y)=72 to double your money. It answers both questions, how much time do I need, given a rate, and how much return do I need, given a time?  Logic tells me that if 72 is the number to double, 144 is the 4X. But I'm a math guy, and my logic might not be logical to OP. So -  Let's take the 20th root of 4.   This is the key to use. 4, (hit key) 20, equals. The result is 1.07177 or 7.177%. And this is the precise rate you'd need to quadruple your money in 20 years) Now (n)(y)= 20* 7.177 = 143.55 which rounds to 144. ""Rule of 144"" to quadruple your money.  This now answers OP's question, ""How to derive a Rule of X"" for a return other than doubling.  One more example? I want 10X my money. Of course I need the initial guess to enter one calculation. People like 8%, in general. It's a bit below the 10% long term S&P return, and a good round number. The Rule of 72 says 9 years to double, so, 18 years is 4X, and 36 years is 8X. For my initial calculation, I'll use 40 years. The 40th root of 10. I get 5.925% (Again the precise rate that gives 10 fold over 40 years) and multiplying this by 40, I get a ""Rule of 237"" which I'm tempted to round to 240.  At 6%, 237/6= 39.5 yrs,  1.06^39.5 = 9.99 At 6%, 240/6= 40.0 yrs,  1.06^40.0 = 10.29 You can see that you lose some accuracy for the sake of a number that's easier to remember, and manipulate. 72 to double is pretty darn accurate, so I'll stick with ""Rule of 237"" to get 10X my money.   To close, the purpose of these rules is to create the tool that lets you perform some otherwise tough calculations away from any electronic device. Of course I know how to use logs, and in real life I'm paid to explain them to students who are typically glad when that chapter is over. I've shown above how the ""Rule of X"" can be formulated with a power/root key, which, for most people, is simpler. Ironically, log calculations as @jkuz offered, force a continuous compounding which may not be desired at all. It would give a result of 230 for my 10X return example, and the following (using the first equation he offered) - At 6%, 230/6= 38.3 yrs,  1.06^38.3 = 9.31 which is further away from the desired 10X than my 237 or rounded 240." The point of a total return index is that it already has accounted for the capital gains + coupon income. If you want to calculate it yourself you'll have to find the on-the-run 10y bond for each distinct period then string them together to calc your total return. Check XLTP if they have anything Aah... well, then in that case you should actually integrate a monte carlo return scenario with your equity values.  Ultimately it's not going to matter what your RFR is because it's going to be equal in both cases, so you're really just talking about return differences.  Again, it would be impractical for these two options.  Maybe just look at cash outlay - an amortization table of sorts, and that's how you'd calculate the breakeven point.  You could inflate the unspent cash difference by a small margin (perhaps 0.5% to account for something like a CD or an ARS) but the big difference is going to be the interest rate with respect to a mortgage vs. a perpetual cash payment of rent that never attains any real value. You are comparing a risk-free cost with a risky return. If you can tolerate that level of risk (the ups and downs of the investment) for the chance that you'll come out ahead in the long-run, then sure, you could do that.   So the parameters to your equation would be: If you assume that the risky returns are normally distributed, then you can use normal probability tables to determine what risk level you can tolerate. To put some real numbers to it, take the average S&P 500 return of 10% and standard deviation of 18%.  Using standard normal functions, we can calculate the probability that you earn more than various interest rates: so even with a low 3% interest rate, there's roughly a 1 in 3 chance that you'll actually be worse off (the gains on your investments will be less than the interest you pay).  In any case there's a 3 in 10 chance that your investments will lose money. Well, one can easily have rates below -100%.  Suppose I start with $100, and end up with $9 after a year. What was my rate of return? It could be -91%, -181%, -218%, or -241%, or something else, depending on the compounding method.  We always have that the final amount equals the initial amount times a growth factor G, and we can express this using a rate r and a day count fraction T. In this case, we have T = 1, and B(T) = B(0) * 0.09, so: So, depending on how we compound, we have a rate of return of -91%, -181%, -218%, or -241%. This nicely illustrates that: such analysis helps you keep track of the percentage of income that you invest You could apply 1:1:1 ratio. i.e spend 1/3, save 1/3 and invest 1/3. Hope that helps I use the following method. For each stock I hold long term, I have an individual table which records dates, purchases, sales, returns of cash, dividends, and way at the bottom, current value of the holding. Since I am not taking the income, and reinvesting across the portfolio, and XIRR won't take that into account, I build an additional column where I 'gross up' the future value up to today() of that dividend by the portfolio average yield at the date the dividend is received. The grossing up formula is divi*(1+portfolio average return%)^((today-dividend date-suitable delay to reinvest)/365.25) This is equivalent to a complex XMIRR computation but much simpler, and produces very accurate views of return.  The 'weighted combined' XIRR calculated across all holdings then agrees very nearly with the overall portfolio XIRR. I have done this for very along time.  TR1933 Yes, 1933 is my year of birth and still re investing divis! I hope that there are no significant differences between the things you list once the formulas for compounding interest are understood. I will, again, lay out these formulas below.  First, definition of the variables: R means Total Return ratio. The sum of all money you get, both dividends (or interest payments) and return of initial capital.  I is a ratio. It is the percent (10.4%) divided by 100 (0.104) then added to one (1.104). P means the number of periods in which the interest rate is paid and compounded.  R = I^P I = R^(1/P) P = log(R) / log(I)  Once you have R you multiply it by the amount of your initial investment to find out how much total money is returned.  For simplicity the following amounts are approximate: 2 = 1.104^7 1.104 = 2^(1/7) 7 = log(2) / log(1.104)  So to double your money in seven years you need a yearly interest rate of 10.4, if compounded yearly. There is a fundamental flaw in this statement:  For example, a home bought cash $100,000 would have to be sold $242,726.247 30 years later just to make up with inflation, and that would be a 0% return. You forgot to deduct rent from your monthly carrying costs.  That changes the calculations significantly.   Your calculations are valid ONLY if you were to buy a house, and let it sit empty, which is unlikely.  Either you are going to live in it, and save yourself $1000 a month in rent, or, you are going to rent it out to someone, and earn an income of $1000 a month. Either way, you're up $1000 a month and this needs to be included. I believe the following formula provides a reasonable approximation. You need to fill in the following variables: The average annual return you need on investing the 15% =  (((MP5 - MP20) * 12) + (.0326 * .95 * PP / Y)) / (PP *.15) Example assuming an interest rate of 4% on a 100K home: If you invest the $15K you'll break even if you make a 9.86% return per year on average. Here's the breakdown per year using these example numbers: Note this does not consider taxes. As Chris pointed out: If your expenses are covered by the income exactly, as you have said to assume, then you are basically starting with a $40K asset (your starting equity), and ending with a $200K asset (a paid for home, at the same value since you have said to ignore any appreciation). So, to determine what you have earned on the $40K you leveraged 5x, wouldn't it be a matter of computing a CAGR that gets you from $40K to $200K in 30 years? The result would be a nominal return, not a real return. So, if I set up the problem correctly, it should be: $40,000 * (1 + Return)^30 = $200,000 Then solve for Return. It works out to be about 5.51% or so. Interesting. The answer can be as convoluted/complex as one wishes to make it, or back-of-envelope.  My claim is that if one starts at 21, and deposits 10% of their income each year, they will likely hit a good retirement nest egg. At an 8% return each year (Keep in mind, the last 40 years produced 10%, even with the lost decade) the 10% saver has just over 15X their final income as a retirement account. At 4% withdrawal, this replaces 60% of their income, with social security the rest, to get to nearly 100% or so replacement. Note - I wrote an article about Social Security Benefits, showing the benefit as a percent of final income. At $50K it's 42%, it's a higher replacement rate for lower income, but the replacement rate drops as income rises. So, the $5000 question. For an individual earning $50K or less, this amount is enough to fund their retirement. For those earning more, it will be one of the components, but not the full savings needed.  (By the way, a single person has a standard deduction and exemption totaling $10150 in 2014. I refer to this as the 'zero bracket.' The next $8800 is taxed at 10%. Why go 100% Roth and miss the opportunity to fund these low or no tax withdrawals?) "It can be pretty hard to compute the right number. What you need to know for your actual return is called the dollar-weighted return. This is the Internal Rate of Return (IRR) http://en.wikipedia.org/wiki/Internal_rate_of_return computed for your actual cash flows. So if you add $100 per month or whatever, that has to be factored in. If you have a separate account then hopefully your investment manager is computing this. If you just have mutual funds at a brokerage or fund company, computing it may be a bunch of manual labor, unless the brokerage does it for you. A site like Morningstar will show a couple of return numbers on say an S&P500 index fund. The first is ""time weighted"" and is just the raw return if you invested all money at time A and took it all out at time B. They also show ""investor return"" which is the average dollar-weighted return for everyone who invested in the fund; so if people sold the fund during a market crash, that would lower the investor return. This investor return shows actual returns for the average person, which makes it more relevant in one way (these were returns people actually received) but less relevant in another (the return is often lower because people are on average doing dumb stuff, such as selling at market bottoms). You could compare yourself to the time-weighted return to see how you did vs. if you'd bought and held with a big lump sum. And you can compare yourself to the investor return to see how you did vs. actual irrational people. .02, it isn't clear that either comparison matters so much; after all, the idea is to make adequate returns to meet your goals with minimum risk of not meeting your goals. You can't spend ""beating the market"" (or ""matching the market"" or anything else benchmarked to the market) in retirement, you can only spend cash. So beating a terrible market return won't make you feel better, and beating a great market return isn't necessary. I think it's bad that many investment books and advisors frame things in terms of a market benchmark. (Market benchmarks have their uses, such as exposing index-hugging active managers that aren't earning their fees, but to me it's easy to get mixed up and think the market benchmark is ""the point"" - I feel ""the point"" is to achieve your financial goals.)" "There is a very simple calculation that will answer the question: Is the expected ROI of the 401K including the match greater than the interest rate of your credit card? Some assumptions that don't affect the calculation, but do help illustrate the points. You have 30 years until you can pull out the 401K. Your credit card interest rate is 20% compounded annually.  The minimum payoffs are being disregarded, because that would legally just force a certain percentage to credit card. You only have $1000.  You can either pay off your credit card or invest, but not both.  For most people, this isn't the case.  Ideally, you would simply forego $1000 worth of spending, AND DO BOTH  Worked Example: Pay $1000 in Credit Card Debt, at 20% interest.   After 1 year, if you pay off that debt, you no longer owe $1200.  ROI = 20% (Duh!) After 30 years, you no longer owe (and this is pretty amazing) $237,376.31.  ROI = 23,638% In all cases, the ROI is GUARANTEED. Invest $1000 in matching 401k, with expected ROI of 5%. 2a. For illustration purposes, let's assume no match After 1 year, you have $1050 ($1000 principal, $0 match, 5% interest) - but you can't take it out. ROI = 5% After 30 years, you have $4321.94, ROI of 332% - assuming away all risk. 2b. Then, we'll assume a 50% match. After 1 year, you have $1575 ($1000 principle, $500 match, 5% interest) - but you can't take it out. ROI = 57% - but you are  stuck for a bit After 30 years, you have $6482.91, ROI of 548% - assuming away all risk. 2c. Finally, a full match After 1 year, you have $2100 ($1000 principle, $1000 match, 5% interest) - but you can't take it out. ROI = 110% - but again, you are stuck. After 30 years, you have $8643.89, ROI of 764% - assuming away all risk. Here's the summary - The interest rate is really all that matters.  Paying off a credit card is a guaranteed investment.   The only reason not to pay off a 20% credit card interest rate is if, after taxes, time, etc..., you could earn more than 20% somewhere else.   Note that at 1 year, the matching funds of a 401k, in all cases where the match exceeded 20%, beat the credit card. If you could take that money before you could have paid off the credit card, it would have been a good deal. The problem with the 401k is that you can't realize that gain until you retire. Credit Card debt, on the other hand, keeps growing until you pay it off. As such, paying off your credit card debt - assuming its interest rate is greater than the stock market (which trust me, it almost always is) - is the better deal. Indeed, with the exception of tax advantaged mortgages, there is almost no debt that has an interest rate than is ""better"" than the market."