Which r in perpetuity formula to pricing a business?
In the equity markets, the P/E is usually somewhere around 15.  The P/E can be viewed as the inverse of the rate of a perpetuity. Since the average is 15, and the E/P of that would be 6.7%, r should be 6.7% on average. If your business is growing, the growth rate can be incorporated like so: As you can see, a high g would make the price negative, in essence the seller should actually pay someone to take the business, but in reality, r is determined from the p and an estimated g. For a business of any growth rate, it's best to compare the multiple to the market, so for the average business in the market with your business's growth rate and industry, that P/E would be best applied to your company's income.
There is no formula for calculating a stock price based on the financials of a company. A stock price is set by the market and always has a component built into it that is based on something outside of the current valuation of a company using its financials. Essentially, the stock price of a company per share is whatever the best price it can get on the open market. If you are looking at how to evaluate if a stock is a good value at the current price, then look at some of the answers, but I wanted to answer this based on the way you phrased the question. If annualized rate of return is what you are looking for, using a tool would make it a lot easier.   In the post I've also explained how to use the spreadsheet.   Hope this helps. The book value is Total Assets minus Total Liabilities and so if you increase the Total Assets without changing the Total Liabilities the difference gets bigger and thus higher. Consider if a company had total assets of $4 and total liabilities of $3 so the book value is $1. Now, if the company adds $2 to the assets, then the difference would be 4+2-3=6-3=3 and last time I checked 3 is greater than 1. On definitions, here are a couple of links to clarify that side of things. From Investopedia: Equity = Assets - Liabilities From Ready Ratios: Shareholders Equity = Total Assets – Total Liabilities OR Shareholders Equity = Share Capital + Retained Earnings – Treasury   Shares Depending on what the reinvestment bought, there could be several possible outcomes. If the company bought assets that appreciated in value then that would increase the equity. If the company used that money to increase sales by expanding the marketing department then the future calculations could be a bit trickier and depend on what assumptions one wants to make really. If you need an example of the latter, imagine playing a game where I get to make up the rules and change them at will. Do you think you'd win at some point? It would depend on how I want the game to go and thus isn't something that you could definitively say one way or the other. I was merely trying to be helpful - Conceptually, you have dump this idea that something is skewed. It isn't. Firm A sold for $500 (equity value aka purchase price to shareholders) + debt (zero) - cash (50) for 450.  Enterprise value is the cash free, debt free sale price.  The implied ev multiple is 4.5x on A - that is the answer. The other business sold for a higher multiple of 5x.  If you would pile on more cash onto A, the purchase price would increase, but the EV wouldn't.      The idea is to think hard about the difference between equity value and enterprise value when examining a transaction. Let P denote the amount of the investment, R the rate of return and I the rate of inflation.  For simplicity, assume that the payment p is made annually right after the return has been earned.  Thus, at the end if the year, the investment P has increased to P*(1+R) and p is returned as the annuity payment. If I = 0, the entire return can be paid out as the payment, and thus p = P*R. That is, at the end of the year, when the dust settles after the return P*R has been collected and paid out as the annuity payment, P is again available at the beginning of the next year to earn return at rate R. We have P*(1+R) - p = P If I > 0, then at the end of the year, after the dust settles, we cannot afford to have only P available as the investment for next year. Next year's payment must be p*(1+I) and so we need a larger investment since the rate of return is fixed.  How much larger?  Well, if the investment at the beginning of next year is P*(1+I), it will earn exactly enough additional money to pay out the increased payment for next year, and have enough left over to help towards future increases in payments. (Note  that we are assuming that R > I. If R < I, a perpetuity cannot be created.)  Thus, suppose that we choose p such that P*(1+R) - p = P*(1+I)   Multiplying this equation by (1+I), we have [P(1+I)]*(1+R) - [p*(1+I)] = P*(1+I)^2 In words, at the start of next year, the investment is P*(1+I) and the return less the increased payout of p*(1+I) leaves an investment of P*(1+I)^2 for the following year.  Each year, the payment and the amount to be invested for the following year increase by a factor of (1+I). Solving P*(1+R) - p = P*(1+I) for p, we get  p = P*(R-I)  as the initial perpetuity payment and the payment increases by a factor (1+I) each year.  The initial investment is P and it also increases by a factor of (1+I) each year. In later years, the investment is P*(1+I)^n at the start of the year, the payment is p*(1+I)^n and the amount invested for the next year is P*(1+I)^{n+1}. This is the same result as obtained by the OP but written in terms that I can understand, that is, without the financial jargon about discount rates, gradients, PV, FV and the like. "The fallacy in your question is in this statement: ""The formulas must exist, because prices can be followed real time."" What you see are snapshots of the current status of the stock, what was the last price a stock was traded at, what is the volume, is the price going up or down.  People who buy and hold their stock look at the status every few days or even every few months. Day traders look at the status every second of the trading day.  The math/formula comes in when people try to predict where the stock is going based on the squiggles in the line. These squiggles move based on how other people react to the squiggles.  The big movements occur when big pieces of news make large movements in the price. Company X announces the release of the key product will be delayed by a year; the founder is stepping down; the government just doubled the order for a new weapon system; the insiders are selling all the shares they can. There are no formulas to determine the correct price, only formulas that try to predict where the price may go." For the constant growth problem, I don't see why the answer would be anything other than the stock price grown for 3 years at the cost of equity determined from CAPM. Someone can correct me if the dividend is relevant.  So:  rs = rf + beta x (market premium)  rs = 0.046 + 0.9 x (0.06) = 0.10  price now = $40.00  price after 3 years' growth = $40.00 x (1+0.10)^3 = $53.24  EDIT: https://www.quora.com/Why-are-dividend-yields-factored-into-beta  two conflicting answers so you'll just have to figure it out The equation is the same one used for mortgage amortization.  You first want to calculate the PV (present value) for a stream of $50K payments over 20 years at a10% rate.  Then that value is the FV (future value) that you want to save for, and you are looking to solve the payment stream needed to create that future value. Good luck achieving the 10% return, and in knowing your mortality down to the exact year. Unless this is a homework assignment, which need not reflect real life.  Edit - as indicated above, the first step is to get that value in 20 years:  The image is the user-friendly entry screen for the PV calculation. It walks you though the need to enter rate as per period, therefore I enter .1/12 as the rate. The payment you desire is $50K/yr, and since it's a payment, it's a negative number. The equation in excel that results is: =PV(0.1/12,240,-50000/12,0) and the sum calculated is $431,769 Next you wish to know the payments to make to arrive at this number:  In this case, you start at zero PV with a known FV calculated above, and known rate. This solves for the payment needed to get this number, $568.59 The excel equation is: =PMT(0.1/12,240,0,431769) Most people have access to excel or a public domain spreadsheet application (e.g. Openoffice). If you are often needing to perform such calculations, a business finance calculator is recommended. TI used to make a model BA-35 finance calculator, no longer in production, still on eBay, used.  One more update- these equations whether in excel or a calculator are geared toward per period interest, i.e. when you state 10%, they assume a monthly 10/12%. With that said, you required a 20 year deposit period and 20 year withdrawal period. We know you wish to take out $4166.67 per month. The equation to calculate deposit required becomes - 4166.67/(1.00833333)^240= 568.59  HA! Exact same answer, far less work. To be clear, this works only because you required 240 deposits to produce 240 withdrawals in the future. I was emailing back and forth with a manager in a different department on how real returns are being calculated, and he said that the industry standard is 1 + real returns*(1+inflation) - fees, and to not use my formula because it can double count inflation, making fees lower. However, real returns are not observable in the future, and I do not why he uses that formula. The returns were used in an Excel spreadsheet.   What are your thoughts about this? 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 find the higher estimates a bit unbelievable. A big part of my job is liability valuation and small assumption changes can have a huge impact on results.  They may be right (future) dollar value wise but the proper way to think about this stuff is in present value terms.  This could actually be a really interesting study - you all just gave me a great idea for a potential masters thesis :) "Complex matter that doesn't boil down to a formula.   The quant aspect could be assessed by calculating WACCs under various funding scenarii and trying to minimize, but it is just one dimension of it.  The quali aspects can vary widely depending on the company, ownership structure, tax environment and business needs and it really can't be covered even superficially in a reddit comment...  Few examples from the top of my mind to give you a sense of it:  - shareholders might be able to issue equity but want to avoid dilution, so debt is preferred in the end despite cost. Or convertible debt under the right scenario.  - company has recurring funding needs and thinks that establishing a status on debt market is worth paying a premium to ensure they can ""tap"" it whenever hey need to.  - adding debt is a way to leverage and enhance ROI/IRR for certain types of stakeholders (think LBOs)  - etc etc etc  Takes time and a lot of experience/work to be able to figure out what's best and there isn't always a clear answer.  Source: pro buy side credit investor with experience and sizeable AuMs." Average rates of return usually assume compounding, so your formula would be for annual compounding ,or for continuous compounding. dafuq did I just read.  I think this would be better explained with an excel chart because I have no idea what you just wrote.   Can you please use real numbers in a possible hypothetical case study.  I'm absolutely terrible at thinking theoretically. I was looking for ideas on what the usual figures are in such positions. Something like market value, or other terms such as changing slab percentages in compensation. Not sure what the best practices are in this situation.  Not sure what you are referring to.