What is the equation for an inflation adjusted annuity held in perpetuity?
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. 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. The question lacks specificity, i.e. when does the initial investment occur, now or one period from now?  If now then it is a perpetuity due.   I  will consider under 2 scenarios, A and B, relating to the size of the initial investment. A. Assuming that the initial investment (C_0) occurs now and each payment thereafter has the relationship (1+g) with this investment then the relevant base equation is that for the present value of a growing perpetuity due, expressed in terms of C_0, i.e.  PVGPD=  [C_0*(1+g)*(1+i)]/(i-g). Now, to suit the question asked, we can see that i=fixed rate of return (f) and g = expected inflation rate (e) such that we can rewrite the equation as PVGPD = [C_0*(1+e)*(1+i)]/(i-e].   We know that f = is a fixed nominal rate and must be adjusted for e to calculate the real rate (r) according to the equation f=(1+r)*(1+e)-1. Therefore  PVGPD = [C_0*(1+e)(1+(1+r)(1+e)-1)]/((1+r)*(1+e)-1-e] Tidying up  PVGPD = {C_0*(1+r)(1+e)^2}/[r(1+e)] PVGPD = [C_0*(1+r)*(1+e)]/r B. Assuming that the initial investment (X) is not equal to each subsequent perpetual payment (C_1) then the relevant base equation is that for the the initial investment plus the present value of a growing perpetuity, i.e.  PVGP=  X + [C_1/(i-g)] Rewriting  PVGP = X + [C_1/(f-e)]    Substituting PVGPD = X + {C_1/[(1+r)*(1+e)-1-e]} Tidying up  PVGPD = X + C_1/[r*(1+e)]
"Can you give the context? A bond usually only has one ""tenor,"" so weighted average tenor could apply to all the bonds that a single firm owes? Or an industry or the market or a specific bond rating or whatever. But we usually use duration as it's more useful" Say you buy a bond that currently costs $950, and matures in one year, at $1000 face value. It has one coupon ($50 interest payment) left. The coupon, $50, is 50/950 or 5.26%, but you get the face value, $1000, for an additional $50 return. This is why the yield to maturity is higher than current yield. If the maturity were in two years, the coupons still provide 5.26%, and the extra 1000/950 is another 5.26% over 2 years, or (approx) 2.6%/yr compounded, for a total YTM of 7.86%. This is a back-of envelope calculation, the real way to calculate is with a finance calculator. Entering PV (present value) FV (future value) PMT (coupon payment(s)) and N (number of periods). With no calculator or spreadsheet, my estimate will be pretty close. "This is really two questions about yield and contents. Content As others have noted, an annuity is a contractual obligation, not a portfolio contained within an investment product per se. The primary difference between whether an annuity is fixed or variable is what the issuer is guaranteeing and how much risk/reward you are sharing in. Generally speaking, the holdings of an issuer are influenced by the average ""duration"" of the payments. However, you can ascertain the assets that ""back"" that promise by looking at, for example, the holdings of a large insurance or securities firm. That is why issuers are generally rated as to their financial strength and ability to meet their obligations. A number of the market failures you mentioned were in part caused by the failure of these ratings to represent the true financial strength of the firm. Yield As to the second question of how they can offer a competitive rate, there are at least several reasons (I am assuming an immediate annuity) : 1) Return/Depletion of Principal The 7% you are being quoted is the percent of your principal that will be returned to you each year, not the rate of return being earned by the issuer. If you invest $100 in the market personally and get a 5% return, you have $105.  However, the annuity's issuer is also returning part of your principal to you each year in your payment, as they don't return your principal when you eventually die. Because of this, they can offer you more each year than they really make in the market. What makes a Ponzi scheme different is that they are also paying out your principal (usually to others), but lie to you by telling you it's still in your account. :) 2) The Time Value of Money A promise to pay you $500 tomorrow costs less than $500 today A fixed annuity promises to pay you a certain amount of money each year. This can be represented as a rate of return calculated based on how much you have to pay to get that annual payment, but it is important to remember that the first payment will be worth substantially more in real purchasing power than the last payment you get. The longer you live, the less your fixed payment is worth in real terms due to inflation! In short, the rate of return has to be discounted for inflation, it is not a ""real"" rate of return. In other words, if you give me $500 today and I promise to pay you $100 for the next 5 years, I am making money not only because I can invest the money between now and then, but also because $100 will be worth less five years from now than it is today. With annuities, if you want your payment to rise in step with inflation, you have to pay more for that (a LOT more!). These are the two main reasons - here are a few smaller ones: 3) A very long Time Horizon If the stock market or another asset class is performing well/poorly, the issuer can often afford to wait much longer to buy or sell than an individual, and can take better advantage of historical highs and lows over the long term. 4) ""Big Boy"" investing A large, financially sound issuer can afford to take risks that an individual cannot, such as in very large or illiquid assets, such as a private company (a la Warren Buffet). 5) Efficiencies of scale Institutional investors have a number of legal advantages over individuals, which I won't discuss in detail here. However, they exist. Large issuers are also often in related business (insurance, mutual funds) such that they can deal in large volumes and form an internal clearinghouse (i.e. if I want to buy Facebook and you want to sell it, they can just move the stock around without doing any trading), with the result that their costs of trading are lower than those of an individual. Hope that helps!" There is always some fine print, read it. I doubt there is any product out there that can guarantee an 8% return.  As a counter example - a 70 yr old can get 6% in a fixed immediate annuity. On death, the original premium is retained by the insurance company.  Whenever I read the prospectus of a VA, I find the actual math betrays a salesman who misrepresented the product. I'd be really curious to read the details for this one. Beware of surrender charges also Surrender Charges Many annuities   will impose a surrender charge if the annuity is cashed in before a   specific period of time. That period may run anywhere from 1 to 12   years. A typical surrender charge is one that starts at 7% in the   first year of the contract, and declines by 1% per year thereafter   until it reaches zero. The charge is made against the value of the   investment when the annuity is surrendered, and its purpose (other   than simply to make money for the insurance company) is to discourage   a short-term investment by the purchaser. For that reason, an annuity   should always be considered a long-term investment. In the typical   fixed annuity, though, this charge will not apply provided no more   than 10% of the investment is withdrawn per year. source: http://www.fool.com/retirement/annuities/annuities02.htm If you've held it for 10 years as you claim, you may not owe any or much in surrender charges, but you definitely want to know what the situation is before you make a move. The example from the following website: Investopedia - Calculating The Present And Future Value Of Annuities  specifically the section 'Calculating the Present Value of an Annuity Due' shows how the calculation is made.  Using their figures, if five payments of $1000 are made over five years and depreciation (inflation) is 5%, the present value is $4545.95  There is also a formula for this summation, (ref. finance formulas) I came up with a real way. I saw once the market be so dumb as to allow this to work. Inflation rate = 2.5%. Home interest rate = 3%. Tax deduction = 1%. Money spent on inflation-adjusted I bonds (at the time these paid 0% net, that is 2.5% gross). Result, .5% profit after accounting for inflation. The kicker: Uncle Sam's I bonds are tax free. Sure it's not possible today, but the rates occasionally drop low enough. "There is no formula to answer the question. You have to balance return on investment with risk. There's also the question of whether you have any children or other heirs that you would like to leave money to. The mortgage is presumably a guaranteed thing: you know exactly how much the payments will be for the rest of the loan. I think most annuities have a fixed rate of return, but they terminate when you both die. There are annuities with a variable return, but usually with a guaranteed minimum.  So if you got an annuity with a fixed 3.85% return, and you lived exactly 18 more years, then (ignoring tax implications), there'd be no practical difference between the two choices. If you lived longer than 18 years, the annuity would be better. If less, paying off the mortgage would be better. Another option to consider is doing neither, but keeping the money in the 401k or some other investment. This will usually give better than 3.85% return, and the principal will be available to leave to your heirs. The big drawback to this is risk: investments in the stock market and the like usually do better than 3 or 4%, but not always, and sometimes they lose money. Earlier I said ""ignoring tax implications"". Of course that can be a significant factor. Mortgages get special tax treatment, so the effective interest rate on a mortgage is less than the nominal rate. 401ks also get special tax treatment. So this complicates up calculations trying to compare. I can't give definitive numbers without knowing the returns you might get on an annuity and your tax situation." The solution is x = 8.92. This assumes that Chuck's six years of deposits start from today, so that the first deposit accumulates 10 years of gain, i.e. 20*(1 + 0.1)^10.  The second deposit gains nine years' interest: 20*(1 + 0.1)^9 and so on ...    If you want to do this calculation using the formula for an annuity due, i.e. http://www.financeformulas.net/Future-Value-of-Annuity-Due.html  where  (formula by induction) you have to bear in mind this is for the whole time span (k = 1 to n), so for just the first six years you need to calculate for all ten years then subtract another annuity calculation for the last four years.  So the full calculation is:  As you can see it's not very neat, because the standard formula is for a whole time span.  You could make it a little tidier by using a formula for k = m to n instead, i.e.  So the calculation becomes  which can be done with simple arithmetic (and doesn't actually need a solver). You need to solve the money-weighted return equation.  It cannot be expressed as a formula for the interest, but it can be solved numerically as shown here. Using the OP's figures, with monthly withdrawals of $100,000. The summation for the withdrawals can be replaced with the standard annuity formula.  The resulting monthly return is converted to a nominal annual return compounded monthly. Money-weighted return equation with start and end balances s0 and s1  The interest is 2.63282 % per annum, nominal compounded monthly. Selectable equation Dealhoud, You are on the right path but this is actually an Equivalent annual cost problem. You have to take it a step further and divide the NPV by the Annuity factor.  You need to calculate the annuity factor and divide the investment by that amount.    Annuity Factor = (1-(1/(1+r)^n )) /r   We know the annual cost of the the current forklift = $5K. We need to figure out the Equivalent annual cost of the new forklift.  Investment / Annuity Factor + annual Cost = EAC.   Therefore $20K / (1 - (1/(1.04^10 )) / 0.04 = $2,465.82 + annual Maintenance = $4,465.82 therefore it makes sense to replace the old machine. Ok so I used Excel solver for this but it's on the right track.  Latest price = $77.19  Latest div = $1.50  3-yr div growth = 28%  g = ???  rs = 14%  So we'll grow out the dividend 3 years @ 28%, and then capitalize them into perpetuity using a cap rate of [rs - g], and take the NPV using the rs of 14%. We can set it up and then solve g assuming an NPV of the current share price of $77.19.  So it should be:  NPV = $77.19 =   [$1.50 / (1+0.14)^0 ] +  [$1.50 x (1+0.28)^1 / (1+0.14)^1 ] +  ... +  [$1.50 x (1+0.28)^3 / (1+0.14)^3 ] +  [$1.50 x (1+0.28)^3 x (1+g) / (0.14-g) / (1+0.14)^4 ]  Which gives an implied g of a little under 9%. Let me know if this makes sense, and definitely check the work... Yes - it's called the rate of inflation. The rate of return over the rate of inflation is called the real rate of return.   So if a currency experiences a 2% rate of inflation, and your investment makes a 3% rate of return, your real rate of return is only 1%. One problem is that inflation is always backwards-looking, while investment returns are always forward-looking. There are ways to calculate an expected rate of inflation from foreign exchange futures and other market instruments, though. That said, when comparing investments, typically all investments are in the same currency, so the effect of inflation is the same, and inflation makes  no difference in a comparative analysis.  When comparing investments in different currencies, then the rate of inflation may become important. The formula for determining the number of payments (months) you'll need to make on your loan is:  where i=monthly interest rate (annual rate / 12), A=loan amount (principal), and P=monthly payment. To determine the total interest that you will pay, you can use the following formula:  where P=monthly payment, N=number of payments (from above formula), and A=loan amount (principal). A quick example: using the numbers in the screenshot above ($10,000 loan, $500 monthly payment, 10% APR), the number of payments ends up to be 21.97 (which means that payment number 22 is slightly less than the rest).  In the second formula, you take that number times your $500 payment and determine that you have paid $10,984.81 over the course of the entire loan period.  Subtracting the principal, you have paid $984.81 in total interest. On your spreadsheet, the function you are looking for is NPER: NPER(rate, payment_amount, present_value, [future_value, end_or_beginning]) rate - The interest rate. (This should be the monthly rate, or the annual rate divided by 12.) payment_amount - The amount of each payment made. (For a loan payment, this should be a negative number.) present_value - The current value of the annuity. (The initial principal of the loan) future_value - [ OPTIONAL ] - The future value remaining after the final payment has been made.  (This should be 0, the default if omitted.) end_or_beginning - [ OPTIONAL - 0 by default ] - Whether payments are due at the end (0) or beginning (1) of each period. Annuities, like life insurance, are sold rather than bought.  Once upon a time, IRAs inherited from a non-spouse  required the beneficiary to  (a) take all the money out within 5 years, or  (b) choose to receive the value of the IRA at the time of  the IRA owner's death in equal installments over the expected  lifetime of the beneficiary.  If the latter option was chosen, the IRA custodian issued the  fixed-term annuity in return for the IRA assets. If the IRA  was invested in (say) 15000 shares of IBM stock, that stock  would then belong to the IRA custodian who was obligated to  pay $x per year to the beneficiary for the next 23 years (say). There was no investment any more that could be transferred  to another broker, or be sold and the proceeds invested in Facebook stock (say).  Nor was the custodian under any  obligation to do anything except pay $x per year to the  beneficiary for the 23 years. Financial planners loved to get at this money under the old IRA rules by suggesting that if all the IRA money were taken out and invested in stocks or mutual funds through  their company, the company would pay a guaranteed $y  per year, would pay more than $y in each year that the  investments did well, would continue payment until the  beneficiary died (or till the death of the beneficiary  or beneficiary's spouse - whoever died later), and would return the entire sum invested (less payouts already made, of course) in case of premature death. $y typically would be a little larger than $x too, because it factored in some earnings of the investment over the years. So what was not to like? Of course, the commissions earned by the planner and the lousy mutual funds and the huge surrender charges were  always glossed over.