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)]
"No, virtually ever item in the CPI is adjusted using hedonics, which by definition can only be used to lower inflation, not adjust it up.  Hedonics is not necessarily bad, but it doesn't actually reflect inflation, it reflects standard of living vs. purchasing power, which is not useful for a purely monetary measure.  It also does not correctly reflect that despite the fact that a 1970's 20"" TV cannot from an economic standpoint be directly compared to a 2012 20"" TV, if the price is the same (adjusted for inflation), inflation represents how the value of the dollar has accrued vs the value of a 20"" TV, which is what the definition of inflation is.  So, no, I don't think you're correct on this. This is also kind of glossing over the fact that the CPI essentially makes the bold argument that energy and transportation prices never affect inflation." I have answered your question in detail here https://stackoverflow.com/questions/12396422/apr-calculation-formula The annuity formula in FDIC document is at first finding PVIFAD present value annuity due factor and multiplying it with annuity payment and then dividing it by an interest factor of (1+i) to reduce the annuity to an ordinary annuity with end of period payments They could have simply used PVIFA and multiplying it with annuity payment to find the present value of an ordinary annuity In any case, you should not follow the directions in FDIC document to find interest rate at which the present value of annuity equals the loan amount. The method they are employing is commonly used by Finance Professors to teach their students how to find internal rate of return. The method is prone to lengthy trial and error attempts without having any way of knowing what rate to use as an initial guess to kick off the interest rate calculations So this is what I would suggest if you are not short on time and would like to get yourself familiar with numerical methods or iterative techniques to find internal rate of return There are way too many methods at disposal when it comes to finding interest rates some of which include All of the above methods use a seed value as a guess rate to start the iterative calculations and if results from successive calculations tend to converge within a certain absolute Error bound, we assume that one of the rates have been found as there may be as many rates as the order of the polynomial in this case 36 There are however some other methods that help find all rates by making use of Eigenvalues, but for this you would need a lengthy discourse of Linear Algebra One of the methods that I have come across which was published in the US in 1969 (the year I was born :) ) is called the Jenkins Traub method named after the two individuals who worked jointly on finding a solution to all roots of a polynomial discarding any previous work on the same subject I been trying to go over the Jenkins Traub algorithm but am having difficulty understanding the complex nature of the calculations required to find all roots of the polynomial In summary you would be better of reading up on this site about the Newton Raphson method to find IRR Retirement calculation, in general, should be based on the amount of money needed per year/month and the expected life expectancy. Life expectancy, if calculated to 90 years (let's say) indicates that post retirement age (60 yrs.) your accumulated/invested money should generate adequate income to cover your expenses till 90 years. The problem in general is not how long you shall live but what would be your expected spending from retirement to end of life expectancy. The idea is at the minimum your investments should generate income that is inflation adjusted. One way to do this is to consider your monthly expense now i.e. the expense that is absolute minimum for carrying on (food, electricity, water, medicines, household consumables, car petrol, insurance, servicing, entertainment, newspaper etc.) this does not contain the amortizable liabilities (home loan, child's education, other debts). It is better to take this amount per family rather than per person and yearly rather than monthly (as we tend to miss a lot of yearly expenses). This amount that you need today will increase at a Compounded Annual Growth Rate (CAGR) of the average inflation. For example, if today you spend 100 per year in 7 years you will need to spend appx. 200 at 10% inflation. Now, your investments will not increase post your retirement, so your current investment needs to do two things (1) give you your yearly requirement (2) grow by a fixed amount so that next year it can give you CAGR adjusted returns. In general, this kind of investment grows by high net amounts initially and slowly the growth decrease. The above can be calculated by Net Present value (NPV) formulae (http://en.wikipedia.org/wiki/Net_present_value). The key is to remember that the money that is invested when you retire should be able to give you inflation adjusted returns to cover your yearly expenses. How much money you need depends on your life style/expectation and how much return is received depends on the instruments that you invest on. As for your question above on the difference between the age of you and your spouse, it better to go with the consolidated family requirement and get an idea of how much investment is necessary and provision the same as soon as possible from your as well as your spouse's income. Hope this helps.- thanks I don't like it using percentages makes no sense.  Find out what market value is for rent and pay 1/2 of that to your partner, adjust annually.   You partner should be protected from inflation if he is going to invest in real estate. I don't understand the logic of converting a cost of funds of 4% to a monthly % and then subtracting that number from an annual one (the 1.5%).  Unfortunately without seeing the case I really can't help you...there was likely much you have left out from above. "How will contribution limits rise?  Contribution limits are raised based on COLA. COLA (Cost of Living Adjustment) is a number based on CPI (Consumer Price Index) which is published by the Bureau of Labor Statistics. The IRS publishes these limits annually and a simple search term to find them each year is ""415 limits"" because section 415 lists how much the various qualified plans can set aside each year. CPI is a heavily managed and very political number. It is the number that is cited when the media talks about ""inflation."" Since it drives increases in salaries, deductions, Social Security and pension payments, there is very heavy pressure to keep the number smaller than it really is.  My next step was to calculate how long that money will last  One traditional rule-of-thumb is to withdraw 4% of your balance each year. Another alternative is to purchase annuities. While younger people think that annuities are horrible investments, they appeal to older people because they protect the annuitant from excessive risk of losing your capital. When you are 30, and another 2008 comes along wiping out half your savings, you have time to re-earn that money. When you are 60, and another 2008 comes along wiping out half your savings, you have very few years to recover that money. So an annuity pushes that risk onto insurance companies. If you think you will die in your 90s, then an annuity is going to be a good investment (the insurance company will be betting that you won't be living that long).  I have a simple spreadsheet that I use to calculate estimated projected balances and compare them to actual performance. Don't forget that when you reach 50, the amount you can contribute goes up due to ""catch up contributions.""   There are 2 views of the same tab in this picture. There are 3 growth rates: pessimistic, nominal and optimistic. You can change the numbers to suit your own projections of future growth. Other tabs on this spreadsheet include measurements of actual performance by my 401k and IRA accounts. At the end of the year, I replace the numbers in columns F, G & H with the actual end-of-year dollar amounts. This way, future estimates do not get too far unhinged from reality.  If you need more sophisticated planning, such as Monte Carlo analysis to attempt to cope with inflation, I recommend the book Engineering Your Retirement. The book is aimed at younger engineers, so there is a bit more math than the average person would want." That's pretty typical for introductory problems.  It's leading you into an NPV question.  They're keeping the cash flows the same to illustrate the time value of money to show you that even though the free cash flow is the same in year 1 and year 4 or whatever when you discount it to present value today's stream is worth more than tomorrow's The article links to William Bernstein’s plan that he outlined for Business Insider, which says: Modelling this investment strategy Picking three funds from Google and running some numbers. The international stock index only goes back to April 29th 1996, so a run of 21 years was modelled.  Based on 15% of a salary of $550 per month with various annual raises: Broadly speaking, this investment doubles the value of the contributions over two decades. Note:  Rebalancing fees are not included in the simulation. Below is the code used to run the simulation.  If you have Mathematica you can try with different funds.    Notice above how the bond index (VBMFX) preserves value during the 2008 crash.  This illustrates the rationale for diversifying across different fund types.