Annuities question - Equations of value
These are the steps I'd follow: $200 today times (1.04)^10 = Cost in year 10.  The 6 deposits of $20 will be one time value calculation with a resulting year 7 final value. You then must apply 10% for 3 years (1.1)^3 to get the 10th year result.  You now have the shortfall. Divide that by the same (1.1)^3 to shift the present value to start of year 7. (this step might confuse you?) You are left with a problem needing 3 same deposits, a known rate, and desired FV. Solve from there.  (Also, welcome from quant.SE. This site doesn't support LATEX, so I edited the image above.) 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 don't mention what kind of insurance you're talking about, but I'll just address one angle on the question. For some kinds of insurance, such as health insurance (in the US), auto insurance, and homeowner's insurance, you may be insuring against an event that you would not be able to pay for without the insurance.  For instance, if you are at fault in a car accident and injure someone, they could sue you for $100,000.  A lot of people don't have $100,000.  So it's not even a matter of ""I'll take the risk of having to pay it when the time comes""; if the time comes, you could lose virtually everything you own and still have to pay more from future earnings.  You're not just paying $X to offset a potential loss of $Y; you're paying $X to offset a potential derailment of your entire life.  It is plausible that you could assign a reasonable monetary value to that potential ""cost"" that would mean you actually come out ahead in the insurance equation. It is with smaller expenses (such as insuring a new cellphone against breakage) that insurance becomes harder to justify.  When the potential nonfinancial ""collateral damage"" of a bad event are less, you must justify the insurance expenses on the financial consequences only, which, as you say, is often difficult." "Say you need to use Einstein's field equations at work. If your job is to find the ""one equation to rule them all"" than you should ask questions concerning the validity of the equations and try to unify them in the quantum realm. But like most of us, if your job is find tune the many knobs that govern the output, you don't need to use math beyond what is already packaged in libraries. You still need to know what the equations are and what they do, but your interaction with them on a daily basis is call function X on object Y." "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." "Not to state the obvious, but whenever an investment is being made, the ""nuts and bolts"" is your return on investment.  Analyzing the rate of return on an investment is the primary factor in any decision.  Ideally, once the actual mechanics of investment and side ""benefits"" are factored out, the goal is to be able to analyze the pure financial return. Usually the biggest problem faced in analyzing various investments is comparing the Present Value of an investment to a series of payments that may be made or received in the future.  When considering the purchase of a large equity, for example, you might be looking at what series of payments are required to purchase the asset.  You can also reverse this and ask, ""What amount of money is equivalent to this series of payments?"" Ultimately, the Present Value of an Annuity is the way to make these comparisons equal.   Fundamentally, the Present Value of an annuity is an amount of money that should, in theory, be equivalent to a series of payments.  There is, for example, technically no difference between $1064.94 today and $100 a month for a year, at an interest rate of 1% per month.  Grant you, most people would be happier with the money now, but that is what interest does - it compensates you for waiting on your money.   You can fire up a spreadsheet and calculate the Present Value as long as you have the monthly payment, interest rate, and number of periods.  Alternatively, you can calculate any one of those missing four variables - and the key is usually to understand what that rate would be in order to compare the investments. Finally, the taxable implication is really just an adjustment to the rate of return. Imagine the following three scenarios: (Obviously the rates are fictional - the goal is to show they are the same). Scenarios 1 & 2 are really just two sides of the same coin.  Using the Future Value formula in Excel = FV(0.5%, 12, -100), you get $1233.56.  In scenario 1, you would have $1233.56 in your bank account.  In scenario 2, your bank would have $1233.56 from you, and you would have $100 less debt per month.  They are equivalent transactions.   Scenario 3 is really just a variation on scenario 2, localized to the United States.  Because the interest is tax deductible, however, the rate of 6% isn't really accurate.  Assuming you had a 25% tax bracket, you'd actually be getting back one quarter of your interest.  Put another way, 7.5% mortgage interest costs you as much as 6% credit card debt. This is how you compare apples and organges - just turn everything into an annuity or a lump sum, using Present Value calculations.   Finally, quick rule of thumb - if you owe taxes in both Canada and the US, your Canadian taxes are probably higher than your American ones.  As such, any tax incentives will be concomitantly higher.  If you only can only use Canadian tax incentives, then look to those incentives, other things being equal." Damn, helpful Harry above me.   So, in general, when compounding the value of an investment, if you're seeing an annualized interest rate of 4%, and the interest compounds monthly (or n number of times per year), you're going to multiply the Principal P by the growth rate (the interest rate), adjusted for the number of periods that your investment grows in a year. P_end = P * (1 + 0.04/n)^(n * t), where n = number of periods, and t = number of years. If the interest compounds annually, you earn P *(1.04), if it compounds monthly, you earn (1 + 0.04/12)^(12 * 1).   Apply this logic to discounting future cash flows to their net present value. When discounting future cash flows, you're essentially determing the opportunity cost of now being unable to put your investment elsewhere and earning that corresponding interest (discount) rate.   Thus, you would discount $1000 by (1 + 0.08/12)^1, and $2000, $3000 in a similar fashion. Then, as icing on the cake, sum up to get your cumulative net present value.  Please let me know if any portion of my explanation is unclear; I would be happy to elaborate! 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)] "The answer to almost all questions of this type is to draw a diagram.  This will show you in graphical fashion the timing of all payments out and payments received.  Then, if all these payments are brought to the same date and set equal to each other (using the desired rate of return), the equation to be solved is generated. In this case, taking the start of the bond's life as the point of reference, the various amounts are: Pay out = X Received = a series of 15 annual payments of $70, the first coming in 1 year.  This can be brought to the reference date using the formula for the present value of an ordinary annuity. PLUS Received = A single payment of $1000, made 15 years in the future.  This can be brought to the reference date using the simple interest formula. Set the pay-out equal to the present value of the payments received and solve for X I am unaware of the difference, if any, between ""current rate"" and ""rate to maturity"" Finding the rate for such a series of payments would start out the same as above, but solving the resulting equation for the interest rate would be a daunting task..." If you're in the USA and looking to retire in 10 years, pay your Social Security taxes? :P Just kidding. Do a search for Fixed Rate Annuities.