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).
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. "An annuity is a product. In simple terms, you hand over a lump sum of cash and receive an agreed annual income until you die. The underlying investment required to reach that income level is not your concern, it's the provider's worry. So there is a huge mount of security to the retiree in having an annuity. It is worth pointing out that with simple annuities where one gives a lump sum of money to (typically) an insurance company, the annuity payments cease upon the death of the annuitant. If any part of the lump sum is still left, that money belongs to the company, not to the heirs of the deceased. Fancier versions of annuities cover the spouse of the annuitant as well (joint and survivor annuity) or guarantee a certain number of payments (e.g. 10-year certain) regardless of when the annuitant dies (payments for the remaining certain term go to the residual beneficiary) etc. How much of an annuity payment the company offers for a fixed lump sum of £X depends on what type of annuity is chosen; usually simple annuities give the maximum bang for the buck. Also, different companies may offer slightly different rates. So, why should one choose to buy an annuity instead of keeping the  lump sum in a bank or in fixed deposits (CDs in US parlance), or  invested in the stock market or the bond market, etc., and making periodic withdrawals from these assets at a ""safe rate of withdrawal""? Safe rates of withdrawal are often touted as 4% per annum in the US,  though there are newer studies saying that a smaller rate should be used.  Well, safe rates of withdrawal are designed to ensure that the retiree does not use up all the money and is left destitute just when medical bills and other costs are likely to be peaking.  Indeed, if all the money were kept in a sock at home (no growth at all), a 4% per annum withdrawal rate will last the retiree for 25 years. With some growth of the lump sum in an investment, somewhat larger withdrawals might be taken in good years, but that 4% is needed even when the investments have declined in value because of economic conditions beyond one's control. So, there are good things and bad things that can happen if one chooses to not buy an annuity. On the other hand, with an annuity, the payments will continue till death and so the retiree feels safer, as Chris  mentioned.  There is also the serenity in not having to worry how the investments are doing; that's the company's business. A down side, of course, is that the payments are fixed and if inflation is raging, the retiree still gets the same amount. If extra cash is needed one year for unavoidable expenses, the annuity will not provide it, whereas the lump sum (whether kept in a sock or invested) can be drawn on for the extra expense. Another down side is that any money remaining is gone, with nothing left for the heirs. On the plus side, the annuity payments are usually larger than  those that the retiree will get via the safe rate of withdrawal method from the lump sum. This is because the insurance company is applying the laws of large numbers: many annuitants will not survive past their life expectancy, and their leftover monies are pure profit to the insurance company, often more than enough (when invested properly by the company) to pay those old codgers who continue to live past their life expectancy. Personally, I wouldn't want to buy an annuity with all my money, but getting an annuity with part of the money is worthwhile. Important: The annuity discussed in this answer is what is sometimes called a single-premium or an immediate annuity.  It is purchased at the time of retirement with a single (large) lump sum payment. This is not the kind of annuity that is described in JAGAnalyst's answer which requires payment of (much smaller) premiums over many years. Search this forum for variable annuity to learn about these types of annuities." 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. The general concept is that your money will grow at an accelerating rate because you start getting interest paid on your returns in addition to the original investment.  As a simple example, assume you invest $100 and get 10% interest per year paid annually.   -At the end of the first year you have your $100 + $10 interest for a total of $110. -So you start the second year with $110 and so 10% would be $11 for a total of $121. -The third year you start with $121 so 10% would be $12.10 for a total of $133.10 See how the amount it goes up each year increases? If we were talking a higher initial amount or a larger number of years that can really add up. That is essence is compound interest. Most of the complicated looking formulas you see out there for compound interest are just shortcuts so you don't have to iteratively go through the above exercise a bunch of times to find out how much you would have after some number of years. This formula tells you how much you would have(A) after a certain number of years(t)  at a given interest rate(r) assuming they pay interest n times per year, for example you would use 12 for n if it paid interest monthly instead of yearly. P represents the  amount you started out with.   If you keep investing monthly (as shown in your example) instead of just depositing it and letting it sit, you have to use a more complicated formula. Finance people refer to this as calculating the future value of an annuity.  That formula looks like this: A = PMT [((1 + r)N - 1) / r] x (1+r)  A : Is the amount you would have at the end of the time period.   N : The number of compounding periods (months if you get interest calculated monthly)   PMT : The total amount you are putting in each period (N)    r: Just like before, the interest rate you are getting paid. Be sure to adjust this to a monthly number if N represents months (divide APR by 12)* *Most interest rates are quoted as APR, which is the annualized interest rate not counting compounding. Don't confuse this with APY, which has compounding built into it and is not appropriate for use in this formula. Inserting your example: r (monthly interest rate) = 15% APR / 12 = .0125 n = 30 years * 12 months/year = 360 months   A = $150 x [((1 + .0125)360 - 1) / .0125] x (1+.0125) A = $1,051,473.09 (rounded) "The extent that you would have problems would depend on if the annuity is considered Qualified or Non-Qualified. If the annuity is qualified that means that the money that was put into it has never been taxed and a rollover to an IRA is simple.  The possible issues here are tax issues and a CPA is likely the best person to answer this question.  Two other things to consider in such an event is the loss of any 'living benefit' or 'death benefit'. Variable annuities have been through quite the evolution in the last 15 years. Death benefits have been around longer than living benefits but both are usually based on some derrivitive of a 'high water' mark of the variable sub accounts. You might want to ask Hartford the question ""...how will my living or death benefits be affected if I roll this over""." "Sometimes an assumptions is so fundamentally flawed that it essentially destroys the relevance and validity of any modelling outputs.  ""Obviously, we're assuming the company can pay it back""  Is one of those assumptions.  The person gets a notes stating that they will get $525 'IN ONE YEAR' You need to divide $525/(1+Cost of Capital)^n n being the number of periods to find out what the note will be worth today.  Google 'Present Value of an Annuity' to deal with debt that is more complex than you have $500 now and give me $525 in a year..." "An index annuity is almost the same as Indexed Universal Life, except the equity-index annuity is an investment with a guaranteed minimum return, with sometimes a higher return that is a function of the gain in the stock market, but is not associated with a life insurance policy.  After a time, you can convert the EIA to a lifetime income (the annuity part) or just cash it out.  They often are very complicated, but are constructed by combining bonds with index options (puts) just like indexed universal life. Unfortunately these tend to have high fees and/or commissions, and high (early) surrender charges, which can make them a poor investment.  Of course you could just ""roll your own"" by buying bonds and puts FINRAS bulletin on EIAs, pdf warning. Here's a description of one of these securities: pdf." Let's break this into two parts, the future value of the initial deposit, and the future value of the payments: D(1 + i)n For the future value of the payments A((1+i)n-1) / i) Adding those two formulas together will give you the amount of money that should be in your account at the end.  Remember to make the appropriate adjustments to interest rate and the number of payments.  Divide the interest rate by the number of periods in a year (four for quarterly, twelve for monthly), and multiply the number of periods (p) by the same number.  Of course the monthly deposit amount will need to be in the same terms. See also: Annuity (finance theory) - Wikipedia "Hey guys I have a quick question about a financial accounting problem although I think it's not really an ""accounting"" problem but just a bond problem.  Here it goes    GSB Corporation issued semiannual coupon bonds with a face value of $110,000 several years ago. The annual coupon rate is 8%, with two coupons due each year, six months apart. The historical market interest rate was 10% compounded semiannually when GSB Corporation issued the bonds, equal to an effective interest rate of 10.25% [= (1.05 × 1.05) – 1]. GSB Corporation accounts for these bonds using amortized cost measurement based on the historical market interest rate. The current market interest rate at the beginning of the current year on these bonds was 6% compounded semiannually, for an effective interest rate of 6.09% [= (1.03 × 1.03) – 1]. The market interest rate remained at this level throughout the current year. The bonds had a book value of $100,000 at the beginning of the current year. When the firm made the payment at the end of the first six months of the current year, the accountant debited a liability for the exact amount of cash paid. Compute the amount of interest expense on these bonds for the last six months of the life of the bonds, assuming all bonds remain outstanding until the retirement date.  My question is why would they give me the effective interest rate for both the historical and current rate?  The problem states that the firm accounts for the bond using historical interest which is 10% semiannual and the coupon payments are 4400 twice per year.  I was just wondering if I should just do the (Beginning Balance (which is 100000 in this case) x 1.05)-4400=Ending Balance so on and so forth until I get to the 110000 maturity value. I got an answer of 5474.97 and was wondering if that's the correct approach or not." "Be very careful about terminology when talking about annuities.  You used the phrase ""4% return"" in your question.   What exactly do you mean by that?   An annuity that pays out 4% of its principle is not giving you a ""4% return"" in the sense of ROI, because most of that was your money to begin with.   But to achieve a true 4% return in the current environment where interest rates are at historic lows on anything safe (10 year UK Gilts at 0.91%) would make me very nervous about what the insurance company is investing my annuity in." First let's define some terms. Your accrued benefit is a monthly benefit payable at your normal retirement age (usually 65). It is usually a life-only benefit but may have a number of years guaranteed or may have a survivor piece. It is defined by a plan formula (ie, it is a defined benefit). A lump sum is how much that accrued benefit is worth right now. Lump sums are based on applicable interest rates and mortality tables specified by the IRS (interest rates are released monthly, mortality annually). Your plan can either use the same interest rates for a whole year, or they can use new ones each month. Affecting your lump sum is whether your accrued benefit is payable now (immediately, you are age 65), or later (deferred, you are now age 30). For example, instead of being paid an annuity assume you are paid just one payment of $1,000 on your 65th birthday. The lump sum of that for a 65 year old would be $1,000 since there would be no interest discount, and no chance of dying before payment. For a 30 year old, at 4% interest the lump sum would be about $237 (including mortality discount). At age 36 the lump sum is $246. So the lump sum will get bigger just because you get older. Very important is the interest discount. At age 30 in the example, 2% interest would produce a $467 lump sum. And at 6% $122. The bigger the rate, the smaller the lump sum because interest helps an amount now grow bigger in the future. To complicate things, since 2008 the IRS bases lump sums on 3 different interest rates. The monthy annuity payments made within 5 years of the lump sum date use the 1st rate, past 5 and within 20 years use the 2nd rate, and past that use the 3rd rate. Since you are age 30, all of your monthly annuity payments would be made after 20 years, so that makes it simple since we'll only have to look at the 3rd rate. When you reach age 45 the 2nd rate will kick in. Here is the table of interest rates published by the IRS: http://www.irs.gov/Retirement-Plans/Minimum-Present-Value-Segment-Rates You'll find your rates above on the 2013 line for Aug-12. That means your lump sum is being made in 2013 and it is being based on the month August 2012. Most likely your plan will use the same rates for its entire plan year. But what is your plan year? If it is the calendar year, then you would have a 5 month lookback for the rates. But if is a September to August plan year with a 1 month lookback, the rates would have changed between August and September. Your August lump sum would be based on 4.52%, your September on would be based on 5.58% (see the All line for Aug-13). For comparison, a 30 year old with a $100 annuity payable at age 65 would have a lump sum value of $3,011 at 4.52%, but a lump sum value of $1,931 at 5.58%. The change in your accrued benefit by month will obviously have some impact on the lump sum value, but not as much as the change in interest rates if there is one. The amount they actually contribute to the plan has nothing to do with the value of the lump sum though. 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. "There is no equation. Only data that would help you come to the decision that's right for you.  Assuming the 401(k) is invested in a stock fund of one sort or another, the choice is nearly the same as if you had $5K cash to either invest or pay debt.  Since stock returns are not fixed, but are a random distribution that somewhat resembles a bell curve, median about 10%, standard deviation about 14%. It's the age old question of ""getting a guaranteed X% (paying the debt) or a shot at 8-10% or so in the market."" This come up frequently in the decision to pre-pay mortgages at 4-5% versus invest. Many people will take the guaranteed 4% return vs the risk that comes with the market. For your decision, the 401(k) loan, note that the loan is due if you separate from the company for whatever reason. This adds an additional layer of risk and another data point to the mix. For your exact numbers, the savings is barely $50. I'd probably not do it. If the cards were 18%, I'd lean toward the loan, but only if I knew I could raise the cash to pay it back to not default." With the formula you are using you assume that the issued bond (bond A) is a perpetual. Given the provided information, you can't really do more than this, it's only an approximation.  The difference could be explained by the repayment of the principal (which is not the case with a perpetual). I guess the author has calculated the bond value with principal repayment.  You can get more insight in the calculation from the excel provided at this website: http://breakingdownfinance.com/finance-topics/bond-valuation/fixed-rate-bond-valuation/ You need to see that prospectus.  I just met with some potential new clients today that wanted me to take a look at their investments.  Turns out they had two separate annuities.  One was a variable annuity with Allianz.  The other was with some company named Midland Insurance (can't remember the whole name). Turns out the Allianz VA has a 10 year surrender contract and the Midland has a 14 year contract.   14 years!!!   They are currently in year 7 and if they need any money (I'm hoping they at least have a 10% free withdrawal) they will pay 6% surrender on the Allianz and a 15% surrender on the other. Ironically enough, they guy who sold this to them is now in jail.   No joke.