Calculating Future Value: Initial deposit and recurring deposits of a fixed but different Value
But how do I bring the initial deposit into the equation? Basically, you can't.  Unless you combine two different formulas from Math of Finance into a single expression. The single initial deposit of $1000 will compound for 20 years at 5% compounded annually. The final amount for this part of the deposit will be: V1 = 1000 x (1.05)^20 In addition the series of 20 payments will be an ordinary annuity with a regular payment of $100, with the value on the occasion of the 20th payment given by:  So the final total amount in the account at the end of 20 years will be the sum of these two values... If I is the initial deposit, P the periodic deposit, r the rent per period, n the number of periods, and F the final value, than we can combine two formulas into one to get the following answer: F = I*(1+r)n + P*[(1+r)n-1]/r In this case, you get V = 1000*(1.05)20 + 100*[(1.05)20-1]/0.05 = 5959.89 USD. Note that the actual final value may be lower because of rounding errors. Illustrating with a shorter example:  Suppose I deposit 1,000 USD. Every year I deposit another 100 USD. I want to know how much money will be on that savings account in 4 years.  The long-hand calculation is Expressed with a summation  And using the formula derived from the summation (as shown by DJohnM) So for 20 years Note in year 20 (or year 4 in the shorter example) the final $100 deposit does not have any time to accrue interest before the valuation of the account.
First, assuming you are making payments for a savings deposit.  The present value of the deposit is the sum of the all the payments discounted to present value.  In this case they would be discounted by the rate of inflation: £100 deposited next year is worth less than £100 today because it will be eroded by inflation.  With a higher rate of inflation the payments are discounted more heavily, so the present value decreases.  A deposit, or annuity due (see Calculating the Present Value of an Annuity Due), can be expressed mathematically like so:-  ∴ by induction  So for example, the following annuity has a present value of £1,136.76 The total amount that will be paid for the annuity is 12 x £100 = £1,200.  With a higher rate of inflation, say 2% per month, and with the same 12 x £100 payments, the present value of the annuity decreases. In Excel (£1,078.68) A similar case is that for a loan, or ordinary annuity (see Calculating the Present Value of an Ordinary Annuity), except the discounting factor is the loan interest rate rather than inflation and repayments are made at the end of each period rather than at the start. The present value of a loan is the value of the all the future repayments discounted to present value.  With a higher interest rate the payments are discounted more heavily, so the present value decreases.  A loan can be expressed mathematically like so:-  ∴ by induction  So for example, the following values fulfill a loan worth £1,125.51 The total amount that will be paid for the loan is 12 x £100 = £1,200.  With a higher rate of interest, say 2% per month, and the same 12 x £100 repayments, the present value of the loan that can be obtained decreases. In Excel (£1,057.53) The short answer is you'd be much better off paying up front in this case.  The present value of $2,500 plus 12 $500 monthly payments is $8,128 at a 12% discount rate, which is much higher then the $6,000 you could pay now. The long answer is how you get that present value.  How can I use time value of money to find the present value of money if I choose to go with option A? First of all, I'd question your discount rate.  A 12% discount rate means that you can safely reinvest the money that you're not spending today at a 12% annual (1% monthly) rate, which seems very high.  Normally for short-term spending decisions you'd use a risk-free rate, which would be closer to 1%-2%. However, to discount at 1% monthly you'd just divide each monthly payment by 1 plus the discount rate raised to the power of the number of periods until each payment. So the total is which is $8,127.54 You could also use the NPV function in Excel. It seems like to get an accurate answer the calculation of the interest rate should take into account compounding period as well? Correct, and in the example above the compounding is assumed to be monthly since that's the periodicity of the cash flows. You could calculate it with a different compounding period but it gets much more complicated and probably wouldn't make a significant difference. The discount rate does take compounding into effect, meaning if you saved the $5,628 (the PV of $8,128 minus the $2,500 initial payment), you'd earn 1% interest on $5,628 the first month, $5,128 plus that interest the second month, etc. While there are many very good and detailed answers to this question, there is one key term from finance that none of them used and that is Net Present Value. While this is a term generally associate with debt and assets, it also can be applied to the valuation models of a company's share price. The price of the share of a stock in a company represents the Net Present Value of all future cash flows of that company divided by the total number of shares outstanding. This is also the reason behind why the payment of dividends will cause the share price valuation to be less than its valuation if the company did not pay a dividend. That/those future outflows are factored into the NPV calculation, actually performed or implied, and results in a current valuation that is less than it would have been had that capital been retained. Unlike with a fixed income security, or even a variable rate debenture, it is difficult to predict what the future cashflows of a company will be, and how investors chose to value things as intangible as brand recognition, market penetration, and executive competence are often far more subjective that using 10 year libor rates to plug into a present value calculation for a floating rate bond of similar tenor. Opinion enters into the calculus and this is why you end up having a greater degree of price variance than you see in the fixed income markets. You have had situations where companies such as Amazon.com, Google, and Facebook had highly valued shares before they they ever posted a profit. That is because the analysis of the value of their intellectual properties or business models would, overtime provide a future value that was equivalent to their stock price at that time. In Maryland, a landlord must hold your security deposit in an escrow account and pay you interest when returning the deposit.  The interest is simple interest; it does not compound. The interest rate that they must pay has changed over the last 43 years.  Before October 1, 2004, the rate was 4%.  Until January 1, 2015, the rate was 3%.  Currently, the rate is 1.5% OR the simple interest rate accrued at the daily U.S. Treasury yield curve rate for one year, as of the first business day of each year, whichever is greater.  (This year, the rate is 1.5%.)  Maryland's Department of Housing and Community Development has a Security Deposit Calculator for easy calculation of this interest; however, it only works for deposits since January 1, 2015. It is unclear to me whether the interest rate in effect is the one that was in place when the security deposit was made, or if the rate changes over the years.   At most, if you get 4% interest every year, I would expect you to receive $429.76, which is $158 + ($158 * 4% * 43).  The interest is accrued every 6 months, so you would not get any interest for the 3 months that you rented in your 44th year. (With the new law that took effect this year, interest is accrued monthly.) At least, if the interest rate changes with the new laws, I would expect you to receive $413.18, which is $158 + ($158 * 4% * 32.5) + ($158 * 3% * 10.25) + ($158 * 1.5% * 0.5). Some text on the Security Deposit Calculator suggests that the laws for Prince George's County are different than the rest of the state.  If you are in that county, you'll need to check the local ordinances to see what security deposit policies apply. "If it could, it seems yet to be proven.  Long Term Capital Management was founded by a bunch of math whizzes and they seem to have missed something.  I'd never suggest that something has no value, but similar to the concept that ""if time travel were possible, why hasn't anyone come back from the future to tell us"" I'd suggest that if there were a real advantage to what you suggest, someone would be making money from it already.  In my opinion, the math is simple, little more than a four function calculator is needed." Fixed  You are confirming the amount you are going t pay over the term of the loan. Variable: 3.79% over 82mo. The total difference over the life of the loan comes to around $1200 That is the wrong way to calculate the variable portion. The variable is primarily set with a margin over a certain benchmark i.e. Fed rate. Assuming the Fed rate doesn't change over or only goes lower the variable rate is the one to go. If it rises then your payment will increase. And the margin they take over the benchmark rate may increase, so the total amount you pay might increase too.  I would assume a read through the T&Cs should clarify that for you. Is it ever a good idea to choose a variable rate loan? Only if you think we are in a low interest rate environment i.e. the economy is in doldrums and the Feds are trying to simulate the economy by decreasing the benchmark rates. And you are sure that the lender isn't going to increase his margins if the rate remains low for quite a substantial amount of time. And I might assume there will be penalties for paying off a loan quicker. I have found a good explanation here: http://www.contracts-for-difference.com/Financing-charge.html Financing is calculated by taking the overall position size, and multiplying it by (LIBOR + say 2%) and then dividing by 365 x the amount of days the position is open. For instance, the interest rate applicable for overnight long positions may be 6% or 0.06. To calculate how much it would cost you to hold a long position for X number of days you would need to make this 'pro rata' meaning that you would need to divide the 0.06 by 365 and multiply it by X days and then multiply this by the trade size. So for example, for a trade size of $20,000, held for 30 days, the interest cost would be about $98.6. It is important to note that due to financing, long positions held for extended periods can reduce returns. You want the net expense of the surcharge minus the rewards to be no more than the interest that you would pay otherwise.   Where t is the compounding period for the rate D expressed as a fraction of the overall period for D.  So if D is an annual rate (not the APR, the simple rate), it would be expressed as something like 1/365 if compounded daily.  That is the number of years in the compounding period.  If a monthly rate or weekly compounding, that would change.   And p is the number of such time periods in the grace period.  So if the grace period were one month, this might be 30.   Other variables are as used in the question, all expressed as percentages (which is why I'm dividing by 100).  The D rate should be the simple rate, like 6% not the APR of 6.24% or whatever.   Note that I'm saying <=.  When equal, there is no financial advantage or disadvantage.  You could choose either method for the same cost.  Now, one method may be more annoying to implement, in which case you might add a fee for it on one side or the other of the equation.  Or simply change the less than or equal to be just less than.   I may be missing something that you should consider but I don't know.  The problem is generic enough that pertinent details might be hidden.  But hopefully this at least gives you a framework under which to consider it.