So we can express the initial total present value of the annuity as follows:. This gives an expression that can be solved for either the total loan that one can afford with certain payments or the amount that a certain retirement fund can pay out each month for a specified length of time.
With a little more manipulation you can even figure out how long it will take to pay off your student loans with a certain regular payment. Once again, all of this can also be done using Excel. Where things get complicated is when these different concepts are combined in the same problem. For instance, payments could be made toward some future value, and then after a period of time the process could be reversed as you pay yourself back from that amount, now treated as a present value as in a retirement fund.
Or, in the case of either future values or present values, one could go for a period of time without any payments where the balance merely compounds interest. Another interesting question that arises, primarily for tax purposes, is what portion of the payments or change in balance either present or future value in any given period of time was interest?
Often where people mess up is in recognizing which problem they are looking at. Is this a straightforward compound interest problem with a single investment, or is it an annuity with multiple payments? Is this a future value, where the total accumulates toward the end of the payments, or is it a present value, where the payments are later removed from some initial total?
Contact Leo Wibberly at ldwibber vcu. Simple Interest Simple interest assumes that the growth of an investment is approximately linear over short periods of time or for very low rates of interest.
Compound Interest Compound interest assumes that an investment grows based on its current value, rather than the initial value. Future Value of an Annuity An annuity is an investment characterized by regular payments instead of merely a single lump investment most clearly exemplified in various retirement funds.
Present Value of an Annuity While slightly more difficult to get one's mind around, the concept of a present value is actually more frequently encountered in real life. Each successive term affects the sum less than the preceding term. As each succeeding term gets closer to 0, the sum of the terms approaches a finite value. When the sum of an infinite geometric series exists, we can calculate the sum. The formula for the sum of an infinite series is related to the formula for the sum of the first n terms of a geometric series.
What happens for greater values of n? This gives us the formula for the sum of an infinite geometric series. We notice the repeating decimal [latex]0. Looking for a pattern, we rewrite the sum, noticing that we see the first term multiplied to 0. Notice the pattern; we multiply each consecutive term by a common ratio of 0.
So, substituting into our formula for an infinite geometric sum, we have. At the beginning of the section, we looked at a problem in which a couple invested a set amount of money each month into a college fund for six years.
An annuity is an investment in which the purchaser makes a sequence of periodic, equal payments. To find the amount of an annuity, we need to find the sum of all the payments and the interest earned. This is the value of the initial deposit. So the monthly interest rate is 0. We can multiply the amount in the account each month by Let us see if we can determine the amount in the college fund and the interest earned.
How much is in the account right after the last deposit? How much is in the account if deposits are made for 10 years? Improve this page Learn More. Skip to main content. Module Sequences and Series. Search for:. Forward Rate Agreements Interest Rate Futures Interest Rate Swaps Currency Swaps Basis Swaps Interest Rate Options Models Excel Functions Questions Updated: 01 November 01 November
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