Welcome to the finance homework help subject page. Let's take a look at some of the topics covered by a typical course in finance:
The first topic in the list above is Value, which refers to present value and future value problems. You may have seen the following two equations already, the simple derivations of which can be found in any good finance text:
The first equation is the future value formula for compound interest, where A = the future value of the account, P = the present value of the account, often called the principle, r = the interest rate as a decimal, n = the number of compounding periods each year, and t = the time in years. The second equation is the future value formula for an annuity, where m = the amount of the periodic payment. As a reminder, an annuity is any sequence of payments into or out of an interest-bearing account.
A couple of useful finance notes before we continue with this homework help session are as follows: the first equation can easily be solved for P, in which case it becomes a present value formula for compound interest. With that, you can calculate how much you would have to invest now (present value) in order for your account to be worth a specified amount at some time in the future (future value) with a given interest rate. The second equation can easily be solved for m, after which you can calculate, for example, the size of a monthly payment you would need to make in order to have a certain amount of money at some point in the future. When solved for m, the second equation is called the sinking fund formula.
The question we wish to consider now is: How would we calculate the present value of an annuity? Notice that the second equation has no P in it, so we cannot simply solve for P as we did in the first equation. In a sense, we will consider both equations together by thinking about the following situation. Suppose Nick was setting up an interest bearing account at his local bank while his friend Pete was setting up an annuity with his stock broker. Nick's account would be governed by the first equation, while Pete's account would be governed by the second.
Let's assume that we want to have Nick and Pete set up their accounts with the same interest rate (r) and compounding (n), and that after the same amount of time (t), we want the accounts to have the same value (A). Notice that the two variables not mentioned in this scenario so far are the amount Nick will put into his bank account initially (P), his present value, and the size of the periodic payment of Pete's annuity (m). Now, let's choose the size of Pete's annuity payment (m) and let the annuity run its course. Is it not true that we could play around with the size of Pete's initial bank deposit (P) until the two accounts had the same value at the end of this experiment? When this condition is met, we can say that Pete's initial deposit (P), which is a present value, correlates with the present value of Nick's annuity, since it gives the same future value under identical conditions.
This tells us how to calculate the present value of an annuity using the two formulas above. Since the future values will be the same (in our experiment), set the A of one equal to the A of the other. That means we can set the right side of the first equation equal to the right side of the other to obtain:
All we need to do now is solve the equation for P, and with a little algebraic manipulation, we will have the formula for the present value of an annuity:
If you solve the equation above for m, you will be able to calculate the size of a monthly payment when you know the size of a loan (present value) and the other required parameters. When solved for m, this equation is called the payout annuity formula and is discussed in the blog article Landlords and Logarithms. By the way, if you are unable to do the algebra to obtain the various forms of these equations, you need to do some skill building in this subject area.
Now let's see how useful the last equation is. Suppose you decided you could afford $250 per month as a car payment and the best interest rate you could get was 13% over a four-year car loan period. What is the most expensive car you could buy? This is the same as asking what is the largest loan I can afford, and anytime you are trying to determine a loan amount it is a present value problem. (Remember that because of interest you will end up paying much more than the face value of the loan, and that larger amount is the future value of the loan.) If you plug in everything you know to the formula for the present value of an annuity, you find out you can afford a car that costs $9,319. That is a pretty useful calculation for just about anyone.
Let's do one more. Suppose you decide you can afford $1,575 each month as a mortgage payment and interest rates are 10.5% on a 30-year fixed mortgage. How expensive a home can you afford? Once again, you are looking to determine the size of the loan, and that is a present value problem. After plugging in, you find out you can buy a house for $172,180. Congratulations - you now have a house and a car!
Don't be fooled into thinking that finance is a dull subject isolated from math and the sciences. A quick visit to books on Amazon.com and Google will soon dispel that notion.
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