Introduction
-Let’s begin by making a general assumption: It is better to consume something now than it is to wait and consume it later. For example, if you are given the option of receiving $1000 now or $1000 five years from now, chances are that you’d take the money now.
-In economics, there are really only two things you can do with your money: spend it or save it. Should you decide to save your money, you are delaying your ability to consume it now.
-In return for delaying your consumption now, you would want to be compensated by being able to consume more later.
-When we’re talking about money, this compensation takes place in the form of interest.[break]
Compound Interest
-There are several different forms of interest. We’ll focus on compound interest.
-Compound interest pays you not only based on your principle amount (the money you start with), but also on any interest you have earned. In other words, compound interest pays you interest on interest.[break]
Example:
Suppose you have $500 that you put into the bank. The bank is going to pay you 5% interest, compounded every year. Let t = time in years.

Compound Interest Example

So on and so forth.

-There’s a pattern here. As long as we know the amount of money we start with and the interest rate, we can calculate the amount of money we can expect to have for any given year.[break]
Future Value Formula
Proof:
Let i = interest rate
Let t = time in years
Let $X = principle amount of money we start with
Let FVt = future value of the starting principle ($X) after t years

Future Value Proof

-From this we can generalize that when t = n, FVn = $X(1+i)n. (Not the strictest of proofs, but I’m not a mathematician).
-We refer to this formula as the future value formula because it helps us calculate the value of money in the future. (We economists are a pragmatic bunch).[break]
Present Value Formula
-With the future value formula we can found out what our money will be worth for a given interest rate after t years. In other words, we can calculate money going forwards in time.
-Let’s reverse the process to find out how much money needs to be saved now for a given interest rate in order to have a specific amount of money later.[break]
Proof:
Present Value Proof
Example:
Suppose you want to figure out how much money you need to save now at a 7% interest rate compounded yearly to have $5,000 four years from now:
i = 7%
t = 4 years
PV4 = The present value of $5000 four years from now

Present Value Example
So if we save $3814.48 at 7% interest compounded yearly, four years from now we will have $5,000.

-If we take the present value formula and start plugging in numbers into it, we can come to an interesting conclusion.

-Let’s see what happens to the present value of a dollar over time at some interest rate. For this example, we’ll use 5%. (It doesn’t really matter what we pick; the end result will still be the same.)

Dollar Today

-From this we can gather that PV0 > PV1 > PV2 > PV3 > … > PVn.
-In non-mathematical terms, this means that a dollar today is worth more than a dollar tomorrow.
-This conclusion has implications that are widespread throughout all of economics and finance and will be something that we’ll explore more in the future.

Disclaimer

All works here are my own and are considered works in progress and may be subject to change at any time. The opinions expressed here are mine only unless otherwise noted. I am not being paid by a third party to endorse a product of any sort. These writings are written for my own references. I do not claim to be a professional of any kind so follow any information you find here at your own risk. The facts that I post on here are things that I believe to be true, but may not necessarily be so. This is the internet; do your own fact checking and take everything with a grain of salt.

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