A dollar today is worth more than a dollar tomorrow because you can invest it and earn more money.

Key idea: Interest compensates lenders for giving up their money now, and inflation can reduce the buying power of future money.

This idea helps us understand why people pay or earn interest. When you lend money to someone, you are giving up the chance to use that money yourself or invest it to make more. So, the borrower pays you interest as a thank you for letting them use your money and to make up for the opportunity you missed. Similarly, when you put money in a bank, the bank uses your money to lend to others or invest, and they pay you interest for that privilege.

It is not just about making more money. It is also about things like inflation, which is when prices go up over time. If a candy bar costs 1 dollar today, and inflation makes it cost 1.03 dollars next year, then your 1 dollar today can buy a candy bar, but 1 dollar next year might not. So, even if you just keep your money under your mattress, its buying power can shrink over time. This makes money today even more valuable.

Key idea: We use calculations to compare money from different times, either finding its present value or its future value, using an interest rate to account for growth.

So, how do we actually compare money from different times? We use calculations that either bring future money back to today's value, called present value, or project today's money forward to a future value, called future value. Think of it like having a time machine for your money. You can either bring future money back to the present to see what it is worth today, or send today's money into the future to see what it will become.

The core idea is to adjust the value of money based on how long you have to wait for it and how much interest it could earn. This 'adjustment' is done using an interest rate, which acts like the growth rate for your money. A higher interest rate means your money can grow faster, making future money less valuable today, and today's money more valuable in the future.

Key idea: Future value shows what money today will grow into, while present value shows what future money is worth today, both adjusted by an interest rate.

Let us look at a simple example. If you have 100 dollars today and invest it at a 5 percent annual interest rate, what will it be worth in one year? It will be worth 105 dollars. This is the future value. The formula for this is quite straightforward.

Now, what if someone promises you 105 dollars one year from now, and you know you could earn 5 percent interest if you had the money today? What is that 105 dollars in the future worth to you today? It is worth 100 dollars. This is the present value. You are 'discounting' the future money back to today. The higher the interest rate you could earn, the less that future money is worth to you today, because you are missing out on more potential growth.

This concept is crucial for making smart financial decisions. Should you take a lump sum payment now or smaller payments over time? Should a company invest in a project that promises returns in five years? The time value of money helps answer these questions by putting all financial options on an equal footing, comparing their value at a single point in time.

“The time value of money helps answer these questions by putting all financial options on an equal footing, comparing their value at a single point in time.”

Future Value of a Single Amount

Key idea: The future value formula calculates how much a single amount of money will grow over time with compound interest.

The main formula to calculate the future value of a single amount of money is quite simple. It tells you what your initial money will grow into after a certain number of years, given an interest rate.

Let us say you have 100 dollars (PV) and you invest it for 2 years (n) at an annual interest rate of 5 percent (i).

After the first year, your money grows to 100 dollars × (1 + 0.05) = 105 dollars.

After the second year, that 105 dollars then grows again: 105 dollars × (1 + 0.05) = 110.25 dollars.

So, your 100 dollars today will be worth 110.25 dollars in two years. This is the power of compound interest, where you earn interest not just on your original money, but also on the interest you have already earned.

“This is the power of compound interest, where you earn interest not just on your original money, but also on the interest you have already earned.”

Present Value of a Single Amount

Key idea: The present value formula calculates what a future amount of money is worth today by 'discounting' it back.

Now, let us flip that around. If you want to know what a future amount of money is worth today, you use the present value formula. This is like reversing the growth process.

Imagine you are promised 110.25 dollars (FV) two years from now (n), and the interest rate you could earn is 5 percent (i). What is that future 110.25 dollars worth to you today (PV)?

Using the formula, you would find that it is worth 100 dollars today. This calculation is called 'discounting' because you are reducing the future value to find its current worth. The higher the interest rate, the more you 'discount' the future money, meaning its present value will be lower.

This is why a lottery payout of 1 million dollars paid over 20 years is worth less than 1 million dollars paid today. The future payments are discounted, making their total present value less than the advertised amount.

Annuities: Streams of Payments

Key idea: Annuities are a series of equal payments over time, and special formulas help calculate their total present or future value.

Sometimes, you do not just have one payment, but a series of equal payments over time. This is called an annuity. Think of regular rent payments, loan repayments, or pension payouts. There are special formulas to calculate the present value or future value of these streams of payments.

For example, if you receive 100 dollars at the end of each year for 5 years, and the interest rate is 5 percent, you can use a specific annuity formula to find out what that entire stream of payments is worth to you today. It will be less than 500 dollars (5 × 100 dollars) because each future payment is discounted.

These formulas are very useful for evaluating things like mortgages, retirement plans, and insurance policies, where money changes hands in regular, predictable ways over many years.