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Compound interest is the addition of interest to the principal sum of a loan or deposit, or in other words, interest on interest. It is the result of reinvesting interest, rather than paying it out, so that interest in the next period is then earned on the principal sum plus previously accumulated interest.

Let's break it down with a simple example: Suppose you deposit \(1,000 into a bank account offering an annual compound interest rate of 5%. After the first year, you would earn \)50 in interest, making your total balance \(1,050. In the second year, you would earn interest on your new balance, which includes the initial principal plus the interest earned the previous year, leading to more than \)50 in interest for that year. This process would continue each year, with the interest earning interest of its own.

Daily compounding means that the interest amount is calculated and added to the principal balance daily. The daily compound interest is smaller than the annual one, but as it gets added every day, it adds up and compounds quickly.

The frequency of compounding has a substantial impact on the total amount of interest accumulated over time. With more frequent compounding intervals, such as daily, the interest is calculated on an updated principal that includes the previous days' interest, hence the balance grows at a faster rate compared to annual compounding.

The future value (FV) is a critical concept in finance that refers to the value of a current asset at a future date based on an assumed rate of growth over time.

With respect to compound interest, the future value demonstrates how much an investment made today will grow over a period. It is important to note that the growth is not linear due to the effect of compounding. The formula for the future value of an investment subject to compound interest is: \[ FV = PV(1 + r)^{nt} \]
where PV is the present value or initial amount, r is the daily interest rate, n is the number of compounding periods per year, and t is the total number of years.

The time value of money is the concept that money available now is worth more than the same amount in the future due to its potential earning capacity. It provides a basis for comparing investment alternatives and understanding the long-term impact of financial decisions.

The core principle is that a dollar today can be invested and earn interest, making it worth more than a dollar tomorrow. Future value and present value calculations like the one in our example are used to determine how much future money is worth today or how much an amount of money today will be worth in the future. This idea is fundamental to all of finance and is the reason why interest is charged on loans and expected on investments.