#### Time Value of Money Explained with Formula and Examples – Investopedia

Investopedia / Mira Norian

The time value of money (TVM) is the concept that a sum of money is worth more now than the same sum will be at a future date due to its earnings potential in the interim. The time value of money is a core principle of finance. A sum of money in the hand has greater value than the same sum to be paid in the future. The time value of money is also referred to as the present discounted value.

Investors prefer to receive money today rather than the same amount of money in the future because a sum of money, once invested, grows over time. For example, money deposited into a savings account earns interest. Over time, the interest is added to the principal, earning more interest. That’s the power of compounding interest.

If it is not invested, the value of the money erodes over time. If you hide $1,000 in a mattress for three years, you will lose the additional money it could have earned over that time if invested. It will have even less buying power when you retrieve it because inflation reduces its value.

As another example, say you have the option of receiving $10,000 now or $10,000 two years from now. Despite the equal face value, $10,000 today has more value and utility than it will two years from now due to the opportunity costs associated with the delay. In other words, a delayed payment is a missed opportunity.

The time value of money has a negative relationship with inflation. Remember that inflation is an increase in the prices of goods and services. As such, the value of a single dollar goes down when prices rise, which means you can’t purchase as much as you were able to in the past.

The most fundamental formula for the time value of money takes into account the following: the future value of money, the present value of money, the interest rate, the number of compounding periods per year, and the number of years.

Based on these variables, the formula for TVM is:

$begin{aligned}&FV = PV Big ( 1 + frac {i}{n} Big ) ^ {n times t} \&textbf{where:} \&FV = text{Future value of money} \&PV = text{Present value of money} \&i = text{Interest rate} \&n = text{Number of compounding periods per year} \&t = text{Number of years}end{aligned}$

Keep in mind, though that the TVM formula may change slightly depending on the situation. For example, in the case of annuity or perpetuity payments, the generalized formula has additional or fewer factors.

The time value of money doesn't take into account any capital losses that you may incur or any negative interest rates that may apply. In these cases, you may be able to use negative growth rates to calculate the time value of money

Here’s a hypothetical example to show how the time value of money works. Let’s assume a sum of $10,000 is invested for one year at 10% interest compounded annually. The future value of that money is:

$begin{aligned}FV &= $10,000 times Big ( 1 + frac{10%}{1} Big ) ^ {1 times 1} \ &= $11,000 \end{aligned}$

The formula can also be rearranged to find the value of the future sum in present-day dollars. For example, the present-day dollar amount compounded annually at 7% interest that would be worth $5,000 one year from today is:

$begin{aligned}PV &= Big [ frac{ $5,000 }{ big (1 + frac {7%}{1} big ) } Big ] ^ {1 times 1} \&= $4,673 \end{aligned}$

The number of compounding periods has a dramatic effect on the TVM calculations. Taking the $10,000 example above, if the number of compounding periods is increased to quarterly, monthly, or daily, the ending future value calculations are:

This shows that the TVM depends not only on the interest rate and time horizon but also on how many times the compounding calculations are computed each year.

Opportunity cost is key to the concept of the time value of money. Money can grow only if it is invested over time and earns a positive return. Money that is not invested loses value over time. Therefore, a sum of money that is expected to be paid in the future, no matter how confidently it is expected, is losing value in the meantime.

The concept of the time value of money can help guide investment decisions. For instance, suppose an investor can choose between two projects: Project A and Project B. They are identical except that Project A promises a $1 million cash payout in year one, whereas Project B offers a $1 million cash payout in year five. The payouts are not equal. The $1 million payout received after one year has a higher present value than the $1 million payout after five years.

It would be hard to find a single area of finance where the time value of money does not influence the decision-making process. The time value of money is the central concept in discounted cash flow (DCF) analysis, which is one of the most popular and influential methods for valuing investment opportunities. It is also an integral part of financial planning and risk management activities. Pension fund managers, for instance, consider the time value of money to ensure that their account holders will receive adequate funds in retirement.

The value of money changes over time and there are several factors that can affect it. Inflation, which is the general rise in prices of goods and services, has a negative impact on the future value of money. That's because when prices rise, your money only goes so far. Even a slight increase in prices means that your purchasing power drops. So that dollar you earned in 2015 and kept in your piggy bank buys less today than it would have back then.

The time value of money takes several things into account when calculating the future value of money, including the present value of money (PV), the number of compounding periods per year (n), the total number of years (t), and the interest rate (i). You can use the following formula to calculate the time value of money: FV = PV x [1 + (i / n)]^{ (n x t)}.

The future value of money isn't the same as present-day dollars. And the same is true about money from the past. This phenomenon is known as the time value of money. Businesses can use it to gauge the potential for future projects. And as an investor, you can use it to pinpoint investment opportunities. Put simply, knowing what TVM is and how to calculate it can help you make sound decisions about how you spend, save, and invest.

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