The term time value of money (TVM) infers that the value of money is related to time. Another way to say it is "the value of money over time". How about the popular expression time is money. However you say it, the bottom line is the same. A sum of money is worth more today than that same sum is worth in the future.
Given the choice of receiving $1000 today or $1000 in two years, which would you choose? Right! $1000 today. How about $1000 today or $2000 in two days?
Assuming we knew the $2000 would be paid, hopefully everyone would choose the $2000. Why, what changed? The change is the amount of time that has passed.
If we didn't care about the value of money over time, then the old adage of keeping our money in our mattress would be our guide. Our principal would always be there but would never grow and unfortunately its purchasing power would be less over time due to inflation. It would be the same as getting a sum of money today or getting the same or less money at some future time period.
The concept that a given amount of money is worth more now than that same amount of money will be in the future is due to its earning potential. In other words its ability to earn interest. This principle of the time value of money explains why interest is paid or earned. Interest, whether it is on a bank deposit or debt, compensates the depositor or lender for the value of money over time.
Money today can be invested and potentially grow into a larger amount in the future. The longer the time period money has to grow the
more it will be worth at the end of the time period.
We can use this core principle of finance to help us calculate what a sum of money will be worth in the future and also how much a future sum of money has to be discounted to equal todays dollars. Why is this important? It helps you determine how well a potential investment will perform. To gauge the soundness of an investment.
Computing the time value of money involves 5 components.
1. Present Value (PV) = the amount of money to be invested today.
2. Future Value (FV) = what the amount of money invested will be worth at some point in the future.
3. Interest or Yield (r) = the amount of money the investment can earn or the annual rate of return.
4. The Number of Compounding Periods (n) = how often the money invested is compounded each year.
5. The Number of Years (t) = the time period for the future value to be realized.
Let's look at each component to see the role each plays in computing the time value of money. In order to solve for any of the 5 components we
need to know at least 3 other components.
1. Present Value (PV) This component calculates what something at a future date is worth today. More specifically, it's the current value of a future sum of money or a stream of cash flows assuming a specific rate of return. For those that need to see a mathematical formula:
PV = FV / (1 + r)t where FV is the future value, r is the rate of return, n is the number of compounding periods and t is the number of years.
For example, if we knew we would receive $10000 (FV), in 3 years (t), with a guaranteed interest rate of 10% (r), and was compounded annually (n) then we can
compute what the PV would be:
PV = $10000/(1 + .10)3 = $7513.15. So the present value of $10,000 earning 10% interest, compounded annually for 3 years would be $7513.15 or
the present value of $10,000 earning 10% interest, compounded annually for 3 years would be $10,000. If you invested $7513.15 today for 3 years at 10% interest compounded yearly you would have $10,000 at the end of 3 years.
Please note that when a formula has a number to the "t power" as we see in our formula, it's the same as saying we multiply the (1 + r) by itself by however many times the "t power" is. In our example, the "t power" is 3 so we would multiply (1 + .10) x (1 + .10) x (1 + .10) = 1.331. If the "t power" were 20 then we would multiply the (1 + .10) by 20 times = 6.727.
2. Future Value (FV) This component calculates what something will be worth at a future date. It's the value of a current asset at a future date based on an
assumed growth rate. Before FV can be calculated, we must know if the interest rate will be based on simple interest or compounded interest as the FV will
be different.
We will discuss both simple and compounded interest in greater detail as this article progresses, so bear with me for now. Any questions you may have
concerning interest will hopefully be answered in that section.
The formula to calculate future value assuming SIMPLE interest is as follows:
FV = PV x (1 + (r x t)) Where PV is today's investment amount, r is the interest rate and t is the time period in years. The investment amount will be compounded annually (n).
For example, we have $5000 to invest and are guaranteed to earn 10% for 3 years compounded annually using SIMPLE interest we will solve for FV.
FV = $5000 x (1 + (.10 x 3)) = $5000 x (1 + .30) = $6500. The FV of our $5000 would be $6500 ($5000 + $1500 in simple interest).
The formula to calculate FV using COMPOUND interest is:
FV = PV x (1 + r)t Where PV is our initial investment amount, r is the interest rate and t is the period of time in years. The investment will be compounded annually (n).
Using our above example we have $5000 invested at 10% for 3 years compounded annually using compound interest. Solve for FV
FV = $5000 x (1 + .10)3 = $5000 x (1.33)3 = 6655.00. The FV of our $5000 would be $6655 ($5000 + $1655 in compound interest).
What does this tell us about the time value of money as it relates to present value and future value? If we compare PV and FV we see that PV attempts to calculate what a sum of money in the future will be worth today and FV attempts to calculate the value of a sum of money in the future. Notice how the formulas work. To calculate PV, we DIVIDE FV by the interest and time calculation. To calculate FV we MULTIPLY the PV by the interest and time calculation.
3. Interest or Yield (r) Another component of calculating the time value of money is interest. Interest can be defined as the cost of borrowing money. It's the price we either pay OR earn for borrowing OR lending money. As we discussed above, there are two types of interest. Simple interest and compound interest.
a. Simple Interest: The formula to calculate simple interest is P x r x t where P = the principle amount, which is the PV (present value, the amount of money we have to borrow or invest today), r is the interest rate, and t is the term of the loan or income in years.
To illustrate, we are investing $5000 at 10% interest for 3 years. Using the P x r x t formula, $5000 x 10% x 3 = $1500 in simple interest. The interest for year 1 would be $500 ($5000 x .10 x 1). The interest earned in year 2 would also be $500 ($5000 x .10 x 1) or $1000 at the end of year 2 ($5000 x .10 x 2 = $1000) and $1500 total interest earned in year 3 ($5000 x .10 x 3 = $1500)
The important takeaway for simple interest is that the interest is only calculated on the principle amount each year. In our example, the interest paid each year would always be $500 whether the loan is for 3 years or 20 years.
b. Compound Interest: The formula to calculate compound interest is P[(1 + r)t - 1] where P is our principle amount, r is the interest rate, t is the time period we are compounding and n is the number of compound periods.
To illustrate, we are investing $5000 at 10% interest compounded annually for 3 years. Using the P[(1 + r)t - 1] formula, $5000[(1 + .10)3 - 1] = $1655 in compound interest. The interest for year 1 would be $500 ($5000[(1 + .10)1 - 1]). The interest at the end of year 2 would be $1050 ($5000[(1 + .10)2 - 1]). The interest at the end of year 3 would be $1655 ($5000[(1 + .10)3 - 1]).
When we compare the interest earned in our example between simple ($1500) and compound interest ($1655), we see the compound interest is greater. Why? With simple interest, the interest is only calculated on the initial principle amount. With compound interest, the interest is calculated on both the initial principle plus the accumulated interest from previous periods. The greater number of compounding periods the greater the compounded interest will be. Compounding interest generates interest on interest.
4. The Number of Compounding Periods (n) This time value of money component has to do with a time frame frequency referred to as a compounding period. A compounding period can be defined as the span of time when interest was last compounded and when it will be compounded again. The following are some popular compounding periods.
1. Compounded annually = a full year will pass before interest is compounded again.
2. Compounded monthly = a full month will pass before interest is compounded again.
3. Compounded quarterly = a full 3 months will pass before interest is compounded again.
4. Compounded semi-annually = a full 6 months will pass before interest is compounded again.
5. Compounded daily = a full day will pass before interest is compounded again.
6. Compounding continuously = interest can compounded without a time limit, the balance is earning interest continiously or at all times.
The compounding periods most often used are:
Here is the key takeaway regarding compounding frequency: The more frequently interest is compounded, the amount of interest earned in each compounding increment of time becomes smaller,
but the total amount of accumulated interest grows faster.
5. The Number of Years (t) This component of the time value of money represents the length of time a payment stream is active. A good illustration is a 30 year home mortgage. The length of time the payment stream is active would be 30 years assuming this loan goes the distance. Even though this loan is paid monthly, the (t) value is the total length of time the loan is set to run. Think of the this length of time a payment is active as the "macro" and the monthly compounding periods as the "micro". The micro happens within the macro period.
Just as self interest makes the world go round, the time value of money makes the financial world go round. It's all about comparing dollars we possess today versus dollars we have to wait to receive at some time in the future. This comparison involves two building blocks, compounding and discounting.
Compounding is about moving today's money into the future. We solve for a future value for money we have today by compounding our investment with an interest rate.
Discounting is about moving future money back to present day dollars. We solve for the present value of a future lump sum of money or future cash flows by applying a discount rate.
The bottom line: Money invested today is compounded to the future and money to be received in the future is discounted to today... hence the time value of money. If you invested today how much would it be worth in 5 years or if you had to wait 5 years to get your money, what would that be worth today? If you can master the concept of the time value of money, you are on your way to making sound investments.
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