Time Value of Money Finance - TVM Formulas & Calculations - Annuities, Present Value, Future Value

The time value of money affects all aspects of business in every industry. The reason why we have the time value of money is due to interest. Because of interest, the value of a dollar amount is different depending on the point in time that it is paid or received. For example, a dollar received today is more valuable than a dollar received in five years or any time in the future for that matter. This is because a dollar today can earn interest over the next five years and would therefore be worth more than the dollar in the future that is being compared to. In this time value of money lesson, we will go over simple interest, compound interest, future value, annuities, present value, entry year compounding interest, future value of annuities, present value of annuities, and perpetuities. So what is simple interest? Interest can be thought of as rent to borrow money. You can either receive interest or rent when you lend out money, or you can pay interest or rent when you borrow money. Simple interest is when the interest received or paid is based solely on the amount of money that was initially invested. So if you invested $100 and it earned simple interest at an annual rate of 10% for five years, then your investment would earn $10 each year for the next five years, totaling in $50 in interest. The formula to solve the interest rate is the initial investment times one plus the interest rate times the number of periods that the investment will be held for. Therefore, if we invest $100 in an account for five years that earns 10% simple interest per year, then we would have $150 in five years. The ending balance in five years would consist of our initial $100 investment and $50 in interest earned over the five-year period. Our formula would be 100 times one plus the interest rate of 10% times five years, which is five periods. What is compound interest? Compound interest is much different than simple interest. Compound interest is the kind of interest you would like to receive in an investment, but definitely not the kind of interest that you want to pay. Why? Because the interest rate is based on the balance of the investment when it is calculated, not just the initial investment. What this means is that interest is being earned on both the investment and interest is being earned on the previous period's interest. This may be easier to understand with an example. Let's say you invested $100 for five years at a compounding interest rate of 10%. At the end of this first year, your investment would be worth $110 because it earned 10% interest on $100. At the end of the second year, your investment would be worth $121. This means that it earned $11 in interest the second year, which is different from the $10 in interest earned in the first year. Why? Because the second year your investment earned 10% interest on the initial investment and 10% interest on the $10 interest earned in the first year. The easiest way to understand the compounding interest concept is to understand the interest is being earned on the initial investment and the interest earned in previous periods. The best way to calculate what an investment earning compound interest will be worth at some point in the future is to use the future value formula. The future value formula is future value equals present value times one plus the interest rate to the nth power. So for our investment, we would calculate the future value of $100 earning 10% interest over a five-year period. Notice that our investment of $100 will be worth $161.05 in five years. This means that it earned $61.05 in interest or an interest than if it earned simple interest. Now you can see why compound interest is the kind of interest that you want to receive, but not the kind of interest you want to pay. So what is future value? Future value is what a dollar today will be worth in the future. This is because of interest that the dollar can earn over time, therefore making it more valuable in the future. If someone offered to give you $100 today or $100 in the future, you would obviously take the $100 today. Why? Because even if you didn't need the $100 today and you knew that you would need it in the future, you could simply invest it and earn interest over that year. Then a year from now, when you did need the money, you can cash out and have the $100 plus any interest that it earned. So if it earned 10% interest, your $100 today would be worth $110 one year in the future. This means in theory that the future value of money is always worth more. Again, the future value formula is future value equals present value times one plus the interest rate to the nth power. Suppose you invested $1,000 in an investment that was expected to earn 10% annual interest for the next 10 years. What would the future value of your investment be worth in 10 years? Let's plug our figures into the future value formula to find out. We do $1,000 times one plus 0.1 to the 10th power, giving us a future value of $2,593.74. This would mean that $1,000 today, the future value of $1,000 is $2,593.74. What is an annuity? An annuity is a series of equal payments that are either paid to you or paid from you. Annuities can be cash flows, paid such as monthly rent payments, car payments, or they can be money received such as semi-annual coupon payments from a bond. Just remember, for a series of cash flows to be considered an annuity, the cash flows need to be equal. An annuity due is when a payment is made at the beginning of the payment period, rent for example, where you're usually required to pay at the first of every month. An ordinary annuity is a payment that is paid or received at the end of the period. An example of an ordinary annuity would be a coupon payment made from bonds. Usually bonds will make semi-annual coupon payments at the end of every six months. What is present value? Present value is today's value of money from some point in the future. For example, $100 received a year from today is worth less than $100 received today. This is because of interest earned over time. For example, if we invested $90.91 in a fund that earned 10% interest, then our $90.91 would be worth about $100 one year in the future. This means that according to an available 10% interest rate, the present value of $100 a year from today is worth $90.91 today. The present value formula for a lump sum of money in the future is shown here. The interest rate is also known as the discount rate since you will be discounting the future sum of money by the interest rate. Suppose you expected to receive $100 in one year and there were currently several investments offering 10% interest. What would the present value of your investment be? Just plug in the proper figures to the present value formula and you will see that the present value comes out to $90.91. How to find the present value of a series of cash flows? Let's assume you have an ordinary annuity that pays you $100 at the end of each year for the next three years and is coming from an investment that is earning 5% interest. What is the present value of your annuity? The way you would solve the present value of this annuity is by solving the present value of each payment or cash flow individually. You would do this by using the present value formula for each cash flow that is to occur in the future. Look at the example here. This means that if you invested $272.32 in a fund that earns 5% interest, you would withdraw $100 for the next three years. This also means that if you receive $100 at the end of the next three years, you could discount each cash flow back. You would sum them up and give you a present value of $272.32 for these cash flows. Entry-year compounding interest. Entry-year compounding interest is when interest is compounded more frequently than one time per year. This means that there are multiple compounding periods per year. For example, some interest rates are compounded semi-annually, which is two times per year, monthly, which is 12 times per year, etc. To find out the interest rate that is being earned or paid for each compounding period, you'll need to divide the annual interest rate by the number of compounding periods per year. If you have an annual interest rate of 10% and interest is compounded monthly, then you would divide 0.10 by 12, giving you a rate of 0.083333. Suppose you invested $10,000 at 6% interest that compounded semi-annually, two times per year, and held it for five years. What will the future value of your investment be? First, we'll need to solve for the interest rate for each compounding period. Again, we do this by dividing the annual interest rate by the number of compounding periods per year. Since our annual interest rate is 6% and interest is compounded semi-annually, then we would divide 6% by 2, giving us a 3% rate per compounding period. Now we need to solve for the number of compounding periods for the total life of the investment. To do this, we multiply the number of years that we would hold our investment by the number of compounding periods per year. Since we're holding the investment for five years and our investment is compounded semi-annually, we would multiply five years by two compounding periods, giving us a total of 10 compounding periods over the life of the investment. Now we can plug our values into the future value formula to find out what the value of our investment will be in five years. Again, the future value formula is future value equals present value times one plus the interest rate to the nth power. So for our investment, our future value would be 10,000 times 1.03 to the 10th power, giving us a total of $13,439.16. Hopefully you have a good understanding of compounding interest. Just remember that when an investment is compounded more than one time per year, then you will need to solve the rate per compounding period by dividing the annual interest rate by the number of compounding periods per year. And you will need to find the number of compounding periods for the life of the investment by multiplying the number of years by the number of compounding periods per year. Since the concept of present value with entry year compounding is so important, let's try another more complicated example. Let's assume that at the beginning of the year, you purchased an investment that will pay you $1,000 per month at the end of each month for the next six months and is invested in a fund that earns 10% annual interest. What is the present value of your payments? The present value of this series of cash flows would be about $5,828.91. We are using the present value formula to solve the present value of each cash flow individually. We then sum up the values of each cash flow to find the present value of the series of cash flows. It is important to understand that we divide our interest rate by 12 since it is an annual interest rate and we are receiving our payments monthly. Imagine you paid $1,000 into a fund that earned 5% interest at the end of every year for the next five years. What would the future value of this ordinary annuity be? First, let's look at our cash flows on a timeline so we have a visual understanding. Here, zero represents today. One represents a year from today. Two represents two years from today, and so on. To solve the future value of an ordinary annuity, you would solve the future value of each payment individually. The payment made in period one will earn interest over four periods until it is withdrawn, so the future value would be 1,000 times 1.05 to the fourth power, giving us a future value of $1,215.51. The payment made in year two will earn interest over three periods, so the future value will be 1,000 times 1.05 to the third power, giving us a future value of $1,157.63, and so on. This means that period five's payment will be held for zero periods and will therefore not earn any interest, so the future value will be 1,000 times 1.05 to the zero power, giving us a future value of $1,000. If we sum up the future value of all payments, then we find that the future value of our ordinary annuity is $5,525.64. Notice that over the next five years, we've received five $1,000 payments, except this time they're being paid at the beginning of each period. Now let's solve the future value of an annuity due of $1,000 held for five periods in an account that earns 5% interest. Since the first payment is made at the beginning of the first period, it will earn interest over five periods until it is withdrawn five years from today, so the future value will be 1,000 times 1.05 to the fourth power, giving us a future value of $1,276.28. The second payment will earn interest over four periods, so the future value will be 1,000 times 1.05 to the fourth power, giving us a future value of $1,215.51, and so on for each payment. We'll solve the future value of each cash flow individually and then sum them up to find the future value of our annuity due. The future value is $5,801.92. Notice that this annuity has identical characteristics, such as time, interest, and the payment amount, except that it's an annuity due, and our other example is an ordinary annuity, so both annuities are identical, but the future values are different. The future values are different because the payments are made at different times, meaning that they earn interest over a different number of periods. The future value of our ordinary annuity is $5,525.64, which is $276.28 less than the future value of our annuity due. Solving the present value of an ordinary annuity. Remember, ordinary annuity means that the payments are made or received at the end of each payment period. Let's assume we have an ordinary annuity that pays $1,000 at the end of each year for the next five years. Let's also assume that the money is coming from a fund that earns 5% interest. What is the present value of this annuity? To solve the present value of our annuity, we will simply solve the present value of each payment and sum them up to find the present value of our annuity. Since our first payment is received one year from today, our present value formula will be 1,000 divided by 1.05 to the first power, giving us a present value of $952.38. Since our second year payment is received two years from today, our present value formula will be 1,000 divided by 1.05 to the second power, giving us a present value of $907.03. Remember, in our present value formula, we are raising the denominator, which is 1 plus the interest, to the number of periods from today that the payment will be received. After we sum the present value of all five payments, we find that the present value of our ordinary annuity is $4,329.48. Now let's assume we have an annuity due that makes five payments of $1,000 over the next five years. Let's also assume that it's in a fund that earns 5% interest. What is the present value of our annuity due? Remember, since this is an annuity due, our payments will be received at the beginning of the next five periods. This means that our first payment will be received at zero, which is today on our timeline. Since our first payment was received today, it didn't have time to earn interest. So the present value of our first payment is $1,000. Our second year payment is paid at the beginning of the second year, which is one year from today, meaning it has one year to earn interest. So the present value of our second year payment is $952.38. We did this by dividing $1,000 by 1.05 to the first power. If we sum the present value of all of our payments, we find that the present value of an annuity due is $4,545.95, which is $443.77 more than our ordinary annuity. Our annuity due is worth more than our ordinary annuity because with an annuity due, the payments are received sooner, which means they are worth more. Remember, money today is worth more than an equal amount of money received in the future. Perpetuities. A perpetuity is a fixed cash flow received over an indefinite number of periods. For example, if someone were to receive $1,000 per month until they died, that would be a perpetuity. The formula to solve the present value of a perpetuity is shown here. The present value would be cash flow divided by the interest rate. Let's assume we were to receive $1,000 per year for the rest of our lives and could earn interest in other investments at a rate of 5%. What would the present value of this perpetuity be? To solve the present value of this perpetuity, we would simply divide $1,000 by 0.05, giving us a present value of $20,000. Now, let's make this a little more complicated. What if we were to receive $1,000 per month for the rest of our lives and we could earn 5% interest in other investments? What would the present value of our perpetuity be? It would be $1,000 divided by 0.05 divided by 12, giving us a present value of $238,095.24. Notice that we divided the interest rate by 12 since our perpetuity pays monthly. In this time value of money lesson, we covered all the basic time value money concepts. I hope I was able to help you in some way. Thanks for watching.