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 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 and this time value of money lesson we will go over simple interest compound interest future value annuities present value and treat your 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 an interest the formula to solve the interest rate is the initial investment times 1 plus the interest rate times a 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 one hundred and fifty dollars and five years the ending balance in five years would consist of our initial $100 investment and fifty dollars an interest earned over the five year period our formula would be 100 times 1 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 periods 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 dollars because it earned 10 percent interest on $100 at the end of the second year your investment be worth 121 dollars 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 one hundred and sixty one dollars and five cents and five years this means that it earns sixty one dollars and five cents an interest or an interest than if it earned simple interest now you can see why compound interest is a 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 it would obviously take the $100 today why because even if you didn't need the $100 today and you near 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 could cash out and have the $100 plus any interest that it earned so if our 10% interest your $100 today would be worth 110 dollars 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 $1000 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 $1000 times 1 plus point 1 to the 10th power giving us a future value of 2,590 $3.74 this would mean that $1000 today the future value of $1000 is 2 thousand five hundred ninety three dollars and seventy four cent 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 ninety dollars and ninety one cents and a fund that earned 10% interest that hour ninety dollars and ninety one cents 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 ninety dollars and 91 cents today the present value formula for a lump sum of money and 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 percent 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 dollars and 91 cents 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 it's coming from an investment that is earning five percent 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 270 dollars and 32 cents in the fund that earns 5% interest you would withdraw $100 for the next three years this also means that if you would 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 dollars and 32 cents for these three cash flows enter a year compounding interest ensure your 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 0.1 zero by 12 giving you a rate of point zero zero eight three three three three suppose you invested $10,000 at 6% interest that compounded semi-annually two times per year and held it for five years what would 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 5 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 one point zero three to the tenth power giving us a total of 13 thousand four hundred and thirty nine dollars and 16 cents 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 your 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 ten percent annual interest what is the present value of your payment's the present value of this series of cash flows would be about five thousand eight hundred twenty-eight 91 cents 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 twelve since it is an annual interest rate and we're 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 is withdrawn so the future value would be 1,000 times 1.05 to the fourth power giving us a future value of one thousand two hundred and fifteen dollars and fifty one cents the payment made in year two will earn interest over three periods so the future value will be one thousand times one point zero five to the third power giving us a future value of one thousand one hundred fifty seven dollars and 63 cents and so on this means that period 5s payment will be held for zero periods and will therefore not earn any interest so the future value will be one thousand times one point zero five to the zero power giving us a future value of one thousand dollars if we sum up the future value of all payments then we find that the future value of our ordinary annuity is five thousand five hundred twenty five dot sixty-four cents notice that over the next five years we receive five one thousand dollar 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 five percent interest since the first payment is made at the beginning of the first period it will earn interest over five periods until does withdrawn five years from today so the future value will be one thousand times 1.05 to the fifth power giving us a future value of one thousand two hundred and seventy six dollars and twenty eight cents the second payment will earn interest over four periods so the future value will be one thousand times 1.05 to the fourth power giving us a future value of one thousand two hundred and fifteen dollars and 51 cents and so on for each payment will solve the future value of each cash flow individually and then sum them up to find the future value of our new 'ti due the future value is five thousand eight hundred and one dollars and ninety two cents notice that this annuity has identical characteristics such as time interest in 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 five thousand five hundred twenty-five dollars and 64 cents which is two hundred and seventy 6.28 cents less than the future value of our new 'ti do 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 five percent 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 dollars and 38 cents since our second year payment has received two years from today our present value formula will be one thousand divided by 1.05 to the second power giving us a present value of nine hundred seven dollars and three cents remember and our present value formula we are raising the denominator which is one 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 four thousand three hundred twenty nine dollars and forty eight cents 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 five percent interest what is the present value of our annuity do remember since this is an annuity do 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 dollars and 38 cents we did this by dividing $1000 by 1.05 to the first power if we sum the present value of all of our payments we find the present value of an annuity due is four thousand five hundred and forty-five dollars and ninety-five cents which is four hundred forty three dollars and seventy seven cents more than our ordinary annuity our annuity do is worth more than our ordinary annuity because with an annuity do the payments are received sooner which means they are worth more remember money today is worth more than an equal amount of money receiving the future perpetuities a perpetuity is a fixed cash flow received over an indefinite number of periods for example if someone were 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 point zero five giving us a present value of $20,000 now let's make this a little more complicated what if we were to see $1000 per month for the rest of our lives and we could earn 5% interest in other investments what would the pressive value of our perpetuity be it would be 1,000 dollars divided by 0.05 divided by 12 giving us a present value of two hundred and thirty eight thousand ninety five dollars and twenty four cents notice that we divided the interest rate by twelve 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