How to Calculate Net Present Value, Annuity, & Perpetuity

welcome to our first module in our financial maps course discounted cash flows in this session we will be covering the following the time value of money how to calculate present values using the discounted cash flow methodology also how to calculate the present value of an annuity and to calculate the present value of constant and growing perpetuity let's start with the basic idea the concept of discounted cash flows relates to what somebody today would be willing to pay to receive a cash flow in the future the underlying principle is that the sooner somebody is to receive a cash flow and the more certain they are receiving it the more valuable it is today we call money we receive in the future the future value and its value today is called the present value the concept of present value relates the idea that money you have now is worth more today then an identical amount you would receive in the future why there are three reasons one opportunity we can invest the money today and receive more for it in the future to risk there is no risk in having the money today whereas if we wait until the future there is a risk we may not receive it and three inflation because of inflation many today buys more than an identical amount in the future in order to determine the present value of a future cash flow we need to discount future cash flows by a discount rate this discount rate is often referred to as the required rate of return and is expressed as a percentage the higher the discount rate the more we discount the future value by and the lower the present value this is why the discount rate is also referred to as the required rate of return the less we pay today to receive a cash flow in the future the larger our returners so here's a question for you which is more valuable to you $100 today or $100 in one year's time if you are like most financially rational investors you would prefer to have $100 today remember our three reasons why $100 today is worth more than $100 in the future this idea that money has different values to us today depending on when we receive it is called the time value of money a simple example will help illustrate this further let's say you had $100 today and you invest it for three years in a bank receiving an interest rate of 10% every year interest is paid to at the end of each year and you choose to leave this interest in the bank after one year you would have a hundred and ten dollars after two years you would have one hundred twenty one dollars and after three years you would have a hundred and thirty three dollars and ten cents another way to think about this is the funkiest rates were 10% it would be financially equivalent to you if you received $100 today or one hundred and thirty three dollars and ten cents and three years time they both have the same value to you today we can extend the previous simple example to derive a general formula linking the future value of a cash flow with its present value where the interest rate is our to increase the present value of $100 by 10 percent we simply multiply the present value by 1 plus 10% which we write as 1 point 1 0 we can do this for each of the three years which is equivalent to multiplying $100 by one point 1 0 to the power of 3 this gives us a future value of one hundred and thirty three dollars and ten cents and provides us with some equation linking the present value with the future value when we start with the present value and calculate the future value of their cash flow as we've just shown this is known as compounding when we know the future value and want to it backwards to determine its present value this is known as discounting hence the term discounted cash flows we can rearrange the previous equation so that we are calculating the present value rather than the future value rearranging this equation we can isolate a term called the discount factor it is this discount factor which we multiply the future cash flow by to determine its present value now let's look at a very common application of discounted cash flows called net present value or NPV analysis managers are often required to make decisions on whether a particular project is profitable the typical project consists of an initial investment in this case a thousand and a stream of cash flows resulting from the investment the cash flow at the end of year one is 400 at the end of year two is 600 and at the end of year three is 200 the net present value or NPV methodology sums the present value of the initial investment which is minus a thousand as it is cash we are spending and the present values of all the future cash flows we can see the calculation of the year to discount factor that is one divided by one plus the discount rate of 10% all ^ - to get the NPV we add all the present values together if the NPV is positive the project is profitable if it is negative the project is unprofitable in this example the NPV is positive so this project is indeed profitable we can use Excel to do our NPV analysis let's write our required rate of return in cell b1 remember this is our in the discount factor formula the number we're using is 10% set up an Excel worksheet with four rows these four rows will be time future value discount factor and present value populate the time and future value rows using the information on the previous slide now we need to do the calculation for the discount factor I'm going to link my formula with cell b1 we absolute reference this by pressing f4 then simply drag the formula across the formula reads equals 1 divided by bracket 1 plus B 1 which we absolute reference close bracket all ^ time calculating each cash flows present value is now simply a matter of multiplying the future value by the discount factor and I can calculate the NPV for this project by summing together all the present values we can see the present value at time 0 remains the same this makes sense as we are paying a thousand dollars today therefore the present value of a thousand dollars today is a thousand dollars with the required rate of return of 10% the NPV is positive therefore based on this we would invest in the project if the required rate of return rises to 11 percent what is the new NPV of the project would you invest in this project or not as you can see when the required rate of return of the project rises to 11 percent our NPV becomes negative so we would not go ahead and invest in this project look at what has happened to each of the present values they've become smaller this is because we are discounting their future cash flows using a higher discount rate now it's your turn to calculate the NPV to go through our first exercise click on the attachment link entitled in PB exercise once you've had a go compare your answers with your test NPV solution good luck and annuity is a series of equal payments at equal time periods and guaranteed for a fixed number of years usually this fixed time period is one year which is why it's called an annuity but the time period can be shorter or even longer understanding annuities is crucial for understanding loans and investments that require or yield periodic payments while we can calculate the present values of each future cash flow of an annuity to arrive at its present value we can also calculate an annuity factor which makes our task a lot easier let's look at an example say we hit a three year annuity receiving $100 starting one year from today for each of the next three years how much would we be willing to pay today to receive a seniority of interest rates with 10% using our discount factor formula where r is 10% we can work out each of the three discount factors remember the formula is 1 divided by 1 plus R all to the power of n we can then easily calculate the present value of the annuity by first multiplying the future values by their discount factors and then by adding all the present values together so the present value of this annuity is 248 dollars and sixty nine cents which is how much we would be prepared to pay to receive it if he sum these discount vectors what do you notice and 3 discount vectors we get when we calculate the previous annuity a zero point nine zero nine one zero point eight two six four and zero point seven five one three four years one two and three respectively if we add the discount vectors we get what is known as the annuity factor in this case two point four eight six this is very useful as it provides a shortcut to calculate the present value of an annuity notice if we multiply our annuity payment of $100 buy the annuity vector we get the same answer as before two hundred and forty eight dollars and sixty nine cents we can also use annuity tables to work out our annuity vector to save even more time annuity vector tables can easily be used to save us from having to add the individual discount vectors to derive an annuity vector typically as in the case of this table the number of years the annuity lasts for or n is placed down the left-hand side of the table along the top row of the table are the various discount rates or required rates of returns as you can see from this annuity table if we went down to the third row and across to the tenth column the annuity factor is two point four eight six nine let's now look at a practical application of annuities annuities often sold as investments and Bester's pay a lump sum today to receive a fixed amount at fixed time intervals in the future you can find a PDF copy of the annuity tables under the attachments tab this particular annuity pays investors $15,000 every year for 10 years and offers a return of 7% to find the present value or the price of the sonority set up an Excel spreadsheet with a discount rate in cell b1 setup four columns time future value discount factor and present value respectively time will go from 1 to 10 in the future value is 15,000 each of these ten years we need to insert the discount factor formula which reads equals 1 divided by bracket 1 plus B 1 close bracket all ^ a4 we can now pull this formula down finally we can calculate our present value by multiplying the future value by the discount factors summing these gives us a present value or price of the annuity of a hundred and five thousand three hundred and fifty four dollars adding the discount vectors we get an annuity factor of seven point zero two three six which when we multiply by our annual payment of $15,000 gives us the same price of a hundred and five thousand three hundred and fifty four dollars okay now it's your team to try this out yourself click on the attachment link entitled annuity exercise once you're finished check your attempt with the attached and your teeth solution good luck a perpetuity is an investment that has no definite end a stream of cash payments that continues forever mathematically the sum of an infinite stream of numbers is undefined however the present value of a perpetuity is not undefined as eventually the future cash flows are received so far into the future they have no value to us today a real-life example of a perpetuity air war loans bonds issued by the UK government with no redemption date also income from real estate can be considered a perpetuity if we assume we will receive rent from it into the foreseeable future we will look at two types of perpetuity z' constant and growing perpetuity z' let's say we one hundred dollars today if interest rates at ten percent and one is time I would have a hundred and ten dollars I could withdraw ten dollars leaving one hundred dollars I leave this 100 dollars in the bank for another year and at the end of year two I would again have a hundred and ten dollars I could withdraw ten dollars leaving one hundred dollars and so on in other words I've created a stream of ten dollars every year forever starting with one hundred dollars therefore the present value of a perpetuity if interest rates with ten percent paying ten dollars every year as one hundred dollars which we can also write as ten dollars divided by ten percent so from our example we can see that it is possible to calculate the present value of a cash flow which will occur in perpetuity the present value of a constant perpetuity is the payment we are to receive into perpetuity divided by the discount rate or the required rate of return using the figures from our previous example you can see that this formula gives us the same present value of a hundred a growing perpetuity is again a stream of cash flows that start one year from today and go on forever however the amount that we receive grows by a constant growth rate G we can calculate the present value of a growing perpetuity by using the following formula the present value is the first payment we received one year from today divided by the discount rate minus the constant growth rate as a demonstration on the use of a growing perpetuity put yourself in the position of an analyst trying to value a dividend paying stock let's assume that the price of the stock is equal to the present value of all the future dividends you will receive from owning the stock this is a fair assumption to make if you intend to buy and hold the stock for a long period of time let's now assume that the dividend you will receive one year from today is $5 but the company's dividend policy leads you to believe this will grow at a steady rate of 1% per year let's insert these two variables in cells b1 and b2 finally let's say your required rate of return as 10% let's and set this value enter cell b3 and cell b5 I'm going to type my growing perpetuity formula which will read equals 5 divided by bracket b3 minus b2 closed bracket this gives us a present value of 55 dollars and 56 cents if the stock is trading below this figure this may represent a buying opportunity for us for our final exercise we'll look back at everything that we've discussed in this module click on the attachment link entitled prize draw exercise once you've had a go take a look at the attached prize draw a solution to see if you get the same answer as me good luck that concludes our first module in this financial math series on discounted cash flows there are some key messages for you to take away from this module the time value of money sex money received today as with more than an equivalent amount received in the future the discounted cash flow methodology is the discount factor to find the present value of a future cash flow single cash flows and new tees and perpetuate these are all types of investments we can calculate the present value of using the DCF framework and module two we extend our discussion on discounted cash flows by applying some of the same techniques the price bonds I hope you can join me again goodbye

Channel: Corporate Finance Institute

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Keywords: how to calculate net present value, how to calculate NPV, how to calculate annuity, how to calculate perpetuity, how to calculate net present value annuity & perpetuity, corporate finance, corporate development, finance training, finance education

Id: iXh5tX-wuoI

Length: 17min 46sec (1066 seconds)

Published: Sat Oct 22 2016