How To Do NPV’s ©2007 Dr. B. C. Paul Note – The principles covered in these slides were developed by people other than the author, but are generally recognized as common knowledge to those studied in the field. Teaching methodologies and suggestions represent the views of the author.

Note Items These slides suggest tactics for doing NPV’s that have proven effective to the author and should be effective in increasing points earned in this class The tactics and suggestions would be expected to work best for visual learners as opposed to people favoring theory The tactics are not all part of official procedures for doing NPVs

What is an NPV? NPV is Net Present Value (also called Net Present Worth – NPW) Money in the future does not have as much buying or motivating power as money right now. An NPV reports the amount of cash right now that is equivalent to the money in the cash flow You get an NPV by multiplying future cash flow amounts by “magic numbers” that convert them into an equal amount of money right now You then sum up all the money once it is converted to present dollars. The sum is the NPV

Steps to Get the NPV Step 1 – Make a Visual Graphic of the Cash flow Step 2 – Obtain the interest rate to use for the NPV Step 3 – Determine which magic numbers you will use Step 4 – Write out the solution approach –For each magic number draw on the graphic where the cash flow number is being moved to –This help you see that all the money did get moved back to time 0

Continuing Solution Steps Step 5 – Obtain the magic numbers needed to solve the problem you wrote out Step 6 – Perform the multiplication for the problem you wrote out Step 7 – Add up all the cash numbers moved to time 0. Report the sum as the NPV

The Starting Point It is assumed that you have a cash flow and have determined that you need to do an NPV on that cash flow. Suggestion #1 – Draw out the cash flow as a visual graphic such as shown below Visual graphics help you to see a big picture of what Is happening which can help avoid mistakes.

Example The following cash flow is given on a quarterly basis Time 0 -50,000,000 Time 1 -50,000,000 Time 2 -50,000,000 Time 3 -50,000,000 Time 4 -50,000,000 Time 5 +55,000,000 Time 6 +55,000,000 Time 7 +55,000,000 Time 8 +55,000,000 Time 9 +55,000,000 Time ,000,000 Time ,000,000 Time ,000,000 Time ,000,000 Time ,000,000 Time ,000,000 Time ,000,000 Time ,000,000 Time ,000,000 Time ,000,000 Time ,000,000 Time ,000,000 Time ,000,000 Time ,000,000

The Cash Flow Can Be Drawn as -50,000, ,000, ,000, ,000,000 Note that this drawing uses large boxes that span several time Periods to represent repeating payments. This short-hand Makes the drawing fast and highlights where annuities may be Found.

Now Its Your Turn Saluki Saver opens a discount brokerage account with Spot Trade for $1,000 which he puts into Goober stock Every month for the next 5 years Saluki Saver puts another $200 into Goober stock At the end of 5 years (5 years 1 month) Saluki Saver sells his Goober stock for $25,000. Draw out this cash flow Do Not Go to the Next Slide till you have drawn the cash flow.

Does Your Cash Flow Look Like This? ………………………………………………………… ,000

Step 2 Ask Questions About the Interest Rate to Use –Interest rates are standardly reported on an annual basis Does the problem say another time period for the interest rate? If not the default is that the interest rate is given yearly –Check the cash flow – is the cash flow on an annual basis? (many are not) –Check the interest compounding period in the problem The compounding period often matches the cash flow – ie a quarterly cash flow and quarterly compounding often go together If the compounding period is different than the cash flow reporting period what is the compounding period?

Consider the Possibilities Possibility #1 –The cash flow is reported annually –The compounding period is annual –The interest rate is reported annually –Use the interest rate reported for the problem Ie the problem says the interest rate is 7.9%, the cash flow is annual, the compounding period is annual so do the NPV at 7.9%

Possibility #2 The cash flow is for a time interval shorter than a year. The compounding period is for the same shorter time period The interest is reported annually You must first obtain the “period interest rate” –Divide the annual interest rate by the number of compounding periods in one year Please note this is not likely to be the same number as the total number of compounding periods in the total cash flow. Example – Quarterly Compound, Annual Interest rate of 7.9% Do the NPV Calculation using a 1.975% interest rate

Possibility #3 The cash flow is reported for a longer time interval than the interest compounding period –Version #1 – the compounding period is shorter than one year but the cash flow is for time intervals of a longer period. –Version #2 – the compounding period is annual, but the cash flow is for a time interval longer than one year. (This is not a common situation)

Getting the Interest Rate for Version #1 You must first obtain the “period interest rate” –Divide the annual interest rate by the number of compounding periods in one year Please note this is not likely to be the same number as the total number of compounding periods in the total cash flow. –Example 7.9% annual interest rate daily compounding Next Obtain the Yield for the time period on which your cash flow is given

Obtaining the Yield How Many Compounding Periods Occur in the interval? Calculate the Yield using the equation –Where i is the period interest rate converted to decimal form and n is number of compounding periods in one cash flow time interval Example – 7.9% interest with daily compounding and an annual cash flow

What to Do With the Yield Now use the yield as your interest rate for getting the NPV on the cash flow

Possibility 3 Version 2 The interest rate is annual The compounding period is annual The cash flow time interval is greater than 1 year (say a 5 year plan cash flow) To obtain the interest rate for the cash flow –You do not need to calculate the period interest rate (since the compounding period is one year and interest was reported for one year) –You do need to get the yield over 5 years

Example 7.9% annual interest with a cash flow with 5 year time intervals Get the 5 year yield Use % interest for the time intervals on the cash flow (Note – this type of situation rarely occurs)

So Lets Get The Right Interest Rate The annual interest rate is 7.9% The compounding period is 1 year The cash flow is written out over 1 year time intervals Calculate the interest rate you should use for this NPV calculation Don’t go to the next slide till you have done the problem and have an interest rate

What did you Get? You should have got 7.9% –The reported interest rate, the compounding period, and cash flow interval are already the same.

Try Again The cash flow is monthly The compounding period is monthly The interest rate is 7.9% annually What interest rate should be used to get the NPV Don’t go to the next slide till you have done the problem and have an interest rate

Did You Do This? Obtain the Period Interest rate The compounding period is shorter than the time period for which interest and the cash flow is reported. You need to get the interest rate for your compounding period.

One More Try The interest rate is 7.9% annual The cash flow is reported annually The compounding period is daily What interest rate should you use to get the NPV on the cash flow? Don’t go to the next slide till you have done the problem and have an interest rate

Did You Do This? First get the period interest rate since the compounding period is less than the 1 year for which interest is reported. Next obtain the Yield for the 1 year time interval of the cash flow Did You Say %?

Step 3 Determine What Magic Numbers to Use To obtain an NPV you will use magic numbers that move money backwards in time to time 0. There are 3 possibilities –The money in the cash flow is already at time zero You don’t need to do anything to move this money – its already there. –A single cash flow interval’s money can be moved back with a P/F –An annuity exists and the entire annuity can be moved back with a P/A

Lets Look At Our Cash Flow to See What We Have -50,000, ,000, ,000, ,000,000 We have -50,000,000 at time 0 – it needs no magic number at all We could move all the other sums of money back With P/F values. Or – We could look for annuities (helps us reduce Our work) Our First time zero # -50,000,000!

Look at Moving Each Number Back with a P/F P/F is a function of i, and n –i is the interest rate we figured out –n is the number of cash flow periods a sum of cash must move In this case the number of cash flow periods it must move to get to time 0

Lets Go to Step 4 using P/F values to get our solution -50,000, ,000, ,000, ,000, ,000,000 At time 0 1 We have -50,000,000 at time 1 – it needs to move One period to get to 0 Write -50,000,000*P/F i,1 We have -50,000,000 at time 2 – it needs to move two Periods to get to 0 Write -50,000,000*P/F i,2 We have -50,000,000 at time 3 – it needs to move three periods To get to 0 Write -50,000,000*P/F i,3 We have -50,000,000 at time 4 – it needs to move four Periods to get to 0 Write -50,000,000*P/F i,4

Consider Drawing in Arrows to Show Where the Magic Number Has Move the Money +55,000, ,000, ,000, ,000,000*P/F i,1 -50,000,000*P/F i,2 -50,000,000*P/F i,3 -50,000,000*P/F i,4 One of the advantages of drawing in arrows it allows you to see where You are moving the money. (Hopefully if you mistakenly used an F/P you would see your money Moving into the future instead of back to time 0 – take time to think where The magic number is moving your money too)

Lets Look at Moving Some Positive Dollars +40,000, ,000, ,000, ,000,000*P/F i,1 -50,000,000*P/F i,2 -50,000,000*P/F i,3 -50,000,000*P/F i,4 +55,000,000*P/F i,5 +55,000, ,000,000*P/F i,6 +55,000,000*P/F i,7 +55,000,000*P/F i,8 +55,000,000*P/F i,9

Something to Notice Note that at this point I am making no attempt to get the number for P/F – I am only writing them down After we have written out our solution method – then in step #5 we will get the numeric value of our magic numbers Remember – most of the points in this class come from knowing how to do the problem – not crunching numbers. –Take time to write out your solution method –Even if you mess up your numbers you can get most of the points for knowing what you are doing.

Now Its Your Turn! ,000, ,000, ,000,000*P/F i,1 -50,000,000*P/F i,2 -50,000,000*P/F i,3 -50,000,000*P/F i,4 +55,000,000*P/F i,5 +55,000,000*P/F i,6 +55,000,000*P/F i,7 +55,000,000*P/F i,8 +55,000,000*P/F i,9 Write Down the notation for moving the Cash flow items in years 10 to 14 Don’t Advance to the Next Slide Until you have finished writing down The notation and showing the arrows For what you did +40,000,000

Did You Write Something Like This? ,000, ,000,000*P/F i,1 -50,000,000*P/F i,2 -50,000,000*P/F i,3 -50,000,000*P/F i,4 +55,000,000*P/F i,5 +55,000,000*P/F i,6 +55,000,000*P/F i,7 +55,000,000*P/F i,8 +55,000,000*P/F i,9 +40,000,000*P/F i,10 +40,000,000*P/F i,11 +40,000,000*P/F i,12 +40,000,000*P/F i,13 +40,000,000*P/F i,14

We Haven’t Yet Done Years 15 to 23 Already you can see the problem with doing an NPV using all P/F values –You will wear out the key pad on your calculator punching in all those numbers Rather than finish with years 15 to 23 lets look for an easier way to do this.

Back to Step 3 and considering using P/A P/A is useful if the cash flow contains an annuity –An annuity is a repeating series of payments of the same amount –The payment occurs at the end of every compounding period. –The payments start one compounding period into the future. P/A magic numbers let us move an entire series of payments with a single multiplication –Which is probably looking really good now after writing out even part of the NPV calculation using P/F

How P/A Works P/A moves an annuity of same payments to the point in time one period before the first payment i is the interest rate (which we have already figured out) n is the number of payments that are the same amount in the annuity. Multiplying just one payment out of the annuity by P/A will move the whole annuity back to one time period before the first payment.

Lets Look for Things that Might be Annuities -50,000, ,000, ,000, ,000, ,000,000 At time 0 How about this series of -50,000,000 from time 1 To time 4 - Number is the same each time - We have made our interest rate match the time periods on our cash flow so the payments are every compounding period - The payments do start at time 1 which is one time period ahead of time 0 Yup – its an annuity – lets use our P/A on it.

Moving the Annuity with P/A -50,000, ,000, ,000, ,000, ,000,000 At time 0 -50,000,000*P/A i,4 Note we have 4 payments of -50,000,000

Identify All the Places You Still See Repeating Cash Flow Elements +55,000, ,000, ,000, ,000, ,000,000*P/A i,4 Do not advance to the next slide Until you think you have identified Any remaining series of same Cash flow elements

Did You Identify These Three Things? +55,000, ,000, ,000, ,000, ,000,000*P/A i,4 A repeating series Of +55,000,000 Cash flows from Times 5 to time 9 A repeating series Of +40,000,000 Cash flows from Times 10 to time 14 A repeating series Of +20,000,000 Cash Flows from time 15 to Time 23

The Problem with P/A +55,000, ,000, ,000, ,000, ,000,000*P/A i,4 +55,000,000*P/A i,5 P/A only moves the money to one time period before the Cash flow started – ie we only got the money to time 4. All money must be to time zero before we can count it up And call it an NPV

Lets Use P/F to move the now lump sum back the rest of the way. +40,000, ,000, ,000, ,000,000*P/A i,4 +55,000,000*P/A i,5 +55,000,000*P/A i,5 *P/F i,4

Now You Move the +40,000,000 Annuity Back to Time 0 +40,000, ,000, ,000, ,000,000*P/A i,4 +55,000,000*P/A i,5 * P/F i,4 Don’t Turn to the Next Slide Until you have written out and Drawn in what it will take to Move those repeating payments Of +40,000,000 back to time 0.

Did You Get Something Like This? +40,000, ,000, ,000, ,000,000*P/A i,4 +55,000,000*P/A i,5 * P/F i,4 +40,000,000*P/A i,5 +40,000,000*P/A i,5 *P/F i,9 Note that the little arrows showing where You have moved money too can help you Keep straight how much you have already Done and how much you still have to do.

Now Finish Writing Out the Problem by moving that last 20,000,000 series of payments ,000, ,000, ,000,000*P/A i,4 +55,000,000*P/A i,5 * P/F i,4 +40,000,000*P/A i,5 * P/F i,9 You Guessed it. – Write out The rest of the problem before Going to the next slide.

Does Your Problem Look Something Like This? ,000, ,000,000*P/A i,4 +55,000,000*P/A i,5 * P/F i,4 +40,000,000*P/A i,5 * P/F i,9 +20,000,000*P/A i,8 *P/F i,14

Now We Are Ready for Step #5 Step #5 is to get the values for the magic numbers Right now we have just written in a notation that shows how the problem is going to be solved. We don’t have the numbers to solve it.

There 3 Ways to Get the Numeric Value of the Magic Numbers We Need Method #1 – We can use the formula for P/A and P/F Method #2 – We can use Class Assistant and let it calculate P/A and P/F Method #3 – We can look up the values for P/A and P/F in a table. (Of course any of the methods will get us the same number)

The Formula Approach The Formula for P/F is The Formula for P/A is Remember that when you plug in for i your interest rate must be Expressed as a decimal – lets say we are using 8% so i=.08

Lets Plug in and Get P/A 8%,4 Now lets substitute it into our problem set up.

Substituting In ,000, ,000,000*P/A i,4 +55,000,000*P/A i,5 * P/F i,4 +40,000,000*P/A i,5 * P/F i,9 +20,000,000*P/A i,8 *P/F i,14 Substitute in Now its Your Turn – Calculate and Substitute in P/A 8,5 - Oh Yes and don’t go to the next slide Until you have done it

Did You Substitute in ? ,000, ,000,000* ,000,000*P/A i,5 * P/F i,4 +40,000,000*P/A i,5 * P/F i,9 +20,000,000*P/A i,8 *P/F i, Lets see how we could get P/A 8%,8 from Class Assistant Or an interest table.

Using Class Assistant Plug in our 8% interest And 1 compounding period per time interval Plug in that there are 8 payments in our annuity Read off that P/A 8%,8 is

Now Lets Try it With the Interest Table (available under Resources on the Web Site) First – Find the P/A Column – Here it is Second – Find the Number of Payments in The Annuity Third – Read over to The P/A column Note that you read off 5.747

Lets Substitute it in ,000, ,000,000* ,000,000* * P/F i,4 +40,000,000* * P/F i,9 +20,000,000*P/A i,8 *P/F i,

Now Try to Formula Approach to get P/F 8, ,000, ,000,000* ,000,000* * P/F i,4 +40,000,000* * P/F i,9 +20,000,000*5.747*P/F i,14 The formula is (1+i) -n Calculate the Value and Substitute It in before going to the next slide

Did You Get 0.735? ,000, ,000,000* ,000,000* * P/F i,4 +40,000,000* * P/F i,9 +20,000,000*5.747*P/F i,

Now Try for P/F 8%, ,000, ,000,000* ,000,000* * ,000,000* * P/F i,9 +20,000,000*5.747*P/F i,14 This Time Use Class Assistant To get the number As Always Do Not Advance to The Next Slide Till You have Done the work.

Did You Set Class Assistant Up Like This? 8% interest Compounded once per time interval Moving Money 9 compounding periods Did you read off ?

Lets Substitute it in ,000, ,000,000* ,000,000* * ,000,000* * P/F i,9 +20,000,000*5.747*P/F i,

Now Go After P/F 8%, ,000, ,000,000* ,000,000* * ,000,000* * ,000,000*5.747*P/F i,14 This time use the interest table on The next slide to get the value

Do Not Go to the Next Slide Until You have picked off P/F 8%,14

What Value Did You Pick? Did you pick ?

You Are Now Ready for Step #6 (Multiply out the terms) ,000, ,000,000* ,000,000* * ,000,000* * ,000,000*5.747* Example -50,000,000* = -165,605,000 Now You Multiply out all the other Terms before going to the next Slide.

Did You Get These Answers? ,000, ,605, ,404, ,885, ,137,070 Now for Step #7 – Add all these numbers up. Try it and then go to the next Slide.

And the Answer is! NPV 0 …………………………………………………………………………………23 $114,822,909 We used our magic numbers to move all the Money to a common point in time (time 0), and Then we counted it. That’s an NPV