### What is the definition of IRR-NPV?

The new Item is the IRR; IRR stands for the internal rate of return. The internal rate of return is the interest rate that makes the net present value =0. But first, it is important to know what is the NPV or the net present value.

### What is the NPV?

The definition is quoted from Investopedia, net present value -NPV is the difference between the present value of cash inflows and the present value of cash outflows over a period of time.

The NPV could be positive in case your investment is good, else if negative your investment Is bad. But in this post, we are interested in finding the rate i that makes this NPv=0. But we will have two examples of the positive and negative NPV quoted from, the (math is fun ) site.

The first example is that for someone who wants to get an investment of 10%, he lends his friend $500 now, and he will receive after one year $570. It is required to check the NPV and find out whether is positive or not.

We draw the time scale from 0 to 1, and the vertical axis will be cash in-out, the $500 will be drawn pointing downwards since it is outflow.

After a one-year payment of $570 is received, a vertical arrow to be drawn by that value. The Idea of estimating the NPV is to return all the future inflows to the corresponding present value.

These inflows will be pointing upwards. The NPV is estimated by summing all the cash in and cash out at time t=0. We consider i=10% , the present value of $570 is =(0.909*570), this value will be added to ($-500). As we can see in the next slide, the NPV value is equal to $18.18. The investment of 10% is achieved as required.

*The second case is when the investor wants to get 15% as interest value.*

The sample example will be used, but this time, the interest i% value will be taken as 15%. The present value of $570 is obtained and will be =(570/1.15). The NPV can be found to be =$-4.35. This value will indicate that this investment is not good.

### A Solved problem for IRR.

Let us review this example. One invests $2000 now and receives 3 yearly payments of $100 each, and in the last year, he receives $2500. We draw the diagram as follows, $2000 is drawn downwards, the Time is drawn at 0,1,2,3 year intervals, at the Time 0 $2000 is drawn downwards at the end of the first year, and $100 is drawn upwards.

Same at 2nd year and also at the third year plus $ 2500 is drawn upwards at the end of the third year. We will estimate the net present value =0, we will convert the payment to the present value, and estimate the net present value.

Here the interest is not given, we are going to find out the value of I. For every future worth, we will convert it back to its present worth, and the Net present value =0, then $2000 =100*(1/1+i)+100*(1/1+i)^2+100*(1/1+i)^3+2500*(1/1+i)^3

### Selecting different values for IRR.

** Guessing I=10%**.

We use trial and error, starting by assuming that i=10% and then substituting. The left-hand side in this case =2000, while the RHS, the right-hand side is =$ 2127.17, and the net present value is =$127.17.

But we need The NPV to be zero. This is the case of i=10%.

** Guessing I=12%**..We will change the value of I as a second trial, try i=12%. The net present value is $6.325, we are approaching the zero value.

** Guessing I=12.4%**..Try i=12.4%. After substituting. The net present value is $0.937.

### Use an Excel sheet to get the value of IRR.

Now we open an Excel sheet, in a tabulated form and check the steps we have done. First in the first column put the value of Time as 0,1,2,3. In the second column, we list the values of the cash flow as shown in Excel, for the third year we put the value of $2600.

For the present value equation, the value equals cash flow/(1+i). The value is = cell B5/(1+c3), first trial as 10%, as referential cell $C$3, because we will repeat the value with common i, first estimate the c5=B5/(1+$c$3)^A5, then make control+D.

The present value at the cash flow, for instance, when the cash flow =$100, at time =1 year.

The present value=100/(1.10)^1=$90.901, when the cash flow =$100 at time =2 year, the present value=100/(1.10)^1=$82.6446 and etc.

We can use the Excel built function and directly estimate the I; there is a function, which is called the Pv, present value we use the negative sign to get the required value, the present value=-Pv($C$3, A5,0, B5,0).

The first item is I value $c$3. While the second item, A5, is Time in Years, the third item is B5, which is cash flow value at the year.

The fourth item is 0, the PV function gives -$2000 for cell D5 and at D6 the value of $90.91, open one bracket (interest, the year,0, cash flow,0), and the same values are shown in column D from row 5 to row 8, the value of sum for C11=D11.

For the Present Net Value. We get the sum from time 0 to Time =3, and the NPV =126.39772. We have solved our previous problem by setting i=10%, then i=12%, then 12.40%, for which we have obtained $-0.938.

For the IRR directly, we made trial and error to get a closer value to the IRR and then check the value that approaches zero.

### Use the built-in function in Excel to get the value of IRR.

We use a function IRR at cell F31 as F31=IRR (B30: B33) gives the value of the internal rate of Return as i=12.3816% directly; then we get the internal rate of Return.

This is the pdf file used in the illustration of this post.

For the next posts: 8A-What is Capital Recovery?

This is a link to post 5 –Step-by-step illustrations of solved problems for P-F value.

For a useful external resource, Engineering Economy this is a link: Applying Theory to Practice.