Notebook
P.W.Sims Business Program 410-650-LW Finance Teacher: Dominic Fournier
Table of Contents
Simple Interest When money is borrowed, interest is charged for the use of that money for a certain period of time. When the money is paid back, the principal (amount of money that was borrowed) and the interest is paid back. The amount of interest depends on the interest rate, the amount of money borrowed (principal) and the length of time that the money is borrowed. The formula for finding simple interest is: Interest = Principal * Rate * Time. The formula to find the final amount paid (Where Pt is the principle at time t) : Pt = Principal + ( Principal x rate x time) this is equal to (Principle x 1) + (Principle x rate x time) Since principle appears in both terms we can simplify Therefore : Pt = Principal x ( 1+ rate * time) Or Pt = P0(1 +r*t) where P0 is Principle at time 0
Ex: If $100 was borrowed for 2 years at a 10% interest rate, the interest would be $100*10%*2 = $20. The total amount that would be due would be $100+$20=$120 or P2 = 100 x (1+0.1*2) = 100 x (1.20) = $120 Simple interest is generally charged for borrowing money for short periods of time. The interest is SIMPLE if it is calculated on the initial amount during the whole length of time. We usually use simple interest on relatively short term transactions. Simple interest becomes complex to calculate in investment projects such as mortgages or funds. For example, many non voting preferred shares are calculated using simple interest. The interest will be distributed periodically according to the contract. The amount of interest distributed for each period will always be the same. Certificate of deposits are also managed in the same way. Note: All financial transactions of this nature involve a borrower and a lender (or investor). So, when you borrow, you are paying interest, and when you invest you receive the interest. The math in both cases is the same.
Commercial Discounts are the reverse process of calculating simple interest. When we want to receive before the due date a certain amount, it is normal not to receive the full amount, since a percentage should be withheld to compensate the period of time.
In many cases, the interest will be calculated on the original amount and then subtracted (such as commercial credit discounting). P-t = Principal - ( Principal x rate x time) or P-t = P0(1 -r*t)
Example: An invoice for 1,000$ is due in 30 days. Your supplier mentions a 2% discount for the whole period if you pay right away. P-30 = P0 x (1-2% x 1) or 1,000 x 98% = 980$ A more precise way would be to calculate what is the P0 that would grow to the Pt at a given rate of r and time of t. Pt = P0(1 +r*t) is to accumulate interest to value so P0 = Pt / (1 +r*t) would discount the value
Example: You have a document that guarantees you a payment of 25 000$ in 8 months. You want to get the money right away. The financial institution could accept to ‘discount’ this amount at a certain rate. It will then give you an amount inferior to the initial amount and will retain a discount calculated as simple interest. If the rate is 11%, the amount of the discount would be : $25,000 = P0 x (1+11%*8/12) or P0 = $25,000 / (1 + 11% * 8/12) = $25,000 / (1.0733) = $23,291.93 Note: In the examples, the frequency of r is always 1 year whereas t is always expressed at the same frequency (years). In real cases, r might be expressed on a different frequency than t, you must always t’s frequency to correspond with r’s frequency. Ex: t = 4 months, r= 11%/year So convert T in years = 4/12 year = 1/3 year = 0.33 years
Compound Interest Compound interest is interest figured on the principal and any interest owed from previous period. The interest charged the first period is just the interest rate times the amount of the loan. The interest charged the second period is the interest rate, times the sum of the loan and the interest from the first period. The interest charged the third period is the interest rate, times the sum of the loan and the first two periods interest amounts. Continue figuring the interest in this way for any additional periods of the loan. Example: A bank charges 8% compound interest on a $600 loan, which is to be paid back in two years. It will cost the borrower 8% of $600 the first year, which is $48. The second year, it will cost 8% of $600 + $48 = $648, which is $51.84. The total amount of interest owed after the two years is $48 + $51.84 = $99.84. Note that this is more than the $96 that would be owed if the bank was charging simple interest. Example: A bank charges 4% compound interest on a $1000 loan, which is to be paid back in three years. It will cost the borrower 4% of $1000 the first year, which is $40. The second year, it will cost 4% of $1000 + $40 = $1040, which is $41.60. The third year, it will cost 4% of $1040 + $41.60 = $1081.60, which is $43.26 (with rounding). The total amount of interest owed after the three years is $40 + $41.60 + 43.26 = $124.86. Compound interest is similar but the total amount due at the end of each period is calculated and further interest is charged against both the original principal but also the interest that was earned during that period. If r is the rate per period P0 is the initial amount n is the number of periods during the time the rate applied Pn is the final amount (or after n periods)
Then, we can calculate the amount after each period, using the result of the preceding period (capital + interest) : After 1 period: After 2 periods: After 7 periods: After n periods:
P0 (1+r); P0 (1+r) (1+r) or P0 (1+r)2 P0 (1 + r) (1+r) (1+r) (1+r) (1+r) (1+r) (1+r) or P 0 (1+r)7 P0 (1 + r)n
And the final result will be:
Pn = P0 (1 + r)n
This relation also works when we are looking for the initial amount, if we divide each side by (1+r)n , we will obtain: P0 = Pn/(1 + r) n or P0 = Pn(1 + r) -n
Compounding interest rates Rates (r) are not always expressed at the same frequency as the compounding period. Ex: 12% annual rate compounded quarterly. This means that the rate will be divided to match the compounding periods then compounded. Ex: 12% annual rate compounded quarterly for 3 years (12 quarters) = 1+(12%/4)12 = (1.03) 12 = 1.42576 Using your Financial calculator to find the answers: MAKE SURE YOUR CALCULATOR IS IN FINANCE MODE! With model EL-733A.For EL-738, it is always in finance mode by default. PV = initial amount. Since the initial amount is the money borrowed or the money invested, you must enter it negatively. FV = Final amount n = number of periods. i (I/Y) = interest. The interest entered must be for the same time frame determined with n. IF n is in months; the interest must be converted in months. If n is in years, the interest must be in years, etc.
Example: you place 100 $ for 3 years at a yearly compound rate of 13,5%. Enter PV : - 100 Enter n : 3 Enter i : 13,5 COMP FV And the result should give you : 146.21$ Ex. 2 : at a rate of 18% capitalized quarterly, $20,000 invested for 5 years will give what results ? PV = -20 000 n = 5 years in quarters = 5 x 4 = 20 i = 18/4 = 4.5
COMP FV = $48,234.28
Time Line
Compounding: The process of determining the future value (FV) of a payment or a series of payments when applying the concept of compounding interest. This process is the opposite of discounting. Discounting: The process used to determine the present value (PV) of a payment or a series of payments.
Nominal and Effective rates of interest Both consumers and businesses need to make objective comparisons of loan costs or investment returns over different compounding periods. As mentioned before, rates (r) are not always expressed at the same frequency as the compounding period. In order to put interest rates on a common basis, to allow comparison, we distinguish between nominal annual and effective annual rates. The nominal, or stated, annual rate (APR) is the contractual annual rate charged by a lender or promised by a borrower. It is the advertised rate by lenders. The Effective periodic rate (EFF), or Effective annual rate (EAR) is the rate of interest actually paid or earned during a period. It reflects the impact of compounding, whereas the nominal annual rate does not. In terms of interest earnings, the EAR is probably best viewed as the annual interest rate that would result in the same future value as that resulting from application of the nominal annual rate using the stated compounding frequency. It increases with increased compounding frequency. Examples: • • •
A nominal rate of 12% capitalized semi-annually [12(2)%] will translate in an effective semestrial rate of 6% (compounded semi-annually) and 12.36% effective annual rate. (1+6% x 1 + 6%) 15% [15(1)%] is 15% if the interest is compounded once per year. A nominal rate of 1,25% per month will translate to 16.0755% if the interest is compounded.
Equivalent Rates: It becomes obvious that many rates that may seem different at first are in fact equivalent and will produce the same results using the same amounts after the same time. In a case where there is no compounding, the nominal rate and the effective rate are the same. So for the rates r(m)% (r the rate and m the compounding frequency) and i(k)% (i the interest rate and k the compounding frequency) and r≠i and m≠k We will get : P(1 + r/m% )m = P(1 + i/k% )k
If we simplify the equation: since P will be the same amount in both cases we can remove it without changing the equation We will get: (1 + r/m% )m = (1 + i/k% )k (1 + r/m%)m = (1+ i/k%)k
and if k = 1 year
(1 + r/m%)m = (1+ i/%) and
i% = (1 + r/m% )m – 1 = EAR
Example: 16(2) % = (1 + 16/2%)2-1 = 16.64% annual effective rate and 8% effective semestrial rate 16(4) % = (1 + 16/4%)4-1 = 16.98% annual effective rate and 4% effective quarterly rate 16(12) % = (1 + 16/12%)12-1 = 17.23% annual effective rate and 1% effective monthly rate Equivalent rates may also be calculated between to different compounding frequencies. Example: What quarterly rate would be equal to a 12(2)% interest rate? In this case, you must calculate 2 equal effective rates, one a compounding frequency of 2 and and second at a compounding frequency of 4. (1 + 12/2%)2 = (1+ i/4%)4 so 1.1236 = (1+ i/4%)4 so ((1.1236)1/4-1) x 4 = i = 11,82%
How to calculate nominal or effective rates with your calculator: WITH SHARP EL-733A
1.
If we are looking for the effective rate: enter number compounded annually, press 2ndF, then EFF, enter nominal rate.
1.
If we are looking for a nominal rate: enter number compounded annually, press 2nd F, then APR, enter effective rate. WITH SHARP EL-738
1. If we are looking for the effective rate: enter number of compounding periods 2. press x,y, then, enter nominal rate.
3. 1. 2. 3.
Press 2ndF EFF, If we are looking for the nominal rate: enter number of compounding periods press x,y, then, enter effective rate. Press 2ndF APR,
RESEARCH EXERCISE: Go on Wikipedia.com, and read the section on APR and EFF! It is very well done and can complete your readings for this concept.
NOTE: Only NOMINAL rates may be divided by a denominator equal to the compounding frequency. This operation would result in an effective periodic rate.
The concept of Present Value and Future Value. The interest is a way of evaluating what an amount of money would be worth at a future date or a previous date. When we calculated simple interest or Compound interest, we evaluated what those amounts were worth at a later date or at a previous date. In other words, the current value of a future amount is called the present value. It represents the amount of money that would have to be invested today at a given rate of return over a specified period to equal the future amount. Present value depends largely on the investment opportunities of the recipient and the point in time at which the amount is to be received. This section explores the present value of a single amount. The process of finding present values is often referred to as discounting cash flows. It is concerned with answering the question: If I can earn r percent on my money, what is the most I would be willing to pay now for an opportunity to receive FV dollars n periods from today. This process is actually the reverse of compounding interest. Instead of finding future value of present dollars invested at a given rate, discounting determines the present value of a future amount, assuming the opportunity to earn a certain return, r, on the money. This annual rate of return is variously referred to as the discount rate, required return, cost of capital, or opportunity cost.
In the case of Compound interest, we have already established that FV = PV (1+r)n and that there is therefore a relation with PV = FV (1+r) -n
Example using a time line: Pam Valenti wishes to find the present value of $1,700 that will be received 8 years from now. Pam’s opportunity cost is 8%.
PV = FV (1+r) -n so PV = $1 700 (1+ 8%)-8 = $ 918.42
Annuities With A Fixed Date And Perpetuities
Annuity A series of fixed payments paid at regular intervals over the specified period of the annuity. The fixed payments are received after a period of investments that are made into the annuity.
An annuity is essentially a level stream of cash flows for a fixed period of time. It is most often used as a form of income during retirement.
Annuities At some point in your life you may have had to make a series of fixed payments over a period of time - such as rent or car payments - or have received a series of payments over a period of time, such as bond coupons. These are called annuities. If you understand the time value of money and have an understanding of future and present value, you're ready to learn about annuities and how their present and future values are calculated. What Are Annuities? Annuities are essentially series of fixed payments required from you or paid to you at a specified frequency over the course of a fixed period of time. The most common payment frequencies are yearly (once a year), semi-annually (twice a year), quarterly (four times a year) and monthly (once a month). There are two basic types of annuities: ordinary annuities and annuities due: Ordinary Annuity: Payments are required at the end of each period. For example, straight bonds usually pay coupon payments at the end of every six months until the bond's maturity date. Annuity Due: Payments are required at the beginning of each period. Rent is an example of annuity due. You are usually required to pay rent when you first move in at the beginning of the month, and then on the first of each month thereafter. Since the present and future value calculations for ordinary annuities and annuities due are slightly different, we will first discuss the present and future value calculation for ordinary annuities.
End of period payments VS beginning of period payments (or cash flows). Up to now, we have learned how to calculate present and future values. The cash flows were paid or received at the end of the period. However, in certain circumstances, we must take into consideration that payment we make or receive is performed at the beginning of a period, as for a residential rent for example. Those are called beginning of period cash flows. The calculator is set by default in end of period mode. When faced with calculating beginning of period cash flows or payments, we must change the mode by pressing on BGN/END. You will notice that BEG will appear on your calculator screen. The rest of the operations are performed as usual, but we will ultimately gain one extra month in interest.
Calculating the Future Value of an Ordinary Annuity If you know how much you can invest per period for a certain time period, the future value of an ordinary annuity formula is useful for finding out how much you would have in the future by investing at your given interest rate. If you are making payments on a loan, the future value is useful for determining the total cost of the loan.
Let's now run through Example 1. Consider the following annuity cash flow schedule:
In order to calculate the future value of the annuity, we have to calculate the future value of each cash flow. Let's assume that you are receiving $1,000 every year for the next five years, and you invested each payment at 5%. The following diagram shows how much you would have at the end of the five-year period:
Since we have to add the future value of each payment, you may have noticed that, if you have an ordinary annuity with many cash flows, it would take a long time to calculate all the future values and then add them together. Fortunately, mathematics provides a
formula that serves as a short cut for finding the accumulated value of all cash flows received from an ordinary annuity:
C = Cash flow per period i = interest rate n = number of payments If we were to use the above formula for Example 1 above, this is the result:
= $1000*[5.53] = $5525.63 Note that the $0.01 difference between $5,525.64 and $5,525.63 is due to a rounding error in the first calculation. Each of the values of the first calculation must be rounded to the nearest penny the more you have to round numbers in a calculation the more likely rounding errors will occur. So, the above formula not only provides a short-cut to finding FV of an ordinary annuity but also gives a more accurate result.
Calculating the Present Value of an Ordinary Annuity If you would like to determine today's value of a series of future payments, you need to use the formula that calculates the present value of an ordinary annuity. This is the formula you would use as part of a bond pricing calculation. The PV of ordinary annuity calculates the present value of the coupon payments that you will receive in the future. For Example 2, we'll use the same annuity cash flow schedule as we did in Example 1. To obtain the total discounted value, we need to take the present value of each future payment and, as we did in Example 1, add the cash flows together.
Again, calculating and adding all these values will take a considerable amount of time, especially if we expect many future payments. As such, there is a mathematical shortcut we can use for PV of ordinary annuity.
C = Payment or Cash flow per period i = interest rate n = number of payments The formula provides us with the PV in a few easy steps. Here is the calculation of the annuity represented in the diagram for Example 2:
= $1000*[4.33] = $4329.48
Calculating the Future Value of an Annuity Due When you are receiving or paying cash flows for an annuity due, your cash flow schedule would appear as follows:
Since each payment in the series is made one period sooner, we need to discount the formula one period back. A slight modification to the FV-of-an-ordinary-annuity formula accounts for payments occurring at the beginning of each period. In Example 3, let's illustrate why this modification is needed when each $1,000 payment is made at the beginning of the period rather than the end (interest rate is still 5%):
Notice that when payments are made at the beginning of the period, each amount is held for longer at the end of the period. For example, if the $1,000 was invested on January 1st rather than December 31st of each year, the last payment before we value our investment at the end of five years (on December 31st) would have been made a year prior (January 1st) rather than the same day on which it is valued. The future value of annuity formula would then read:
Therefore,
= $1000*5.53*1.05 = $5801.91
Calculating the Present Value of an Annuity Due For the present value of an annuity due formula, we need to discount the formula one period forward as the payments are held for a lesser amount of time. When calculating the present value, we assume that the first payment was made today. We could use this formula for calculating the present value of your future rent payments as specified in a lease you sign with your landlord. Let's say for Example 4 that you make your first rent payment at the beginning of the month and are evaluating the present value of your fivemonth lease on that same day. Your present value calculation would work as follows:
Of course, we can use a formula shortcut to calculate the present value of an annuity due:
Therefore,
= $1000*4.33*1.05 = $4545.95 Recall that the present value of an ordinary annuity returned a value of $4,329.48. The present value of an ordinary annuity is less than that of an annuity due because the further back we discount a future payment, the lower its present value: each payment or cash flow in ordinary annuity occurs one period further into future. Conclusion: Now you can see how annuity affects how you calculate the present and future value of any amount of money. Remember that the payment frequencies, or number of payments, and the time at which these payments are made (whether at the beginning or end of each payment period) are all variables you need to account for in your calculations.
Perpetuities There will be situations in life where the number of annuities is infinite. The annuities start at a given moment but go on indefinitely. That is the case for example for sums of money that a foundation wishes to give annually for a cause. It is also useful in situations where it is impossible to predict the exact end of the instalments. Period For an end of period perpetuity, a period before the instalment is easy to establish. The first way is to say we need C = P0 x i The sum P0 at an interest i per period during each period. We make P0 x i and this amount is destined for paying exactly the instalments C It is necessary because if C > P0 x i, the interests gained by P0 will not be enough to pay C and we will then have to use part of P0 to pay Câ&#x20AC;Ś we canâ&#x20AC;&#x2122;t do that indefinitely! If C < P0 x i, the interest gained by P0 will be more than sufficient to pay C and therefore the following instalments. The value of P0 increases and becomes infinite, which is impossible! IF C is neither smaller nor bigger than P0 x i, it must therefore be equal. Therefore : C = P0 x i or
P0 = C/i
Growing perpetuities In some cases, the perpetuities will want to increase its instalments to compensate for the effect of inflation. Or an investment in a business will increase because of the strong growth. This relationship will play on the interest rate directly, where g = growth rate (or inflation rate)
Therefore : P0 = C/(i-g)
where g = 0, the Annuities are constant.
Present value of a mixed stream The internal rate of Return to evaluate projects To find the present value of a mixed stream of cash flow, we determine the present value of each future amount, as described in the preceding section and then add together all the individual present values. From now on, we will distinguish the entries with a + sign and the exit amounts as â&#x20AC;&#x201C; signs. Example : We must give an amount of $1,000 in 6 months and in 9 months, and a sum of $2,000 in one year. It is also planned that we will receive $3,000 in 10 months. What would be the present value of this project if the interest rate is 9(12)% ?
Present value of mixed streams : -$1,000( 1 + 9%/12)-6 -$1,000( 1 + 9%/12)-9 -$2,000( 1 + 9%/12)-12 = - $3,719.60 The sum of $3,000 is worth : $3,000( 1 + 9%/12)-10 = $2,784.01 So the total PV = - $935.58
(we could have obviously calculated all in one operation)
Test your knowledge: try the same calculation using your calculator.
The Net Present Value The net present value is a particular case of the present value calculated in certain circumstances. As all present value calculations, we need an interest rate to calculate it. We add the word NET to remind us that all the amounts presented in a problem have been taken into consideration. The NPV depends on the rate chosen to calculate it. We will use the most appropriate rate which could be the one suggested by the investor or the borrowing rate, the rate that can be obtained by other investments, which is often called the value of money. For the same project, the NPV will be positive or negative depending on the rate. We will examine the most frequent case, when an investment will necessitate an initial down payment and will generate stream of cash flows. Example: We are contemplating investing in a project that would last 7 years and generate the following amounts after each year: 1. $7,500 2. $10,000 3. $2,000$
4. $14,000 5. $20,000 6. $17,000
7. $3,000
An investment of $45,000 is needed at the beginning of year 1 and this amount will not be given back at the end of the project. We want to compare the value of the streams with the $45,000 invested. We will calculate the PV of the streams at a rate that seems acceptable and we will compare this value with the $45,000. We can also calculate the PV of the inflows and outflows at the same rate, considering the outflow as a negative amount. We work with annual streams of cash flow, considering an initial amount invested and the streams of cash flow : (-45,000, 7,500, 10,000, 2,000, 14,000, 20,000, 17,000, 3,000) If the NPV for a certain rate is positive, we could say that the project is interesting. If the NPV is negative, it would not be an interesting opportunity to invest in. At a rate of 12,5%, NPV12,5% = -45,000(1+12.5%)0 + 7,500 (1+12.5%)-1+ 10,000 (1+12.5%)-2+ 2,000(1+12.5%)-3 + 14,000 (1+12.5%)-4+ 20,000 (1+12.5%)-5+ 17,000 (1+12.5%)-6+ 3,000 (1+12.5%)-7 = $512.25 NOTE: (1+12.5%)0=1, a power of 0 will always give a result of 1
We could have invested $45,512.25 and the project would have generated 12,5%. At 12,5%, the project will generate more than what it costs. How to calculate the NPV with your financial calculator (Model 733A) :
Enter interest rate i , enter all cash flows in order, starting with period 0. Make sure to enter the right sign, minus for money you invest in and plus sign for stream of cash flows generated. Once all cash flows have been entered, press COMP, and then NPV. BE PATIENT! And the result will come.
NOTE : MAKE SURE YOU ERASE USING CA BEFORE STARTING ANOTHER NPV CALCULATIONS. The calculators tend to keep data stocked. How to calculate NPV with your financial calculator (MODEL EL-738) VERY IMPORTANT: MAKE SURE NO CASH FLOWS ARE STILL IN MEMORY BEFORE STARTING! If when pressing on ENT, you are not at 0.00, that means you must erase memory firstâ&#x20AC;Ś. PRESS CFi then 2ndF, then CA. To calculate NPV: Start by entering cash flow amount, then press ENT for each new cashflow. Once all your cashflows are entered, press 2ndF then CFi. Enter rate and press ENT. The rate will stay in memory and can be changed by entering another one, without affecting the cash flows previously entered. Press on down arrow. NET_PV will appear. Press COMP to get final answer of NPV.
Internal Rate Of Return (IRR)
The discount rate often used in capital budgeting that makes the net present value of all cash flows from a particular project equal to zero. Generally speaking, the higher a project's internal rate of return, the more desirable it is to undertake the project. As such, IRR can be used to rank several prospective projects a firm is considering. Assuming all other factors are equal among the various projects, the project with the highest IRR would probably be considered the best and undertaken first. IRR is sometimes referred to as "economic rate of return (ERR)".
You can think of IRR as the rate of growth a project is expected to generate. While the actual rate of return that a given project ends up generating will often differ from its estimated IRR rate, a project with a substantially higher IRR value than other available options would still provide a much better chance of strong growth. IRRs can also be compared against prevailing rates of return in the securities market. If a firm
can't find any projects with IRRs greater than the returns that can be generated in the financial markets, it may simply choose to invest its retained earnings into the market. Ref: Investopedia.com
An Inside Look At Internal Rate Of Return by Linda Grayson (Contact Author | Biography) The internal rate of return (IRR) is frequently used by corporations to compare and decide between capital projects, but it can also help you evaluate items in your own life, like lotteries and investments. The IRR is the interest rate (also known as the discount rate) that will bring a series of cash flows (positive and negative) to a net present value (NPV) of zero (or to the current value of cash invested). Using IRR to obtain net present value is known as the discounted cash flow method of financial analysis. Read on to learn more about how this method is used. (For more insight, read the Discounted Cash Flow Analysis tutorial.) IRR Uses As we mentioned above, one of the uses of IRR is by corporations that wish to compare capital projects. For example, a corporation will evaluate an investment in a new plant versus an extension of an existing plant based on the IRR of each project. In such a case, each new capital project must produce an IRR that is higher than the company's cost of capital. Once this hurdle is surpassed, the project with the highest IRR would be the wiser investment, all other things being equal (including risk). IRR is also useful for corporations in evaluating stock buyback programs. Clearly, if a company allocates a substantial amount to a stock buyback, the analysis must show that the company's own stock is a better investment (has a higher IRR) than any other use of the funds for other capital projects, or than any acquisition candidate at current market prices. (For more insight on this process, read A Breakdown Of Stock Buybacks.) Ref: http://www.investopedia.com/terms/i/irr.asp
Example: What would be the IRR of this series of cash flows? NPV(-100;30;35;40;45) NPV =0 IRR = i, IRR = 17.09%
[ SOME RESERVATIONS ON THE USE OF IRR As an investment decision tool, the calculated IRR should not be used to rate mutually exclusive projects, but only to decide whether a single project is worth investing in.
Two mutually exclusive projects
In cases where one project has a higher initial investment than a second mutually exclusive project, the first project may have a lower IRR (expected return), but a higher NPV (increase in shareholders' wealth) and should thus be accepted over the second project (assuming no capital constraints). IRR makes no assumptions about the reinvestment of the positive cash flow from a project. As a result, IRR should not be used to compare projects of different duration and with a different overall pattern of cash flows. Modified Internal Rate of Return (MIRR) provides a better indication of a project's efficiency in contributing to the firm's discounted cash flow.
The IRR method should not be used in the usual manner for projects that start with an initial positive cash inflow (or in some projects with large negative cash flows at the end), for example where a customer makes a deposit before a specific machine is built, resulting in a single positive cash flow followed by a series of negative cash flows (+ - - - -). In this case the usual IRR decision rule needs to be reversed. If there are multiple sign changes in the series of cash flows, e.g. (- + - + -), there may be multiple IRRs for a single project, so that the IRR decision rule may be impossible to implement. Examples of this type of project are strip mines and nuclear power plants, where there is usually a large cash outflow at the end of the project. USING YOUR FINANCIAL CALCULATOR TO FIND IRR with EL-738-A: Example : what is the IRR of NPV(-100;30;35;40;45) Enter series of cash flows as you would normally do to find NPV: •
-100 , ENT ; 30, ENT ; 35, ENT ; ; 40, ENT ; 45, ENT
•
2ndF , CASH
•
Rate (I/Y) should appear. Press COMP
•
And you will get the IRR as a result for this given NPV.