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Claim My Discount!Today we are going to talk about energy economics. (Refer Slide Time: 0:23) We are going to talk in terms of looking at the viability of energy efficiency or a renewable project. And the subject that we are covering is also going to be amenable for any project, any project whether it deals with energy or not, the principles will remain the same, but we are going to focus basically on energy-related projects. (Refer Slide Time: 0:47) So, what are the types of decisions that one takes? If you are in an industry or you are in a company, we want to decide there are two types of decisions. One is, you can think in terms of a Yes, No kind of decision. So, for instance, there is a boiler or a furnace in industry and from the exhaust gases which are going out, they are going out at some temperature. So, we may decide, should we have a waste heat recovery system where we recover the energy from that boiler. We have a pump where we are looking at pumping water, should we have a controller or a variable speed drive. So, it is a yes-no kind of decision, a particular option whether we should go for it or not go for it. There could also be another type of a decision, for instance, you are looking at a remote village or you are looking at an island and you want to electrify that island you are looking at let us say Elephanta Island you can, you have an option where you can connect from the mainland you can have a pipe electricity supply going under the water and then you have a grid-based supply coming from the mainland. You could also think in terms of a diesel engine, or a solar photovoltaic, or a biomass gasifier engine. And so there are a whole host of options and we may want to look at out of all these options, which is the best possible option. So, we want to rank or choose between the different possible options. So, whether it's a yes-no decision or a decision where we are choosing between several options, the basis and the economic calculations are the same. In this lecture, we will assume that all technically feasible options have been included and they are all equivalent in terms of their performance. So, whenever we look at different options, there are multiple criteria on which we can be compared. One is there could be the cost criteria, which could be the initial cost or the operating costs. We can think in terms of reliability, we can think in terms of emissions, in terms of operational flexibility or convenience. So, usually, whenever we make decisions, it is a combination of a variety of things, but for this lecture, we will presume that there are many different options which are being considered all these options have equivalent performance and we are only making the comparison based on the economics. (Refer Slide Time: 3:26) So, let us look at the factors which determine the cost-effectiveness of an additional investment we are looking at something where we are putting in a waste heat recovery boiler or a variable speed drive and energy-efficient equipment on what basis should we decide whether it is cost-effective. So, there are many different parameters which we will consider. (Refer Slide Time: 3:52) One of the parameters which affect the decision is what is the amount of investment so, if we have to invest more, then we would expect that we, we may need to look at what kind of savings are obtained. The other parameter is the amount of energy-saving and most of the cases we are looking at fossil fuels being replaced by renewable or fossil fuels being saved. So, how much is the amount of energy-saving? What is the price of energy? So, that the amount of energy saving into the price of energy will give you the annual savings and then you compare the investment with the savings, there is also in these, the life of the equipment or the project will be involved. And then the time value of money. The time value of money is a concept that we need to understand and based on that concept, everything else in this lecture we can then calculate the parameters. So, we will first start with the kind of different indices. So, we said, the amount of investment, the amount of energy saving, the price of energy, the life of the equipment, all of these affect the decision and then there is the time value of money. Typically, when we think in terms of renewables, usually they are higher initial costs, and they have lower operating costs. So, of course, now the costs are coming down, but in general, as compared to fossil, fossil will have an operating costs renewables almost as zero operating costs, but they have a higher initial cost and we usually make this kind of trade-off. (Refer Slide Time: 5:29) So, we have different indicators that we compute when we calculate the economic criteria. Some of the indicators are mentioned here. The indicators that you see the simple payback period, we will define that this is the one which is the payback simple payback period, very commonly used, and based on its simplicity, ease of calculation, then we have these three indicators the net present value, the benefit by cost ratio and the internal rate of return. All these three indicators use the time value of money. Most companies use one or more of these measures the NPV, the B by C ratio or the internal rate of return. So we will first talk about the simple payback period, then we will introduce the concept of the time value of money and the discount rate. And then we will define these three indices, the NPV, the B by C ratio, and internal rate of return. After doing that, for many large projects, societal projects and government projects, we also look at life cycle costing, and where we will look at life cycle cost or the annualized life cycle costs. So, these are all the different criteria and we will see how we derive thesecriteria, we will then take some examples and calculate these criteria and use it to make our decisions. (Refer Slide Time: 6:51) So, let us start with the first index, which I am sure most of you are already familiar with. This is the simple payback period. The simple payback period, as the name suggests, is an index which just reflects the number of years in which the investment will pay back for itself. So, in terms of a definition, it will be the initial investment by the annual savings. So, very straight forward calculation, we just look at whatever is the initial investment divided by the annual savings and we will get the payback period. (Refer Slide Time: 7:50) So let us take an example, let us consider an example when an energy auditor has done an audit of a boiler, and that auditor has found that there is some insulation which can be improved and by doing this insulation you are, on an annual basis based on the way the boiler operates, we get a saving of five kilolitres or 5000 litres of light diesel oil. The price of light diesel oil is 50 rupees per litre. So, we want to calculate what is the simple payback period for this energy conservation opportunity. You can do this, this is very straight forward. (Refer Slide Time: 8:02) SPP= Co AS 300, 000 5×1000×50 1.2YEARS(1YEAR 3 MONTHS) SPP<SPPacceptable We can just take simple payback period is the initial investment by the annual saving, initial investment here is 3 lakhs and the annual saving is we have 5-kilolitres, five into 1000 one kilolitre is a thousand litres and we are paying 50 rupees per litre. So, we get an annual saving is 2.5 lakhs. And simple payback period is simply 3 lakhs divided by 2.5 which is nothing but 1.2 years or roughly 1 year 3 months. Now, we have calculated the index the simple payback period, how do we use this for making a decision? The first thing is the simple payback period must be less than the life of the equipment of the project. So, in this case, the installation is going to last for 10 years or 20 years. The second thing is the company which is making that investment will decide what is the maximum acceptable simple payback period. So, for instance, if the company says any project, which has a payback of fewer than two years, it is willing to accept then we compare this 1.2 with two and we find that the simple, the payback period is less than the minimum or the maximum acceptable payback hence, we can go ahead and invest. So, this SPP must be less than SPP acceptable. And the company who is investing will decide what is an acceptable payback. So for instance, if you have a project where there is a payback of three years, and the company wants paybacks only less than two years the company will not go for it even if the project will give benefits for more than 10 years. So, we have to, the decision will be taken by the company which is investing. So, this is what we look at in terms of the simple payback period. (Refer Slide Time: 10:48) Now, let us talk about what are the limitations of this simple payback period. For doing this, let us take a simple example. We have these two options, Option A and option B for the same application in the case of A, there is an investment of one lakh and the saving of 50,000. So, if we look at this if we just divide one lakh by 50,000 we will get a simple payback period of two years for A. So, SPP A is two years and in the case of B investment is higher which is 1.2 lakhs and the saving is lower which is 40,000. So, the simple payback period for B is three years. (Refer Slide Time: 11:39) SPPA=2YEARS SPPB=3YEARS So, if you write this we will see that SPPA is 2 years and SPPB is 3 years. And if the company has any project which is less than equal to three years it is willing to go, when you compare these two it looks like the project with the lower simple payback period is the one that we should opt for. So, we should opt for A but if we are told that for instance the life of A is 3 years and the life of B is 8 years, then immediately you will see that the decision changes. And it is more rational to go for B because we are getting payback for a long, we are getting the savings for a longer period. So, one of the limitations of the simple payback period is that does not cash consider the cash flows after the payback is achieved. The second limitation is it considers all cash flows as equivalent. So, that means, whether the return cash flow is in this year or it is in the next year all of them are considered equivalent. There is no concept of the time value of money. Despite these limitations, the simple payback period is an index that is widely used because of its simplicity and especially if it is, for any project which has relatively low investments, and it has the, it has quick paybacks. So, if you are calculating something where you are getting a payback of six months or a year, the simple payback period may be sufficient for you to make the decision. However, if you are looking at a large power project, which has significant investments, and you are talking of payback periods of 4 years or more, you need to look at the time value of money and other issues and then some other criteria would be more suitable. So, as I told you earlier, the main concept that we need to understand is the time value of money and to look at the concept of the time value of money we have to understand that individuals, companies, industries, we all prefer money today compared to money in the future. What is the reason for that? The reason for that is mainly because anything associated with the future is uncertain. There is a risk associated with the future. And because of that, all LIFE 3 YEARS 8 YEARS individuals prefer to have the money today compared to money in the future. This preference that individuals and companies have for money today as compared to money in the future, is something that we would like to understand and incorporate in our calculations. To do that, we introduce a concept called the discount rate. (Refer Slide Time: 15:00) And the discount rate is a basis by which we compare investments today with the expected future benefits let me just show you. (Refer Slide Time: 15:06) So, for instance, what we will do is that if you have in different years. (Refer Slide Time: 15:14) We are talking of 2019, 2020, 2019 plus k, if we had the value in the year and the present value. So, if we have one unit, one rupee, 1000 rupees, one lakh that in 2019 that is the same in 2019. If we talk about one unit one rupee in 2020 that has less value for us today. So, that would be reduced by a factor which is one by one plus d, where d is the discount rate. It is a positive number discount rate and you can, we can put it as a percentage also or as a fraction. And so, this is we are discounting the future we are reducing any future cash flow to bring it into an equivalent value with equivalent present value. So, suppose we had it in the kth here, then this will be one by one plus d raise to k, okay, so, this just means that we take any future cash flow and we bring it into its equivalent present value by dividing by this one plus d we are discounting it or reducing it to bring it into the present value. Now, we can look at this as let us try to understand what does this value of the discount rate mean? (Refer Slide Time: 17:24) So, typically what happens is that suppose consider a company which has many different projects, which it can invest in and each of these projects has a rate of return on the project and it has an investment which is there. So, suppose we have these different investments and we arrange these projects in terms of, from the highest rate of return to the lowest rate of return. So, that means there is a project which is giving us the highest rate of return, we would like to go for it first and for that, we would have to. (Refer Slide Time: 18:08) So, let us make it so that r1 c1, r2, c2 and so on torn, and we arrange it so that r1 greater than r2 greater than and so on to rn. So, the idea is that we arrange these projects in terms of the amount of return that we are getting. So, we will first invest in the project which gives the highest rate of return in that process we will use up c1 then we will use up c2 we can keep on doing this till our entire budget gets used up. (Refer Slide Time: 18:58) So, suppose we have this rate of return here. This one is r1 and we have put c1, then r2 c2, r3, c3 and so on. Till the time that the total amount that we are investing sigma ci will be equal to the C total or the total amount of money that I have to invest. So, that means this value of the rate of return any project which has a rate of return greater than or equal to this is what I am going to invest in. So, this value then becomes my discount rate. So, that this will mean that suppose the company had half that amount of money instead of CT which we have here if it had half the amount of money what will happen is, this point will shift here and your discount rate will be higher. If it had more money, then the discount rate would below. So, the discount rate reflects the scarcity of capital. In another sense, if we look at it, suppose you were thinking in terms of investing hundred rupees in a bank or an institution that you have faith in what is the minimum amount of annual return that you expect before you make that investment. So, if you think about it, you can put down the value and you will see that that value suppose you say that value is 20 rupees, that means that you will invest hundred rupees if and only if you are getting 20 rupees or more every year, your principal is gone, but every year you get a fixed amount of return. That value 20 is your discount rate. (Refer Slide Time: 20:52) So, typically what happens is, if we go back the discount rate represents how money today is worth more than money in the future, there is no theoretically correct value it reflects the scarcity value of capital it also reflects how people what kind of, how do you treat future risks and what is the key, what is your risk aversion, the lower bound will, of course, be the bank interest rates so you will expect at least a minimum which will be the bank interest rate that you get. But in societies where capital is scarce in developing countries, you usually have a higher discount rate. (Refer Slide Time: 21:38) So, in the, in typically if you see, we will look at a discount rate of 10 to 12% which will be like a society discount rate. And if you look at 15 to 20% are the discount rate for the public sector companies. Also, the companies which are investing in the infrastructure sector have this kind of and 20 to 30% of the private companies, private industry, these are the kind of discount rate. These are typically the discount rates for in Indian context. If you look at households and you look at low-income households you may find that the discount rates are quite high 40%, 50% 60%. So, now that we have looked at this concept of the discount rate, let us see how we can use this to look at the decision where you are investing today and you are getting the benefits in the future. (Refer Slide Time: 22:51) So, we would like to now look at a situation where we are looking at, you are making an upfront investment Co and we are getting benefits over the life of 30 equipment in different years A1, A2, Ak to An where n is the life of the equipment or the project. Now, the question is how do we put this all together? So, let us look at a way in which we can take all of these cash flows and bring them up into a present value, equivalent present value. (Refer Slide Time: 23:49) So, when we would like to do that, let us take this and we will derive that we have a present value we want to replace all of these A1, A2, An. So we will try to, we will see that. (Refer Slide Time: 23:56) P=∑ k=1 n Ak (1+d) k ¿ A11+d + A2 (1+d) 2 +... Ak (1+d) k +... A ( 1+d) n Ak=constant=A Now, there can be a special case in many situations where we have Ak is equal to constant. Constant in terms of this is the constant cash flows, which is A and this is often the case because what we are doing is we are making a calculation today about the future, we are looking at a project where you are going to get the same amount of electricity generated or the same amount of energy generated, if we do all the calculation based on today's prices, then you could have constant annual cash flows. (Refer Slide Time: 25:37) So, when we have constant annual cash flows, this will reduce we can see this, this becomes a geometric progression, this becomes P is equal to A by 1 plus d plus A by 1 plus d square k plus A by 1 plus d raise to n. So, we can take this and we can divide this by 1 plus d and we'll get a by 1 plus d squared plus and so on a by 1 plus d raise to n plus 1. So, you can now subtract 1 and 2, if we just subtract 1 minus 2 we get p minus p by 1 plus d is equal to A by 1 plus d minus A by 1 plus d raise to n plus 1. So, you get this you can simplify it 1 minus 1 plus d take A by 1 plus d common here and you get 1 minus 1 plus d raise to n we took 1 plus d common. (Refer Slide Time: 27:15) So, when we simplify this further, we can write this as P into 1 plus d minus 1 by 1 plus d equal to A by 1 plus d and I can take 1 plus d raise to n common, this becomes 1 plus d raise to n minus 1. Now, 1 plus d is not equal to 0. So, I can cancel out these two terms, and then I get P into d is A into 1 plus d raise to n minus 1 and 1 plus d raise to n, so P is equal to A into 1 plus d raise to n minus 1 divided by d into 1 plus d raise to n. This factor which we have is called the uniform present value factor. This factor is what we multiply the annual cash flow stream to get it into the equivalent upfront present value. This is called the uniform present value factor, and we will use the inverse of this the uniform present value factor. (Refer Slide Time: 29:23) UNIFORM PV FACTOR= P A CAPITAL RECOVERY FA(CRF) = A P = d(1+d) n (1+d) n−1 CRF(d, n) So, uniform present value factor as we said in uniform present value factor is uniform PV factor is equal to P by A. And the inverse of this is the capital recovery factor, then that is the factor that we will be using in most of our calculations. Capital recovery factor also is known as CRF in a short form is A by P, which is d 1 plus d raise to n by one plus d raise to n minus 1. And so, this is a factor of two variables, discount rate and life and if you see this, this is what we are talking of d into 1 plus d raise to n by 1 plus d raise to n minus 1. (Refer Slide Time: 30:24) UNIFORM PV FACTOR= P A CAPITAL RECOVERY FACTOR (CRF) = A P = d(1+d) n (1+d) n−1 CRF(d, n) And this gives us the way to calculate the annualized investment corresponding to a particular investment. So, if you have an initial investment, we can convert that into what does it mean in terms of an annualized investment. Let us take an example. CRF (0.12,10) ¿ d(1+d) n (1+d) n−1 0.12(1.12) 10 (1.12) 10−1 ¿0.177 So, consider an investment in a piece of equipment with a life of 10 years and a real discount rate of 12 per cent. So, the question is what does this signify, what is this 0.177? So, this is to give you an idea let us think in terms of investing a 1000 rupees in a piece of equipment or a project which has a life of 10 years and the company, or the individual investing has a real discount rate of 12 per cent. This will mean that that thousand rupee is equivalent to an annualized the investment of 177 rupees each year over the life of the equipment, that is what this 0.177 means, which means that if in this project if I am getting a benefit of 200 rupees every year, then it is worthwhile to go for it. So, I can compare this annualized investment with the actual benefit that we are getting. So, this is the significance of the capital recovery factor. (Refer Slide Time: 2:10) CRF (0.12,10) ¿ d(1+d) n (1+d) n−1 0.12(1.12) 10 (1.12) 10−1 ¿0.177 CRF (0.3, 10)=0.323 And this implies as we said, this implies that an investment of rupees 1000 today is equivalent to annual investments of 177 rupees, if the life, over the lifetime of the equipment. What happens if the discount rate is higher? If the discount rate for instance increases to 30 per cent you can plug in the values, you will find that now the capital recovery factor increases. So, that in this case when you say CRF 0.3 or 30 per cent and 10 years you will find that that comes out to be 0.323, same investment 1000 rupees, same life n 10 years but discount rate is 30 per cent. Which means that the same investment looks more costly because now the annualized investment is 323 rupees. So, then in that case where if you are getting a benefit of 200 rupees per year, you will not make that investment because your capital is more scarce and you expect a higher return, you discount the future with the higher value and that is why this is. So, this is the this is one parameter. The second thing is what happens if life increases? If the life increases then obviously the capital recovery factor will decrease and because so that it gets distributed over a smaller point of time. So, you can see that the capital recovery factor depends on the discount rate and the life of the equipment. So, now we are we have understood the concept of the discount rate we are now ready to look at the different indicators that we talked off, then we will start with these all these three indicators are coming from the same equation. (Refer Slide Time: 4:11) NPV =PV of BENEFITS−PV of COSTS ¿∑ k=1 n Ak (1+d) k −C0 NPV >0 NPV =∑ A (1+d) k −¿C0 ¿ ¿ A CRF(d, n) −C0 The first indicator is the net present value, net present value is the present value of benefits minus the present value of cost, and this will be in money terms, in rupees, dollars, whatever is your currency. So, in the case where we had an upfront investments C0 and we had benefit stream which is AK this becomes sigma AK by 1 plus d, K is equal to 1 to N minus C0 and what is the criteria? The criteria are that net present value should be positive, benefits must exceed the cost, NPV greater than 0 is our criteria if we now had a situation where the special case where A K is constant then NPV will be equal to sigma A by 1 plus d raise to K minus C0 which will be A by CRF in minus C0. So, this is the net present value and this is commonly used by many of the companies for the decision making, and so if you looked at these two examples that we talked of the way where we had A and B and A had a life of 3 years, and B had a life of 8 years, you will find that when we calculate the NPV, we find that the NPV of B is greater than A, of course, it will depend on the discount rate but you can meet that calculation, have an example and you can do this yourself where you can do this calculation. (Refer Slide Time: 6:34) B C = PV of BENEFITS PV of COSTS B C =1 B C = ∑ k=1 n Ak (1+d) k C0 A C0CRF(d,n) Another possibility instead of looking at in the case of net present value it is B minus present value of benefit minus the present value of cost, instead of that some companies use the indicator called the B by C ratio which is NP is the present value of benefits divided by present value of cost and the criteria are B by C must be greater than 1, benefits must exceed the cost. So, this B by C ratio will be nothing but AK by 1 plus d, K is equal to 1 to N divided by C 0 and in the case of constant cash flows, then this will become A by C 0 CRF d, n. So, these are the two indicators net present value and benefit by cost ratio. (Refer Slide Time: 7:37) There is a third indicator which comes from the same equation but slightly different. So, in the case of net present value or benefit by cost ratio, we have to take what is the discount rate of the company, or the individual who is making the decision and based on that then we make the calculation based on that discount rate and find what is the net present value of the project, or the benefit by cost ratio, then we check if that net present value is positive or the B by C is greater than 1 and use that to decide on a yes, no kind of decision. (Refer Slide Time: 8:41) NPV =0=∑ k=1 n Ak (1+r) k −C0 0= A[(1+r)n−1] r(1+r) n −C0 In the case of the internal rate of return we do not assume a discount rate, we look at that equation of the cash flows which are coming from the project and we say what is if we take that equation and we solve for the rate of return that means we set NPV is equal to zero and solve for. So, if you see this, instead of taking the discount rate we make this as an unknown, we set NPV is equal to zero and we solve for r, the r-value that we get is called the internal rate of return, and then we compare this internal rate of return to the minimum return which the company expectson the projects which are equivalent to its discount rate, it is also called the hurdle rate. So, in effect IRR should give you the same result as the NPV or the B by C ratio, but the calculation is different, this is a polynomial equation. So, if you see, now we can simplify this in the case suppose we take the special case where AK is constant that means this is now A by if we write down the equation it is r into 1 plus r raised to n 1 plus r raised to n minus 1 minus C0. (Refer Slide Time: 10:12) C0= A r [ 1− 1 (1+r) n ] rj+1 A C0 [ 1− 1 ( 1+rj) n ] |rj+1−rj|≤TOL Now, we can simplify this by putting this as C0 is A by r, I can divide this and I can get this as. So, now I can solve this equation is a polynomial equation in r, we can do it one of the simplest ways, of course, you can use bisection method, you can find many ways in which you can solve this through but one of the simplest methods is I can take this as r and put this as r is equal to A by C0, 1 minus 1 by plus r raised to n. So, we can start with this equation and start with an assumed value of r, so let us take rj and then update it to get the new value of rj plus 1 and keep iteratively solving this till the modulus of this difference is less than or equal to sum tolerance value. So, this is one way in which we can solveand get the internal rate of return. Of course, in many of you know you have the IRR even in your excels, there is an IRR function, you can calculate and see that it brackets the roots and do this but this is a simple way of doing this. So, we have seen how the three methods the net present value, benefit by cost ratio, and internal rate of return, and now let us do one example.
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