noc18-ce39-Lec 04-Time Value of money

Namaskar and welcome to this lecture in this course on Accounting and Finance for Civil Engineers. So, today we will get started with the concept of time value of money and do some simple interest calculations to illustrate the point. Now getting started, the concept is derived from the cash flow diagram, you will recall that we constructed a cash flow diagram last time, when the inflow and outflow of funds is expected or planned at different points in time. So, what we are trying to say is that if on the time axis, we expect outflows at different points, we expect certain inflows at different times and if this axis is long enough, that is the time is long enough that warrants the study on interest calculations, how the value of money changes over a period of time and so on, then we need to use this concept of time value of money. We will initiate our discussion from the concept of interest on a bank deposit that we are familiar with. Now, what is this concept of interest on a bank deposit? We know that if we deposit 100 rupees in the bank, at the end of 1 year, the bank may give us 105 or 107 or 109 depending on the rate of interest that is applicable and the ways in which it is calculated. So, that is something which we are familiar with. Another concept that we are familiar with and we will use that point today is the fact that we borrow from the banks. We take a house loan, we may take a car loan and so on and those loans have to be repaid and obviously, when we take 100 rupees from the bank, what we pay back to the bank is not 100 rupees. It is a little more than 100 rupees and that is something which we will need to understand a little bit before we get started with our discussion as far as financing is concerned. These concepts are very important to understand when we are talking of trying to organize finances for a project and that is what we have talked about in the first few lectures when we said that one of the reasons why civil engineers should be aware or should be well versed with these methods is because they need to plan and organize, make sure that funds, the right amount of funds are available to them at the right point in time. So, moving forward, this is the example that we gave last time that here is a construction project which involves an expenditure of 100, 150 and 100 crores that is a total expense or an investment of 350 and we expect certain revenues. Now what we have to see and that is what we had mentioned last time also, how do we calculate the present value of all these revenues or all the investments or the future value because at the end of it if we want to compare this 100 with this 170, we have either to bring this 170 to this point or take this 100 to this point. Then only we can make a rational comparison. These are the kind of things which will become handy when we try to compare options and that is what we will see in the next class. So moving forward, let us take another look at the cash flow diagrams. Cash flow diagrams I have already talked to you is a visual representation of inflow and outflow funds which are either received or spent during the project's lifetime and this is another example where we expect 10, 50, 150,000 being spent at different points in time and certain amounts of revenue. So what we are trying to understand is how do we make a comparison between this cash flow and another cash flow which may have different inflow and outflow even if the total inflow and outflow is the same. So of course we followed the same convention as far as inflow and outflow is concerned being plotted on the top half and the bottom half of the time axis. So now coming to the formal definition of time value of money, the value of money changes with time and interest represents the earning power of money and can be looked upon as the premium paid to compensate the owner for the loss of use of the loaned money. So our deposits in the bank can be looked upon as loans to the bank and because we do not have the freedom to use those funds, the bank pays us the interest. With those funds, the bank loans to other people, maybe some people take house loans, some industries take loans to set up industry and so on and the bank charges them interest part of which it repays to us and part of it goes to sustain the bank's expenses and the bank's profits. So that is how the money goes around and thus there exists the concept of a present value and a future value for any investment. So that is something which we are going to look at formally. Now we are already familiar with the concept of simple interest and that is the interest computed only on the original sum but not on the accrued interest. That is what we know from high school and when it comes to compound interest, it is the interest computed on the unpaid balance due which includes the unpaid interest from the preceding period. Now unless otherwise mentioned, the remaining discussion in this course is based on compound interest calculations. Now let us try to define the concept of nominal and effective interest rates. The nominal rate denoted by r is the annual interest rate for a 1 year period without any compounding. Now assume that a bank compounds an interest biannually. Now if the compounding is such that it is at the rate of 2.5%. So what it means is, it is a matter of terminology, what it means is that the interest rate for the interest period denoted by i is 2.5% but the nominal rate is 5% because it is being compounded biannually. So it is done twice a year, therefore this 2.5 becomes 5. So it is 2.5 times 2 coming from the fact that it is being compounded biannually. So now in this case, what is the effective annual interest? We know this formula and we understand that the effective annual interest rate in this case is 1 plus r divided by m to the power of m which in this case will be taken as 2 because we are trying to calculate the effective annual interest rate and we are talking of a biannual system and this is how we will calculate it. So if we look at an example or an illustration, suppose a savings bank pays an interest of 2% every 3 months. Now we are compounding it quarterly. What are the nominal and effective rates of interest per year? So for that purpose, m becomes 4 because we are doing it quarterly. The nominal interest rate is 4 into 2 because 2 is the interest being paid every 3 months. So the nominal interest rate is 8 and the effective interest rate turns out to be 8.24 and not 8. I would leave it to you as an exercise to calculate the effective interest rate in the previous case where we had a 2.5% biannual interest. So that is something which I am leaving out to you to do on your own and we will move forward. So let us look at this whole concept once again from the point of view of repayment of a loan. Now these loans are taken and have to be repaid over a period of time which oftentimes is substantially long. So it depends on how much we are repaying, what we have taken from the loan, what is the interest payable and the biggest thing is the payment that we are making, what is it being used for. So let me clarify this. Repayment basically means paying back the money previously borrowed from a lender. Usually this is done in the form of periodic payments and can be viewed differently for adjustments such as repayment is used initially against the principal, repayment is used initially against the interest and repayment is used to pay both the principal and the interest. This is something which I will try to explain to you through a illustrative calculation. Now depending on the conditions of repayment whether it is this or it is this or it is this one, the time that it takes to repay that loan is quite different and let us try to see this through an example. Let us take an illustrative example on the condition that the repayment will be used to pay the interest first. If we take 10 lakhs of money from a financial institution with an annual rate of interest being 10% and we make an annual repayment of 90,000. The condition being that this 90,000 will first be used to service the interest on that loan. What happens? Here is a table which tells us that at the end of year 1, the principal on which the interest is going to be counted is 1000. We are always talking in terms of thousands. So the interest that accrues on this amount is a 100 and the repayment is only a 90. So now this 90 being less than this 100 obviously means that the principal at the end of the second year becomes this 1000 plus this interest which accrues minus the repayment which has been made and 1010. And this 1010 will attract an interest of 101 in the next year and since we are making an equal payment of 90 all the time, we find that the principal is always increasing. And therefore, the interest accrued is also increasing and therefore we are in a situation that the loan can simply never be repaid. Even though we are repaying a certain amount of money every time, the condition being that the repayment is being used for the interest servicing first simply means that unless we at least clear this interest, there is no way that the principal is going to go down. So that is one condition that we have. So this was one case. Now we come to a second case where still we are paying the interest first but the payment is a 110,000. So in that case if we look at the same table, now what we are doing is we are paying a 110. So we find that the principal is gradually reducing and at the end of say 25, 26 years we have been able to service this loan. So I am leaving out the spreadsheet kind of calculation to you to do as an exercise. You can change these values and try to come up with your own numbers. And let me repeat this exercise with a different amount of repayment annually, let us try to make it 120,000 and then we find that we are able to service the loan in about 19, 20 years. So this really shows that if the condition was that the repayment will be used to service the interest first, the obvious thing is that the repayment amount should be at least equal to the interest that the loan attracts in a given year. And after that so long as that is met, if we keep on increasing the amount of repayment being made, the time it takes to repay the loan could be reduced. Now instead of the repayment being used to pay the interest, if the condition was that it will be used to pay the principal first, then let us try to see what happens to the numbers. So if we have the same conditions, 10 lakhs of money borrowed, 10 percent of interest applicable and the yearly payment is made for 90,000 as before, we find that we are able to service the loan in about 19, 20 years. But this is again a very simplistic example because what we are saying is that this 90 is being used to reduce the principal all the time and the interest is calculated only on the principal of that particular year and this interest does not attract any interest. So if these conditions are what are operating, then yes we can at this point say that well the sum of all the interest up to this point is 506 and in order to service this 506 without any interest being accumulated in this period, we are able to clear the loan. So these are some simple assumptions that we are making, of course they are not realistic at times. The moment we introduce more complications, you need to go back to the spreadsheet and try to do your own calculations. Now instead of 90,000, if we were to use the example of 110 which we also did last time, we find that we are able to complete the payment in 15 years and moving forward if we were to pay 120, then the loan can be serviced in 13 years. Now let us look at a situation where the repayment is adjusted partly for the principal and partly for the interest. What we have done so far is that whether it was 90,000, whether it was 110,000 or 120,000 that money was used to either pay the interest that was accruing or to reduce the principal. It was a 0, 1. Now let us try to obviously combine the two and try to see that if partly it was used for the principal and partly for the interest, then how much time does it take. So for that, the first example is we take 50, 50 that is 50% of the repayment is adjusted for the principal and 50% for the interest. So this is what is happening in this case. We have a principal of a 1,000, we pay back 90 out of which 45 is counted towards the principal and 45 towards the interest. So interest that has accrued here is 1,000 into 0.1 which is 10% of a 1,000 which is a 100. So the total interest is here. So this 45 here goes to reduce my principal here to 955. Since I have a 100 as the interest, I have repaid 45 on that. So my interest remaining to be paid is 55. This 955 attracts an interest of 95.5. This 55 plus 95.5 is this interest here which is the total interest that needs to be paid and therefore the total payable here becomes 955 plus 150.5 which is 1105.5. So this is how we will continue to do the calculations and in a generic sense what we have is this. These are the columns that we are talking about 2, 3, 4, 5, 6, 7 and 8 and we can calculate how to make the entries in the columns 2, 3, 4, 5, 6 and so on and of course 7 and 8 are given to us because we are just paying 90 every time. So this calculation if we were to repeat on the spreadsheet that we have been following, we find that we are actually able to clear the loan in about say 26 years. Now you see that this 90 was not sufficient to repay the loan when it was being used only to service the interest but so long as it is being used to service part of the principle it still becomes possible with the assumption that the interest does not attract interest. So you would notice in the last slide that we did not have compounding interest on the interest. So with that simplified assumption it is possible to complete the payment on this loan in say 26 years. Now moving forward if the amount again was a 110,000 which is what we have used in the previous examples and we still stick to the principle of 50-50 then we find that we are able to pay the loan in 19 years using the same arithmetic that is this 110 gets divided into 2 parts, paid towards the interest, paid towards the principle. This component we keep reducing all the time, this component we keep reducing all the time and we find that the total interest here is a sum of this plus this and the total payable is this number here plus this 890. So if we keep doing this arithmetic with the 110,000 being paid every year we are able to clear the loan in 19 years. Similarly if you were to repeat this exercise with 120 you will find that we are able to pay it in 17 years. So now this slide summarizes what happens if we have 90, 110 and 120 being used to service a loan amount of 10 lakhs at an annual interest of 10% with this repayment being used for the interest only, for the principle only and equally for the interest and principle with the assumption that the interest does not attract interest. So this table is only a summary of the amount of time that it takes to repay the amount. What I have not done here is try to calculate that how much money have we actually paid. What I mean is that if there was a calculation that 110,000 is being paid for 20 years then what you are paying totally is 22 lakhs. We were talking in terms of thousands and therefore 2200 and 3 zeros here means 22 lakhs. So what you should realize is that in taking a loan of 10 lakhs what you landed up paying was 22 lakhs except the fact that it was spread over 20 years and you are making a payment of 1.1 lakhs every year. So this is something which we have to understand. We can always try to translate that into a cash flow diagram and say that if we take a loan of 10 lakhs keep repaying 1.1, 1.1 every time it will take 20 years provided the conditions that we talked about were being operated whether it is going to service the interest first, whether it is servicing the principal first and so on and so forth. Now let us try to look at some ready reckoners. Ready reckoners are easy to use tools which help us do these calculations without for example the spreadsheet. Now what is the kind of interest computations that are of interest to us? As far as single payment is concerned what we would like to know is the compound amount factor and the present worth factor. If it is an equal payment we would be interested to know what is called the compound amount factor, the present worth factor, the sinking fund deposit and the capital recovery factor and if it was unequal payments we are interested in an arithmetic gradient factor and a geometric gradient factor. So these terminologies is a matter of words. We do not really want to insist that you should remember them right away but the principle of it is the following. At the end of it there is a principle involved, there is a time of repayment or a time which is involved and there is a rate of interest involved and there is a final value F which is involved. Now there can be different combinations when out of these 4 variables the P, the I, the N and the F, 3 are given and 1 is unknown. In the case of the single payment compound amount factor which is called the SPCAF what we talk about is what is this F given P, I and N for that the formula is this. This is basically just the compound interest kind of a formula and the illustrative example is how much does a deposit of 10 lakhs grow into in 5 years at a rate of 8% compounded annually. We are not getting into complications of biannual or quarterly compounding but if you make an illustrative example here the F is 14.69328 lakhs which means that this here is the formula and the factor that we are talking about is 1.469 that is if we have the P we just need to know this value and we will be able to calculate the F provided we also know the I and the as far as a single payment present worth factor is concerned what we are talking about is what is the present value or present worth of something which will have a value of F at the end of N years with a yearly interest rate being I and the time period being N. In that case it is just the other way around it is the reciprocal of the previous discussion and what we are looking for is what is the P if the F, I and N are known. The illustrative example is given here what is the present value in INR of an asset which is expected to have a worth of 20 lakhs of Indian rupees in 10 years. Assume that the rate of interest compounded annually is 10% and we are talking of the answer to be 7.71 lakhs. What it means is that if we have 7.71 lakhs here it will grow to a value of 20 lakhs that is what is important to understand from a construction perspective or from the construction industries perspective as a contractor if you know for example that you will need to buy an equipment which is going to cost 20 lakhs in 5 years or 7 years what is the kind of money that you need to set aside today. If you are setting aside a single installment here so that is the single deposit that you are creating here with the intention that this will grow into a certain amount of money here which will enable you to buy a crane or a bulldozer or whatever it is. So that is the kind of application that we are talking about here as far as a single payment present worth factor is concerned. Now let us try to look at an equal payment compound amount factor. So in that case what we want to know is what is the F if we keep A, I and N as knowns that is if we set aside an amount of A annually at an interest of I over a period of N years what will be the final value that it will grow to and here is the Redi-Reckoner formula for that and the illustrative example is that what is the worth at the end of 10 years for an ordinary annuity of 20,000 operated at an annual compounding rate of 8%. So if we do that so what we are doing is every year we are setting aside 20,000 why you want to know what will it grow to if this I was 8% and we are talking in terms of N is equal to 10. So if that is what we want to do we find that the F turns out to be 2,89,732 which means that the first 20,000 will attract an interest for 9 years, the next 20,000 will attract for 8 years and so on if we keep doing that we find that finally the amount that we get is 2,89,000 against what we have paid is 2,00,000. So this 8,9,732 is what is the cumulative interest on all these deposits that we have made every year. So coming to the equal payment sinking fund deposit factor we are talking of if we want a certain F we know the I, we know the N how much should we set aside every year. So this factor really talks in terms of what is the A if F, I and N are known and this is the formula. For those of you who are more academically inclined I would think that you should try to derive these formulae it is not so difficult we do not want to do this in this course because we do not have the time but you should be able to derive these formulae it is not so difficult you just need to use the concept of compound interest how much times does this attract interest how many times does this attract interest and so on and try to do a summation and so on and so forth and you should be able to come up with these formulae on their own. Now these formulae have been tabulated and tables are available in order to help you get the numbers and that is why I am calling them ready reckoners. Now as an illustrative example how much INR should be invested in a bank every year to make a sum of 20 lakhs at the end of 20 years assume that the bank offers an interest of 8% compounded annually. So if that happens then you find that an amount of 43,705 would suffice. So you deposit 43,705 every year for 20 years and what you will get at the end is 20 lakhs. Moving forward let us try to look at the equal payment present worth factor which means that what is the given A, I and N. So in that case what is the illustration that we want to make what is the present worth of an ordinary annuity of 20,000 invested over a period of 10 years and being operated at an annual compounding rate of 8%. So in that case what we are looking at is this that if we invest 20,000 for 10 years at the rate of 8% the present worth of this investment is 1,34,260 and as a corollary to this formula for a sufficiently large N the factor becomes pretty close to 1 upon I. You can try to check it out on your own we move forward and try to talk in terms of the equal payment capital recovery factor and there what we are talking about is given a P what should be the A if the N and I are known. This is the factor that we use for this particular calculation and what it says is that what should be the annual installment for a period of 6 years that a lender has to fix to recover a total sum of 2 lakhs at an operating interest of 4%. So if that is what is the condition then the A is 38.15. So this 38.15, 1000 will pay a loan of 2 lakhs in 6 years at the rate of 4%. Now having completed all these factors very quickly one slide after another what I would request you to do is to look at the tables try to do some calculations on your own and also try to compare the numbers. For example we said that if we have a certain annuity over a period of time and what was the F try to see what is the P involved with that for the same conditions of I and N. Once you do that you will get more insight into what really is being talked about. Now we are coming into a situation where the cash flow series may not consist only of constant amounts. The increase and decrease in installments which is payments or disbursements may follow a pattern. Again these patterns are also illustrative in nature. We are going to talk of arithmetic and geometric progressions but of course life is not as simple as that sometimes there will be more distributions, there may be more complications and then you will have to find out the answers on your own. Now coming to the arithmetic gradient factor what we are talking about is what is the present worth P if this A keeps increasing every year for a period of N years and the rate is I. It is like saying that well today I have the paying capacity of 100 but I know that I will be able to start paying more and more I can give 110 next year, 120 next year and so on and so forth. Then if that is the kind of series that we are able to follow what is the present worth of that investment. Now in order to do that this is broken up into 2 parts this A remains constant that is we take the A part of it out from here we just draw the line and this then becomes G, 2G, 3G, N-2G, N-1G and so on. So if we do that then the present worth P is also broken up into 2 parts or can be looked upon as having 2 components one is P1 arising from here and P2 which is arising from here. If we look at those 2 numbers P1 we already know and P2 is given by this formula and where do we use this kind of concept. A company has purchased a new equipment it wishes to set aside enough money in a bank account to pay the maintenance on the equipment for the first 5 years and the estimated cost of maintenance of equipment is shown below. So we see that it is a 120, 150, 180, 210, 240 so we see that this is increasing every year at the rate of 30,000 rupees and that is a fair assumption we can assume that or it sounds logical that as the equipment becomes older the maintenance cost of that equipment increases. So if we model it like this then what should be the kind of company deposit in the bank now. So what should be the funds that are set aside now. Since this question is coming under the food for thought scheme we are not creating the answer for this in this slide we will probably do it in an assignment at some point in time but not now. What you need to do is to go back to the previous slide and try to break it up into A plus G, A plus 2G and so on. Try to get the A's at one place as we did try to get the G's increasing like this and use the formula which was given and try to come up with the P1 and P2 add them and compare notes add them and determine your answer. Now as far as the geometric gradient factor is concerned here it is not arithmetic it is geometric which means that we are talking of what is the P for C, C times 1 plus G, C times 1 plus G square cubed and so on till it reaches N years and again we are talking of a rate of interest being I. So for that the increment or decrement occurs at a uniform rate and that is exactly what is geometric progression that we talk about or the present worth is given by Pn is equal to Cn times 1 plus i to the power of minus n where cash flow Cn in a given year is C times 1 plus G to the power of n minus 1 and then the net present worth which is what we are looking for that is this P here can be determined using this formula. So here again it is a food for thought we are not trying to solve this problem for you what the statement says is that the maintenance cost of an equipment during the first year of operation is estimated to be 1 lakh and it increases uniformly at the rate of 10% per year assuming that the bank pays 8% interest calculate the amount of money that has to be deposited in the bank now and that is where the P comes in. So what should be deposited now if we expect the expenditure to start at 1 lakh increase at 10% when the bank is giving us only 8% and the equipment has to be maintained for 5 years. So now you have to formulate this problem in the manner which was given in the previous slide and try to get the answer for P. So with this we come more or less to the close of our discussion today it has been a long day and we started with the principle of interest calculations we talked a little bit about loan repayment we talked about the idea of time value of money and we talked about some relevant ready reckoners which are available in the form of tables which will enable you to determine the answers pretty quickly rather than having to go to a spreadsheet all the time except that you should remember that those ready reckoners or the kind of spreadsheet calculations that we did make certain assumptions and those assumptions are what you should be very careful about whether something is uniform or it is not uniform and so on and that is what is going to make the calculations more tortuous for you and with this we come to an end of the discussion today. But before that here is a list of references which you may like to use sometime to enhance your understanding. Thank you.